SEMICONDUCTORS AND SEMIMETALS VOLUME 24 Applications of Multiquantum Wells, Selective Doping, and Superlattices
Semiconductors and Semimetals A Treatise
Edited by R . K. WILLARDSON
ALBERT C. BEER
WILLARDSON CONSULTING
BATTLE COLUMBUS LABORATORIES
spomm,WASHINGTON
COLUMBUS, OHIO
SEMICONDUCTORS A N D SEMIMETALS VOLUME 24 Applications of Multiquantum Wells, Selective Doping, and Superlattices
Volume Editor RA YMOND DINGLE G A W ELECTROMCS CORPORATION SOMERVILLE, NEW JERSEY
PUBLISHED BY ARRANGEMENT WITH AT&T
ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers San Diego New York Berkeley Boston London Sydney Tokyo Toronto
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Contents PREFACE................................................................
ix
Chapter 1 Fundamental Properties of I11. V Semiconductor TwoDimensional Quantized Structures: The Basis for Optical and Electronic Device Applications
C. Weisbuch I. Introduction ......................................................... i I1 The Electronic Properties of Thin Semiconductor Heterostructures ........... 9 111 Optical Properties of Thin Heterostructures ............................... 46 IV. Electrical Properties of Thin Heterostructures ............................. 78 V. Conclusion.......................................................... 115 Selected Bibliography ................................................. 116 References .......................................................... 117
. .
Chapter 2 Factors Affecting the Performance of (Al. Ga)As/GaAs and (A). Ga)As/InGaAs Modulation-Doped Field-Effect Transistors: Microwave and Digital Applications
H . Morkoc and H . Unlu I. Introduction ......................................................... I1. How Modulation Doping Works ........................................ 111. FETFabrication ..................................................... IV. Principles of Heterojunction FET Operation .............................. V. Optimization ........................................................ VI . Performance in Logic Circuits .......................................... VII. Microwave Performance............................................... VIII . Anomalies in the Current -Voltage Characteristics ......................... IX . Advanced Technology Requirements .................................... X. Pseudo-morphic MODFET-(In, Ga)As/(Al, Ga)As ........................ XI . Remaining Problems and Projections .................................... XI1. Summary and Conclusions............................................. References .......................................................... V
135 136 138 140 149 155
160 168 175 180 191 196 198
vi
CONTENTS
Chapter 3 Two-Dimensional Electron Gas FETs: Microwave Applications
Nuyen T. Linh I . Introduction ......................................................... I1. TEGFET Structures .................................................. 111. Transport Properties in TEGFETs ...................................... IV . Device Modeling ..................................................... V . TEGFET Microwave Performance ...................................... VI . Conclusion.......................................................... References ..........................................................
203 206 210 216 229 243 245
Chapter 4 Ultra-High-speed HEMT Integrated Circuits
M . Abe. T. Mimura. K . Nishiuchi. A . Shibatorni. M. Kobayashi. and T. Misugi I. Introduction ......................................................... I1. Technological Advantages of HEMTs .................................... I11. HEMT Technology for VLSI ........................................... IV. HEMT Integrated Circuits ............................................. V. Future HEMT VLSI Prospects.......................................... VI . Summary ........................................................... References ..........................................................
249 250 255 264 274 216 277
Chapter 5 Nonlinear Optical Properties of Multiple Quantum Well Structures for Optical Signal Processing
D . S. Chemla. D . A. B. Miller. and P . W. Smith I. Introduction ....................................................... 11. Linear Absorption in Multiple Quantum Well Structures. . . . . . . . . . . . . . . . . . 111. Excitonic Nonlinear Optical Effects in Multiple Quantum Well Structures.... IV. Variation of Optical Properties Induced by a Static Field .................. V . Conclusion........................................................
References ........................................................
279 282 291
304 314 316
Chapter 6 Graded-Gap and Superlattice Devices by Bandgap Engineering
Federico Capasso I . Introduction: Bandgap Engineering...................................... I1. Real-Space Transfer Structures ......................................... 111. Channeling Diodes ................................................... IV. Low-Noise Multilayer Avalanche Photodiodes and Solid-state Photomultipliers .
319 320 331 338
CONTENTS
V. Other Device Applications of Staircase Band Diagrams and Variable-Gap Superlattices......................................................... VI . New Heterojunction Bipolar Transistors.................................. VII. Sequential Resonant Tunneling and Effective Mass Filtering in Superlattices.... VIII. Doping Interface Dipoles: Tunable Heterojunction Barrier Heights and Band-Edge Discontinuities ............................................. References ..........................................................
vii 352 361 384 387 392
Chapter 7 Quantum Confinement Heterostructure Semiconductor Lasers
W. T. Tsang I. Introduction .........................................................
I1. Theory of Quantum Confinement Heterostructure Lasers:Quantum Well. Quantum Wire. and Quantum Bubble Lasers ............................. I11. Short-Wavelength (-0.68-0.85 pm)Quantum Well Heterostructure Lasers .... IV. Long-Wavelength (A - 1.3- 1.6 pm) Quantum Well Heterostructure Lasers..... V . Very-Long-Wavelength (A 2.5 -30 pm) Quantum Well Heterostructure Lasers VI . Summary ........................................................... Appendix ........................................................... References ..........................................................
-
397 397 409 434 439 443 443 455
Chapter 8 Principles and Applications of Semiconductor Strained-Layer Superlattices
G. C. Osbourn. P . L. Gourley. I . J . Fritz, R . A4. Biefeld. L. R . Dawson. and T. E . Zipperian I. Introduction ......................................................... I1. Background ......................................................... 111. Electronic Properties.................................................. IV. Applications of Strained-LayerSuperlattices .............................. V. Summary ........................................................... References ..........................................................
INDEX...................................................................
459 459 467 490 499 500
505
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Preface This volume is devoted to the properties and applications of ultrathin layers of I11 - V semiconductor heterostructures grown by modem epitaxial techniques. Much of the pioneering work in the area was performed at AT&T Bell Laboratoriesand at the IBM Thomas Watson Research Center. While the early work is of a very fundamental nature, interest has quickly moved to applications, devices, and early products. As is often the case, the original interests of the pioneering researchers were very different and led to different schools of investigation. It is indeed fortunate that several of those early workers have agreed to contribute to this collection. Although not encyclopedic, the chosen topics give a good coverage of the field in a number of areas. The work described here began in the early 1970sand was spurred on by the development of molecular beam epitaxy and the continuing interest in heterostructures and thin films. So rapidly is the field developing that two of the authors have already formed commercial companies whose products are based on this technology. In addition, major industrial laboratories have committed to the development of the technology, and commercial products are now available from several independent sources. Against this background, it is very appropriate that a volume devoted to the applications of ultrathin 111-V heterostructures, written by those who developed the field, is now available. This book is divided into three parts. The first, consisting of a single chapter, describes the basic phenomena, materials, and optical and electrical properties of various structures and establishes the foundation for the subsequent sections of the book. Written by Claude Weisbuch (Thomson -CSF, Central Research Laboratories),who pioneered many of the developments in the optical properties of superlattices and quantum well materials, this chapter is the most comprehensive work ever written on the fundamental properties of these materials. It should become the standard reference work in this area for many years to come. The second part comprises Chapters 2-4 and focuses on electronic devices and circuits based on quantum well, superlattice, and single, selectively doped heterostructure interface structures. Beginning with the chapter by MorkoG and Unlu (University of Illinois at Urbana-Champaign), which discusses basic parameters and device performance for both microix
X
PREFACE
wave and digital applications, the next two chapters describe in detail microwave (N. T. Linh of Picogiga) and digital integrated-circuit (M. Abe et al. of Fujitsu Laboratories) applications. In the latter chapter, reference is made to a 4-kbit SRAM, which consists of 26,864 transistors and has a minimum access time of 2.0 nsec. It is expected that fully functional 16-kbit SRAMs, with over 100,000 transistors and an access time of less than 5.0 nsec, will be announced during 1987. This rate of increase in complexity is far in excess of the well-known Moore’s law, which predicts a factor of two growth in complexity per year for silicon-based integrated circuits. The phenomenal growth in chip complexity is paralleled by an equal growth in commercial interest in early development products. Millions of dollars have been committed by major and start-up organizations in the belief that these devices have a major commercial future. The following three chapters focus on the generation and detection of light using single or multiquantum well structures. Chapter 5 by Chemla, Miller, and Smith (AT&T Bell Laboratories and Bell Communications Research) focuses on multiquantum well structures (MQWSs) and their nonlinear optical properties as prototypes for optical signal processing applications. Of major importance here is the existence of a well-defined excitonic state at room temperature and the electric field dependence of the MQWS optical absorption characteristics. These effects are direct consequences of the ultrathin nature of the layers in the MQWS and can be understood in terms of confined particle electronic properties as originally described in 1974. As the authors point out in their closing statements, developments are so rapid that the future of systems based on optical bistability in MQWSs is very difficult to assess in any quantitative manner. Their expectations are that these advances will have ramifications far beyond the field of semiconductor physics. In Chapter 6, Federico Capasso (AT&T Bell Laboratories) explores a range of devices based on multilayer structures which contain both sharp and graded interfaces. “Bandgap engineering” is stressed, and the application of graded and sharp heterointerfaces to device development is clearly established. The flexibility of design, with the attendant control over optical and electronic properties that these man-made materials provide, expands dramatically the range of device types that may be fabricated. Research activities in this area abound, and novel structures are constantly proposed -already going beyond the advances reported in the present volume. In Chapter 7, W. T. Tsang (AT&T Bell Laboratories) continues the exploration of the bandgap engineering concept in MQW lasers. Based upon the two-dimensional nature of electron motion in quantum well heterostructures, Tsang characterizes quantum wire lasers and quantum
PREFACE
xi
bubble lasers in which the particle motion is fully quantized and the density of states consists of discrete energy functions. The discussion covering short-, long-, and very-long-wavelength quantum well heterostructure lasers pinpoints new material combinations that are becoming important in this field and focuses attention on the emerging new epitaxial technique of chemical beam epitaxy. The final chapter of this volume describes the extension of the MQW/ superlattice structures from closely lattice-matched materials (e.g., GAS/ AIGaAs) to combinations in which the individual materials have lattice constants that differ by 1% or more. This concept of the semiconductor strained-layer superlattice has been pioneered by the Sandia Laboratories group led by G. C . Osbourn. The key observation, that the lattice mismatch can be accommodated by uniform elastic strain in ultrathin layers of less than some critical thickness rather than by the formation of misfit dislocation networks, has been known for some years; however, the application of this knowledge to real materials for fundamental and device studies is very recent. This book has been written for the expert as well as for the novice who wishes to become familiar with the potential of this rapidly developing field. The individual authors have admirably fulfilled their charter and the success of this volume, as with the success of the field itself, will be largely due to their outstanding contributions.
RAYMOND DINGLE
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SEMICONDUCTORS AND SEMIMETAU, VOL. 24
CHAPTER 1
Fundamental Properties of I11 - V Semiconductor Two-DimensionalQuantized Structures: The Basis for Optical and Electronic Device Applications C.Weisbuch LABORATOIRECENTRALDERECHERCHES
-
THOMSON CSF DOMAINE DE CORBEVILLE, 9 140 1 ORSAY, FRANCE
I. Introduction 1. THEADVENTOF ULTRATHIN, WELLCONTROLLED
SEMICONDUCTOR HETEROSTRUCTURES Although the search for ultrathin materials can be traced quite far back,lS2 the motivation for their production went up sharply when new types of devices3v4were predicted, such as the Bloch oscillator. At the same time, the advent of a new growth technique, molecular beam epitaxy (MBE),5-12opened the way to the growth of semiconductors atomic layer upon atomic layer. In 1974 two basic experiments were carried out: Esaki and Chang reported the oscillatory behavior of the perpendicular differential conductance due to resonant electron tunneling across potential barr i e r ~ and , ~ ~ the optical measurements of DingleI4 showed directly the quantization of energy levels in quantum wells, the well-known elementary example of quantization in quantum mechanics textbooks.'" Studies of ultrathin semiconductor layers have since then proliferated at an explosive rate. Owing to progress in crystal availabilityand control, basic understanding of low-dimensional systems, and applicability of heterostructure concepts, the recent years have also seen the emergence of a wide family of structures and devices, which can be classified into four main (overlapping) families, as shown in Table I. At this point it seems worthwhile to emphasize the various structures that will be described or mentioned in this review, as their abundance can sometimes be confusing. They are depicted in Fig. 1 by means of their band diagrams. In many of these structures, we will be 1 Copyright 0 1987 Bell Telephone IaLmmtorie%Incorporated. AU rights of reproduction in any form reserved.
2
C. WEISBUCH
TABLE I
THEFOURMAINFAMILIES OF DEVICES ORIGINATING ULTRATHIN, WELL-CONTROLLED SEMICONDUCTOR HOMO-AND HETEROSTRUCTURES
FROM
JTWO-DIMENSIONAL I
SYSTEMS] I
SDHT-TEGFET-HEMT-MODFET NPI Quantum Wells Quantum Hall Devices CHARGE TRANSFER SYSTEMS I IONE-DIMENSIONAL SYSTEMS
SDHT-TEGFET-HEMT-h4ODFET MPI Real SDaCeTrTranSfer Devkes
3
Tunneling Structures Superlattices h perpendicular NPI transport Quantum-Well Wires
1 BAWGAP ENGINEERU, STRUCTURES] all of the above plus non-quantized-motion structures: DarbleHeterostruciure Lasers GradedGap APD Heterostructure Bipolar Transistors (gaded base or not ) Separate Absorption- Multiplication APD Staircase Solid State Photomultiplier
Note that the same structures can belong to several of the families and that, using the term bmdgap engineeringin its most general description of engineered structures with desired properties obtained by a tailoring of the band structure, all of the structures can be considered “bandgapengineered.”
interested in quasi-two-dimensional properties; the free motion of the carriers occurs in only two directions perpendicular to the growth direction, the motion in the third direction z being restricted to a well-defined portion of space by momentum, energy, and wave-function quantizations. Compared to “classical” heterostructures like double-heterostructure (DH) 1asers,15J6the “quasi-2D’ term means that the z motion is defined by one or a few quantum numbers, which is only the case in ultrathin structures and/or at low enough temperatures. We use here the word quasi to mark the difference with exact 2D systems in which the wave function is exactly confined in a plane, with no extension outside of that plane. In the
1. 111-v SEMICONDUCTOR QUANTIZED STRUCTURES
@SINGLE TYPE INTERFACE
i
@SINGLE BARRIER TUNNELING STRUCTURE
@SINGLE TYPE II INTERFACE
@SINGLE QUANTUM WELL
@MULTIPLE QUANTUM WELL
@DOUBLE BARRIER
@ INCOHERENT MULTILAYER
TUNNELING STRUCTURE
TUNNELING STRUCTURE
@TYPE I SUPERLATTICE
3
@TYPE I1 SUPERLATTICE
FIG.1. The various types of heterostructures discussed or mentioned in this chapter. The widely used type-I heterostructure is shown in (a), with the band discontinuities such that both band edges of the smaller gap material are below those of the wide-bandgap material. In the type-I1 interface (b), the band structure is such that the top of the valence band of one of the compounds lies above the bottom of the conduction band of the other compound. Charge transfer occurs,leading to a conducting heterostructure. The type-I quantum well is shown in (c). The multiplequantum-well structure [MQW, (d)] is such that &, is large enough to prevent tunneling. Conversely, in the single barrier (e), double barrier (f), type-I incoherent tunneling (g) and superlattice (h) structures is small enough to allow carrier tunneling across the barrier material. The difference between these two latter structures, (9) and (h), is that in the superlattice structure disorder and scattering are low enough to allow the coherent superlattice band states to build up, whereas in the incoherent tunneling structure scattering by disorder (here disordered interface fluctuations) destroys the phase coherence between the tunneling states. As charge transfer occurs in type-I1 multiplequantum-well structures (i), these are considered as semimetallic superlattices, with the exception of ultrathin structures where energy quantization is so large that energy levels are raised enough in the respective bands to prevent any charge transfer.
rest of this chapter we shall refer to our quasi-2D systems merely as 2D systems. The most widely known devices exploiting 2D motion are the quantumwell lasers”J8and the SDHT-TEGFET - HEMT - MODFET heterostructure transistor^.'^-^* The parallel transport properties of n - i - p - i structuresZ2 might prove useful in some devices like the heterojunction modulation superlattice. Due to the extraordinary properties of the quantum Hall effect,23some applications might be found in high-performance
4
C. WEISBUCH
gyromagnetic devices. Quantum Hall structures are already being widely used as standard resistors in numerous national standards laboratories. In some cases, we will be interested in the one-dimensional phenomena occurring along the z direction, either due to our search for perpendicular properties (i.e., perpendicular transport) or due to the unconfined extension of the wave functions in the z direction (superlattices or type-I1 multiple wells). Devices using these one-dimensional properties rely on tunneling or superlattice transport. Whereas tunneling devices such as the tunneling transistorsu or negative differential resistance (NDR) tunneling diodes25have been demonstrated, clear superlattice effects have so far remained elusive. Great efforts are being devoted to the fabrication and understanding of true quantum one-dimensional systems best viewed as quantum well The third family of devices shown in Table I relies on charge transfer, either static or dynamic. In the static case the charge transfer occurs between heterodoping and/or heterocomposition structures, leading to the appearance of electrostatic confining potentials due to depleted charges. Some of the 2D systems discussed above rely on this charge-transfer effect. Dynamic charge transfer occurs when electric-field-heated carriers can overcome potential bamers in heterostructures, leading to diminishing conductance and thus to NDR.2S-30 Bandgap engineerin?' consists of the tailoring of an association of materials in order to custom design the structure for some desired properties unattainable in homostructures. A very good prototype of such structures is the double-heterostructure l a ~ e r , ' ~where , ' ~ one increases both the carrier confinement and optical wave confinement by using a heterostructure. It is clear that all the devices described above can be viewed as being due to bandgapengineered structures. A number of other structures have been recently developed that do not involve space quantization in ultrasmall structures. These are shown in the lower part of Table I. As can be seen in Table I, the variety of devices which have now been demonstrated is quite overwhelming, although the first devices (quantumwell lasers and modulationdoped structures) only appeared in the late 1970s. The present review aims at presenting the basic physical phenomena encountered in these devices. The field is already so large, however, that we have concentrated on the basic phenomena encountered in the simplest and most widely used semiconductor pairs, the so-called type-I quantum wells and interfaces, where the small-bandgapmaterial has both its electron and hole levels confined by the wider-bandgap materials. The other configurations (type-I1 quantum wells) have been thoroughly r e v i e ~ e d . ~ ~ - ~ ~ More details on strained-layer superlattices and their applications can be
1. II1-v SEMICONDUCTOR QUANTIZED STRUCTURES
5
found in reviews of this young but rapidly developing field,36-38including that by Osbourn et in this volume. The new field of amorphous s e m i c o n d u ~ t o r s ~is- ~too ~ far afield and will not be considered here, although many of the tools developed here can be applied to that subject. Bandgap-engineered structures are reviewed in this volume by cap ass^.^ * We do not cover the basic phenomena (nonlinear absorption and dispersion, electrooptic effects, etc.) to be used in optical signal processing devices, since they are described in a definitive manner along with applications in the chapter by Chemla et aLU 2. A PREREQUISITE: THEMASTERING OF SEMICONDUCTOR PURITYAND INTERFACES
The mastery of layer growth is a prerequisite to all the structures which will be discussed in this chapter. We therefore wish to give an overview of the achievements in that field, referring the reader to more specialized texts for details. Quite different techniques have been used to grow quantized structures such as MBE,5-'2 metal- organic chemical vapor deposition (MOCVD),4s,46 hydride vapor transp~rt,~'~~* hot-wall epitaxf9 (HWE), or even liquid- phase epitaxySO(LPE). One can even trace through time how progress brought about by such a near-perfect growth technique as MBE has induced parallel spectacular progress in other growth techniques by demonstrating new and attainable goals. The highly detailed control of crystal growth in MBE has been crucial to its progress and is due to the UHV environment, which allows for the implementation of powerful in situ analytical techniques. The growth sequence in an MBE chamber uses specificcharacterizations to ensure that each growth step is correctly carried out: before growth has started, mass analysis of residual molecules in the chamber detects any unwanted molecular species. Molecular beam intensities are precisely controlled by ion gauges. Substrate cleaning is checked by Auger electron spectrometry, which analyzes the chemical nature of the outer atomic layer. Reflection high-energy electron diffraction (RHEED) patterns monitor surface reconstruction after ion cleaning, annealing, and also during atomic layer growth. Studies of atomic layer growth through desorption measurements and RHEED analysis have provided a detailed understanding of MBE growth rne~hanisms.~'-~~ RHEED oscillations due to recurrent atomic patterns in the layer-after-layer growth mode provide a very useful means of measuring layer thickness and are being more and more widely TEM measurements of grown films have evidenced the smoothing effect of MBE growth on the starting substrate's roughnesss7(Fig. 2). Although the growth kinetics of the MOCVD process is not as well monitored as that of MBE, recent progress leads to believe that MOCVD
6
C. WEISBUCH
FIG.2. Smoothing action of MBE quantum-well growth on interface roughness as observed in a dark-field transmission electron micrograph. The roughness of the starting GaAs surface is smoothed out by the growth of 3 to 5 quantum wells (courtesy of P.M. Petroff, AT&T Bell Laboratories).
growth leads to similar control of impurity content and interface a b r u p t n e ~ s (Fig. ~ ~ - ~3). ~~ A vast amount of effort has also been devoted to characterization of interfaces, using various ex siiu techniques such as chemical etching,62 beveling,62SIMS,63Auger,MTEM,65and x ray^.^^,^' The latter two tech-
1. 111-v SEMICONDUCTOR QUANTIZED STRUCTURES
7
AL,,Ga,,As AlAs -GaAs -Al As
AlAs AleGa ,AS
Al , F a Al ,Go Al 65Ga3 5 A ~ Al ,Ga ,As -GaAs
GaAs substrate
FIG. 3. (a) TEM characterization of a test sample grown by MOCVD. The growth sequence and the structure are shown in (b). The remarkable features are the sharpness of the very narrow GaAs layers (minimum -25 A) appearing at the lower right-hand side comer, the interface roughness showing up at the uppermost interface of the A l A s layer, and the subsequent smoothing of this roughness by the multilayer growth (upper left-hand side corner) (after Leys el aL6').
niques have been shown to yield extremely precise information on a microscopic scale (Fig. 4). It has been thus shown that the preferred growth techniques, MBE and MOCVD, which are far from equilibrium growth processes, allow very low growth rates and thus good control for desired abrupt changes. Hot-wall epitaxy, an evaporation method, also leads to good interface control but has been used much less, due to the required high-purity bulk material. VPE and LPE are near-equilibrium methods with large growth rates and instabilities in the regime where redissolution (LPE) or etching (chlorine VPE) could diminish the total deposition rate. Stringfellow68 also involved C1 absorption in the C1-VPE method as a limitation to atomic in-plane motion and hindered coalescence of islands during atomic layer formation. Frijlink et pointed out the strong reactivity of aluminum chloride with reactor material, forbidding growth of Al-containing structures with the Cl method.
8
C. WEISBUCH
FIG. 4. High-resolution electron micrograph of a Gas-GaAlAs interface. The arrows point to the interface plane, which appears very smooth on the atomic scale (courtesy P. M. Petroff, AT&T Bell Laboratories).
In terms of purity, the two techniques have now emerged as those yielding the best bulk material ever grown by any technique; high mob& ties and sharp luminescence intrinsic peaks attest the high quality of MOCVD69,70and MBE-grown GaAs7'Recently, MOCVD-grown InP72 has given mobilities of 195,000 cm2 V-' s-' at 77 K. The steady progress in recent years can be traced to the availability of purer source materials and to a better control of the growth environment. The latter point is especially well documented for MBE, where introduction of better pumping systems, liquid-nitrogen shrouding of the growth space, and vacuum interlock transfer of substrates each brought significantly improved material properties. The amount of effort still being made in the basic understanding of growth methods, the better quality of starting substrates, and the availability of ever-purer starting materials should lead to still increasing material quality. The range of materials now grown in ultrathin layers is extremely wide, and we shall not attempt to list them, as the rate of appearance of new ultrathin materials is still high. It has been widely believed that high-quality material could only be grown with layers perfectly lattice matched to the substrate, although it was remarked very early73.74 that no misfit dislocation generated by the mismatch would occur if the epitaxial layers were sufficiently thin, allowing the mismatch to be fully accomodated by elastic strain. The realization of this effect led to the consideration of ultrathin
1. 111- v SEMICONDUCTOR
QUANTIZED STRUCTURES
9
multilayer structures with a much wider set of materials than with latticematched ~ombinations.7~ Within the allowed range, the choice of layer thickness allows one to select a strain value which offers an additional parameter for the tailoring of electronic properties. The most promising recent systems are at present HgTe/CdTe,75%bwhere the superlattice growth should allow an easier control of bandgap than in LPE-grown alloys7$ InAsSb/InAsSb, where the lattice strain should permit one to decrease the bandgap in the 10 p m range77;CdMnTe/CdTe,78*79,79a which has fascinating magnetic properties; and Ge,Si ,-x/Si,80*80a where the strain could allow one to reach the 1.77 pm range for photodetectors and might also lead to direct-gap material on a Si substrate. This chapter is organized as follows: the electronic properties of thin heterostructures (quantum wells, selectively doped interfaces, etc) are first described. We then analyze the current understanding of their optical and transport properties.
11. The Electronic Properties of Thin Semiconductor Heterostructures
3. QUANTUM WELLENERGY LEVELS
a. Conduction Electron Energy Levels The simplest quantum situation to be dealt with consists of a single layer of material A embedded between two thick (thickness much greater than the penetration length of the confined wave function) layers of material B, where B has a bandgap larger than A and where the band discontinuities" are such that both types of carriers are confined in the A material (Fig. lc). This is the situation exemplified by the pairs of materials GaAs/GaAIAs, GaInAsP/InP, GaInAs/AlInAs, GaSb/ AlSb, etc. The energy levels in the conduction band can be calculated quite easily in the approximation ofthe envelope wave function,82-82dusing a Kane models3 for describing the electron and hole states of the parent A and B materials.84The approximation assumes (1) an interface potential strongly localized at the A-B interface, which means that on the scale of variation of the envelope wave function the interface potential is well localized at the geometrical interface, and (2) an interface potential which does not mix the band-edge wave functions but only shifts them, which is plausible due to the very different symmetries of the conduction and valence bands. It can then be shown82 that the electron wave function takes approximately the form
10
C. WElSBUCH
where z is the growth direction, kl is the transverse electron wave vector, u&(r) is the Bloch wave function in the A or B material, and x,(z) is the envelope wavefunction, determined to a good approximation by the Schrodinger-likeequatioP
where m*(z) is the electron effective mass of the A or B material, Vc(z) represents the energy level of the bottom of the conduction bands, and E, is the so-called confinement energy of the carriers. Therefore, the early description of the energy level scheme by simple confinment in a quantum well due to energy-band discontinuities in Dingle's work1,14can be well justified. The continuity conditions at the interfaces are that x,(z) and [ l/m*(z)][dx,(z)/az] should be continuous. In the earlier works, the usual continuity condition of the derivative of the wave function was used. That condition is derived in elementary quantum mechanics textbooks148by assuming a constant free-electron mass throughout space. It was showns2 that in the semiconductor QW case the particle current is not conserved and one rather has to assume continuity of ( I / m * ) ( d X / a ~ ) . ~ ~ 9 ~ ~ In the injinitely deep well approximation, the solution to Eq. ( 1 ) is very simple, as the wave function must be zero in the confining layer B, and therefore also at the interface because of the continuity equations. Taking the z abscissa origin at one interface (Fig. 5), the solution of Eq. ( I ) can evidently only be - sin(naz/l), n being an odd or even integer. The confining energy E , is then simply n2(a2h2/2m*L2)from Eq. (1).
PARTICLE IN A BOX
ONE DIMENSIONAL CASE : VO= CO
-
n=3
(g)
d2*/dZ2 = EJI
tl2
En=%
(F)
n m I#,= Asin LZ
2
n=1,2,3---
n=2
n=l
FIG.5. Infinitelydeep quantum-well energy levels and wave functions (after Dingle').
1. 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES
11
The Schrodinger-like equation (1) in the finite-well case with the aforementioned boundary conditions can be exactly solved to yield the wave functions and energies. Noting that the problem has an inversion symmetry around the center of the well now taken as the center of coordinates (Fig. 6), the solution wave functions of (Eq. I ) can only be even or odd. Therefore, they can be written as (writing for simplicity kl = k) for Iz(< L/2
xn(z)= A cos kz, = B exp[-K(z = B exp[+ ~
for z > L/2 for z < -L/2
- L/2)], ( zL/2)],
+
(2)
or x,(z)
=A
for lzl< L/2,
sin kz,
for z > LJ2 for z < -L/2
- L/2)], = B exp[+ ~ ( zL/2)], = B exp[- K(Z
+
(3)
where A2K2 A 2k2 vo, En=--Vo<&
En=--
A cos(kL/2) = B
(kA/m2)sin(kL/2) = IcB/m;
TE ,-sin
I
-COS
-L/2
0
+
Ll2
kz KZ e-
.
kz
2
FIG.6 . First two bound energy levels and wave functions in a finite quantum well.
12
C. WEISBUCH
Therefore (kA/rn2)tan(kL/2) = Ic/ms
(5)
Similarly, Eq. (3) yields k/rn,* cotan(kL/2) = - Ic/ms
(6)
The equations can be solved numerically or graphically. A very simple graphical type of solution can be developed if rnz = rnz. Then, using Eq. (4), Eqs. ( 5 ) and (6) can be transformed into implicit equations in k alone: cos(kLf2)= k/ko,
for tan kL/2 > 0
(7)
sin(kLf2)= k/ko,
for tan kL/2 < 0
(8)
where ko = 2rn* Vo/fi2
(9) These equations can be visualized graphically (Fig. 7). There is always one bound state. The number of bound states is 1
25zL2)”2]
+ Int [(
where Int[x] indicates the integer part of x. The important limiting case of the infinitely high barriers (Fig. 5 ) can be found again by putting ko = 00 in Fig. 7. There is then an infinity of bound states with k = m/L. Even solutions arex, cos kz, with kL = (2n 1)n; odd solutions are x,,- sin kz, with kL = 2nn. X, even and X, odd are the
-
+
FIG.7. Graphical solution for a s . (7) and (8). Solutions are located at the intersections of the straight line with slope with curves y = c~~kL/2 (with tan kL/2 > 0; -; even wave functions) or y = sin kL/2 (with tan kL/2 < 0 ---; odd solutions).
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
13
usual solution of the infinitely deep well xn = sin(naz/L), n integer even or odd, in the natural choice of coordinate origin z = 0 at one interface.
b. Hole Energy Levels Turning to the hole quantization problem, the situation is much more complicated in usual semiconductor materials. The bulk hole bands are described in the Kane model by basis functions with angular momentum J = 3 symmetry, i.e., 4-fold degeneracy at k = 0 (neglecting the spin-orbit split-off valence band). The dispersion near k = 0 can be described by the Luttinger Hamiltonian?
H=-
fi2
2m0
KY 1 + 3r2)k2- 2YZ(k$E + k$G + k:J3
- 4Y3({k,
* kyHJx
-
Jy
+
* *
- 11
( 1 1)
where yI , y z , y3 are the Luttinger parameters of the valence band and the symbol { * > represents the anticommutation
In the bulk, propagation in a given direction can be described in terms of heavy- and light-hole propagation. Taking as a quantization axis z for the angular momentum the direction of propagation of the hole, the levels J, = f4 and J = -t 4 give a simple dispersion relation from Eq. ( 1 1). Taking for example k, in a [ 1001 direction, the kinetic energy of holes is
for J, = k $ for J, = &+
One obtains the usual [ 1001 heavy-hole mass mo/(y,- 27,) and lighthole mass mo/(yl 27,). For hole levels in a quantum well, in a successive perturbation approach, one first treats the quantum-well potential as a perturbation to the k = 0 unpertur,bedstates, then adds the Luttinger interaction as a new perturbation to the quantum-well levels.85As a first perturbation, the quantum-well potential lifts the degeneracy between the J, = kj and &+ bands as they correspond to different masses. According to the Luttinger equation, Eq. (1 l), inserting the values k, = k l , k, = k, = 0, the k dispersion in a [ 1001
+
14
C. WEISBUCH
direction perpendicular to z is then given by
E
The transverse dispersion equation corresponding to J, = k$(heavyhole band along the z direction), now has a light mass (m&, yz), whereas the J, = level now has a heavy muss (Fig. 8). This situation is quite similar to that developed under a uniaxial compressive stress in the [ 1001direction.86The difference here is that the $ band is the higher-lying one. Due to the lighter mass of the $ band, one initially expects a crossing of the two bands. However, higher-order k p perturbation terms lead to an anticrossing behavior, which increases the “heavy-hole” band mass and decreases the “light-hole” band mass. Actually, the above procedure, which describes qualitatively the complicated valence-band effects, is not correct. One has to treat on equal footing the k p perturbation, which yields the dispersion, and the dimensional
+
++
E
E
lE
l
Kv
0 FIG.8. Hole dispersion curve in a simple-mindedsuccessive perturbative calculation.The quantum-well potential lifts the 4-fold degeneracy of holes [in 3D, (a)] at k = 0. (b). The k -p interaction term as described by the Luttinger Hamiltonianthen yields the dispersion in the y direction (for example) (c); finally, higher-order terms lead to an anticrossing behavior, (d).
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
15
perturbation introduced by the quantum well. In the degenerate set of valenceband levels at k = 0, one has to diagonalize a perturbative Hamiltonian:
H = HL.) + Hqw
(13)
As is well known, the first-order solutions are linear combinations of the
k = 0 valence-band wave functions [when HQw = 0, they are the J, = k + and 314 functions with the dispersion given by Eq. (12)l. Complications arise here because of the boundary conditions which have to be simultaneously satisfied for the quantum well. The set of functions which diagonalize: is not a basis set for H,,, and strong mixing of the J, = kt and 23 bands is required to satis@the boundary conditions, as was recognized as early as 1970 for the infinitely deep well.87More recent works have dealt with various band situations,88finite wells,89and the additional influence of magnetic field^.^^.^ One should note in addition that this effect strongly influences the value of the exciton Rydberg. Also, the strong nonparabolicity of the valence bands should influence dramatically the valence-band density of states from an exact steplike shape to more complicated shapes, which require a detailed knowledge of the valence-band levels. The case of an infinitely deep well has been treated a n a l y t i ~ a l l y . ~ ~ ~ ~ ~ Neglecting band warping in the spherical approximation (equality of the Luttinger parameters y2 = y3 = F), the energy levels at k = 0 are given by the usual uncoupled levels series:
The dispersion for kL = k,, # 0 is given by the dispersion equation
+
4[k&kL k;(Pb +6kj%&dI
+ k;) + 4k;] sin k& - cos k&
sin k,J
cos k&) = 0
where
It is then possible to derive effective masses in the layer by
One finds then that for GaAs some of the heavy-hole subbands have positive (i.e., electronlike) masses, independently of the width of the well.
16
C. WEISBUCH
Only numerical results were obtained in the finite-well calc~lations,8~ but the features obtained in the infinitely deep well approximation (nonparabolicity, positive hole masses) are retained or even emphasized. Such effects have been considered to explain magnetic field measurements of absorption spectra,g0l u r n i n e ~ c e n c e , ~ and - ~ ~cyclotron resonance of holes in modulation-doped heterojun~tions?~-~~ Tight-binding calculations have also led to nonparabolicities of hole dispersion C U I - V ~ S , ~in ~ ~close ~’ agreement with the envelope wave-function approximation calculations (Fig. 9). The effect of the symmetry of the confining potential on hole levels has been shown by Eisenstein et aL9*by comparing modulation-doped single or double (quantum-well) heterostructures. The asymmetric single heterostructure reveals in magnetotransport a lifting of the spin degeneracy of hole bands.
\
u -0.02
J
.
-0.03I
I
0.04
0.03
0.02
-[I101
0.01
0
F+Ef)
0.01
0.02
0.03
0.04
[loo]--
Transverse Electron Wave Vector
FIG.9. Calculated transverse dispersion curves in a GaAs/GaAlAs MQW in an LCAO model. The QW and bamer thicknesses are, respectively, 68 and 71 atomic layers. The double curves correspond to spin-orbit-split bands as the Kramers degeneracy is lifted at k f 0. Note the negative masses of some heavy-hole bands and the strong nonparabolicity (from Chang and S ch ~ I ma n ~~ ) .
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
17
A number of properties of quantum wells show different behavior than in 3D structures, thanks to their bidimensionalityWJw;we shall discuss them in the following paragraphs. 4. TWO-DIMENSIONAL DENSITY OF STATES
Besides the energy quantization along the z axis, the main property of thin quantizing films is the bidimensionality in the density of states (D0S).'OoAs the motion along the z direction is quantized (k, = nn/L in the limit of an infinitely deep well), an electron possesses only two degrees of freedom along the x and y directions. The spin-independent k-space density of states per unit area transforms into an! E-space density of states through the usual calculation of k states allowed between the energies E and E dE
+
In the parabolic approximation E = fi2k$/2m*,Eq. ( 18) yields p2D = m*/di2
(19)
The density of states of a given quantum state En is therefore independent of E and of the layer thickness. The total density of states at a given energy is then equal to Eq. (19) times the number n of different k, states at that energy (Fig. 10). The 2D DOS shows discontinuities for each En.It is
v)
W I-
< Iv)
ti >
k
v)
2
W
n
3rn'lnh2
P N
I El
€2
€3
E4
ENERGY
FIG. 10. 2D density of states (DOS) and comparison with the 3D DOS calculated for a layer with a thickness equal to that of the quantum well (after Dingle').
18
C. WEISBUCH
interesting to compare the 2D density of states with the 3D areal DOS, calculated for a thickness equal to that of the layer. From p3D= 21/2m*3/2n-2fi-3E1/2
(20)
one finds that, in the infinitely deep well approximation, using the expression of Enand (19), = P2D; as shown in Fig. lo, the 3D and 2D densities of states are equal for energy En. Two remarks can be made: (1) One should not conclude that there is little difference between 2D and 3D systems, even though one can always find an energy in a 3D system for which the DOS is equal to that of a 2D system. The important point here is that the DOS isfinite even at the bottom of the 2D level, whereas it tends towards zero in the 3D system. This has fundamental consequences on the properties of 2D systems as it means that all dynamic phenomena remain finite at low kinetic energies and low temperatures, such as scattering, optical absorption, and gain. (2) On the other hand, when numerous levels are populated or when one looks at transitions involving large values of n, such as in thick layers, nothing can distinguish between 2D and 3D behaviors, analogous to the correspondence principle between quantum and classical mechanics.
5 , EXCITONS~O~ AND SHALLOW IMPURITIES'o2 IN QUANTUM WELLS'O3
From the expression of the Bohr radius in semiconductors
where eRis the dielectric constant, p is either the effective mass (impurities) or the reduced mass (excitons), one infers that the wave function and energy levels of excitons and impurities are quite modified in a quantum well where the thickness is usually of the order of or smaller than the Bohr diameter 2a,. In the limiting exact 2D case where L << aB, one should obtain the usual 2D Rydberg value RzD= 4R3Dfor the infinitely deep The energy levels are then given bylWJo5
For finite thickness, exciton binding energies have been calculated through a variational method. The perturbative exciton Hamiltonian is,
1. 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES
19
REDUCED WELL THICKNESS ( L / a B ) FIG. 1 1 . Exciton binding energy and Bohr radius in the infinite-well approximation. Curves ( 1 ) and (2) display calculationsfor, respectively, separated or nonseparated [Eqs. (23)
and (23a)l exponential factors in the wave function. The nonseparated wave functions of Eq. (23a) give back the usual 3D exciton quantities at large L. (a) Exciton binding energy; (b) ( d )'I2, (c) ( z, z, ) (after BastardlOg).
-
20
C. WEISBUCH
assuming nondegenerate, isotropic bands,
where m:, m;,z,, z h are the masses and z position of the electron and hole, respectively,P, and P,,projections on the x and y axes of the center-of-mass exciton momentum, p x and p,, the relative-momentum projections, and p the reduced mass. Bastard et ~ 1 . used ’ ~ variational wave functions totally confined in the well such as W A ( ~= ) N(L, 2) cos(nze/l) cos(azh/L) ex~[-(p/A)l
(23)
or ty{(r) = N(L, A’) cos(nz,/l) cos(nzh/L)exp{- [p2
+ (2, - Z ~ ) ~ ] ~ / ~ / A ’ ) (234
and N(L, A), where A and A’ are variational parameters, p = (x2+ y2)1/2y N(L, A’) are the normalizing coefficients. The nonseparated exponential factor in the spatial coordinates’06of Eq. (23a) ensures some amount of Coulombic binding even when the quantum well is wide as compared to a variable-separated factor. Binding energies and reduced Bohr radii are shown in Fig. 1 1 as a function of the reduced well thickness. At vanishing L, RZDextrapolates to 4R3D and p to a,J378. More accurate exciton energies taking into account the well finiteness have recently been calculated, however using simple parabolic hole bands’,’ (Fig. 12). The increase in exciton binding energy (Fig. 13) has a profound influence on quantum-well properties. It allows GaAs-based quantum wells to have their optical properties dominated by exciton effects even at room temperature. This is a rather unique instance in standard semiconductors? Usually large exciton binding energies are associated with large reduced masses,Io1i.e., large gaps (according to Kane’s model), then to large ionicity (Phillips’ theory of ionicitylo8)and therefore to strong LOphonon coupling which ionizes excitons at room temperature. The roomtemperature excitons in quantum wells allow very promising features such as optical bistability, four-wave mixing, and large electrooptic coefficients which are developed in the article by Chemla et u1.Shallow impurity effects have been widely calculated using a number of approximations.Im-‘l 1 It should be noted that the problem is somewhat complicated by the degree of freedom brought about by the position of the impurity relative to the well interfaces. First, variational calculationslW
1. 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES
13
I
\
11
I
I
I
I
I
21
I
\ \ \ \
\
\\\
\
9
7
I 5
I
0
I
I
100
I
I
I
300
200
WELL THICKNESS
I
400
(A)
FIG. 12. Exciton binding energy for finite quantum wells as a hnction of well thickness. Three barrier potentials are shown corresponding - to x
rier case. Heavy-holeexcitons (-) et aI.'O').
-
0.15 and 0.30 and the infinite-barand light-hole excitons (---) are displayed (from Greene
considered an infinite potential at the interface. In this approximation, the wave function of bound particles must vanish in the barrier. The wave function for impurity atoms at the bamer must therefore be a truncated plike state, whereas for impurity atoms located at the center of the well s-like wave functions are allowed. One expects that the ground-state binding energies for these two states in the large-well limit to be Rhp/4 and Rhp (Rhp being the 3D impurity binding energy).
22
C . WEISBUCH
N-2
N-3
fl
3 D Sornrnerfeld F a c t o r
/* 0
--*----
-
without exciton effect
/’ #
P
v
4
t
a
(b)
2D EXCITON N- 1
N-2
N-3 r / 2D Sornrnerfeld F a c t o r
----------2 without exciton effect E FIG. 13. Comparison of the absorption coefficientsdue to (a) 3D or (b) 2D excitons. The characteristic energy is = 4 times larger for 2D excitons. Oscillator strengths are increased (- a i 3 in 3D, G~in 2D). For continuum states, the absorption coefficient is increased over the excitonless value (-) by the Sommerfeld factor, determined by the continuum wave functions of the hydrogen atom, which represents the effect of electron-hole correlation in unbound states.
The calculation starts from the perturbation Hamiltonian:
H = - -P-2 2m*
e2
+ +
[x2 JJ’ (Z - ~ i ) ~ ] - ” ’ 4 7 1 ~ ~ ~ ~
+ Vm,X~)
(24)
zi being the impurity atom position and Vmdz)the quantum-well confining potential (defined by energy-band discontinuities).
1. 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES
23
In the infinite-well approximation, trial wave functions are taken as
+
yA(r)= N(L, A,, zi) cos(nz/l) exp{ - [x2 y 2 -I-(z - Z ~ ) ~ ] ~ / * / (25) A) if JzJ< L/2, and yA= 0 otherwise. The binding energy measured from the confined quantum state is given bY E(L, zi) = n2A2/2m*L2- min,( yAIHlwA) where minAmeans the minimum value of (w,lHlw,) with respect to the variational parameter 1. Exact solutions are found when L - 0 or 03. For L - w , one finds E(L, 0) = Rimpand E(L, 2L/2) = Rimp/4. When L 0, one finds the usual 2D result:
-
E(L, zi)
-+
4Rkp;
(n
(
yA(r)-+ 1 8)'12 exp -2[x2
+
y']'/2)
aB More detailed calculations' lo-'' ' have taken the finite bamers into account. Mailhot et al."' have also considered ion image charges due to the different dielectric constants of GaAs and GaAlAs. In the limit of vanishing well thicknesses, one expects to recover the 3D GaAlAs donor energy, since in that case the confinment effect of the wave function due to the GaAs well becomes vanishingly small for finite barriers. Measurements of the donor energy levels by electronic Raman scattering112and infrared absorption' l 3 are in good agreement with the theoretical evaluations. As in 3D, the calculation of the energy levels of the acceptor impurities is much more complicated than for donors due to the degeneracy of the valence band. Masselink et al. has recently provided a detailed calculation. l4 Various other situations have been studied Cha~dhuri"~ considered the influence on the binding energy of the spreading of the impurity wave function in superlattices. The influence of high camer densities on the impurity binding energies in modulation-doped QWs' l6 and superlatt i c e ~ "was ~ also calculated. Finally, the impurity bound states associated with excited quantum-well subbandswere analyzed by Priester et a/."*and observed by Perry et al. l9 in Raman scattering studies. aB
'
6. TUNNELING STRUCTURES, COUPLED QUANTUM WELLS, AND SUPERLATTICES Tunneling phenomena across bamers open the way to many fascinating effects, the most eagerly expected one being the Bloch oscillator (to be described in Section 19). The renewed interest in transmission across simple systems such as single barriers, double bamers, etc. also lies in the
24
C. WEISBUCH
recent availability of the high-performance growth techniques developed for the multiple heterojunction superlattice. The advances in growth techniques are evidenced by the symmetric I- V characteristics now observed. Whereas transport properties and related structures will be discussed with experimental results in Section 19, we develop here the energy-level schemes of these communicating multiheterointerface devices. Several calculation techniques can be used, such as the Kronig-Penney successive multiple tunneling model, I2O perturbative tight-binding model,121or the LCAO model.122We will use here the simplest descriptions by a tight-binding model and also give the results of Kronig-Penney calculations. a. The Double- Well Structure
Beyond the double-bamer single-well structure (Fig. If) which leads to zero-bias electronic properties very similar to the single QW previously described, the simplest structure is the double-well configuration (Fig. 14), which can be easily analyzed by the usual tight-binding perturbation model. As the barrier thickness is decreased, the exponentially decaying wave function in the barrier can have some finite value in the next well. Treating this wave-function overlap as a perturbation, one finds the perturbation matrix element to be, in a two-well configuration,
(26) where H is the electronic Hamiltonian, y, and y2 the unperturbed wavefunctions of single wells 1 and 2, and V2the confining potential of well 2. Due to this interaction between the wells, the two degenerate well levels are split into a symmetric and antisymmetric compound wave function with energies E l - V,, and E l VI2, re~pectively.'~~ It should be noted that the double-we11 Schrodinger equation has an exact solution given by124 v 1 2 = (WIIHlY2/2>= (YIIv21w2>
+
+ (<- I/<) sin kL f (g + 1/0sin kL e-& = 0 ( 2 7 ) is the barrier thickness, < = K / k , K = [ - 2 r n ~ / f i ~ ] and '/~ k = 2 cos kL
where b [2m(e Vb)/fi2]'/'.The (k) sign corresponds to the antisymmetric and symmetric states relative to the center of the structure, respectively. The energies are counted from the top of the wells. When b m, Eq. (27) gives back the isolated well equation [in a form which can be transformed into Eq. (5)i. b. The Communicating Multiple-Quantum- Well Structure or Superlattice
+
Introducing more wells leads to the creation of a continuous band of states. The transition from single wells to multiply connected wells, as
1. 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES
25
COUPLED WELLS
_--_---------- - ---------------'
-
-
I
-
--
*-.
---- ------_ -
I
I
I
v2
2 DEGENERATE STATES
W,
AND W2
PERTURBATION MATRIX ELEMENT V12 + ANTISYMMETRIC STATE
E=E1tV12
- SYMMETRIC STATE
FIG. 14. Double-well structure: the unperturbed potentials (V,, V,) and wave functions and (-) respectively.
(r,, v2)of the separate wells are represented by (-)
revealed by optical absorption, has been studied by Dingle et al.Iz3For N wells, the N-degenerate levels give rise to bands with 2 N states. The simplest way to analyze this is to consider a tight-binding model of the N-well chain125(Fig. 15). The Bloch-like envelope wave function can be written as
where xfbc(z- nd)is the ith wave function of the quantum well centered at z = nd and g is the Bloch wave vector. Assuming a nearest-well interaction, the energy is
+ + 2ti cos qd
&,.(q)= Ei si
(29)
with si= E & ( z - d ) V ( z ) ~ [ &- d ) dz
ti = ~ & ( z ) V ( Z ) ~ $-(dZ) dz
(30)
The factor of 2ti in Eq. (29) that yields a bandwidth of 4ti as compared with 2 V,, in the double-well case comes from the interaction of one well with its two neighbors in the chain. The variation of the electronic bandwidth in GaAs MQW is shown in Fig. 16.
26
C. WEISBUCH
SUPERLATTICES
d
loc N WELLS N-DEGENERATE GROUND STATE TIGHT-BINDING APPROXIMATION
I E = E,+S +2Tcosqd(
2N states usin
BVK
boundary condigions
I I
lid
q
FIG. 15. Tight-binding model of superlattices.
Assuming usual Born - von Karman periodic conditions, one finds that q can only take discrete values which are integer numbers of l / N d Therefore, the superlattice band can accommodate 2 N electrons with different quantum states [different q’s in Eq. (28)]. The superlattice effect introduces a profound change in the 2D DOS. The dispersion of the N states in a band destroys the steepness of the square density of states. From the energy E,*(q, k,) = fi2k:/2m + E,(q)
1. 111- v
SEMICONDUCTOR QUANTIZED STRUCTURES
27
c
5
E
c d
I
t 0
z
z a
D
w
0
8 8 100 A
1 LA = 30
II-
a
2 LA
a
w
I
3 LA =
n
50
3
v)
0
50
100
BARRIER THICKNESS
150 LB
(8)
FIG.16. Tight-bindingmodel of GaAs/GaAIAs superlattices: Variation of the fundamental state bandwidth [4ti of Eq. (30)] in the tight-binding model as a function of barrier thickness for three different well thicknesses. X = 0.2; V, = 212 meV (after Bastard*25).
one finds a density of states as represented in Fig. 17:
This is obtained by summing over q values and over the various bands the 2D DOS corresponding to the transverse free motion of a single q state. There are still two singularities of the density of states at both extrema of each band. One sees that the 2D limit for N independent wells is retained when the bandwidth goes to zero, i.e., when the overlap matrix element vanishes due to wide barriers. More precise calculations of the band structure of superlattices have been carried out in the envelope wave function approximation, using the Kane model to describe the band structure within each well and barrier. BastardE2has shown that for kl = 0, in the parabolic band approximation,
28
C. WEISBUCH
-
a b/
11
c /I/-SUPERLATTICE
4
I
ENERGY E FIG. 17. Comparison ofthe DOS of a superlattice with that of a 2D system (--) and a 3D isotropic system. Note the broadening of the superlattice band with band index as the overlap of wave functions increases with energy E in the tight-binding description, increasing the transfer matrix element ti(from Esaki').
the equation yielding the values of q takes the simple Kronig- Penney form COS
qd = COS kALA cash 7C&g - +(I/< - r ) Sin kALA sinh IC&g
<
(32)
with = mfkA/mgkB. The allowed energy bands are given as usual by - 1 5 cos qd 5 1 (Fig. 18).
The solutions of unbound states (E > 0) are similarly given by COS qd = COS kALA COS kBLB - f(I/<
+ r ) Sin kALA Sin kBLB
(33)
Equations (32) and (33) can be solved graphically. The limit of noncommunicating wells is found in Eq. (32) for K ~ L 00, which ~ leads to cos kALA- f(l/( - r ) sin kALA= 0 , the usual single-well equation [Eq. (5)]after simple transformation. For the hole bands, Bastard82considered uncoupled hole bands and showed that they obey Eq. (32) with a corresponding change of parameters. The situation is much more complicated if kl f 0, as in the single quantum well. The heavy- and light-hole states are mixed by the boundary conditions. Only numerical calculations have been camed ~ ~ t . ~ h ~ The limiting cases of very thin layers, where the envelope wavefunction approximation tends to break down, can be calculated using LCAO methods. Such calculations have been performed by Schulman et ~ l . , ' ~ ' , ' ~ ~ yielding results similar to the envelope approximation when the layer thickness is 2 6 - 8 monolayers. Superlattice effects involving band extrema other than at the r point ( X point of GaAs, for instance) have been shown by Mendez et a1.126 -+
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
29
SUPERLATTICE
0
10
20
30 40 50 60 70 WELL OR BARRIER WIDTH a IN A
80
90
100
FIG. 18. Allowed energy bands E , , E2, E,, and E4 (hatched) calculated as a function of well or bamer width (Lz= & = a) in a superlattice with a barrier potential V = 0.4 V. Note the existence of forbidden gaps even above the bamer potential (from Esaki4).
7. CONTINUUM STATES
It is a classic textbook14aexample that continuum states ( E > 0) in a quantum well can play an important role in the dynamics of incident particles. The transmission and reflection coefficients of a quantum well display resonances every time the condition kL = na is fulfilled. This is the quantum analog for the electronic de Broglie waves of Fabry-Perot resonances in classical wave optics. The particle spends a longer time in the quantum-well region, which should have important consequences regarding the particle capture by the well. One should note in Fig. 7 that, with decreasing well thickness, a new resonant continuum state pops out of the well whenever a bound state reaches the well top for kL = na.These states have been calculated by Bastard'*' in the envelope wave function framework, and by Jaros and Wong'28using pseudo-potential calculations. The
30
C. WEISBUCH
resonant continuum states should be of greatest importance in the camer capture of QW, as the reflection coefficient is near unity for such states.Iz9 The continuum states have recently been observed in optical studies of coupled wells124and by resonant Raman scattering.I3O 8. MODULATION DOPINGOF HETEROSTRUCTURES130a A major advance for potential high-performance devices was made when Stormer and Dingle et a1.131J32 introduced n-type modulation-doped samples (Fig. 19). The underlying idea is that, at equilibrium, charge transfer occurs across a heterojunction to equalize the chemical potential (i.e., the Fermi on both sides. Doping the wide-bandgap side of a GaAlAs/ GaAs heterojunction, electrons are transferred to the GaAs layer until an equilibrium is reached; this occurs because electron transfer raises the Fermi energy on the GaAs side due to level filling and also raises the electrostatic potential of the interface region because of the more numerous ionized donors in the GaAlAs side. The charge-transfer effect makes possible an old dream of semiconductor technologists, i.e., getting conducing electrons in a high-purity, high-mobility semiconductor without having
FVACUUM LEVEL
(I
jEgZ
FIG. 19. Schematics of the energy-band diagram of a selectively doped GaAlAs/GaAs heterostructure before (left) and after (right) charge transfer has taken place. The relative energy bands are, as usual, measured relative to the vacuum level situated at an energy Q (the electron affinity) above the conduction band. The Fermi level in the Ga,-,AI,As bulk material is supposed to be pinned on the donor level, which implies a large donor binding energy (x > 0.25).
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
31
to introduce mobility-limiting donor impurities. Since then, modulation doping has been applied to a number of situations involving various semiconductor pairs and also to hole modulation doping. The impressive development of the subject is due to the applicability both to basic science (2D physics, quantum Hall effect, . . .) and to very high-performance devices called equivalently HEMT (high-electron-mobility transistor), TEGFET (two-dimensional electron-gas FET), SDHT (selectively doped heterostructure transistor), and MODFET (modulation-doped FET).The pace of progress is shown by today’s performance (as of 1985): electron mobilities of 2 X lo6 cm2/V s, hole mobilities of 83000 cm2/V s, research production of ICs such as 4k RAMS,. . . . We will concentrate in this article on the basic properties of MD structures, referring the reader to other chapters of this book for more details on device applications. a. Charge Transfer in Modulation-Doped Heterojunctions The understanding of the mechanism of charge transfer in heterojunctions is of the utmost importance, as it determines the GaAs channel doping and sets the design rules for the growth sequence of the doped and undoped layers. There are three main phenomena to be determined in a self-consistent manner to calculate the Fermi energy throughout the structure and therefore the charge transfer. (1) The electric charges and field near the interface determine the energy-band bending in the barrier and in the conducting channel. (2) The quantum calculation of the electron energy levels in the channel determines the confined conduction-band levels. (3) The thermodynamic equilibrium conditions (constant Fermi energy across the junction) determine the density of transferred electrons. A very crude calculation can show the interplay of the various factors in a simple situation (Fig. 20). Assume that before the charge transfer occurs the potential is flat-band; after charge transfer of Nselectrons, the electric field in the potential well created can be taken as constant to first order, given by F = Nse/coc, (Gauss’s law). The electrostatic potential is then 4 ( z )= -Fz for z > 0. The Schriidinger equation for the electron envelope wave function is then
[p2/2m*- e4(z)lx(z)= q ( z )
(34)
A quantum calculation of the energy levels in the infinite triangular quantum well gives the ground state134,135
El = ( ~ 2 / 2 m * ) 1 / 3 ( 3 ~ e 2 N , / ~ 0 & ~ ) 2 ~ 3 (35)
32
C. WEISBUCH
/
AT
/
I---
-l
'-
E,
I
b
0
2
FIG.20. Scale-up of the right-hand side of Fig. 19. The various symbols are defined in the text.
One usually writes phenomenologically El = yMd3,with y to be determined experimentally. As charge transfer increases, the electrostatic confining potential created by the transferred electrons also increases, leading to the raising of the bottom of the conduction band E of channel electrons. Equilibrium at T = 0 occurs when the top of the filled states, given by (if only one confined subband is occupied) NS
E=E,+-=EI+P2D
Irh2Ns m*
is equal to the Fermi level on the GaAlAs side of the heterojunction. That level is equal to the bulk GaAlAs Fermi energy level, pushed downwards by the electrostatic potential Vdepbuilt up at the interface due to the depleted donor atoms. Assuming a constant doping in the GaAlAs, V,, is
where W is the depleted thickness. Counting the energies from the bottom of the conduction band at z = 0, one finds that AE,=E,+-
afi2N,
m*
+ed+eV,,
where E,, is the donor binding energy in GaAlAs. This assumes that the donor level in GaAlAs is sufficiently deep so that the Fermi level is pinned there (In other cases one has to calculate through standard procedures the Fermi level position in GaAlAs). Remembering that Ns = N,W, one ob-
I. III -V
SEMICONDUCTOR QUANTIZED STRUCTURES
33
tains the implicit equation in N,:
Various more exact calculations have been provided, relying on more or less rigorous bases and providing analytical or numerical results. They, however, rely on the very simple, although well-justified, assumption that the GaAs electron wave function has a negligible penetration in the barrier. Thus the electron energy levels are unaffected by the possible changes of the barrier electrostatic potential induced by charge transfer. As will be seen below, the wave function penetrates at most 20 A in the barrier, which is much less than the depleted thickness in the barrier. Therefore, the wave function is only determined by the barrier height, and only at second order by the electric field at the interface due to the depletion charges. It is then possible to uncouple the equilibrium conditions from the electrostatic and energy-level calculations. The additional ingredients of the more exact calculation follow.
b. Electrostatic Potential The different parameters entering the calculations of the electrostatic potentials are as follows: (1) The various layers have some degree of compensation that must be taken into account for evaluations of the charge transfer. As will be discussed in Section 17, these uncontrolled ionized impurities play a crucial role in the ultimate performance of devices. (2) Residual doping in the GaAs layer creates an electrical field in the resulting depleted region of GaAs, but also contributes to the potential in the barrier.135aFor ptype residually doped GaAs, this doping is described by the depletion charge due to the interface band bending in the depletion length LAof the material:
As the doping is usually quite small, the depletion width in GaAs is much larger than all other dimensions in the system and the potential due to these charges can be considered triangular (constant E field) in the region of interest. In the limit of large charge transfer, this potential can be almost neglected when compared to the field F, of transferred electrons: For NA= lOI4 ~ m - Nacp ~ , = 4.6 X 10" cm-2, L A = 4 X lo-* cm, Fdcp= 7.5 X lo3V/cm, whereas for N, = 5 X 10" cm-*, E, = 7.5 X lo4 V/cm. On the other hand, the detailed knowledge of the residual impurities potential is
34
C. WEISBUCH
extremely important in the case of small charge transfer. It also dominates in all cases for the determination of the electron excited states, as their wave function is very sensitive to the potential away from the interface due to the orthogonality of the excited-state wave functions with the groundstate wave function. (3) The case of n-type residual doping in GaAs is not so easy to sohe, as the Fermi level away from the interface cannot be evaluated independently of the charge transfer occurring at the interface. One usually treats this case as a quasi-accumulation case, considering that the situation is the limit of either a very small p-type residual doping (NdW = lo9 cm-2 in the case of and^'^^,'^') or that the Fermi energy far away from the interface lies 1 eV below the conduction band. Detailed calculations actually show that one or the other of these choices does not influence the calculated energy level. (4) An undoped GaAlAs “spacer” layer of thickness WV is usually used to separate the ionized donor atoms further from the channel electrons: increasing this spacer layer diminishes the Coulomb interaction between the ionized donors and the electrons, resulting in an increased m ~ b i l i t y . ’ ~There * J ~ ~is, however, a limit: since the electric field is constant in the spacer layer (no space charge in the absence of ionized impurities), the electrostatic potential builds up there, although it does not correspond to transferred charges. The consequence is that increasing the spacer layer width W, tends to decrease the channel electron density N,. An additional effect is the diminution of the impurity screening in the channel with decreasing N,, which should also diminish the mobility when limited by residual channel impurities. At low temperatures, the equilibrium Eq. (38) can be rewritten as
+
AE, = eVdW eV,
+ cd + El + nA2Ns/m*
(41)
with Vsp= e W # S / & & R . c. Energy-Level Calculation
The electron energy levels can be self-consistently calculated using approximations of various degrees of sophistication. The most widely used scheme is the Hartree approximation calculated using variational FangHoward-type wave functions. At lowest order, the electron-electron interaction Vee(z)is described by the Hartree approximation; i.e., Vee(z)is given by
1 . 1II-V
SEMICONDUCTOR QUANTIZED STRUCTURES
35
which expresses that an electron feels the average electrostatic field created by all others electrons. The Schrodinger equation is then
where Vo and Vi, are, respectively, the heterojunction and channel ionized impurity electrostatic potentials. The usual Fang-Howard" functions used for the Si-SiO, case (no penetration in the Si02 barrier) are modified to account for the penetration in the GaAlAs b a ~ ~ i e r I ~ ' * ' ~ ' ~ ( z= ) Bb'I2(bz
+ p) exp(- bzlz),
= B'bf1J2exp(b'z/2),
if z < 0
if z > 0
(44) (45)
where B, B', b, b', and p are variational parameters. The usual boundary and normalization conditions leave only two variational parameters, b and
FIG. 21. Calculated energy levels and wave functions of the GaAlAs/GaAs selectively doped interface. The Fang-Howard variational wave function (no penetration in the barrier) is shown (---). The variational wave functions [Eqs. (44) and (45)] are also shown (-). The line ( . . .) represents the numerical calculation, which includes correlz$on effects. The different confining potentials V(z)are shown. The spacer thickness is 50 A and the donor binding energy of GaAlAs has been chosen as 50 meV (from and^'^^).
36
C . WEISBUCH
\
(a)
>
0
0
1 o ! I
I
I I
I
L 0
I
FIG.22. Energy-level schemes in SDHT/TEGFET/HEMT/MODFET structure. (a) Band diagram for normally-off (top) and normally-on (bottom) transistors. (b) Energy levels and wave functions in a highly doped normally-on structure. The Ga,,,8Al,,,2As layer thickness is 600 A, ND = 2 X loL8~ m - AEc ~ , = 0.22 eV, and the potential at the GaAlAs surface is taken as 0.33 eV. The total number of electrons in the structure is N, = 8.45 X loi2crn-,, but most of them are in the GaAlAs banier (Nus = 6.90 X 1OIz cm-2). The Fermi energy EF is counted as the zero energy and is well above the bottom of the conduction band of GaAlAs as the donor level is shallow for the actual A1 concentration (see below, Fig. 25). Two subbands belonging to the channel are occupied, corresponding to the energy levels El and E,. Five subbands are occupied in the GaAlAs barrier (after VinteriM).
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
DISTANCE
FROM THE METAL-Ga APAs INTERFACE FIG.22 (Continued)
37
(A)
b'. A very good approximation for b' is actually b' = 2A-*(2m* AE)'l2, the standard wave-function penetration in the barrier. More refined values of the electron - electron interactions have been considered, such as the local exchange correlation potential. Ando gave numerical solutions for the Schrodinger equation in that case137(Fig. 21). Very complete calculations, including finite-barrier effects, effective mass and dielectric constant discontinuities,interface grading, and nonzero temperature, were performed by Stem and Das Sarma.14* d. ThermodynamicEquilibrium In a real device situation one has to consider the finite temperature through Fermi- Dirac distribution functions for level occupancy. The determination of the relation between the Fermi energy and the channel
38
C. WEISBUCH
density N, is very simple because of the constant DOS in 2D. From the usual expression
JE,
wheref(E) is the usual Fermi- Dirac probability function. One deduces
[+ (
N , = - k ~ m In * 1 nh2
exp
E ~ i E 1 ) ]
which can be used in Eq. (41). The finite extension of the GaAlAs layer also plays an important role in devices (Fig. 22): The full MESFET-type structure must therefore be analyzed, taking into account the Schottky built-in voltage at the metalGaAlAs interface. Vinter has made quantum-mechanical calculations of the gated h e t e r o J ~ n c t i o n ' ~As ~ Jcan ~ : be seen in Fig. 22b, quantum levels exist both in the GaAs channel and in the barrier material. The potential barrier at the GaAs/GaAIAs interface is actually so thin that coupling of the quantum states occurs between the two regions. Technology interesting solutions of the whole structure can arise, such as the two situations of normally-on or normally-off GaAs channel (Fig. 22a); the latter situation occurs because the rather large Schottky voltage (-0.33 eV) can completely deplete the GaAlAs layer and the GaAs channel for thin enough GaAlAs layers. One can therefore have on the same chip normally-on and-off devices by controlled etching of some GaAlAs thickness from a normally-on layer. The voltage control of devices is also tailored by the layer thickness and doping. One of the challenges raised by LSI and VLSI components is actually the required layer uniformity for constant switching characteristics across the whole circuit. Only MBE at its best seems now acceptable for the needed level of control. The device characteristics of TEGFET/HEMT/SDHT/MODFET structures due to the gate control of the charge densities, which in turn induce such quantities as transconductance, etc., have been calculated using I5O various models and approximations for various (Fig. 23). Several authors have addressed the problem of the design rules needed to obtain optimum performance of single heterostructures: Increasing the doping level NDof the barrier material increases charge transfer as Wvaries as N,1/2 [Eq. (37)], leading to a charge transfer WN, 0~ N g Z .Such a high transfer leads to efficient channel impurity screening, but it also leads to more scattering by the impurities located in the barrier. Inclusion of an undoped spacer layer leads to a decrease of this last scattering mechanism,
1. 111- v
SEMICONDUCTOR QUANTIZED STRUCTURES
39
x 10”
GATE VOLTAGE
V,
(V)
FIG.23. Measured gate-voltage dependence of the channel density of 2D electrons in the GaAUs/GaAs system. W, represents the spacer thickness.All samples have ND = 4.6 X 1017 except sample #R-76 A, which has ND = 9.2 X 1017~ m - The ~ . channel carrier density saturates with V,, as electrons tend to accumulate in the barrier subbands once they can be populated (see also Fig. 22) (from Hirakawa et ~ 1 . ’ ~ ~ ) .
but also diminishes charge transfer. Considerations on the optimal doping have been produced by Stern.lS1The experimental determination of the density of transferred electrons as a function of the doping level Nd and the spacer thickness W , is shown in Fig. 24a. As can be seen, there is not cm-2 in single GaAs/ much room for obtaining densities larger than loL2 GaAlAs interfaces. In order to obtain higher camer densities in the channel, several authors1S2J53 have studied the double-heterojunction field-effect transistor where the active layer consists in a wide GaAs undoped layer imbedded between two selectively doped GaAlAs barriers. This configuration produces a double hetero-interface situation, thus allowing one to double the channel carrier density (Fig. 24; see also Section 17). More recent studies have been made on modulation-doped multiquantum wells which have still higher densities and allow high-current operation.153a It might well seem that we have at hand enough theoretical mastery to be
40
C . WEISBUCH
I
I h
3v 2 1
cn
01
A
\
'
I
I
I
I
-
ND NA ( cm-3)
- c o ~ ~ 0 . 3
I
0
I
100
I
I
200
1 xl018
I
I
I
300
AlGaAs
1
Wsp (
GoAs
I I
I
NEUTRAL , DEPLETION~SPACER~ j LAYER ; LAYER: LAYER 1
7
,
I
I
400
d)
;SPACER j LAYER
, I
AlGoAs DEPLETION LAYER
-:-I
-
x-x,
x=xt
-
lNEUTRAL ;LAYER -8
I
x-0
I
I
500
I
i
-
I
I
n - GaAlAs I G a A s
SPACER LAYER THICKNESS
( b,
I
x-XI
FIG.24. (a) Sheet electron concentration N, at an n-type GaAs/GaAlAs single interface as a function of the spacer layer thickness W,p and of the doping level in the doped portion of the GaAlAs layer. The lines are theoretical calculations for the three doping levels indicated (after Hirakawa et al.'"). (b) Schematicsof the double-heterojunction transistor (from Miyatsuji et ~ 1 . l ~ ~ ) .
1. 111-v SEMICONDUCTOR QUANTIZED STRUCTURES
41
I-
0
0.1
02
0.3
0.4
Al MOLE FRACTION x
FIG.25. Thermal ionization energy of the Si donor as a function of the Al mole fraction in Gal-,Al,As (from Schubert and Pl00g'~~).
able to design heterostructureswith great precision. Unfortunately, a number of precautions must be taken, in order to precisely design a desired structure: (1) The donor energy levels Ed in Gal-,Al,As are quite unreliable, changing from only 6 meV at x 5 0.1 to more than 160 meV for indirectgap m a t e ~ i a l ' ~ (Fig. ~ - ' ~25). ~ The transition from shallow to deep donor level occurs at x = 0.235. This increase in Ed is very detrimental to the transfer of large charge densities [see Eq. (39) and Fig. 201 and to good operation at low temperatures (77 K), because of camer freeze-out. On the other hand, one requires large values of x in order to increase A E c , thus increasing the charge transfer. A very elegant way to solve this problem has recently emerged15': the charge-transferring side of the heterojunction is made up from a GaAs-GaAlAs superlattice, where only the GaAs layers are strongly n-type doped and the GaAlAs barriers are thin enough to allow charge tunneling. Due to the large carrier confinement effects in the thin (<20 A) GaAs layers, the donor ground level associated with the lowest confined level is raised well above the bulk GaAs conduction-band level, almost to the GaAlAs barrier level. Charge transfer can therefore occur between these GaAs confined donor levels and the GaAs channel (Fig. 26). A number of bothersome low-temperature effects have thus been elimi-
42
C. WEISBUCH
FIG.26. Superlattice selectively doped GaAlAs/GaAs heterostructuretransistor.
nated, such as source-drain polarization effects, persistent conductivity, and electron freeze-out at 77 K.158,159 (2) The number of electrically active donors is not unambiguously related to the number of metallurgical donors. Some Si atoms seem to be associated with deep defects, giving rise to persistent photoconductivity centers. These centers are also responsible for the collapse of device characteristics at low temperatures. (3) Recent detailed experiments160-161 point out the segregation of Si towards the GaAlAs growing front, the more so for larger values of x. This observation tends to provide a satisfactory explanation for a number of hitherto puzzling results: (a) The smaller-than-expected transfer of charges in normal heterostructures (GaAlAs over GaAs) is due to the actual wider spacer than metallurgically grown. This in turn explains the sometimes observed higher mobility than theoretically expected. (b) On the other hand, the past poor performance of inverted structures (GaAs over GaAlAs) can also be explained the segregation of Si impurities to the GaAlAs surface brings impurities near or in the GaAs channel. The excellent characteristics of the inverted MODFET by Cirillo et ~ 1evidence . ~ the high quality of inverted interfaces now attainable. Heiblurnl6lobserved that for x = 0.1 and low growth temperatures the segregation effect was much reduced. (4) The compensation and residual doping of the various layers are not too well controlled even in a given growth chamber, and might not be uniform in a given sample depth or reproducible from sample to sample. This situation is best exemplified in the sequential growth of high-quality samples by Hwang et charge transfer as a function of spacer thick-
~
1. 111- v
SEMICONDUCTOR QUANTIZED STRUCTURES
43
ness follows a reasonable behavior and allows a determination of the donor energy level in the GaAlAs barrier taken as an adjustable parameter. On the other hand, the mobility does not show a maximum as predicted from the decreasing bamer and increasing channel scattering probabilities. This is probably due to the erratic residual doping in the channel, evidencing that this doping is the limiting factor in these high-quality samples.L51 Earlier results on lower-quality samples did exhibit a well-behaved mobility variation due to the more consistently reproducible background channel doping.'63 The progress in growth control is, however, such that one can expect to achieve more systematically the well-behaved charge transfer and mobility obtained recently (see Fig. 56).149J61J638
9. n-i-p-i
StructuresL64
As was proposed in the original paper by Esaki and Tsu,~a spatial modulation of the doping in an otherwise homogeneous lattice can produce a superlattice effect, i.e., a spatial modulation of the band structure which induces a reduction in the Brillouin zone of electrons and new energy bands in the superlattice direction. The realization of such structures was achieved using periodic n-doped, undoped, p-doped, undoped, n-doped, . . . , multilayer structures, hence the acronym n-i-p-i for such doping superlattices (as opposed to compositional superlattices). By comparison with the modulation-doped heterostructure, the appearance of the doping superlattice effect is easy to understand (Fig. 27): charged particles are subject to a self-consistent potential:
where Vimp(z)is the electrostatic potential of the ionized impurities, V,(Z) the Hartree potential of electrons and holes, and Vxc(z)the exchange and correlation potentials. The first term, Vimp(z),can be calculated from the Poisson equation
Similarly, the Hartree potential is
The exchange and correlation terms for electrons have been calculated by Ruden and DOhleP5 in a density-functional formalism: V,,(Z)
21
0.61 1
where N(z) is the electron density.
(49)
44
C. WEISBUCH
t
FIG.27. n-i-p-i band-structure formation. (a) Growth sequence of the structure; electrons from neutral donors recombine with holes located on the neutral acceptors, leaving a net space charge assoCiated with ionized impurities shown in (b); the resulting band-gap variation and carrier confinement are shown in (c).
Energy levels for the z-quantized motion have to be calculated self-consistently through the one-dimensional Schrodinger equation
where xe and x h are the envelope functions of the electron or hole wave functions. A number of straightforward features can be extracted from Eqs. (46) and (50)'% (1) In the case of exact compensation (equal numbers of donors and acceptors), +dl2 +d/2 W Z ) = N A ( Z ) d.2 j-42
j-d/2
where d is the superlattice period, no free camer exists in the unexcited sample.
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
45
(2) For equal uniform doping levels N A = ND and zero-thickness undoped layers, the periodic potential consists of parabolic arcs and has an amplitude
(for GaAs with N A = N D = 1018cm-3 and d = 500& one has V o = 400 meV). The quantized energy levels in the potential wells are approximately the harmonic oscillator levels:
For electrons, for instance, the subband separation is 40.2 meV for the
direction of crystal growth -m FIG.28. Three types of n-i-p-i superlattice: (a) compensated intrinsic superlattice with 2V0 < E, and N,d, = NAd,,; (b) n-type superlattice with 2V0 < E, and N,d, > NAdA;(c) semimetal superlattice with 2 V,, > V, (after Ploog and DohleP).
46
C . WEISBUCH
above parameters. Since the effective bandgap Eeffshown in Fig. 27 is given by Eeff= EG - 2 V, -4- E , -4- &h, it is reduced below the bulk material value. (3) When there is unequal doping, free carriers will accumulate in the corresponding potential well (Fig. 28). Equations (46) and (50) must then be solved self-consistently. The Fermi level can be located at will (Fig. 28b). (4) For large enough spacings and dopings, the effective bandgap can ). then become negative (i.e., d z 700 A for N A = N D = 10l8~ m - ~ There exists charge transfer from hole wells to electron wells until a zero gap is attained due to three factors: band filling, diminishing of the periodic superlattice potential thanks to the charge neutralization by the transferred charges, and quantized energy-level modification (Fig. 28c). ( 5 ) Under nonequilibrium conditions such as photoexcitation or carrier injection, electron and hole populations can build up in the wells, leading to charge neutralization and an effective bandgap increase. (6) Under such nonequilibrium conditions, electrons and holes are spatially separated and the radiative recombination rate is strongly diminished as compared to the bulk case as for an indirect-bandgapsemiconductor. At the same time, nonradiative recombination rates are also strongly decreased, leading to reasonable quantum efficiencies. This justifies the hopes for tunable light sources expected from doping superlattices, even though they are real-space indirect semiconductors. They should also lead to excellent photodetectors, as the photoconductive gain should be very large. Many of the features that are expected from the n - i-p- i structure have indeed been observed: variation of the bandgap with increased excitation, change of absorption features with light intensity, tunable luminescence, etc. The reader is referred to the review articles by Ploog and Dohler165and Abstreiter'& for a very exciting description of doping superlattices. An interesting recent development is the heterojunction doping superlattice (Fig. 29); in the standard doping superlattice the transport of electrons and holes occurs in doped regions and therefore mobility is rather poor. By introducing undoped small-gap semiconductor layers in the middle of the n- and p-doped layers of the superlattice, the carriers are transferred in the undoped small-gap material where they experience high mobilities as in usual modulation-doped structures. 111. Optical Properties of Thin Heterostructures
The most general and surprising feature of the optical properties of quantum wells is the strength of the intrinsic optical effects as compared to
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
47
FIG.29. Heterojunction doping superlattice:Instead of occumng in doped (low-mobility) regions as in the standard structure of Fig. 27, carrier transport occurs here in the high-purity undoped potential wells (after Ploog and Dohler164).
bulk optical properties: in many circumstances one measures features of comparable size for a single quantum well of = 100 A as for bulk samples of thickness of the order of an absorption length, a few 1000 A (Fig. 30).167 In particular, the quantum efficiency of luminescence has been observed to be larger in QW structures for all the systems reported up to now: GaAs/ GaA1As,l6' GaInAs/AlInAs,'69 GaSb/GaAlSb,170 GaInAs/InP,6'" CdTe/ CdMnTe,79GaAsSb/GaAlSb,170a ZnSe/ZnMnSe,170b. . . . 10. OPTICAL MATRIXELEMENT
The interband transition probability for particles confined in quantum wells can be calculated by perturbation theory and is, as usual, the product on an optical matrix element times a density of states. The modification to the usual probability stems from the 2D density of states as it can easily be shown that the optical matrix element is hardly changed as compared to 3 D taking the electric dipolar and the infinite-well approximations, interband optical matrix element15,16 has the form
..
where x,(z) and xh(z)are the electron and hole envelope wave functions, k,, k h are electron and hole wave vectors, q is the polarization vector of
48
C. WEISBUCH
E GaAlAs
1.6
1.4
1.8
2.0
2.2
e
PHOTON ENERGY FIG.30. Electroreflectanceofa double 49 A quantum-well sample.The remarkable feature is the size of the n = 1 QW exciton electroreflectance peak, quite similar (factor of 4) to that of the GaAs substrate. The luminescence spectrum is used for the peak assignment (after Alibert et al.Ib7).
light, uck,(r)and u&) are the usual Bloch functions. The integral contains fast-varying functions over unit cells (uck and unk)and slowly varying functions. Using the usual procedure, one transforms Eq. (53) in a summation of localized integrals involving only Bloch functions over the N crystal unit cells:
J-
dr
-$
~ c k e ( m * ruvkh)
dr
(54)
The only difference between this and the usual 3D summation lies in the z-direction summation, which produces a factor ZX~(R,)X,,(R,) dRi,where the R;s are the lattice cell centers in the z direction and dR, is the distance between two lattice sites, i.e., the lattice constant. Transforming back into an integral J $ x ~ ( z ) x ~dz, ( z )one finds a unity factor for the transitions between electron and hole states with the same quantum number n, as they are identical [-sin(nnz/l)] and normalized to unity. The optical matrix element is therefore the same in 2D and 3D. In the absence of exciton effects, the absorption coefficient should reflect the 2D DOS, i.e., should consist of square steps corresponding to the various confined states. This situation is usually obscured by exciton effects, and has only been observed in standard absorption measurements in the GaSb- AlSb ~ystem.'~'
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
49
From these considerations, one concludes that the band-to-band absorption probability is independent of layer thickness. It has been calculated by V o i ~ i n to ' ~ be ~ =6 X per layer using the known parameters of GaAs. 1 1. SELECTION RULES
One should first notice that the quantum well and superlattice potentials are symmetric under space reflection changing z into - z. Therefore, parity is a good quantum number; i.e., the envelope wave functions are characterized by their even or odd character under space reflection. Considering the electric-dipole matrix element in Eq. (53), the factorization procedure leads to the following results. (1) The usual change of parity of electric-dipole transitions appears in the Bloch integral matrix element. (2) Transitions are then allowed for confined states with the same envelope wave function symmetry under space reflection (even or odd). (3) In his original paper, DingleI4 remarked that in the infinite-well approximation, due to the orthogonality of the envelope wave function, only transitions between confined valence and conduction states with the same quantum number n were allowed (An = 0 rule). It is actually true that these transitions are the strongest observed features in the absorption and excitation spectra.
However, a number of additional transitions have been observed, originating in the breakdown of the simplifying assumptions made. ( I ) For finite quantum wells the envelope wave functions are not exactly orthogonal, which leads to the observation of transitions with differentn (such as the n = 3 heavy hole to n = 1 electron line; see Fig. 36). (2) Even the parity selection rule has been broken in the case of manyparticle spectra either optically created173or due to modulation doping."* In that case, particle- particle interaction breaks the single-particle picture used to derive the selection rules.
Light polarization matrix elements have also been calculated at k = 0, where the quantum-well potential acts as a simple perturbation to the Kane description of the bands. The split-valence states retain their symmetry characterized by the angular momentum of the Bloch wave functions: the heavy-hole level at k = 0 has J, = +$; the light-hole level has J, = f+. The various allowed transitions can be calculated as in the atomic physics case of transitions between ground levels with J = 3, J, = k 3 or J, = k+and excited levels J = +,J, = -C+. The various absorption transitions are shown in Fig. 31 with the corresponding light polarizations,
50
C . WEISBUCH
lo*
ELECTRON S T A T E S
-112
-312
-112
+I12
+3/2
HEAVY-.HOLE S T A T E S
+I12
2t
-112 +I12 LIGHT- HOLE S T A T E S
drection light e m i s s i o r p T
E
FIG.3 1. Optical selection rules for absorption and luminescence between Bloch states of the valence and conduction bands. The usual notations are used. The axis of quantization for angular momentum of electrons and photons is along z, the growth direction. o transitions correspond to electron motion in the x - y plane and polarization vector of the emitted or absorbed photon in that plane (TE polarization). 7~ transitions correspond to electron motion and light polarization along the z direction (TM polarization). The relative values of matrix elements are indicated.
respective to the momentum quantization axis. From the correspondence principle, the polarization vector also describes the electron dipole rnotion. Using the classical description of radiation emission, which states that an electric dipole radiates mainly perpendicular to its own motion and does not radiate in the parallel direction, the following selection rules can be deduced for light absorption or
1. 111- V SEMICONDUCTOR QUANTIZED STRUCTURES
51
a. Light Propagating Perpendicular to the Layers
Only those dipole moments in the plane can absorb or radiate. Free electron- hole absorption (no exciton effect) must be three times larger for the HH band than for the LH band transitions. Under circularly polarized light excitation, 1OOYo spin polarization occurs when electrons are only excited from one of the HH or LH band. This is to be compared with the 50% polarization obtained in the bulk case, where one excites at once both transitions and creates electron spins with opposite directions, the net polarization occurring only because of the unequal transition probabilities.”6 The possibility to obtain 1OOYo electron spin polarization should be of great interest for the production of photoemitted spin-polarized electrons. Preliminary experiments have not yet succeeded in yielding higher free-electron spin polarizations than for bulk GaAs, although photoluminescence measurements of electron spin have evidenced high
t0.2
U+
ABSORPTION LIGHT Amj=+l
4-0. I 0, 2
0 3.0
: I-
4 / 2 1 -+1/2
1
EMISSION
K
U
J
0
-0.1
n
5
4
-0.2
I
0 PHOTON ENERGY
(a)
(ev)
SELECTION RULES
(b)
FIG. 32. Photoluminescence and circular polarization spectrum of GaAslGaAlAs quantum wells under circularly polarized excitation (a). For a+ excitation well above the bandgap, where exciton effects become negligible, one creates three times more spin electrons than Assuming some spin memory at the moment of recombination, these dominant electrons will emit a+ light when recombining with heavy holes and a- light when recombining with light holes (b). The observation of opposite signs for the polarizations of the two lines in part (a) ascertain the 1.522 eV peak as being related to heavy holes and the 1.527 eV peak as being due to light holes. Resonant light excitation experiments were also done to c o n h these assignments (after Weisbuch et a1.’96).
++.
52
C. WEISBUCH
polarizations (- 70%) within the crysta@ l.' ' No specific linear polarization effects are expected. The selection rules have helped to ascertain the quantum states participating in QW luminescence (Fig. 32). b. Light Propagating along the Layers The HH transition can only occur with light polarization parallel to the layer (TE mode). The LH transitions occur both for TE and TM light polarizations (see Fig. 3 1). A remarkable feature is the relative intensities of the TE and TM modes (which from Fig. 3 1 should be in the ratio 1 :$). It was already reported by Dingle that the TE mode luminescence due to the HH transition was much larger than the TM emission.181Although thermalization effects at low temperatures could reasonably explain the effect, it is not so at room temperature, where both the TE-HH luminescence and gain have been shown to be larger ("4 times) than for the TM mode.182-184 This is the more surprising as one could expect from the simplest 2D-DOS analysis the HH transition probability (corresponding to the light-transverse hole mass) to be significantly smaller than the LH transition probability. The analysis clearly requires a more profound analysis of exciton effects, valence-band symmetry, selection rules, and density of states, as the valence-band anticrossing discussed in Section 3 should play an important role. Luminescence of modulation-doped QW samples shows a similar breakdown of selection rules. Pinczuk et a1.'74 were able to show in z propagation a rather strong n = 2 electron to n = 1 HH luminescence, where the ratio between the panty forbidden and allowed transitions amounts to 0.5. More recently, observations of the luminescence along the QW plane in such samples show a strong HH-forbidden TM luminescence, which extends over a wide spectral range. This observation allows the k spectroscopy of the HH valence band (symmetry and DOS) to be performed, since the electron plasma acts as a supply of well-defined excitations (i.e., well-known dispersion curve, DOS, and band filling. 91392
12. ENERGY LEVELS, BANDDISCONTINUITIES, AND LAYER FLUCTUATIONS
As mentioned earlier, the 2D exciton has a stronger exciton - photon coupling due to the increased overlap of the electron and hole wave functions. Refle~tivity,'~'ellipsometric,'67~185~'8sa and p h o t o r e f l e c t a n ~ e ~ ~ ~ ~ ~ ~ measurements evidence this enhancement, as the quantum-well peaks appear strong as compared to the much thicker confining and buffer layers. The well thickness and spectral dependences of the index of refraction have also been observed near resonance. Strong room-temperature excitonic effects have now been reported for a number of semiconductor pairs. One
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
53
of the surprising features of absorption and excitation spectra is the observation of exciton states related to the higher-lying confined states, as they are degenerate with the continuum electron states of lower confined states. This has been theoretically explained by the weak coupling between these states, which leads to long disintegration times of the excitons of the higher confined states and therefore a weak broadening.'I8 Exciton absorption corresponding to the various confined levels has been the first optical evidence of quantum size effects in semiconductor thin layers. The transition energies are given by
where EG is the bandgap, E g d , E$'$ are the electron, heavy-, or light-hole nth confinement energies, respectively, and E,, is the exciton binding energy. Prominent peaks are those with n = m . The influence of layer quality and its progress can be traced through time. Until the middle of 1975, when liquid nitrogen shrouding of the MBE chamber began, no heavy- and light-hole n = 1 transitions could be observed for samples with well thicknesses > 150 A. A major improvement occurred in 1978, when introduction of samples in the growth chamber through UHV interlocks was used. Growth interruption for interface smoothing has led to the present state of the art of atomically flat interfaces and ultrasharp peaks in optical spectra, which, however, can now also be reproduced in high-quality MBE systems without interruption.186a The fit of the early absorption measurements led to the determination of the bandgap discontinuity A E between the conduction bands AE, and valence bands A E,. Calling Q = AEJAE, Dingle14 found that Q = 0.85 -t 0.03, assuming standard values for the [lOO] electron and hole masses, i.e., m, = 0.067 mo, mHH= 0.45 m,,= mzzLH = 0.08mo. In recent similar experiments on squarelS7and parabolic188quantum wells, Miller was led to a reexamination of this partitioning and evaluated Q = 0.60, using a heavy-hole mass mHH= 0.34mo.It is remarkable that the two sets of parameters can explain all the standard features of the A n = 0 transitions for the square wells.IE7It is only for the case of parabolic wells (Fig. 33), and for the n = 1 e to n = 3 HH forbidden transitions in square wells (Fig. 34), that the need arises to consider the newer set of parameters. In an elegant method, using separate-confinement heterostructure QWs, Meynadier et al. were able to measure combination absorption lines between a narrow well embedded in a wider confinement well. Their transition energy is strongly dependent on bandgap discontinuities and allows a determination of Q = 0.6. It should be remarked that a small value of Q, i.e., a small conduction-band discontinuity, tends to account
54
C . WEISBUCH
n
PARABOLIC WELL
E5 E46h
/I'
to
72
I I
I
A
1.52
I.!
u 1.54
llLlLlJ 1.60 1.64 1.68 1.72 PHOTON E N E R G Y (eV) FIG.33. Excitation spectrum of parabolic quantum wells. Note the large number of peaks observed when compared to square wells, due to the relaxation of the An = 0 selection rule. En,,,,refers to a transition from an electron state with quantum number n to a heavy- or light-hole state with same n: EnmLh refers to a transition with a change in quantum number (from Miller ef al. I**). 1.52
1.56
for the smaller-than-expected charge transfer in electron MD heterostructures168J90and conversely the good properties of hole MD heterostructures.Igl Early experiments were also carried out on double and multiple interacting quantum wells.93Interwell coupling leads to the lifting of the degeneracy of the degenerate ground state, evidencing the formation of a superlattice band according to Eq. (29). The data of Fig. 35 show the transitions from the coupled double well with two states, one symmetric and one antisymmetric, to a quasi-continuum of states due to superlattice band formation in a 10 coupled-well sample. More recently, superlattice formation has been shown from electroreflectance measurements for Brillouin zone points away from the center of the zone.126 The data fitting must also include the exciton binding energy [Eq. (55)] as an adjustable parameter, a function of well thickness and exciton (heavy or light hole). Dingle14extracted a value of 9 meV for the exciton Rydberg R* in thin wells (<100 A), which is to be compared to the 4.2 meV value in the bulk. This is, however, a rather imprecise measurement due to its dependence upon the overall fitting procedure. More recently, Miller was
1. 111
-v SEMICONDUCTOR QUANTIZED STRUCTURES
55
200 -
100 -
70 -
50 -
- 30
-
>,
E 20-
S
w I
._ .W
10 -
7-
5-
32-
1'
30
I
I
I
50
70
100
I
$00
-
300
€11
'
E12h
500 7001000
L(A)
FIG.34. Fit of the observed transitions in quantum wells with various thickness. Full lines are obtained with Q = AE,/AE= 0.51, me= 0.0665ma,m, = 0.34ma, m,,, = 0.094. The and is only dashed line is obtained using previous parameters Q = 0.85 and m,,,, = 0.45me, shown for the transition (n = I electron to n = 3 heavy hole), as all other transitions would be satisfactorily fitted by this set of parameters (from Miller ef al.lE7).
56
C. WEISBUCH 1.5
1.0
>
0.5
v)
B
5F $
1m .
0.5
0
1.54
f.55
1.60
1.65
1.70
1.55
1.60
1.65
1.70
PHOTON ENERGY (eV)
FIG.35. Absorption spectra of single (a), double (b), triple (c), decuple (d) coupled quantum wells. The positions of the expected transitions in the perturbative approach [Eq. (26)] are indicated. The appearance of bonding and antibonding states is well evidenced in (b). The inserts show the structures under measurement (after Dingle et ~ 2 . ~ ~ ~ ) .
able to extract similar values from the onset of the n = 2 exciton absorption edge in excitation spectra.192 Another measurement has recently been carried out by Maan et ul.,193in which the unbound electron and hole state levels are determined by extrapolation from their high magnetic field value. The heavy-hole exciton binding energy can be as large as 17 meV for 50 A wells, and 10 meV for the light-hole exciton in 100 A wells. The heavier hole mass of the heavy exciton (as determined from the apparent p), contradictory to the light transverse mass of the heavy-hole band, is a proof of the strong perturbation of the valence band from the simplest pictures. From the exciton radius uB it is clear that one needs to know the dispersion of the valence band up to k = uB = lo6 cm-' to construct the exciton wave functions and deduce the exciton Rydberg. It might be thought that interband transitions should provide a convenient way to measure interface grading. It has, however, been shown t h e ~ r e t i c a l l y ' that ~~.~ grading ~ ~ does not modify the energy-level structure for grading extending up to a few atomic layers, unless the wells are extremely thin. A very convenient way to deduce absorption spectra without any sample preparation (in particular thinning) is the photoluminescence excitation
1. 111 -v SEMICONDUCTOR QUANTIZED STRUCTURES
57
spectra (ES) methodIg6:observing the photoluminescence at a given wavelength, one scans the exciting light wavelength (with a tunable dye laser, for instance). Peaks will appear in the spectrum as a result of increased absorption coefficient (Fig. 36). Actually, there is another contribution to luminescence ES peaks which is due to more efficientrelaxation/coupling to the luminescent level under observation, such as resonant LO-phonon relaxation.’” One should carefully watch whether this effect occurs as it could lead to erroneous assignments in ES peaks. Although very efficient in 11-VI compounds, this mechanism has proven negligible in QW structures, thanks to the smallness of the LO phonon coupling and to the very efficient nonresonant relaxation mechanisms to the luminescent channels. A first use of the ES method was to assess the layer -to - layer thickness reproducibility using a sample with noncommunicating wells (bamers = 150 For such an MQW structure, luminescence peaks and their ES are characteristic of a given well and its thickness. In the case of varying QW thicknesses, the various portions of the overall luminescence spectrum A).’9891w
FIG.36. Excitation spectrum of a multiquantum-well (MQW) sample with 260 A thick barriers and wells. The various observed peaks are labeled according to their origin. Several forbidden peaks (E,) are also observed. The peak labeled E, has since then been assigned to E,3hand yields crucial data for the determination of bandgap discontinuities (see Fig. 34 and discussion in the text). The detection monochromator is set at the energy marked “pump” in the figure, where there is a signal peak due to elastically scattered light (after Miller et ~ 1 . ’ ~ ~ ) .
58
C. WEISBUCH
-
GROWTH DIRECTION
lndlvldual layer luminescence
INTENSITY
t
_-_-_I _______ - - - - _ -r-----
TOTAL LUMINESCENCE INTENSITY
Excltatlon spectra
INDIVIDUAL LAYER ABSORPTION OR EXCITATION SPECTRUM
PHOTON ENERGY
FIG.37. Schematicsof the excitation spectra (ES) to be observed in a multiquantum-well structure where the wells are unequal, leading to an inhomogeneousluminescenceline due to different wells with varying thicknesses. Each recombination wavelength, corresponding to a different well with its own confined energy spectrum, gives rise to different ES.
should be inhomogeneous, i.e., should have different origins in space and have different ES. On the contrary, for equal QW thicknesses, one expects the same ES whatever the luminescence observation energy (Fig. 37). The detection of a single ES in good MQW samples, to a precision better than a tenth of the ES linewidth, allows one to assess the reproducibility of the average layer thickness as better than one-tenth of a monolayer. For such an optimally grown sample, it was observed that the ES linewidth would increase for decreasing layer thickness (Fig. 38). This was interpreted as being due to variations of the confined energies due to intralayer thickness fluctuations.199 In a layer -to-layer growth mode, one expects to find islands where the thickness varies by -0.5 of a monolayer from the average monolayer thickness. Therefore, the various zones correspond to various confinement energies, which leads to broadening of the absorption and excitation spectra due to the spatially disordered exciton absorption band. The simple fit of Fig. 39 represents quite well the results of a series of samples grown sequentially at the optimum temperature. The model assumes that the lateral size of the exciton is smaller than the island size, so that the confinement energy change is that calculated in the usual infinitely wide layer model. Bastard et ~ 1 calculated . ~ the ~ confinement energy variation with the lateral size of islands or holes in otherwise atomically perfect layers. The next step in a detailed analysis would be the determination of the interface topology, either theoretically or experimen-
1 . 111- v SEMICONDUCTOR QUANTIZED I
I
STRUCTURES
I
59
I
145
I
I
1.50
1.55 1.60 1.65 PHOTON ENERGY k V ) FIG. 38. Excitation spectra of various wells with different thicknesses at 1.8 K [C. Weisbuch, R. Dingle, A. C. Gossard, and W. Wiegmann, unpublished (198O)l.
tally. In the Bell Laboratories series of samples,'* the linewidth data could be interpreted assuming a majority of island sizes larger than the exciton diameter (e.g., = 300 A), which is in agreement with X-ray diffuse scattering observationsMand TEM imaging technique^.^^ More knowledge of the topology of the interfaces as revealed by X rays and TEM and of the spatially disordered DOS is required to be able to describe the detailed correlation between the interfaces and the DOS, as revealed by absorption and ES in such samples. In some more perfect crystals, one could expect almost atomically flat layers. A few experiments tend to show such perfection, as deduced from luminescence experimentsm',202 in samples grown by standard procedures. More recently, interrupted MBE growth202a-202c has been used in order to allow for atomic migration and island coalescence at
60
C . WEISBUCH
CONFINEMENT ENERGY (meW FIG.39. Linewidth versus confining energy: The solid line is a fit assuming fluctuations of each interface equal to +u/2, where u is a monolayer. The fit is not good for large confining energies where the energy fluctuations are larger than the exciton Rydberg. At low energies, other broadening mechanisms come into play (from Weisbuch et u Z . ' ~ ~ ) .
interfaces. In that case, the island size can become much larger, and discrete exciton lines display the exact quantized energy corresponding to the various layer thicknesses equal to an integer number of atomic layers. The linewidth assessment has been used to optimize growth conditions. Varying growth temperatures, Weisbuch et al. were able to identify three different growth regimes in an MBE system (Fig. 40); at the optimum temperature (= 690 C), the growth occurs in a layer - to - layer mode, a layer being completed through island extension from nucleation (impurity?) centers and coalescence of the islands into a complete layer. At lower temperatures, surface atom mobilities are not large enough to ensure lateral size growth of islands and instead islands with a height higher than a monatomic height can occur. At the same time, periodic macroscopic fluctuations of the surface can be observed with an optical microscope. At high temperatures, the atom's kinetic energy is large enough to overcome the binding energy at the island coast which leads to a roughening transition, which here appears temperature broadened by impurities instead of abrupt, as in the case of the helium phase t r a n s i t i ~ nThis . ~ ~interpretation is supported by the earlier TEM imaging measurements of ultrathin structures (down to alternate mono layer^)^^ and by Monte Carlo calculations of the growth mechanism.204-206 MOCVD growth exhibits an opposite effect O
1. 111-v SEMICONDUCTOR QUANTIZED STRUCTURES
-z -E
I
\
\
1
I-
-
=
3u X
w
IMPURITY BROADENED INTRINSIC ROUGHENING
\
-1
'-
-i /
/-
CURVATURE
7-----I
FIG.40. ples grown
61
I
t
I
,
I
sam-
of line narrowing with increased Ts,zo6 as an island size decrease leads to fluctuation averaging. Absorption and ES have been used to study various processing methods of GaAs-GaAlAs QWs. Thermal interdiffusion was studied o p t i ~ a l l y , ~ ~ ~ , ~ by x rays,m8 and by TEM.209 In the optical method, one observes the upwards energy shift of the quantum well levels as A1 interdiffises in the well material. Dingle was able to determine a diffusion constant for Al, D = 5 X lo-'' cmz s-I at 9OO0C,in good agreement with X-ray and TEM measurements, which justifies the neglect of interdiffision during MBE growth under standard conditions (T' < 750T). A very promising technique for controlled interdiffusion is provided by Zn-assisted A1 diffusionZlo;during their difision, Zn atoms induce A1 atom diffusion. This opens the way to low-temperature spatially controlled smear-out of quantum wells. This has already been used in the selective protection of laser facets, which enables higher optical powers without facet damage;211the higher-gap regions obtained near the facets confine the recombining electron -hole pairs, thus suppressing radiation-induced defect formation, the main mechanism for facet degradation. Although requiring higher temperatures and longer times, Si-assisted A1 diffusion can also be observed.2 The 2D DOS and the excitonic nature of the absorption coefficient lead to an absorption edge much sharper in QWs than in the usual 3D double
62
C. WEISBUCH
1.4
1.2
-
J. fB
1.0 v) v)
0
0.8
-1
0.6
Room temp. TE mode
"I 0-
A+
-
A\
I"
%J \
-
\cGaAs
h
MOW' 0.4
0.2 0
DH
\ \
-
\
h
I
I
, I S ,
I
I
WAVELENGTH
,
(
*
'\
i)
FIG. 41. Transmission curves for passive waveguides with MQW and conventional DH structures. AROWS A and B denote the spontaneous (higher-energy) and lasing wavelengths, respectively, when each waveguide is current injected (after Tarucha et
heterostructures. This results in a lower transmission loss in MQW waveguides as compared to DH waveguides at the lasing wavelength of the structures213(Fig. 4 1). One therefore expects better performance of monolithic integrated optoelectronic circuits made on QW material. 13. LOW-TEMPERATURE LUMINESCENCE~~~
The luminescence of undoped GaAs/GaAIAs quantum wells at low which is at first quite temperatures consists of a single narrow line,196J98 different from the observed multiple-impurity-related lines observed in bulk material of similar q ~ a l i t y . ~ It ' ~is* also ~ * ~brighter than in thick 3D layers such as typical double heterostructures.'68 A number of factors, occurring simultaneously or not, tend to this single recombination process and large quantum efficiency. (1) Carrier collection in QW at low temperatures is extremely efficient. Carriers created in the overlayer barrier material are largely captured by quantum wells, as shown by the usually small luminescence of the barrier material as compared to quantum-well luminescence. (2) The 2D exciton enhancement leads to efficientexciton formation.
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
63
The accumulation of photocreated carriers in the small phase space of QWs should also increase the bimolecular formation rate of excitons. (3) Exciton luminescence is to first order a forbidden process, as the k selection rule of the optical matrix element only allows excitons with exactly the photon k vector to radiate.215-217 In the pure bulk material, polariton phenomena (i.e., the coupled exciton- photon excitation) relaxes this rule but transforms the exciton fluorescence mechanism into the transport of the coupled excitation to the surface. In this picture, exciton fluorescence is no more an intrinsic phenomenon described by the exciton-photon coupling, but is a transport problem described in terms of excited depth, energy and momentum relaxation, group velocity, etc.216.2'7 In another picture of strong damping (impure material, high temprature), polaritons do not propagate but luminescence occurs due to the scattering of an exciton state to a photonlike state, followed by the transformation of the exciton into a photon thanks to the exciton-photon coupling. In this picture, the exciton luminescence is a second-order process involving exciton interactions with impurities or phonons and the exciton-photon interaction. In quantum wells, excitons cannot propagate along the z axis as they are localized in the well. However, luminescence should be very efficient as the k conservation rule should be lifted thanks to the scattering by confining energy fluctuations. Exciton-mediated luminescence should also play a role at high carrier densities as demonstrated by the sharp ES peaks observed by Pinczuk et ~ 1 . ' 'in ~ modulation-doped samples. (4) Impurity gettering can occur during multilayer formation, which diminishes the number of nonradiative centers. The effect was first evidenced in GaAs-GaAlAs MQWs, where the usual dark spots in photoluminescence of double heterostructures associated with dislocations could not be observed.218The effect was shown to be due to impurity gettering by the GaAlAs barrier material in the first layers.57A smoothing of the interface roughness can also be observed as growth proceeds (see Fig. 2). Similar material improvement was also observed in MOCVD material6' (see Fig. 3). However, material grown in other optimally set systems seems to be exempt of impurities even in single quantum well^?'^^^*^ The free-exciton nature of the pure-material luminescence line was established through a careful study.196 Its energy position coincides almost exactly with the exciton peak observed in ES. The possibility of excitons bound to neutral shallow impurities (donors or acceptors) is ruled out by the spin memory measurements under circularly polarized excitation light, which ascertains the symmetry of the luminescent state as that of a correlated single electron and single hole. The dependence of the high-energy slope of the line on temperature and excitation intensity points out the
64
C . WEISBUCH
free-moving nature of the excitation, ruling out exFitons bound to isoelectronic impurities. For thick enough QWs (> 150 A) the light-hole exciton can also be observed at low temperatures. More higher-lying levels were observed either at higher temperatures or under high-intensity excitation. A careful study of the transition from 3D luminescence features to 2D behavior in a series of samples with varying thicknesses has been given by Jung et d 2 1 9 Theoretical calculations tend to support the dominance of free exciton over bound exciton recombination in quantum wells: Herbert and Rorison220" have shown that, whereas confinement increases free-exciton oscillator strength as 1/L, it slightly decreases that of donor bound excitons due to the decrease in carrier correlation. It is obvious that the quality of interfaces as revealed by ES will influence the luminescence line shape and width (Fig. 42). Whereas the ES directly probes the DOS and marks the peak of the disordered exciton energy band, the luminescence line shape does not represent directly the DOS of the exciton band (see Fig. 44 below). It cannot even be simply positioned relative to the center of the DOS, as the luminescence line shape results from the competition between the energy relaxation time of excitons down the disordered exciton energy band and the recombination time. Therefore, the discussion of the exciton line shape and the shift between luminescence and ES requires a detailed understanding of the disordered exciton band which we lack at the present time. The linewidth of the luminescence peak can however, be used as an indication of the quality of the interface. Sakaki et aLmh have used it to indicate the atomic flatness of growth-interrupted quantum wells. Whereas the early optical measurements were limited to MBE material, measurements in MOCVD material show a very similar quality of such material and interfaces as compared to the best MBE materia1.58-60A very useful scheme has proven to be the sequential growth on the same substrate of several QWs with different layer thickne~ses.~~ This allows one to compare different QWs grown under exactly similar conditions. This is also very useful to ascertain the spatial homogeneity in the case of alloy quantum wells such as Ga1nAs/InP1";when the alloy composition varies across a multiwell sample, it acts as a constant shift of the ground-state energy of the well, independent of the well thickness. If the growth rate is spatially varying, this is reflected in unequal shifts for the various confining energies due to the nonlinear (Emd - Le2 in the infinite-well approximation) relation between well thickness and confining energy. In some high-purity MQW samples a double peak is observed around the n = 1 heavy-exciton position (Fig. 42c). First interpreted as a reabsorption feature, this structure was attributed to the heavy exciton band at high energy associated with a biexciton recombination line at low energy.
1. 111- v
SEMICONDUCTOR QUANTIZED STRUCTURES
I
65
1.639
I
w
I
1.510
I
I
1.520 1.530 PHOTON ENERGY ( e v )
I
1.54
FIG.42. Luminescence of optimally grown quantum wells with varying thicknesses [(a), 5 1 A; (b), 102 A; (c), 222 A]. For the sake of clarity, the energy and count-rate scales have been shifted with respect to one another. Note the log scale for the count rate. The luminescence linewidth is to be compared with the excitation spectrum linewidth of Fig. 39. [C. Weisbuch, R. Dingle, A. C. Gossard, and W. Wiegmann, unpublished (1980)].
Although vanishingly small in undoped structures due to the efficiency of the free-exciton recombination, impurity-related luminescence can easily be observed for deliberately doped samples. Miller et were able to observe the dependence on QW thickness and impurity position of acceptor binding energies through the detection of the electron- to - neutral acceptor transitions. Excitons bound to neutral acceptors were also shown.223Shanabrook and Comas used spike doping at the center of quantum wells to measure donor-related levels.224
66
C. WEISBUCH
14. CARRIER AND EXCITON DYNAMICS
There has recently been a surge of optical transient measurements of ~ a m e and 8 ~ exciton ~ ~ dynamics in QWs. Some care must be used in analyzing the results as compared to the 3D case, as a number of parameters are strongly altered. In particular, the rather small 2D DOS induces band-filling effects even at moderate exciting powers. The pump-and-probe experiments were first performed on MQW structures.225At low densities (Fig. 43) the absorption spectra show a washout of the exciton peaks due to exciton screening. Studies of this bleaching allow one to measure the exciton band filling and ionization time of excitons by hot electron or phonon collisions.226An ionization time of -300 fs due to phonons is deduced at room temperature. The remaining absorption displays the exciton-less absorption curve, i.e., the 2D-DOS step structures. Higher pumping rates show the large amount of band filling and gain at the higher densities. Camer relaxation rates have been estimated and shown to be very similar to those observed in 3D. At still higher excitation rates, energy relaxation was shown to be slower, as evidenced by the hot luminescence correlation-peak method.227It is not yet clear whether this slowing is due to phonon accumulation in the well or band-filling effects. Femtosecond experiments228have recently found similar energy relaxation rates in 2D and 3D, even in the higher-density regime, which, however, does not rule out the phonon-accumulation model227for longer times as the phonon population should be low at the early stages of relaxation. Photoluminescence camer dynamics has been studied through time-resolved l u m i n e s ~ e n c e . The ~ ~ ~relaxation , ~ ~ ~ times of excitons in the disordered band have been measured by Masumoto et dZ3' showing the importance of the spatial disorder that controls the energy migration. Takagahara232made an analysis of the experimental results using a model of energy transfer in a disordered band. The shortening of the exciton lifetime with decreasing well thickness has been traced to the increase of electron- hole wave function overlap in the 2D exciton optical matrix element.229,229a This interpretation, however, supposes that the measured lifetime is directly related to the radiative recombination mechanism, i.e., that the quantum efficiency is near unity. It also supposes that the radiative lifetime is uniquely related to the optical matrix element, although, as discussed above, exciton luminescence is not an intrinsic process described only by the optical matrix element. The increase in luminescence rate with decreasing well thickness may well be due to the scattering by well-thickness fluctuations (inducing an energy fluctuation L-3),although a more detailed study of exciton recombination is required to fully understand
-
1. 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES
67
UNPUMPED ABSORPTION n=3
17.2
A
0.2
1.51
I
I
I
1.54
4.57
1.60
(a)
ENERGY (eV) I
I
I
I I
I
ENERGY (eV) 1.2 -
4.0-
;
0.8-
(b)
I
UNPUMPED ABSORPTION-
0.6-
a
0.4 -
0.2 -
148
1.51
ENERGY l e v ) -
1.54
157 (C)
FIG.43. Subpicosecondpumpand-probe experiment: absorption spectra of a 250 A MQW sample are shown before (-) and at later times (broken curves) following excitation. (c) 10" cm-z. Curve (a) shows quite well the Canier density is (a) 5.1011cm-2, (b) 10l2ar2, almost square-shaped absorption edges of 2D systems when the exciton effects are washed out at the densities used here. Curve (b) shows the large band-filling effects at short times. Curve (c) shows the large structureless gain curve (after Shank et U Z . * ~ ~ ) .
68
C. WEISBUCH
luminescence kinetics. It seems that exciton oscillator strength determination through quantitative reflectivity measurements would be a better way to assign exciton-photon interaction, as is the case in 3D. A detailed and profound analysis of exciton dynamics has been carried out by Hegarty et u1.233-237 Resonant Rayleigh scatteringz3’ has been used as a probe of the homogeneous exciton linewidth within the inhomogeneous exciton absorption band due to interface disorder. As shown in Fig. 44, the exciton DOS as revealed by ES, the luminescence line, and the intensity of elastically (Rayleigh) scattered light are shifted relative to one another. The downward shift of the Rayleigh intensity shows the transition from localized exciton states (undamped, i.e., efficient for light scattering) to delocalized exciton states (less efficient). This is therefore an optical measurement of a mobility edge for excitons on the interface disordered band in the sense of Mott. Hole-burning experiments233in absorption at higher intensities yielded very similar behavior for excitons in the band;
c
L
Luminescence
z 0
tK
zrn 4
2
1.610 PHOTON ENERGY ( e V )
1.615
0
FIG. 44. Absorption peak (-), intensity of elastically (Rayleigh) scattered light (0), and luminescence intensity (--) of a 5 1 A MQW sample. W h e m the absorption peak represents the DOS of the 2D disordered exciton band, the Rayleigh eEciency curve is shifted downward because of the inefficient light scattering by delocalized excitons above the center of the band. The shift of the luminescence peak represents the relaxed state of recombining excitons (from Hegarty et aL235).
1. 111-v SEMICONDUCTOR QUANTIZED STRUCTURES
69
finally, transient grating experiments,2" at still higher densities, directly indicate spatial exciton transport, with again the observation of a mobility edge near the center of the exciton energy band. As expected, the mobility edge appears to be strongly dependent on thermally activated exciton hopping, as revealed by increasingthe temperature?% The whole picture of a disordered 2D exciton band appears well justified from this set of experiments. Quantum wells seem to be a good prototype for 2D disordered systems,237with the strong exciton-photon coupling allowing for the use of very convenient optical probes. 15. INELASTIC LIGHTSCATTERING
Inelastic light scattering by electronic excitations is a very powerful tool for the investigation of 2D systems, although perhaps not widely used. We cannot describe here all the important results which have been obtained with this type of experiment and strongly recommend two recent reviews of the s~bject,2~**'~~ still in active progress. At first glance, it might appear that the electron absolute number at an interface might be too small to allow for any sizable inelastic scattering of light by electrons. However, Burstein et ~ 1pointed . ~out that, ~ thanks to resonant enhancement of the efficiency, signals from the standard electron density N,- 10" cm-' should be observed. The resonance at the spinorbit split-off gap at & + A o is usually used in order to prevent hot luminescence signals to obscure the light scattering spectrum. As in the b~lk,2~~*"' two sorts of signals are to be observed the single-particlespectrum, corresponding to uncorrelated particles, which involves a 'spin-flip and is observed in the orthogonal-polarization configuration. The efficiency is then due to spin-density excitations. Excitations of the collective modes of the electron gas (plasmons) are due to the charge-density fluctuations of the gas and are observed in the parallel-polarizationconfiguration. The theoretical considerations leading to these selection rules have been described in detail by Burstein et al.242 Single-particle intersubband scattering has been studied widely as it provides an excellent tool to directly measure energy levels in modulationdoped heterostructures, either single interface^,"^ quantum wells?44multiple quantum wells or n - i - p - i's.245 The measured energy shifts provide good tests for the evaluation of energy-level calculations. Light Scattering measurements of the 2D hole gas'" confirm the AEc determination of Miller et ~ 1 . They l ~ ~ also indicate the nonparabolicity of the various valence bands (see Section 3). Using modulation-doped MQWs with varying spacer thicknesses, Pinczuk et ~ 1 . ' ~ were ' able to show a striking correlation between light scattering linewidth and electron mobility. This is interpreted by assuming that the same collision mechanisms determining electron
70
C. WEISBUCH
I
0
I
20
,
I
1
I
40
60
80
ENERGY SHIFT (rneV)
FIG.45. Single-particle light scattering spectrum [depolarized backscatteri?g Z ( y ’ x ’ ) a of three MQW MD samples. The varying spacer thicknesses of 0,50, and 15 1 A are correlated withincreasingspectrasharpnessand4.2Kmobilitiesof 12,500,28,000, and 93,000 V2 cm-‘ s-I, respectively. The observed transitions are shown in the insert (after F‘inczuk et ~ 1 . ~ ~ ’ ) .
wave-vector changes are responsible for the mobility value and light scattering k-conservation rules (Fig. 45). Collective excitations observed in the parallel polarization configuration allowed the determination of the LO phonon-plasma coupled modes.248,249 Optically created plasmas have been detected by their induced light scattering in undoped MQW structure^.^^.^^' Carrier densities have been determined from the measured shifts. In n- i-p- i structures, light scattering experiments on photocreated carriers have revealed their 2D character, and the transition to 3D at high intensities when light-induced photoneutralization of impurities destroys the superlattice potential.252 Finally, inelastic light scattering in MQW under strong magnetic fields yields inter-Landau level transitions as observed by Worlock et al.253
1. 111 -v SEMICONDUCTOR QUANTIZED STRUCTURES
71
16. LASERACTION Since a number of recent review articles have given full descriptions of we will only point out here a number of quantum-well their specific properties. Only few papers have tackled the analysis of the operating features of quantum-well lasers.254a-261 The first useful property of quantum-well material is the better spontaneous quantum efficiency than corresponding double-heterostructure material. As discussed in Section 13, this can be due to one or a combination of various effects, such as enhanced 2D radiative recombination, impurity gettering, diminished 2D nonradiative recombination, etc. This enhanced efficiency should lead to enhanced inversion of carriers at a given injection if the relevant mechanisms result in an enhancement of the carrier lifetime. The square shape of the 2D DOS increases the gain at low injection. Figure 46 shows a comparison between two situations, 2D and 3D, for the same layer thickness (see discussion in Section 3): at 300 K, a given gain is obtained in 2D for an equivalent 3D electron density - 30% smaller than in 3D. The gain will also vary more rapidly with injection current (Fig. 47). The square DOS in 2D also leads to a smaller dependence of gain, thus of laser threshold, on temperature. One therefore expects a larger Tovalue262 for quantum-well lasers from this square DOS.260 Due to the 2D DOS, the gain spectrum in QW lasers is steeper than in DH lasers (Figs. 47 and 48).263This leads to the useful additional property of quantum wells already discussed in Section 12 and Fig. 41 that, at the lasing frequency, the unexcited quantum-well layer has a lower absorption coefficient (= lo2 cm-') than in a DH layer (=8 X lo2 cm-l). This property might prove helpful in designing efficient monolithic integrated optoelectronic circuits. The increased exciton effects in 2D could also play some role in laser action. Radiative recombination rates might be increased due to such an effect as the Sommerfeld enhancement factor of the optical matrix element due to the electron - hole correlation in the unbound hydrogenic levels. A very important and certainly main effectexplaining the good performance of QW lasers is the small absolute value of the DOS. In 2D, it is only -3 X 1013 cm-2 eV-l for the conduction electrons, as compared to the typical 3D DOS =2 X 1014cm-2 eV-' in a DH at an energy equal to the room-temperature kT. This means that, in order to fill band states up to an energy of the order of kT to satisfy the Bernard-Duraffourg inversion ~ondition,'~,'~ one will require in 2D =6 times less injected carriers per cm2 than in 3D.264This is, however, to be weighed against an adverse effect of the small energy-independent 2D DOS; the maximum gain will tend to saturate with carrier injection, whereas in 3D the ever-increasing 3D DOS
72
C. WEISBUCH 3D
-=,
n + 2 . 0 x 1 0 ~ ~ c(BULK) m~ n . i . 4 ~10%m-3 (QUANTUM WELL)
W
NEEDED TO REACH THE SAME PEAK GAIN IN (CJ
t'
E
2 0 t
a 0 W
2
w
E
0
h
a
\ I\
W
a 0
L
-E €2
FIG.46. Comparison of gain formation in 2D and 3D systems assuming an equal layer thickness (see discussion in Section 4). The same gain is obtained for a lower equivalent 3D carrier concentrationin the 2D systems thanks to the square DOS.
with carrier population allows one to reach enough gain to obtain laser action (unless the sample is blown out). Dutta's explicit calculations255 (Fig. 49) exhibit the expected quantum-well gain saturation, which implies that, if this intrinsic maximum gain is not large enough to overcome losses due to unfavorable semiconductor parameters (masses, matrix elements, . . .), poor material, or very lossy structure, laser action will not
CURRENT (mA)
FIG. 47. Variation of the maximum gain with injection current for a single QW laser (SQW) and a modified MQW laser (MMQW). The gain versus current slope of a DH laser is also indicated, 2 cm-'/mA (from T~ang*~").
Wavelength (a)
(%I
Current ,
111th
( b)
FIG.48. (a) Gain spectra for MQW structure and (b) dependence of the peak gain on injection intensity with DH comparison (from Kobayashi et ~ 1 . ~ ~ ~ ) .
74
C . WEISBUCH I
I
INJECTED C A R R I E R DENSITY (1048 cm-3) FIG.49. Calculated maximum gain for the electron-heavy-hole transition as a function of injected camer density in undoped material at various temperatures (from D ~ t t a ~ ~ ~ ) .
occur. The alternative to the single QW is then to use multiple quantumwell structures (MQWs), which yield a DOS which is the single QW DOS multiplied by the number of wells in the structure (Fig. 50). The MQW, however, introduces a new unknown: injection of electrons at one side of the structure and holes at the other side leads to a spatially inhomogeneous camer inversion, as the carriers are efficiently captured by the quantum wells nearest to each injection side (Fig. 51b). Observed in some structures, this effect tends to raise the threshold.265Tsang made a systematic study of well and bamer thicknesses and well number in order to optimize the lasing characteristics of these MQW structures.266To overcome the injection inhomogeneity, was led to introduce the modified MQW structure, where lower bamers in a MQW structure allow for more efficient carrier transport across the barriers and therefore lead to a more uniform carrier inversion, leading to lower laser threshold. High-barrier confining material is conserved at both extremities of the lasing region to ensure maximum carrier and optical confinements (Fig. 5 lc). Kroemer and Okamoto have discussed some of the parameters involved in efficient electron transfer from well to
1. 111- v SEMICONDUCTOR QUANTIZED
STRUCTURES
75
I00
L, = IOOA
CURRENT DENSITY ( A / c m 2 ) FIG.50. Variation of gain versus injected current density in N-well quantum-well lasers. The saturation effect due to the square DOS appears clearly in the single (N= I ) well case. Increasing the number of wells increases the threshold current as more states have to be inverted (proportional to N), but it also increases the saturated value of the gain (from Arakawa and YarivZ6’).
In the case of single or few QW structures, one encounters an effect which is quite unfavorable to the QW laser as compared to the DH laser: the optical confinementfactor, i.e., the overlap of the optical wave with the quantum well, tends to be very small, varying as L-2due to two the first is the diminution of overlap between the optical wave and the QW. The second is due to the diminishing confinement of the overall optical wave by the vanishingly small waveguide (in the limit of zero thickness QW, the optical wave would recover its natural size “1, whereas in optimum GaAIAs/GaAs structures its width can be reduced to 800 A). The optical confinement factor enters the gain coefficient as it represents the efficiency for an emitted photon to drive another stimulated e-h pair radiative recombination. One way to keep the optical wave optimally concentrated, independently of QW thickness, is to use a second optical cavity to confine the optical wave. In this manner Tsang developed separate confinement heterostructure quantum-well lasers (SCH-QW). As an =i:
76
C . WEISBUCH
nnn
+
\h
@
@
SCH-QW
@
MQW
@
MMOW
GRIN-SCH-QW
FIG.51. Band-energy levels (under forward bias) of various QW laser structures. (a) Single quantum well, SQW, (b) multiple quantum well, MQW, (c)modified multiple quantum well, MMQW (d) separate-confinement heterostructurequantum well; (e) graded-index separateconfinement heterostructurequantum well GRIN-SCH-QW.
additional improvement, Tsang also proposed the graded-index separate confinement heterostructure laser.268r269 This GRIN-SCH-QW laser structure (Fig. 5 1e) has proven to be remarkably efficient, leading to the lasing thresholds.270This was recently explained by Nagle et dz7@ as due to the small DOS in the confining layers of the GRIN-SCH. A promising feature of QW lasers is their long operating lifetime, which now appears to be e x ~ e l l e n tOne . ~ ~of~the ~~ reasons ~ ~ might be the optical inactivity of dislocations. Another possible mechanism is the lower operating current density, which implies a lower rate of recombination-enhanced defect creation in the bulk. Due to the small overlap of the QW and optical field, crater formation at laser extremities due to carrier-enhanced defect formation should also play a smaller role; the crater should have the QW width, therefore acting as a very small perturbation for the optical field. Catastrophic damage threshold should therefore be higher in QW lasers when compared with DH lasers, as’wasrecently 0bserved.2~~ The observed shift between lasing wavelength and spontaneous or calculated wavelength has been the subject of intense debate. It was widely reported that this shift, as well as the high efficiency of QW lasers, should
1. 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES
I
Ga As / A l Ga As Room temp.
77
MQW
W
V
z
W
% W
2
0
t
z
5 3
K
0
I
I
8000
,
,
,
,
l
l
l
8500
WAVELENGTH
l
l
l
l
5:m a
9000
(A )
FIG.52. Reabsorption effect on the edge luminescence of MQW structures: luminescence emitted perpendicular to the MQW planes [no reabsorption(--) or parallel to the plane )] differ strongly due to the sharp absoxption reabsorption “edge” luminescence (edge (-) (after Tarucha et ~ 1 . ~ ~ ~ ) .
be due to very efficient LO-phonon-assisted radiative recombination in QWs.274,275 However, spontaneous LO-phonon sidebands were neither observed in low-e~citation’~~ nor h i g h - e x c i t a t i ~ n ~photoluminescence ~~.~~~ experiments. The gain structure shown in Fig. 43 and 48a shows no LO-phonon structure. It seems rather that the data reported were either due to impurity-related effects or specific to the structures under scrutiny or to the experimental conditions used or to a combination of these. The observed downward shift (=30 meV) of the laser line from the n = 1 HH confined energy level has been carefully analyzed by Japanese teams.’84~278~279 They first demonstrated in photoluminescenceexperiments that the double-peak line shape of the wave-guided emission along the layer plane (same geometry as that of lasers) is due to reabsorption effects in the layer (Fig. 52). Then, the increasing shift to lower energies of the lower component with increasing injection is supposed to be due to the bandgap renormalization at high carrier densities. This last argument is supported by the luminescence studies of modulation-doped QW samples by Pinczuk et where the analysis of the luminescence line shape yields a bandgap renormalization of 17 meV for Ns= 5.9 X 10” cm-2, a density smaller than that existing at laser threshold. Additional support for the exclusion of the LO phonon mechanism is provided by the time-resolved measurements of Fouquet et ~ 1on MOCVD . ~ ~samples: ~ the feature
78
C. WEISBUCH
observed at the so-called LO-replica position is observed to have a decay time much longer than that of free excitons, whereas an LO-phonon replica would have an equal decay time. Auger recombination, a main limitation in the operation of standard DH GaInAs and GaInAsP lasers,281has created interest in the context of QWs. Several authors have evaluated various Auger probabilities for QW laser structure, with diverging conclusions.281-286 Transient measurements by Sermage et give approximately equal Auger coefficients in 2D and 3D. In that case, single-quantum-well lasers are at a disadvantage: due to the low confinement factor, SQW lasers operate at higher volume gain, thus at higher camer densities, which lead to very high Auger recombinat i 0 n . 2 ~ "This ~ ~ ~is~why only the MQW laser, with its low carrier density, has been operated successfully with sometimes a high TWz8' Additional properties of quantum-well lasers were calculated by Arakawa and Yariv.261They predict a twofold increase in modulation bandwidth and tenfold decrease in spectral linewidth. As a conclusion of this analysis of the elements of laser action in QWs, it appears that the analysis of their excellent operation requires the detailed evaluation of a number of opposing phenomena: low densities for finite gain, poor optical confinement, inhomogeneous carrier injection. . . . On the other hand, the success of present GaAs lasers, together with the potential of impurity-induced interdiffision for fabrication stepsZ'o-Z'2,2B8 should stimulate great activity in the QW laser field in the near future.
IV. Electrical Properties of Thin Heterostructures 17. MOBILITY IN PARALLEL TRANSPORT
The flourishing development of modulation-doped heterostructures is based on the extremely high mobility obtained in such structures. As mentioned above, this arises from the spatial separation between charge carriers in the channel and the impurity atoms from which they originate and which remain in the barrier material. It is, however, important to analyze in more detail the various mechanisms limiting the mobility in order to be able to give interface design rules and predict the behavior of the various semiconductor pairs yielding promising interfaces. We shall follow here an analysis first given by S t i ) ~ m e r . ~ ~ ~ Scattering mechanisms are now quite well understood and are measured in bulk s e r n i c o n d ~ c t o r s , ~although ~ - ~ ~ ~ some higher-order phenomena (such as multiple Coulomb scattering)have never been completely worked The scattering mechanisms in the usual perturbative description are decomposed into five contributions.
1. 111 -v SEMICONDUCTOR QUANTIZED STRUCTURES
79
( I) Optical-phonon scattering (dominant at high temperatures);
(2) Acoustic-phonon scattering due to the deformation potential; (3) Acoustic-phonon scattering due to the piezoelectric field (111-V and 11-VI compounds are piezoelectric due to their lack of inversion symmetry); (4) Scattering by ionized impurities; and (5) Scattering by neutral impurities. The importance of the various mechanisms is shown in Fig. 53 for bulk GaAs as well as experimental results for high-purity VPE GaAs. It is clear that at high temperatures mobility is limited by LO-phonon scattering, very efficient through the Frohlich mechanism, whereas ionized impurity (- N,, - NA)scattering dominates at low temperatures. Two points should be added (i) In doped bulk GaAs, the mobility depends on the shallow impurity concentration, even at room temperature. (ii) In some optical experiments, ionized impurity scattering can be totally suppressed even at
Z t
m
0
I
2
4
6
8 40‘
I 2
I
I 4
I
\
I I I l l 6 8 (0‘
I
I
z
TEMPERATURE ( K )
FIG.53. Experimental temperature variation of the mobility of a high-purity GaAs VPE sample (N,, = 4.80 X lot3~ m - NA ~ ;= 2.13 X 10” crn-’) and calculated mobility curves for each scattering process acting separately and for all scattering processes combined (from Stillman and W01fe~~~).
80
C. WEISBUCH
low temperatures ( T = 4 K) owing to the photoneutralization of ionized impurities by photocreated electrons and h0les.2~~-~% Under such conditions mobilities of -2 X lo6 cm2 V-’ s-’ were observed in bulk GaAs by optically measured electron drift velocity294and by optically detected cyclotron resonance.296Very high hole mobilities were also observed.296Such experiments give direct evidence of the dominant limiting effect of ionized impurities in bulk material at low temperatures, and show that the suppression of this scattering mechanism indeed leads to mobilities similar to those observed in the best MD 2D samples. In heterojunctions or quantum wells, the same five mechanisms apply for carriers in the channel, as well as some additional ones?89
(6) Scattering by GaAlAs phonons (7) Scattering by ionized or neutral impurities located in the bamer material (spatially separated from the channel camers) (8) Scattering by alloy disorder, either in the barrier material such as in the GaAlAs/GaAs case, in the channel when the channel material is an alloy, as in the case of InP/GaInAs or in both, as in the GaInAs/AlInAs case (9) Surface phonon scattering, as new propagating surface modes exist at interfaces. However, since the materials are usually similar in density and dielectric functions, these phonons modes never create large scattering probabilities and will not be considered any further ( 10) Interface roughness scattering (1 I) Intersubband scattering between the quantized levels in the channel These different mechanisms have been analyzed in great detail by varwe~ will only review their main conclusions perious a u t h ~ r s , ~ ~and ’-~ taining to the GaAlAslGaAs selectively doped heterointerface, unless otherwise specified. Mechanisms (1)- (3): The various phonon scattering mechanisms do not change significantly for the quantized channel camers when compared to the bulk situation. The transition rates from a subband state In, k ) to another subband state In’, k’) have been calculated for various phonon scattering mechanisms by P r i ~ e .In~ short, ~ ~ ,the ~ usual 3D momentum conservation in the k direction is replaced by overlap integrals Fn,d(4): Fn,n*(q) =
B
dz dz’ xn(z’),yd(z’)e-d2-’’) xn(z)xn(z’)
(56)
where xn(z)and xn(z’)are the envelope wave functions of the In, k) and In’, k’) states and q = Ik - k’l. A complete calculation by Vinterm shows a mobility reduction lower than 25% at 77 K. One therefore expects high-
1, 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES
81
temperature (>80 K) mobilities of 2D carriers to be comparable to those of 3D electrons when phonon scattering is dominant. The limit of the hightemperature mobility of 2D systems is therefore that of high-purity bulk material if all other scattering mechanisms are small (ionized impurities suppressed by modulation doping and low residual channel doping, low interface roughness, etc.). Actual observationsm show that this is indeed the case and therefore that the LO-phonon scattering rate is similar in 2D and in the bulk. The advantage here is that such values are obtained for highly conductive channels as compared to the low conductivity (low n) of high-purity 3D GaAs.
CHANNEL ELECTRON DENSITY ( 10” ~ r n - ~ )
FIG.54. Calculated low-temperature reciprocal mobility versus channel electron density N, for a GaAs-Ga,-,Al,As heterojunction with 7 X lot7cmV3donors in the barrier with binding energy Eob= 100 meV, a heterojunction barrier height V, = 300 meV, an acceptor doping level Nk = lOI4 ~ m in- the ~ GaAs (for which the density of depletion charges, Nd,is 0.46 X l o l l cm-2), and a residual density of charges in the spacer layer also equal to lot4 cur3. The spacer layer thickness d, is determined for each value of N,. The three sources of scattering, from the barrier doping itself, from the residual doping in the spacer layer, and from the acceptors in the GaAs, are separately shown (from Stern).151
82
C . WEISBUCH
(4) and (5) ionized and neutral impurities in the GaAs channel: Usually the residual doping is quite small (~ m - and ~ ) does not influence the room-temperature mobility. In the case of intentional doping, an impurity contribution to the mobility is ~ b s e r v e d .At ~ ~low ~ .temperatures ~~ and low densities, when all other causes of scattering have been reduced ( p > 2- 3 lo5 cm2 V-' s-'), the limiting factor is still the uncontrolled channel impurity doping.151Progress through the years is well evidenced in the sum-up figure (no. 1) in the review by Mendez.Msa (6) Scattering by GaAlAs phonons does not play a significant role at any temperature. At low temperatures all phonon mechanisms are suppressed; at high temperatures the GaAlAs phonons can be neglected, as the camer wave function penetration in the barrier is negligible. (7) Scattering of channel carriers due to the Coulomb interaction with barrier impurities is an important mechanism of scattering due to the high doping density of the barrier (Fig. 54). Such a mechanism has been calculated in detail and will not be reproduced here.299The most remarkable factor appearing in the scattering time is the form factor of the Coulomb interaction matrix element: m , Z )
=
j-
dz'
lX(Z')I2
exp(-qqlz - 2'1)
(57)
where z is the impurity position, q = 2k sin(8/2) is the scattering vector of the electron with wave vector k, ~ ( zis) the confined electron wave function. As expected, the interaction decreases with increasing impurity channel separation thanks to this form factor. This is directly evidenced in front- and back-gating experiments which change the electron wave function penetration in the barrier (see Fig. 60). Another important effect originates from the form factor: as those electrons being scattered are near EF at low temperatures, one needs to evaluate Eq. (57) for q = 2kFsin(8/2). When kF increases with the channel density N,, this form factor remains significant only for the smaller scattering angles. Such small-angIe scattering events, even though efficient in terms of collision time, can be expected to be less efficient for momentum relaxation time (i.e., mobility increases) because of the factor 1 - cos 8 in the momentum-loss integral. We therefore have the main ingredients of bamer-impurity-limitedmobility: it increases both with impurity-channel separation and with channel density. As these two factors vary in opposite directions with undoped spacer thickness for a given alloy doping, one expects a maximum mobility at some value of the spacer. Assuming now a fixed spacer, one has to change the doping density of vary channel density. In such as case there is also some optimal value of N, (Fig. 55). Such tendencies have been observed experimentally310(Fig. 56).
1. 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES
83
Coulomb Scattering di
(1)
N ~ , , ~ , 5x10'~cm-* E, = 50 meV K = 0.25 I
‘.S
\-I..
,
FIG. 55. Influence of Coulomb scattering as a function of channel camer density N, and spacer-layer thickness d,. The increase in N, is determined by a change in the doping concentration of the GaAlAs bamer. As long as N, is smaller than Ndep,the main effect of increasing N, is to reduce the scattering rate because of increased electron velocity and channel-impurity screening. Above that value, Coulomb scattering by remote donors in the GaAlAs barrier takes over and decreases the mobility (from Ando”’).
The low-temperature behavior of various high-purity samples shows the delicate balance between the various impurity and phonon scattering mechanisms. The temperature dependence of the mobility switches from positive slope to negative slope when the sample mobility increases, i.e., when the impurity-related scattering rate decreases (Fig. 57). This effect was shown by Lin et aL2@to be due to the balance between impurity-limited mobility (from the bamer or in the channel) (positive slope) and acoustic-phonon-limited mobility (negative slope). It is remarkable that for the purest samples studied, even though the main scattering mechanism is due to impurities, the temperature dependence arises from the smaller, but strongly temperature-dependent, acoustic phonon mechanisms. Several authors have used this determination of the acoustic phonon scattering rate to evaluate the various phonon scattering mechanisms.161,300,35,310-313 The form factor in the scattering probability leads to opposite variations of acoustic deformation potential and piezoelectric scattering rates with varying channel electron density N,. In order to obtain a good fit of the observed decrease of the phonon-limited mobility with N, using generally
84
C . WEISBUCH 2.0
I
I
I
I
? NE
0.0
z
8 L
0' 0
I
I
10
I
20
30
I 40
SPACER THICKNESS (
FIG.56. (a) Mobility and (b) channel electron densities as a function of undoped GaAlAs spacer thickness. Fitting curves are calculated after the method described by Stem. NAcand Nspare residual acceptor concentrations in the GaAs and GaAlAs layers, respectively, NDb donor concentration in the doped GaAlAs barrier, Vbbarrier height, and Em, donor energy in the barrier material (from Heiblum et uf.16*).
accepted phonon coupling parameters, Vinter3I3carried out a calculation involving accurate wave functions and screening of the electron- phonon scattering interaction. (8) Alloy disorder scattering is due to the statistical composition fluctuations which are unavoidable even in perfectly grown but fundamentally disordered alloy^.^^^,^'^ Such fluctuations give rise to a random fluctuating
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
2t 1031 I 100 2
I
85
1 I I I
I
4 6810‘
2
I
I l l
I
4 68102 2
I
I
I 1
4 68103
TEMPERATURE (K1 FIG.57. Temperature dependence of electron mobility in a series of GaAlAs/GaAs MD heterostructures(from Lin”’).
potential, well known to limit the mobility in bulk alloy semiconductors. For the GaAs/GaAIAs heterojunction case, there is a weighting factor to the “bulk” alloy mobility given by the channel wave function penetration in the barrier.299As this penetration is typically a few percent, reaching such values as maximum N,, this mechanism is only important at high N,. In the case of an alloy material channel, such as InP/GaJnAs, the alloy potential is undiminished by such a factor and sets a rather low maximum mobility to be expected from the 2D electron^.^^^^^'^ ( 9 ) Scattering by interface roughness. The exact topology of the interfaces is usually unknown. In the case of transport properties, it is modeled by a Gaussian correlation function of the surface position:
-
(A@) A@’))
= Az exp(-lr
- r’12/A2)
(58)
where A@) is the average displacement of the surface height at position r and A represents the lateral decay rate of the fluctuations of the interface. Such changes in the interface position can be modeled to act as a spatial variation of the position of the band discontinuity at the interface. One can calculate the transition probability due to such a variable potential by a
86
C. WEISBUCH
perturbative approach to the otherwise perfectly plane interface used in the calculations of Section 8. Ando299found for the relaxation time 71R(k) due to interface roughness
x (1 - cos 8 ) a(&/(- E/(-J
(59)
where c(q) is the static dielectric function of the 2D electron gas and the interface potential effect is represented by F, the electric field that confines the channel electrons:
The result of the integration over q is shown in Fig. 58, assuming a mean displacement of the interface A = 4 A and a lateral correlation length 1 = 15 A.The effect of such an interaction should be observable, at least at high electron densities for extreme-purity samples. It has, however, not been systematically studied. It has only been indirectly shown, as samples grown outside an optimal temperature of 600- 700°C, dependent on the growth parameters, show poor mobilities.318The high- and low-temperature growth ranges have been correlated by other methods (x rays, TEM,
-Present N~ (cm-2)
FIG. 58. Calculation of the interface-roughness-limited mobility. A is the amplitude of interface position fluctuationsand 1 the lateral correlation length of these tluctuations (from and^"^).
1. 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES
87
optical spectroscopy) to interface roughness (see the discussion in Section 12),which should then be the mobility-limitingfactor if all other scattering causes remain the same. (10) Intersubband scattering occurs at high densities when higher-lying quantum levels (n = 2, 3, . . .) can be p o p ~ l a t e d The . ~ ~ phase space for the final states in scattering events is then larger, increasing the scattering probability and hence diminishing the mobility. Through a study of backgated Hall samples, Stormer et aL319were able to show such a decrease in mobility with an increase in channel electron density, and correlate this effectwith the population of the n = 2 quantum level through the appearance of a double period in Shubnikov-de Haas oscillations (Fig. 59). The effect was also observed by Englert et u I . , ~ ~ Owho used a magnetic field parallel to the layer to change the band separation and show the change in sample resistance when the number of populated subbands changes. The various scattering mechanisms in heterostructures therefore appear rather well understood. The best reported mobility to date for GaAs/ GaAlAs is 2.6 X lo6 cm2V-' s-'. It is not clear whether this value can still be significantly improved, as several scattering mechanisms seem to combine to set a limit mobility at a few lo6 cm2V-' s-'. We have limited our examples to the case of electrons in the GaAs/ GaAlAs system, but a similar analysis of the various scattering mechanisms leading to similar results can be performed on other systems, as has Several been exemplified on holes in the GaAs/GaAlAs teams have successfully operated a p-type FET based on this structure, complementary to the n-type ~ t r u c t u r e . ~ ~ ~ - ~ ~ ~ Besides'the single GaAlAs/GaAs heterointerface, several other systems have been considered. Very early, Mori and Ando326,327 calculated the parallel mobility in modulation-doped superlattices and showed the importance of intersubband scattering. The case of modulation-doped single wells has been considered by Inoue and S a k a I ~ iFor . ~ ~rather ~ wide wells, this situation is equivalent to the double heterointerface situation, but gradually changes to a new situation when the well thickness is decreased. The advantage of this QW structure is the higher electron transfer that can be obtained, about twice when compared to the single interface, which leads to better device characteristics due to the better conductivity. Until recently, this structure could not be grown with good equivalent interfaces (GaAs grown on GaAlAs was bad), but recent progress in growth techniques allowed good symmetric, structures to be g r o ~ n .On ~ the ~ ~other , ~ ~ ~ hand, Sakaki used the different mobilities of the asymmetric interfaces (under bias) to design the velocity-modulation transistor (VMT)331,332: a large change in channel conductivity is controlled by the gate potential, which confines the carriers on one or the other of the two interfaces of the
AREAL DENSITY n~ [lo" 8.5
0.0
I
-600
I
9.0
I
I
400
-200
0
I
I
200
GATE VOLTAGE Vg [V]
FIG. 59. Onset of intersubband scattering in a GaAs/GaAIAs MD heterostructure. The backside-gate voltage dependence of electron density [(a), (b)] and mobility (c) shows the correlation between the decrease in mobility and onset of upper-subband populations. The Shubnikov-de Haas measurement (a) yields the densities of the two subbands from the two oscillation periods observed. It agrees with the Hall measurement of electron density (b) (from StiSnner et aL319).
88
0 G mode
F G mode
(a)
(b) n-AIGaAslGaAs
40
-
I
R-73 FETE
30
Wsp.4.5nm. W~=123nm
al V
Wsub=120pm Ng’4.6 x 1017cm-3 x =0.3
In
2
N
20-
Eu
5)
8-
2 d
1
”
/
-
A 4
1 4
i L’
v
.-2,
1
I
T = 8.9K
U
I
I
I
I
i t A’ ;.‘
10 -
-5-
-
:..’
A’ 1 . // .: FG/’ ...
----.........
/ . . /4 :. EG // .:.
//*
.
,/*
,:. I .
--
Theory
-
NA+Ni= 3 . 2 5 ~ 1 0 ’ ~
cm-3-
K= 0
-
:._. I
I
I
I
I
l
l
,
90
C. WEISBUCH
quantum well, changing their mobility through the deformation of the confined wave function and therefore their interaction with remote ionized impurities in the barrier material (Fig. 60). As one controls the channel conductivity by changing the carrier velocity without changing the carrier density, the switching time of the VMT should not be limited by chargingtime effects and might reach the subpicosecond range. IN PARALLEL TRANSPORT333 18. HOTELECTRON EFFECTS
Since many of the applications of digital heterostructure ICs call for high-speed devices, it is highly desirable to know the high-field properties of such heterostructures. A number of theoretical calculations have been carried out to evaluate the various relaxation rates in 2D systems and the resulting high-field properties297*334-34 (Fig. 6 1). The main result of these calculations using different methods is that the 2D energy relaxation rates should be comparable to the 3D rates. Ridley has predicted that, due to the peculiar 2D momentum, energy relaxation and intersubband scattering, intrinsic negative differential resistance (NDR)could occur.337-339 Hess and his collaborator^^^ have in addition predicted and shown that camer heating in a heterostructure gives rise to a new mechanism of NDR by real-space transfer over the potential barriers. Such concepts of real-space transfer were applied by Kastalsky et al. to design a number of new high-frequency device^?^-^' Room-temperature hot-electron characteristics have been studied by a number of groups using Hall measurements under pulsed applied electric field in order to avoid lattice h e a t i r ~ g . ~ ~Velocities * - ~ @ significantly higher than in bulk GaAs were obtained at 300 K and increased even more at 77 K (Fig. 62).There certainly is a good improvement in performance in TEGFET devices when going from room temperature to 77 K as compared to standard MESFET based on bulk GaAs material. Several additional effects can occur for the hot-electron regime at low temperatures: Schubert and Ploogw observed a decrease in conducting electron density by Hall measurements at 77 K in the GaAlAs/GaAs interface. They explain this effect by the scattering of electrons into the higher-lying confined subband level E2 where they have a low mobility, and by the trapping of hot electrons in localized states situated in the barrier material near the interface. At liquid helium temperatures, the electron density tends to increase in both the barrier and channel due to impact ionization of neutral Si donors in the barrier material in the hotelectron regime. Energy relaxation rates were measured optically at low temperat u r e ~ ~as~in~the , ~3D~case, : carrier heating is deduced from the line-shape analysis of the photoluminescence line (high-energy slope exp(- hv/ kTd), where Td is the carrier effective temperature), the carrier heating
-
1. 111-
v SEMICONDUCTOR QUANTIZED STRUCTURES I
I
I
1
I
I
I
I
I
91 1
51-
----. 2 ----...-.. I-
0
80 0 40 80 ELE C T R 0N K IN E T I C E NE R G Y (meV ) FIG.6 1. Calculated optical phonon scattering rates via the Frahlich mechanism at 300 K. (-*--) represents the 3D scattering rate, (--) includes only the intraband scattering probability (n = 1 to n = 1 confined state) for scattering out of 2D electrons from the lowest (n = 1) subband, while (-) includes inter- and intraband scattering in a GaAs/GaAIAs heterostructure. (a) N, = 4 X 10'0 cm-2; (b) N, = 6.2 X 10" cm-2. The abruptness of the onset of phonon emission at E 36 meV is characteristic of the square 2D DOS. Note the comparable scattering rates in 2D and 3D except near onset (after Vinte?). 40
0
-
being produced by an electric field applied to the illuminated area. Energy relaxation rates are deduced from power-balance equations, which equate the energy loss to the lattice with the energy gained per camer from the amlied field:
These rates have been measured both for the electron and hole gas in the GaAlAs interface. For electrons, camer heating can be detected for fields as low as 0.3 V/cm. Such an efficient electron heating, due to the very high mobility, has also been observed by the Hall e f f e ~ t ~ and , " ~ damping of the Shubnikov- de Haas oscillation^^^^^^^^* (Fig. 63). Comparing electron and hole relaxation rates, Shah et al. found a scattering rate 25 times larger for holes than for electrons.= This difference, which cannot be explained by 2D or coupling effects, has been attributed to the accumulation of hot phonons, well above the thermal number, which interact predominantly with electrons.22" 19. PERPENDICULAR TRANSPORT
As mentioned in the introduction, the hope for new effects in perpendicular transport gave impetus to the development of superlatticesand heterostructures. The semiclassical equations of free motion for electrons in an
C . WEISBUCH
92
As: Si
GaAs/n-AI,Gal-,
.-, 0
20
T=300K
/
/
1
‘BULK
/
n-GaAs
ND=2.5 x 1017,,-3 I
I
I
200
300
LOO
EL. FIELD E CV/cml
MBE # 5182 GaAs I n - A l x G a l - x A s : S i T=77K
(b)
EL.FIELD E I V l c m I FIG.62. Electron drift velocity at (a) 300 K and (b) 77 K under strong applied electric field for bulk or modulationdoped GaAs/GaAlAs heterostructures. The 77 K curves displayed correspond to the fraction of electrons in the 2D lowest subband of the channel and to the whole averaged electron gas. Note the large increase in velocity between the bulk and 2D electrons at 77K (from Schubert and P1oog’).
1. 111- v SEMICONDUCTOR QUANTIZED
-
-50 -
I
I
I
2
at 4.2K dark Ns
-
0,
I-
W
I
93
n -GaAIAs I GaA5
Y
Y
I
STRUCTURES
20-
ia u-333 0 R-6 T R-98 A U-319
(3.5~1011) (4.6~10~’) (7.1 x10”) (8.1~10’’)
:
.
r
I
! ;A
i’ O
-
FIG.63. Electron heating as deduced from the damping of the Shubnikov-de Haas oscillation. (-) represents the results of Shah et al. deduced from the analysis of the luminescence line shape (from Sakaki et d.”’).
energy band E(k) (infinite solid) with an electric field F are
In a steady applied field k(t) = k(0) - eFt/A For electrons in a band, k therefore changes linearly with time. The energy of the electrons also changes according to the dispersion curve E(k), and so does v(k). In the reduced Brillouin zone scheme, once the electron reaches a zone boundary point k, it is Brag reflected in the opposite direction; i.e., it appears at the -km point. Thus v(k) is an oscillatory function of time with a period equal to the time needed for k to cross the Brillouin zone, T = (2~/d)(eF/h)-~, where d is the lattice periodicity (Fig. 64). The motion in real space would have the same frequency, and a very fast oscillator called a Bloch oscillator could be a ~ h i e v e d .However, ~ , ~ ~ ~ the period has to be shorter than the collision time, which is currently impossible when d is an atomic lattice constant (T= lo-” s for F = 10 kV cm-’
94
C. WEISBUCH
I
BLOCH OSCILLATIONS
I
E - E , + E, co s k d v. I/* a E / a k = F
k(t) = k(0)
I
t -to+t,cos
-
sin k d
h
wa
FIG.64. Schematics of Bloch oscillation.
and d = 3.5 A), but should become possible when the lattice constant d is that of a superlattice, about 10 to 50 times larger. The existence of Bloch oscillations was, however, challenged quite early352:the main argument is due to the fundamental modification of the band structure in an electric field, which allows interband transitions at Brillouin zone boundaries rather than Brag reflections. We refer the reader to two recent discussions on the validity and conditions for observation of Bloch oscillation^.^^^^^^^ Taking collisions into account, Esaki and Tsu3,350calculated the drift velocity in an inJnite superlattice using a classical method.35' The velocity increment in a time interval dt is, from Eqs. (61) and (62) eF d2E
do, = --dt k2 dkj The average drift velocity imposed by collisions occumng with a frequency T-* is
As k is changing with time, d2E/dk2is a function of time, and one requires the knowledge of E(k) to proceed further. Assuming a sinusoidal dependence of E on k, E = Eo 2E, cos kd, one finds
+
2.'d =
nk
m,d 1
p
+ n2P
where = eFTd/nfi and 1/msL= ( 1 /fi2)(d2E/dk2).
1. 111- V SEMICONDUCTOR QUANTIZED
STRUCTURES
95
The ud versus F curve has a maximum for a(= 1 and exhibits an NDR beyond this value. The condition to be fulfilled on T to achieve NDR is about 6 times easier than that required to achieve Bloch oscillations. Effects in jnite (i.e., a low number of barriers and wells) heterostructures were also considered very early by Tsu and Esaki using a multibanier tunneling model.120A pioneering theoretical analysis of the I - Vcharacteristics was provided by Kazarinov and Suris as early as 1 972.355a In addition to being very tractable with few-interface problems, such a formalism allows one to treat the case of intersubband electron tunneling transfer which occurs at high electric fields, an effect not easily described in the formalism of Bloch transport [Eqs. (61) and (62)]. The negative differential resistance observed in superlattices by Esaki et a1.13,354,355 was actually explained by resonant electron transfer between adjacent wells due to coincident ground and excited states. The formalism also allows one to take into account the effect of unequal layer thickness and/or interface disorder (caption of Fig. 1) which, in the Bloch oscillator formalism, would lead to untractable scattering events, as they destroy the coherence of the superlatticewave function [of the type described by Eq. (28)]. The electron motion is then most easily described in a hopping model between localized states. The transition between the two types of transport, superlattice or hopping, is discussed by Calecki et a1.355b and compared to experiments. A discussion of the theoretical foundations of quantum transport in heterostructures can be found in Barker.355c The resonant transmission of single and double heterostructures is the subject of renewed interest due to recent advances in growth ~ o n t r o l . ~ ~ ~ - ~ ~ * The origin of the effect is shown schematically on Fig. 65. Resonant transmission of current occurs whenever the Fermi level of the injecting side is resonant (energy matched) with the confined quantum state El in the well embedded between the two GaAlAs barriers. This gives rise to a current maximum occumng at an applied voltage 2E1, as this applied voltage is split into two equal voltage drops at each barrier. These predictions were verified in 1974 by Chang et af.,355 and more recently by Sollner el uI.,*~ who were able to demonstrate the resonance tunneling effect even at room temperature and negative differential resistance with a peak- to valley ratio of 6: 1 at 25 K. High-frequency response with far-ir lasers shows that response times are less than s, consistent with tunneling times which are given by the uncertainty relation T 5 h/AE = s, where A E is the energy imbalance of the tunneling state. Oscillations at 200 GHz and nonlinear harmonic generation above 1 THz are Sollner’s early 1987 state of the art. These results seem to indicate that such systems, much simpler to implement than the superlattice Bloch oscillator, could lead to efficient millimeter and submillimeter amplifiers and oscillators.
96
C. WEISBUCH
Similar resonant tunneling effects were observed in p - n - p GaAs homoThe theoretical analysis of even such a simple heterostructure as the double barrier requires large efforts if one wants to obtain quantitative agreement with the simple bamer-transmission theories predict extremely large peak-to-valley ratios, whereas the best samples only exhibit peak-to-valley ratios = 3 at room temperature. A first correction to simple theories is to use more realistic structures than that shown in Fig. 65: due to the electric fields existing in the structure, one expects charge accumulation at the first heterointerface coming from the left and charge depletion at the last heterointerface at the right of Fig. 65. Such charges play an important role in the operation of devices, in particular in determining their frequency limit as shown by Sollner et a1.,358d but should not change the peak transmission value. One therefore has to take into account transmission channels, competing with the resonant transmission channel, such as nonresonant transmission due to phonon-assisted processes (which do not conserve energy) and to thermoionic emission across the structure. These processes yield a structureless, voltage-increasing current which strongly diminishes the peak-to-valley ratio. Besides these competing transmission mechanisms, one also has to consider the possibility of incoherent scattering during the transmission time in the resonant tunneling process: L~ryi’~& has shown that NDR is not a proof of a resonant, coherent transmission process as NDR can exist when no Fabry-Perot effects due to resonant tunneling are present: NDR only originates from energy and momentum conservation rules for the left-electrode electrons contained in a Fermi sphere and transmitted through the single left barrier. The existence of a coherently transmitted electron wave function through the whole double barrier structure leads to a much higher aspect ratio of the transmitted wave thanks to wave function buildup and interference in the well due to multiple reflections on both bamers, just as in the classical Fabry -Perst optical resonator. This description could therefore provide an alternative explanation of the poor peak-to-valley ratio usually observed: it would then be due to inelastic scattering events of the quantum-well state during the transit time. Such inelastic scatterings as phonon scattering would destroy the charge buildup in the well and therefore would wipe out any sharp resonance. Weil and have, however, shown that under reasonable assumptions, incoherent scattering of the quantum-well state has little effect on the current and that the two possible descriptions of tunneling, i.e., coherent transmission or sequential transmission, lead to equal transmitted currents. Their conclusion supports the model of poor peak-to-valley ratios as being mainly due to non-energy-conserving transmission mechanisms.
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
97
FIG.65. Schematics of resonant tunneling effect. Energy levels in a single-well double-barrier heterostructure (top three drawings) under bias increasing from the top. The electron energy is indicated as a function of pqsition. Parameters are ND, = ND3= 10I8 ND2= 10'' ~ m and - ~W, = W2= W, = 50 A. Resonant transmission occurs for V = 2E/e, when electrons tunnel resonantly into the n = 1 well state from the left electrode (from Sollner et a1.25).
Nonresonant hot-electron tunneling was used in a hot-electron tunneling transistor by Yokoyama et a1.% (Fig. 66). Due to the high kinetic energy of the electrons in the base region, the electron transport is ballistic, leading to transit times well below the picosecond range. Using tunneling358hor thermoionic injection,358ia variety of hot-electron structures have been designed for vertical d e v i c e ~and ~~~ hot-electron j studies, renewing the field of ballistic t r a n s p ~ r t . ~ ~ ~ J * ~ A new optical technique was developed to monitor perpendicular transport: photoexcited carriers moving in conduction and valence bands of
98
C. WEISBUCH
AlGaAs
n-GaAs
Collector EFe--n-GaAs
c
100 nrn
FIG.66. Band diagram (top) and schematic (bottom)of a tunneling hot electron transistor (from Yokoyama ef ~ 1 . ~ ~ ) .
superlattices are trapped and detected in deliberately introduced enlarged quantum wells which act as probes of spatial t r a n ~ p o r t . ~ ~ ~ ~ ~ ~ This field of perpendicular transport is certain to develop significantly, with the bandgap-engineered structures and other real-space transfer devices described in Capasso’s companion ~hapter.~’
20. QUANTUM TRANSPORT36’ As in studies of 3D electrons, transport measurements under strong magnetic fields (in the so-called “quantum regime,” where o,z >> 1, T being the carrier collision time) provide a vast amount of information
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
99
about the parameters of the 2D electron gas. They have also opened the large new area of quantum Hall effects, a major advance in solid-state physics. We therefore devote a detailed description to such studies, broken into three parts: the effect of magnetic fields on 2D electrons, the Shubnikov-de Haas effect, and the quantum Hall effect.
a. Eflect of a Magnetic Field on 2 0 Electrons In the 3 0 quantum-mechanical problem of electrons in a magnetic field B,,361the motion in the x- y plane is described by Landau levels. The wavefunction is given by (in the Landau gauge of the vector potential A = [0, xB, 01) iyynk(r) = (LyL,)-1/2qn(x - Xk)eiky’ eikzr
(66)
where L,, L,, and L , represent the dimensions of a 3D crystal, the qn functions are normalized wave functions of a harmonic oscillator with the quantum number n centered at point X, = -rzky, r, being the classical cyclotron radius of the lowest oscillator (n = 0) orbit,362
r , = W The oscillator quantum number n can take the values 0, 1, 2, . . . . The energy eigenstates are
Ed
= (n
+ +)ha,+ g*pBB, + E ,
where w, is the cyclotron frequency eB/m*, g*pBB, is the spin magnetic energy, g* being the Land6 factor, and E, is the energy associated with the z motion of the carrier. From Eq. (67), it is easily shown that the quantum states in k space are located on cylinders with their symmetry axes along the z direction (Fig. 67a). In the z direction the usual quasi-continuum free-particle DOS has the value LZ/2n.For the x - y motion, states are characterized by the cyclotron energy ( n +)fza,, located on circles with radii k; k: = (2 m */ h2)(n +)hw,. The degeneracy of each single-spin Landau level (i.e., the number of states on each circle) can be found from the number of possible cyclotron orbits in the crystal. One has to ascertain that the center of the quantum state is within sample boundaries, i.e., 0 < x k < L,; this can be transformed into
+
+
+
0 < k,
< mw,L,/A
(68)
From the density of states in the ky direction Ly/2a, this means that the number of possible states in the range [0, rnw,L,/h] is L,Lymo,/2ah; i.e., the DOS per unit area is mwC/2nh= eB/h. Comparing this with the
100
C. WEISBUCH
'
I
I
I
I
I
I
1/2 9/2 512 3J2 112 REDUCED FERMl ENERGY ( EF/*Wc)
,
(a
(€9
FIG. 67. Allowed states and density of states (DOS) for a 3D electron gas in a magnetic field B,. (a) Momentum-space occupied states: Allowed states are characterized by the relation E = (n f)fiwc E,. Such a relation defines cylinders of axis k, and radii k: k: = 2m*/fi2(n thw,). (b) Density of states dN/dE. The change in Landau state degeneracy is smooth, whereas the z-motion DOS diverges each time a new Landau state enters the Fermi sphere, which is reflected in the total DOS. The 3D DOS is shown for comparison.
+
+
+
+
number of states in zero field contained within the energy separation between two Landau states, i.e., (fio,)[rn*/(27rfi2)] = rnoJ27rfi = eB/h
(69) we find the same value! Thus the average density of states in a quantizing magnetic field is unaffected. Instead of having a 2D continuum of states, these states are all collapsed in a single degenerate Landau state. For a 3D electron system, the occupied states within a given Fermi energy EF are contained in a 3D k sphere of radius k, = ( 3 E , / 4 7 ~ ) 'if/ ~no magnetic field is applied. When a quantizing field exists, all the states situated within the sphere on the allowed state cylinders are occupied. The density of states, given by dN/dE, is shown in Fig. 67b. It shows a divergence typical of 1D systems, each time a new cyclotron state comes into the Fermi sphere. For that new Landu state, the number of states is given by the degeneracy of each k Landau state times the density of states for the z motion, i.e.,
which diverges as the cylinder of allowed states is then tangent to the Fermi sphere, yielding numerous new states for a small change in magnetic field (ko,) or charge carrier density (change of EF). In real systems, broadening will wipe out the divergence, but the periodic behavior of the DOS is
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
101
retained. The period is given by the change in the number of cyclotron states in the Fermi sphere, determined by
(n
+ f)hw,= EF
For a fixed number of carriers, it can be shown that the DOS at the Fermi energy oscillates with the magnetic field. As many physical quantities depend on the D O S at the Fermi energy, they will exhibit oscillations with the magnetic field. Such effects such as Shubnikov-de Haas [oscillatory magnetoresistance due to an increase in the scattering rate whenever p ( E ) diverges], de Haas - van Alphen (magnetic susceptibility), etc., have long been observed in 3D systems and have been widely used to analyze the electronic properties of metals and semic~nductors.~~~ In 2 0 heterostructures, with a magnetic field perpendicular to the layer plane, the same Landau quantization occurs (Fig. 68). However, the effect is even more dramatic as the z motion of carriers is also frozen by the confining potential leading to a “completely confined quantum limit” system. The energy-level structure is made up of a ladder of cyclotron levels for each confined state, each level having a singular DOS (Dirac-like function) with a degeneracy of eB/h. As in 3D, this degeneracy is equal to the number of 2D states contained in the energy spacing between two consecutive Landau levels. For real systems, disorder (random impurities, alloy fluctuations, interface roughness, . . .) will broaden this singular
tE
tE Partially filled
P*D(E)
2
0
@
FIG.68. Magnetic field effect in 2D systems. (a) Energy levels and DOS of a heterojunction without magnetic field. (b) Energy levels and DOS of a single quasi-2D level in a magnetic field [scaled up compared to (a)]. Electrons occupy Landau levels up to some last partially filled Landau level.
102
C . WEISBUCH
DOS (Fig. 69). The states in the tails of the levels are localized in space and will be shown below to play an important role in the existence of the quantum Hall effect. The oscillatory behavior of several quantities has been ~ a l c u l a t e d ~ ~ ~ . ~ ~ as in 3D, the Fermi energy oscillates. When a Landau state is not completely filled, the Fermi level lies in that state and has therefore a smooth variation with the magnetic field or electron density. However, when the last occupied Landau state is filled, the next electron must lie in the next Landau state, and the Fermi level jumps there. The result of the calculation of EFincluding a Gaussian broadening of the Landau states is shown in Fig. 70a. Other quantities have been calculated, such as the magnetization (de Haas-van Alphen effect) (Fig. 70b), the specific heat (Fig. ~OC), the thermoelectric power (Fig. 70d), etc. The peculiar shape of the specific heat curve (Fig. 70c) is due to the existence of inter- or intra-Landau state thermal excitations. At finite temperatures and low enough fields, where fimc= kT, intersubband excitations can occur and show up as sharp peaks whenever the Fermi level lies in between two Landau states. Such effects have been observed in heat-pulse experiments by Gornik et al.36sThe
no broadening Gaussian
I I
f
f
I
I
I
1
G9
03
FIG.69. Fermi level in a 2D system. (a) In k,-k, plane. Landau states are all filled up to some fractionally occupied state where the Fermi level lies. (---) represents the Fermi disk, which contains all allowed states when no magnetic field is present. (b) Energy representation: The Landau states are broadened, which smoothes out the transition of the Fermi level from the last fully occupied Landau state to the next empty one when adding an electron or changing magnetic field.
1. 111 - v SEMICONDUCTOR QUANTIZED STRUCTURES
2 E
N
t,
= 2z
103
20 10-
0 -----10-
-20-
-3oc
MAGNETIC FIELD ( k G )
FIG. 70. Oscillatory phenomena in a 2D GaAs/GaAHs system in a magnetic field. (a) Fermi level; (b) magnetization; (c) specific heat; (d) thermoelectric power. A Gaussian broadening of 0.5 meV is assumed (after Zawadski and Lassnig3@).
typical oscillatory behavior of the thermoelectric power has also been demonstrated by Obloh et a1.366 Many spectacular effects have been observed in the quantum regime of 2D heterointerface systems. The 2D cyclotron resonance has been demon-
104
C. WEISBUCH
strated for confined electrons and holes in various heterostructures ( G ~ A ~ A s / G ~GA ~s , I~ ~~ ~A S ~G Ia s~b P/ I ,n A ~ ~~and ~~~ ~has ) allowed the determination of electron and hole93 masses as well as p o l a r ~ n and ~~~,~~ screening effects370in 2D. In particularly pure samples, a specific oscillation of the cyclotron-resonance linewidth has been observed, which has been related to the oscillatory character of the scattering probability with the filling fact09~* or to the softening of the 2D magnetoplasmon mode.372 The de Haas -van Alphen measurement of the oscillatory magnetic susceptibility allows a more direct determination of the 2D DOS as compared to Shubnikov- de Haas magnetoresistance measurements (which involve carrier scattering) and cyclotron resonance (which yields a combined DOS of initial and final states). Such measurements are extremely difficult, as the total number of 2D electrons to be measured is very small, as compared to a 3D case. Nethertheless, using a 272-layer sample, Stormer et were able to measure the 2D electron gas magnetic susceptibility. Switching from a SQUID detection to a torsional balance magnetometer recently allowed a 100-fold gain in sensitivity.374 When the magnetic field B is parallel to the heterostructure layers, the z motion in the confining potential is only slightly perturbed by the applied magnetic fields. Conversely, the usual cyclotron motion is inhibited by the confining potential. Therefore, the above description of 2D or 3D states collapsing into degenerate Landau level is invalid and the only effect of B, is to increase the separation between low-lying confined quantum levels.375 to As mentioned above (Section 16), this effect was used by Englert et study the onset of intersubband scattering. However, in an extremely high parallel magnetic field, a new oscillatory effect sets in when the cyclotron orbit becomes of the order of, or smaller than, the confined wave function. This effect has been observed in QW structures and gives rise to a new form of SdH oscillation.376 We now concentrate on the most widely used magnetic field techniques in physics and assessment of 2D heterostructure systems, the Shubnikovde Haas effect and the quantum Hall effect.
b. The Shubnikov - de Haas Measurements The Shubnikov-de Haas effect, i.e., the oscillations of the longitudinal resistance in a quantizing field ( u c>~1, hut> kT) has long been a premium technique to study 2D systems. In Si MOSFETs, Fowler et a1.377 showed that the oscillation observed with changing electron number (by varying the gate voltage) has a constant period, which proves that each Landau level has the same number of states in 2D. This would not be the case in 3D due to the k, motion, and can provide a signature for the 2D character of the electronic system. Another specific effect is the directional
1. 111-v
SEMICONDUCTOR QUANTIZED STRUCTURES
105
n = 1.7 x tO'7cm-3 p=11400crn2V-'sec-'
Lz = 1844 Lg = 190A
HI LAYERS
HII LAYERS
$1
I
P
0
,
I
2
4
,
,
8
6
, I0
H (TI FIG. 7 1 . Directional dependence of the Shubnikov-de Haas oscillation (from St6rmer et a1.131).
dependence of the SdH effect: only the perpendicular component of the B field confines the x- y motion of carriers and determines the SdH oscillation period, which thus changes as cos B in 2D. This was already observed in early papers on modulation d ~ p i n g ' ~(Fig. ' J ~ ~71). From Eq. (69) one can deduce the carrier density from the period of the SdH oscillation between two adjacent Landau levels A( 1/B):
Ns = (e/N/A( 1/ B )
(71)
These measured values are usually in excellent agreement with those determined by Hall measurements (see Fig. 59), provided that no parallel conductance occurs in the GaAlAs barrier. The cyclotron mass does not enter the value of the oscillation period because the mass factor of ocis cancelled by the mass factor entering the determinationsof EFfrom the 2D density. However, the temperature dependence of the SdH oscillation amplitude allows one to extract an effective mass. Ando et ~ 1 . have ' ~ calculated the low-field (o,z 5 1) oscillatory conductivity as Nse2zr on=-----
m*
1
+
1 (WCQ
1
2(0,z~)~ 2n2k,T (0,7f)2 fim,
+
106
C. WEISBUCH
where z, is the scattering time corresponding to the dephasing of the Landau state. From the temperature and magnetic field dependences of the oscillation amplitude, it is thus possible to extract m* and z., It must however be remembered that this 7, is quite different from that deduced from Hall mobility measurements, as the small-angle collisions can play a much more important role in SdH oscillations, depending on the scattering mechanism. Harrang et carried out a detailed comparison of both determinations. In a number of cases, the spin splitting of the quantized levels has been ~bserved~ (see ' ~ Fig. ~~~ 73~below). The effective g* value, defined as the distance between two spin-split states observed in the SdH measurements, is strongly enhanced as compared to the 3D value of - 0.44.381,382 This has been explained in terms of the electron- electron correlation energy, which depends strongly on the spin of occupied electronic states in the partially filled Landau states. g factors up to 5 in GaAs/GaAlAs have been meareveal an uns ~ r e d , ~ ' whereas ~ , ~ * ~ direct spin-resonance correlated spin splitting with g* = 0.2. In this latter case, the g factor is at variance with the 3D value because of the lifting of the Kramers degeneracy of the conduction band by the confining electric field. When two or more confining levels are occupied, the structure of the SdH oscillations becomes more complex.131If two levels are occupied, two 2218
z
218
n =306 x
10'2 crn-2
> a a a
k m
(L
4
N
I
-u
I
0
I
I
I
I
I
I
I
I
I
I
10
20
30
40
50
60
70
80
90
100
MAGNETIC
FIELD IN kG
FIG.72. Interferenceeffect in the Shubnikov-de Haas oscillations due to the occupancy of two confined subbands. The second derivative of SdH oscillations is shown. After data reduction, calculations show that the two subbands are separated by 8.6 meV (from Stormer et ~ 1 . l ~ ~ ) .
1. 111- v SEMICONDUCTOR QUANTIZED 14,000
1
I
I
I
I
107
STRUCTURES I
I
1
B(kG)
FIG. 73. Normal quantum Hall effect (NQHE) observed in the Hall resistance p, and parallel resistancepu of a selectively doped GaAs/GaALb interface at 50 mK. From the low magnetic field, where p, and pu display a typical “classic” behavior, the NQHE behavior develops from 10 kG.Note the large n = 1 state spin splitting due to the strong electronic correlation (from Paalanen et ai.39’).
-
oscillations will occur with two different periods due to the different densities in the two levels [Eq. (7 I)]. This effect is shown in Fig, 72 and is very useful in ascertaining the number of occupied subband levels. It is also used to differentiate the conducting channels in TEGFET-like structures when parallel conductance in the GaAlAs is present.384 c. Quantum Hall Effect385*386
When observed at high magnetic field, at low temperatures and in high-purity samples, the SdH effect and the Hall effect exhibit a very marked departure from the usual behavior, a linear change of Hall voltage and smooth oscillations of the longitudinal magnetoresistance with mag-
108
C. WEISBUCH
netic field (Fig. 73).387,388 Zeros of the longitudinal resistance are observed, corresponding to well-defined plateaus of the Hall resistance. Also remarkable is that these features exist over a wide range of sample parameters (electron density, mobility, temperature, . . .) and are not dependent on the exact shape of the sample.386Although first reported in Si-MOSFET samples,387this effect, the quantum Hall effect (QHE), has since seen an enormous development in the GaAlAs/GaAs system, the main reason being the lighter electron mass (=0.07moinstead of 0.19mo in Si), which increases by the same amount the cyclotron frequency for a given magnetic field, rendering the extreme quantum limit (low quantum numbers) easier to reach. The higher mobility of GaAlAs/GaAs heterostructures also leads to better resolved plateaus. A standard Hall bar geometry can be used (see inset of Fig. 73). The current I is imposed while the magnetic field, perpendicular to the layer plane, is swept. The Hall resistance p, = V H / Iand the longitudinal resistance p, = g V J I, (where g is a geometric factor depending on the exact geometry) are measured. As usual, the resistivity tensor is related to the conductivity tensor 'a by = Z-l, i.e.
with ayx= -a,; pyx - -pq; on = ;,a p n - p,. The classical Drude model can give a useful physical insight in the problem.385When no collisions are present, an electron moving classically in crossed electric (F,) and magnetic field (B,) describes a cycloid in the x- y plane. The equation of motion is d2r, dr* m- - eF - e -X B dt2 dt
(74)
with the solution dx---F (cos w,t - I); dt B
2
=
$ sin
w,t
(75)
where initial conditions v1 = 0 have been chosen. The time-averaged motion occurs in a direction perpendicular to the electric field, i.e., to the potential drop, and occurs with a constant driji velocity FIB. In that case, a, = a, = 0; , a = -ayx= N,e/B. The Hall voltage is given by pxv = B/N,e. There is no power dissipation in the absence of scattering and the movement of electrons is perpendicular to the electric field. In presence of collisions Eq. (74) can be simply modified by adding a
1. 111- v
SEMICONDUCTOR QUANTIZED STRUCTURES
109
phenomenological friction term mu,/z, where z is the collision time. The time-averaged motion now becomes
from which one deduces
Quantum mechanically, electron motion (in reasonably low electric fields) occurs in Landau levels, i.e., closed cyclotron orbit. The electric field superimposes over this cyclotron motion a drift motion which is given by the same expression as in Eq. (74), but where a, has now its qdantummechanically computed value. The 2D DOS leads to a peculiar situation when the Fermi level is located between two Landau levels numbered i and i 1 (Fig. 69b). In such an occasion, no quasi-elastic scattering can occur at low temperatures; all states below the Fermi level are occupied, and an electron requires an energy Ao, (neglecting broadening) to be scattered to the next empty Landau state. In that case, 0, = 0 and a, is given by the classical collisionless value! From the density of states per Landau level, eB/h, we deduce Ns = ieB/h and therefore
+
- . e2
om-l-j;-;
l h pxV=7i
e2
The Hall resistivity takes quantized values (1 /i)(h/e2)whenever the Fermi level lies in between filled Landau levels. The remarkable feature of Eq. (78) is the fundamental nature of the parameters involved. The particular semiconductor does not even play a role. When compared to the observed SdH and QHE curves, the predicted values p, = 0 and Eq. (78) are extremely well verified386;resistivities as low as lo-'" Q/U, equivalent to Q/cm in 3D, have been measured. This value is three orders of magnitude lower than any other nonsuperconducting material. The accuracy of the corresponding plateau in pw is one part in lo'. Such a high precision is of fundamental physical importance and can be used to calculate the fine-structure constant a = e2/4aeoAc.One also expects to use the QHE to define a new standard of r e s i s t a n ~ e .At ~ ~present, , ~ ~ the precision to the measurement of p, = (l/i)25818.8 - - * R is set to a few parts in lo-', due to limitations in the unprecise value of the reference SI resistor! The QHE resistance from samples with different origins has been mea~ u r e d to ~ ' a~relative experimental uncertainty of 4.6 X
110
C. WEISBUCH
There is, however, a major difficulty in the explanation just given above. It cannot explain the existence of aJinite width for the QHE plateaus and for the zero longitudinal resistance dips: if there are no states between the successive conducting Landau levels, the Fermi level jumps from the last-occupied Landau level to the next higher-lying one. The Fermi level never lies in between conducting Landau levels as the magnetic field is swept and quasi-elastic scattering is always present. Therefore, one has to invoke the existence of localized, i.e., non-current-carrying, states in the tails of the current-conducting Landau levels. The existence of such localized states is well justified within our present understanding of disordered systems, the disorder here being due to random distribution of defects, impurities, or to the random interface topology. When varying the magnetic field or the number of charge carriers, the Fermi energy will either lie in delocalized, current-carrying states where quasi-elastic scatterings are possible, with pxxf 0, or in localized states, in which case the lower-lying current-conducting charges will require a finite energy to be scattered into an empty conducting state. In such a case pxx= 0 at low temperatures and the Hall resistance pv retains a constant value due to the constant number of current-conducting carriers while the Fermi energy is swept through localized states. The new, astonishing phenomenon is the value of pv, exactly equal to ( 1/ i ) h / e 2 ,as if all electrons were in conducting states, independent of the fact that a fraction of them are in localized states, which crucially depends on sample disorder and therefore should vary from sample to sample. Several explanations have been given to explain this amazing result: it was shown by calculating the current carried by delocalized carriers in the presence of disorder that their speed is modified in order to exactly compensate for the lack of conduction of the localized electron^^^'^^^* (Fig. 74). A classical image is that of an obstacle in a pipe carrying a fluid: Around the obstacle, the fluid will flow faster than in the rest of the pipe, in order to conserve a constant fluid flow along the pipe. It is however clear that such an important feature of QHE must be due to first-principles arguments, which were outlined by L a ~ g h l i nHe . ~ showed ~~ the accurate quantization of QHE to be due to two effects: (1) gauge invariance of the interaction of light with matter: and
(2) the existence of a mobility gap.
From these two assumptions, Laughlin was able to demonstrate that, whenever the Fermi level lies within a mobility gap, pxx= 0 and pm =
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
111
Ne B
0
1
LEVEL FILLING FACTOR
FIG.74. Density of states (top), longitudinal (middle), and Hall (bottom) conductivitiesfor a single Landau state as a function of the filling factor p = N/d, where N is the electron density and d the degeneracy of the Landau state. The shaded areas indicate the localized states. Note that the Hall conductivity ofthe filled Landau state is the classical, collisionless value S/h, independent of the fraction of localized states (after Aoki and and^^^').
h/ie2.Detailed discussions of the significance of gauge invariance were given by L a ~ g h l i nA, ~~~ k~ iand , ~H~ a~j d ~ . ~ ~ ~ We therefore now have a satisfying explanation of the QHE: Plateaus are due to Fermi levels situated in localized states due to disorder. Well-defined values of the QHE resistance independent of sample and detailed experimental conditions are due to the adjustment of conducting camers to compensate for localized electrons. The detailed shape of the observed features, however, depends on the sample parameters, and has opened the way to numerous fundamental studies of 2D systems. The localized frac-
FILLING FACTOR, Y
4
2
4
2/3
I
1
I
I
(01
n = 4.48 to'' crn-2
113 I
/
MAGNETIC FIELD B ( k G )
MAGNETIC FIELD B(k6)
FIG.75. Normal (NQHE) and fractional (FQHE) quantum Hall effect for a GaAs/GaAIAs sample. (a) At 4.2 K, only the NQHE is observed, with a small n = 3 dip in the SdH curve, as is expected from odd values of the filling factor (see text). When lowering the temperature, (b) Observed SdH dips develop at fractional values of the filling factor (from Tsui el dW3). and quantum Hall effect at 90 mK (courtesy H. L. Stbrmer, AT&T Bell Laboratories).
112
FILLING FACTOR u
5/3
2l4/3 I I l l I
-
415
Y5 2/31
I
2/5
IA
2/7
I 1
I
a)
I
I00 150 MSNETIC FIELD B
200
1
I
I
I
I
l
250
[kG] FIG. 76. Fractional QHE for various GaAs/GaAIAs samples. Curves (a), (b), (c), and (e) correspond to electron FQHE in different samples, and show a great variety of fractional values of the QHE; (d) corresponds to hole FQHE, in a ptype modulationdoped GaAs/ GaAlAs structure (from Sterner et
113
114
C. WEISBUCH
tion of the DOS determines the width of the plateaus. These have been observed with up to a 95% width with 5% transitions. The width of the plateaus has been correlated with sample mobility. High-precision measurements of pa( T) have demonstrated its dependence upon the residual p,(T) (Eq, 77).396-398 The temperature and voltage dependence of pxx gives information about the transport mechanism in localized states.397 The breakdown of the QHE as a function of applied voltage has also been studied and explained by various heating m e c h a n i ~ r n s .The ~~~ influence ,~ of sample shape and contact interconnections on the sample has been studied in a number of fascinating experiments.401 Going to lower temperatures and in the extreme quantum limit (hm, > E F ) ,novel correlation effects have been observed in the 2D electron gas as the Coulomb interaction between electrons exceeds their kinetic energy, which is almost completely frozen by the magnetic field and the heterojunction confining potentiaLm2The extreme quantum limit is characterized by the filling factor v = N,/d = 1, where d is the degeneracy of Landau levels. In that situation, all electrons are in at most one or two Landau levels. The signals observed under such a situation are shown in Figs. 75 and 76, for highly perfect samples. At even v, the Fermi level is between Landau states of different n, whereas, at odd v, it resides between the spin levels of a given Landau level. The spin splitting in GaAs being much smaller than the cyclotron splitting ha,, the QHE is better observed for even values of v (top curve of Fig. 75a). However, new plateaus appear at the lower temperatures for fractional values of v.m3-405 Many rational values p / q have been observed, all with q being odd. These plateaus can be very well defined, with the v = 3 plateau defined to better than one part in lo4. Such a new effect, named the fractional quantum Hall effect after its resemblance to the integer quantization of the normal QHE (NQHE) described above, cannot be explained in the framework of NQHE. It must, however, rely on a similar type of explanation; i.e., it requires the Fermi level to lie for fractional v's in a true gap or in a mobility kap. The long-predicted Wigner solidification of an electron gas could explain such anomalous behavior if the solid would preferentially form at given fractional values of v. However, all calculations give smooth variations of the cohesive energy of the solid Wigner crystal on the filling factor.406Experiments would also reveal the pinning of the solid at the existing potential fluctuations due to disorder, and yield nonlinear current - voltage characteristics, which have never been observed. A numerical calculation for a finite system of 4,5, and 6 electrons confined in a box in a magnetic field has shown that minima in the total energy could exist a fractional values of v, significantly lower than that of a Wigner crystal which is therefore not
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
115
the ground state of the s y ~ t e m . ~ Laughlin ' has proposed a new many-particle wave function for the description of the ground state, which gives states at filling factors of 1/q, where q is odd.4o8The elementary excitations from this ground state are situated above a gap =0.03e2/rc= 5 K at 150 kG and have e / q charge. These quantum states form an incompressible fluid with no low-lying excitations, implying a flow with no resistance at T = 0. This model, therefore, explains satisfactorily at present the phenomena observed for v = 1/ q and, due to particle- hole symmetry, those observed at v = 1 - 1/ q = ( q - l)/q. Some extensions of the theory have been proposed to explain the other observed fractional values of"'.v The field of FQHE is still very vigorous and aims to attain a fuller basic understanding as well as to explore all its implications: statistics, phase condensations, crystallization, etc. The specific features of the NQHE and FQHE have been observed in several systems other than GaAs/GaAlAs, such as Ga1nAs/InP,4l0GaSb/ I ~ A s . ~They * ' were also observed for holes in the GaAs/GaAlAs system (Fig. 76). A general trend is that the NQHE is better observed (wider plateaus) in samples with some degree of disorder, i.e., nonoptimum mobilities, whereas the FQHE requires samples of the utmost A converging argument is provided by frequency-dependent measurements: Long et a1.413,414 were able to switch from NQHE to FQHE with increasing frequency, which diminishes the length scale of transport and therefore localization effects. The very different origins of NQHE and FQHE are emphasized by the experimental facts, the former requiring disorder and the latter being based on the intrinsic properties of the 2D electron gas in a magnetic field. V. Conclusion
I have tried in this review to give a flavor of the many facets of the basic properties of semiconductor 2D quantized structures relevant to device operations and the characterization of such structures. We are witnessing in this field a rare occurrence, where technological advances driven by the need for ever-better electronic devices have yielded new physical systems which have in turn led to major new advances in fundamental solid-state physics. These 2D structures, first made in U.S. industrial laboratories, and now studied all over the world in many academic institutions. The importance of the field can be well evaluated from the relative space devoted to the subject in such basic conferences as the biennial International Conference on the Physics of Semiconductors. Those attending these conferences know the spectacular impact of 2D systems, witnessed by the very crowded and vivid atmosphere in the specialized sessions. Actually, this exponential
116
C. WEISBUCH
development has caused a major embarassment to this reviewer: the number of signiJicuntpapers in the field is still increasing faster than his ability to grasp them all. I therefore apologize for the many omissions of relevant basic material and refer the reader to the proceedings of the various conferences in our field, past or coming. Single references giving more details on one or other aspect of the physics of 2D structures can be found in the Proceedings of the 1984 and 1986 Maunterndorf Winterscho01,4’~ the 1985 Les Houches Winters~hool,~~ in Lecture notes by G. Bastard,416soon to be published, and in the monumental review by Ando et U L ’ ~ ACKNOWLEDGMENTS I would like to thank my many colleagues in the field. R. Dingle introduced me to the field back in 1979, and has since been more than a colleague, a friend, and a source of major inspiration. H. Starmer was during the Bell Labs years an especially close colleague and friend. Bell Labs was a outstandingly welcoming institution and very fruitful collaborations occurred, principally with A. Gossard, W. Wiegmann, W. Tsang, A. Cho, J. Hegarty, M. Sturge, R. Miller, P. Petrof, C. Shank, R. Fork, B. Greene, A. Pinczuk, and V. Narayanam u d . In Thomson-CSF, B. Vinter, J. Harrang, J. Nagle, A. Tardella, T. Weil, M. Razeghi, and J. P. Duchemin provide a most stimulating scientific environment. B. Vinter, J. Harrang, and C. Hermann read and Criticized some of the early drafts of the manuscript, a main task for which the author is deeply thankful. Many authors have kindly supplied me with preprints and photographs.
Selected Bibliography The field reported here is treated in a very dense form. In addition to the specialized chapters in the present book, the following bibliography provides a recent set of references which are either (i) review papers on some part of the present chapter, (ii) introductory papers at the nonspecialist level, or (iii) recently published specialized papers which bring new light to some of the outstanding issues discussed in the text. The references will be respectively noted as R (review), I (introductory), and S (specialized).
GROWTH AND INTERFACES E. H. C. Parker, ed. (1985). “The Technology and Physics of Molecular Beam Epitaxy.” Plenum, New York. (R) L. L. Chang and K. Ploog, eds. (1985). “Molecular Beam Epitaxy.” NATO AS1 Series, Martinus Nijhoff, Dordrecht. (R) W. T. Tsang (1985). MBE for 111-V Compound Semiconductors. In “Semiconductors and Semimetals” (R. K. Wdardson and A. C. Beer, eds.), Vol. 22A, Lightwave Communications Technology, volume editor W. T. Tsang. Academic Press, Orlando. (R) G. B. Stringfellow (1985). Organometallic W E growth of 111-V Semiconductors. In “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 22A, Lightwave Communications Technology, volume editor W. T. Tsang, Academic Press, Orlando. (R) M. Razeghi (1985). Low-Pressure MOCVD of G a ~ n , - ~ s , , P , - ,Alloys. In “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 22A, Lightwave Communications Technology, volume editor W. T. Tsang, Academic Press, Orlando. (R) D. B. McWhan (1 985). Structure of Chemically Modulated Films. In “Synthetic Modulated Structures” (L. L. Chang and B. C. Giessen, eds.). Academic Press, Orlando. (R)
1. 111- v SEMICONDUCTOR QUANTIZED STRUCTURES
117
BASICCALCULATIONS IN HETEROSTRUCTURES M. Altarelli (1986). In “Heterojunctions and Semiconductor Superlattices” (G. Allan, G. Bastard, N. Boccara, M. Lannoo, and M. Voos, eds.). Springer-Verlag, Berlin and New York. (R) G. Bastard (1987). “Wave Mechahics Applied to Semiconductor Heterostructures.” Editions de Physique, Paris. (I) B. Ricco and M. Ya. Azbel (1984). Physics of resonant tunneling. The one dimensional double-barrier case.. Phys. Rev.B 29, 1970. (S) 3. Barker (1986). Quantum Transport Theory for Small-Geometry Structures. In “The Physics and Fabrication of Microstructures and Microdevices” (M.Kelly and C. Weisbuch, eds.). Springer-Verlag,Berlin and New York. (S) J. Hajdu and G. Landwehr (1985). Quantum Transport Phenomena in Semiconductors in High Magnetic Fields. In “Strong and Ultrastrong Magnetic Fields’’ (F. Herlach, 4.). Springer-Verlag,Berlin and New York. (R) H. L. Stormer (1986). Images of the Fractional Quantum Hall Effect. In “Heterojunctions and Semiconductor Superlattices” (G. M a n , G. Bastard, N. Boccara, M. Lannoo, and M. Voos, eds.). Springer-Verlag,Berlin and New York. (I) (R) R. E. Prange and S. M. Ginin, eds. (1986). “The Quantum Hall Effect.” Springer-Verlag, Berlin and New York. (R)
APPLICATIONS P. M. Solomon (1986). Three Part Series on Heterojunction Transistors. In “The Physics and Fabrication of Microstructures and Microdevices” (M. Kelly and C. Weisbuch, eds.). Springer-Verlag,Berlin and New York. (R) B. de Cremoux (1986). Quantum Well Laser Diodes. I n “Solid State Devices ’85” (P. Balk and 0. G. Folberth, eds.). Elsevier, Amsterdam. (I) C. Weisbuch (1987). The Physics of the Quantum Well Laser. Proceedings of NATO ARW, “Optical Properties of Narrow-Gap Low-Dimensional Structures” (C. Sotomayor-Torres and R. A. Stradling, 4 s . ) . Plenum, New York. (R) S. Luryi and A. Kastalsky (1 985). Hot electron transport in heterostructure devices. PhySica B and C 134,453. (R) S. Luryi (1987). Hot-Electron-Injection and Resonant-Tunneling Heterojunction Devices. In “Heterojunctions: A Modem View of Band Discontinuities and Device Applications” (F. Capasso and G. Margaritondo, eds.). North-Holland Publ., Amsterdam. (I) (R) F. Capasso, K. Mohammed, and A. Y. Cho (1986). Resonant tunneling through double-barriers, perpendicular quantum phenomena in Superlattices, and their device applications. IEEE J. Quantum Electron. QE22, 1853. (R)
WHOLEFIELD The special issue “Semiconductor Quantum Wells and Superlattices: Physics and Applications” of the IEEE Journal of Quantum Electronics (Vol. QE22, September 1986) contains an excellent set of review articles on various aspects of quantum wells and superlattices. (R)
REFERENCES I. For a review of early work up to 1975, see R. Dingle, Festkoerperprobleme 15, 21 (1975). 2. Early Russian work is reviewed by B. A. Tavger and V. Ya. Demishovskii, Usp. Fiz. Nauk96,6 1 (1968) [Sov. Phys. Usp. (Engl. Trans/.) 11,644 (1969)l; A. Ya. Shik, Fiz.
-
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Tekh. Poluprovodn. 8, 1841 (1974) [Sov. Phys. -Semicond. (Engl. Transl.) 8, 1195 (1975)]. 3. L. Esaki and R. Tsu, IBM J. Res. Dev. 14,61 (1970). 4. The subsequent development of the field is reviewed by L. Esaki, in “Recent Topics in Semiconductor Physics” (H. Kamimura and Y. Toyozawa, eds.). World Scientific, Singapore, 1983. 5. A. Y. Cho and J. R. Arthur, Prog. Solid State Chem. 10, 157 (1975). 6. K. Ploog, in “Crystals: Growth, Properties and Applications” (H. C. Freyhardt, ed.), Vol. 3, p. 75. Springer-Verlag, Berlin and New York, 1980. 7. L. L. Chang, in “Handbook of Semiconductors” (T. S. Moss, ed.), Vol. 3, p. 563. North-Holland Publ., Amsterdam, 1980. 8. C. T. Foxon and B. A. Joyce, Curr. Top. Mater. Sci. 7 , 1 (1981). 9. A. C. Gossard, Treatise Mater. Sci. Technol. 24, 13 (1982). 10. A. Y. Cho, Thin Solid Films 100,291 (1983). 11. L. L. Chang and K. Ploog, eds., “Molecular Beam Epitaxy and Heterostructures,” Proc. Erice 1983 Summer School. Martinus Nijhoff, The Hague, 1985. 12. For invaluable information, see the proceedings of the various MBE workshops and conferences, i.e., J. Vac. Sci. Technol.,B [l], 119-205 (1983); B [2], 162-297 (1984); B [3], 511-807 (1985). 13. L. Esaki and L. L. Chang, Phys. Rev. Lett. 33,495 (1974). 14. R. Dingle, W. Wiegmann, and C. H. Henry, Phys. Rev. Lett. 33,827 (1974). 14a. See, e.g., D. Bohm, “Quantum Mechanics.” Prentice-Hall, Englewood Cliffs, New Jersey, 1951; E. Menbacher, “Quantum Mechanics.” Wiley, New York, 1961. 15. H. C. Casey, Jr. and M. B. Panish, “Heterostructure Lasers,” Parts A and B. Academic Press, New York, 1978. 16. G. H. B. Thompson, “Physics of Semiconductor Lasers,” Wiley, New York, 1980. 17. W. T. Tsang, this volume. 18. R. D. Burnham, W. Streifer, and T. L. Paoli, J. Crysr. Growth 68, 370 (1984). 19. H. Morkoc, this volume. 20. N. Linh, this volume. 21. M. Abe, T. Mimura, K. Nishiushi, A. Shibatomi, M. Kobayashi, and T. Misugi, this volume. 22. K. Ploog and G. H. DGhler, Adv. Phys. 32,285 (1983). 23. H. L. Stormer, Festkoerperproblerne24, 25 (1984). 24. N. Yokoyama, K. Imamura, T. Ohshima, H. Nishi, S. Muto, K. Kondo, and S. Hiyamizu, Jpn. J. Appl. Phys. 23, L3 1 1 (1984). 25. T. C. L. G.Sollner, W. D. Goodhue, P. E. Tannenwald, C. D. Parker, and D. D. Peck, Appl. Phys. Lett. 43, 588 (1983). 26. P.M. Petroff, A. C. Gossard, R. A. Logan, and W. Wiegmann, Appl. Phys. Lett. 41,635 ( 1982). 27. A. B. Fowler, A. Hartstein, and R. A. Webb, Phys. Rev. Lett. 48, 196 (1982). 28. A. Kastalsky and S. Luryi, IEEE Electron. Device Lett. EDL-4, 334 (1983). 29. A. Kastalsky, S. Luryi, A. C. Gossard, and R. Hendel, IEEE Electron. Device Lett. EDL-557 (1984). 30. A. Kastalsky, R. A. Kiehl, S. Luryi, A. C. Gossard, and R. Hendel, IEEE Electron. Device Lett. EDL-5, 34 ( 1984). 3 1. F. Capasso, this volume. 32. G. A. Sai-Halasz, Conf:Ser. -Inst. Phys. 43,21 (1979). 33. L. L. Chang, in “Molecular Beam Epitaxy and Heterostructures” (L. L. Chang and K. Ploog, eds.), Martinus Nijhoff, The Hague. p. 461. 34. M. Voos, J. Vac. Sci. Technol. B [l], 404 (1983).
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360. B. Deveaud, A. Chomette, B. Lambert, A. Regreny, R. Romestain, and P. Edel, Solid State Commun. 57,885 (1986). 361. Good reviews of the dynamics of semiconductor electrons in magnetic fields can be found in L. M. Roth and P. N. Argyres, in “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.), Vol. 1, p. 159. Academic Press, New York, 1966; L. M. Roth, in “Handbook on Semiconductors” (T. S. Moss,ed.), Vol. 1, p. 451. North-Holland Publ., Amsterdam, 1982; R. Kubo, S. Miyake, and N. Hashitsume, Solid State Phys. 17, 269 (1965); K. Seeger, “Semiconductor Physics,” Springer Ser. Solid-state Sci. 40, ( 1982). 362. r, can be calculated from the Bohr-Sommerfeld quantization condition JP dr = fi and the dynamic equilibrium condition r n d r , = qwB of classical mechanics. 363. W. Zawadski and R. Lassnig, Su$ Sci. 142,225 (1984). 364. W. Zawadski and R. Lassnig, Solid State Commun. 50, 537 (1984). 365. E. Gornik, R. Lassnig, G. Strasser, H. L. Stormer, A. C. Gossard, and W. Wiegmann, Phys. Rev. Left. 54, 1820 (1985). 366. H. Obloh, K. von Klitzing, and K. Ploog, Surf:Sci. 142,236 (1984). 367. H. L. Stormer, R. Dingle, A. C. Gossard, W. Wiegmann, and M. Sturge, Solid State Commun. 29,705 (1979). 368. M. A. Brummel, R. J. Nicholas, L. C. Brunel, S. Huant, M. Bay, J. C. Portal, M. Razeghi, M. A. Di Forte-Poisson, K. Y. Chang, and A. Y. Cho, Surf Sci. 142, 380 ( 1984). 369. H. Bluyssen, J. C. Maan, P. Wyder, L. L. Chang, and L. Esaki, Solid State Comrnun. 31, 35 ( 1979). 370. W. Seidenbusch, G. Lindemann, R. Lassnig, J. Edlinger, and E. Gornik, Su$ Sci. 142, 375 (1984). 37 1. T. Englert, J. C. Maan, C. Uihlein, D. C. Tsui, and A. C. Gosard, Solid State Cornmun. 46,545 (1983). 372. Z. Schlesinger, S. J. Allen, J. C. M. Hwang, P. M. Platzmann, andN. Tzoar, Phys. Rev. B 30,43 (1984). 373. H. L. Stormer, T. Haavasoja, V. Narayanamurti, A. C. Gossard, and W. Wiegmann, J. Vac. Sci. Technol. B [l],423 (1983). 374. J. P. Eisenstein, H. L. Stormer, V. Narayanamurti, and A. C. Gossard, Proc. Int. Con$ Phys. Semicond., 1984. p. 329 (D. Chadi, ed.). Springer-Verlag, Berlin and New York, 1985. 375. T. Ando, J. Phys. Soc. Jpn. 44,475 (1978). 376. J. Yoshino, H. Sakaki, and T. Hotta, Surf: Sci. 142,326 (1984). 377. A. B. Fowler, F. Fang, W. E. Howard, and P. J. Stiles, Phys. Rev. Lett. 16,901 (1966). 378. J. P. Harrang, R. J. Higgins, R. K. Goodall, P. R. Jay, M. Laviron, and P. Delescluse, Phys. Rev. B: Condens. Matter [3] 31, (1985). 379. T. Englert, D. C. Tsui, A. C. Gossard, and C. Uihlein, Surf: Sci. 113,295 (1982). 380. R. J. Nicholas, M. A. Brummell, J. C. Portal, K. Y. Cheng, A. Y. Cho, and T. P. Pearsall, Solid State Cornmun. 45,9 1 1 (1983). 381. C. Weisbuch and C. Hermann, Phys. Rev. B 15,816 (1977). 382. C. Hermann and C. Weisbuch, in “Spin Orientation” (B. P. Zakharchenya and F. Meier, eds.), p. 463. North-Holland Publ., Amsterdam, 1984. 383. D. Stein, K. von Klitzing, and G. Weimann, Phys. Rev.Lett. 51, 130 (1983). 384. E. F. Schubert, K. Ploog, H. Dambkes, and K. Heime, Appl. Phys. A33,63 (1984). 385. A very simple and elegant description of the Quantum Hall effect using classical and quantum analysis can be found in H. L. Stormer and D. C. Tsui, Science 220, 1241 (1983);another simple description (in French) is given by G. Toulouse, M. Voos, and B. Souillard, C. R. Acad. Sci. (Paris) Vie Sci. 1, 321 (1984).
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C. WEISBUCH
386. For review, see H. L. Stormer, Festkoerperprobleme 24,25 (1984); K. von Klitzing and G. Ebert,Springer Ser. Solid-state Sci. 59, 242 (1984); B. I. Halperin, Helv. Phys. Acta M,75 (1983). 387. K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45,494 (1980). 388. D. C. Tsui and A. C. Gossard, Appl. Phys. Lett. 38, 552 (1981). 389. D. C. Tsui, H. L. Stormer, J. C. M. Hwang, J. S. Brooks, and M. J. Naughton, Phys. Rev. B 28,227 (1983). 390. K. von Klitzing, Festkoerperprobleme 21, (1981). 390a. D. C. Tsui, A. C. Gossard, B. F. Field, M. E. Cage, and R. F. Dziuba, Phys. Rev. Lett. 48,3 ( 1982). 391. H. Aoki and T. Ando, SolidState Commun. 38, 1079 (1981); see also the review by T. Ando, in “Recent Topics in Semiconductor Physics” (H. Kamimura and Y. Toyozawa, eds.), p. 72, World Scientific, Singapore, 1983. 392. R. E. Prange, Phys. Rev. B 23,4802 (1981). 393. R. B. Laughlin, Phys. Rev. B 25, 5632 (1981); see also the illuminating discussion by R. B. Laughlin, Springer Ser. Solid-State Sci. 59,272,288 (1984). 394. H. Aoki, Lect. Notes Phys. 177, 1 1 (1983). 395. J. Hajdu, Lect. Notes Phys. 177, 23 (1983). 396. D. C. S. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. B 25, 1405 ( 1982). 397. M. A. Paalanen, D. C. Tsui, and A. C . Gossard, Phys. Rev. B 25,5566 (1982). 398. M. E. Cage, B. F. Field, R. F. Dziuba, S. M. Girvin, A. C. Gossard, and D. C. Tsui, Phys. Rev. B 30,2286 (1984). 399. H. L. Stormer, A. M. Chang, D. C. Tsui, and J. C. M. Hwang, Proc. Intl. ConJ Phyx Semicond., Sun Francisco, 1984, p. 267 (D. Chadi, ed.). Springer-Verlag, Berlin and New York, 1985. 400. S. Komiyama, T. Takamasu, S. Hiyamizu, and S. Sass, Solid State Commun. 54, 479 (1985). 401. F. F. Fang and P. J. Stiles, Phys. Rev. B 29, 3749 (1984). 402. See, e.g., the discussions in H. L. Stormer and B. L. Halperin’s papers.386 403. D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982). 404. H. L. Stbrmer, A. Chang, D. C. Tsui, J. C. M. Hwang, and A. C. Gossard, Phys. Rev. Lett. 50, 1953 (1983). 405. E. E. Mendez, M. Heiblum, L. L. Chang, and L. Esaki, Phys. Rev. B 28,4886 (1983). 406. See the discussion in B. I. Halperin.386 407. D. Yoshioka, Phys. Rev. B 29,6833 (1984), and references therein. 408. R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983). 409. F. D. M. Haldane, Phys. Rev. Lett. 51,605 (1983). 410. Y. Guldner, J. P. Him, J. P. Vieren, P. Voisin, M. Voos, and M. Razeghi, J. Phys. Lett. (Orsay, Fr.) 43, L6 13 ( 1982). 41 I . E. E. Mendez, L. L. Chang, C. A. Chang, L. F. Alexander, and L. Esaki, Surf: Sci. 142, 215 (1984). 4 12. M. A. Paalanen, D. C. Tsui, A. C. Gossard, and J. C. M. Hwang, Solid State Commun. 50,841 (1984). 413. A. P. Long, H. W. Myron, and M.Pepper, J. Phys. C 17, L433 (1984). 414. C. McFadden, A. P. Long, H. W. Myron, M. Pepper, D. Andrews, and G. J. Davies, J. Phys. C 17, L439 (1984). 41 5. Proceedings of the 1984 and 1986 Maunterndorf Winterschool, “Two-Dimensional Systems, Heterostructures and Superlattices” (G. Bauer, F. Kuchar, and H. Heinrich, eds.), Springer Series in Solid State Sciences, Vols. 53 and 67. Springer-Verlag, Berlin and New York, 1986.
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416. G. Bastard and M. Voos, “Wave Mechanics Applied to Semiconductor Heterostructures.” Les Editions de Physique, Pans, 1987.
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SEMICONDUCTORS AND SEMIMETAU, VOL. 24
CHAPTER 2
Factors Affecting the Performance of (Al,Ga)As/GaAs and (Al,Ga)As/InGaAs Modulation-Doped Field-Effect Transistors: Microwave and Digital Applications H. Morkoc and H. Unlu COORDINATED SCIENCE LABORATORY UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS 61801
I. Introduction
Recent progress combined with a strong need for handling high-speed data have led to increased interest in high-speed devices. In order to operate devices at high frequencies, the transit time through the device and parasitics, or the susceptibility of device to parasitics, must be minimized. One can reduce the transit time by decreasingthe path length, e.g., the gate length in field-effect transistors (FETs), and/or by increasing the carrier velocity. Improved ohmic contacts, metal conductors, and small device geometries are useful in reducing these effects. In digital circuit applications the most common figures of merit are the propagation delay time, zD, i.e., the minimum time required to turn the transistor on and off, and the power dissipated in the transistor during the switching cycle, PD. The former sets an ultimate limit on the speed of the circuit, and the latter sets a limit on how densely the transistors can be integrated before the heat dissipation in the circuit becomes a problem. In microwave amplifiers, the figures of merit are the current-gain cutoff frequency,fT, and the noise figure, NF. The former is a measure of device speed, and the latter is a measure of the minimum signal amplitude required at the input of the amplifier. In both circuit applications, one would like to have a transistor that is capable of carrying large currents. This leads to the requirement that the carrier concentrations, mobility, and velocity must be very high. Having the digital switching device a normally-off type, which implies that the channel must be very thin, will maintain a low power consumption. This is due to the fact that the compound semiconductorgate is based on Schottky 135 Copyright 0 1987 Bell Telephone Laboratories, Incorporated AU rights of reproduction in any form reserved.
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H. MORKOC AND H. UNLU
gates. There is, however, a trade-off in reducing the channel length, since current is proportional to camer density as well as velocity. As the channel length is reduced, the camer density becomes very small and, therefore, the channel transport properties become extremely important. In standard microwave and digital metal - semiconductor field-effect transistors (MESFETs), the channel region of the device is a highly doped (uniform doping) bulk semiconductor. Electrons and donors interact via their Coulomb potential within the channel region, which leads to ionized impurity scattering. The noise of the device starts to increase as both electron and hole mobility and saturation velocity are decreased. In GaAs the peak velocity at room temperature (300 K) decreases from 2.1 X lo7 cm/s (in pure GaAs) to 1.8 X lo7cm/s for a doping level of 1 X lOI7 ~ m - ~ . In order to be able to overcome the aforementioned limitations on device performance imposed by the ionized impurity sites, one must physically separate the electrons from the donors (or holes from acceptors for the hole gas). The new modulation-doped heterostructures allow one to place the dopants (donors or acceptors) only in the layers with wider bandgaps, keeping the layers of narrower-bandgap material as close to intrinsic as possible. Therefore, there is a substantial increase in the twodimensional mobility, because the ionized impurity scattering is reduced by the spatial separation of impurity atoms from the mobile electrons. Mobile carriers in the conducting channels can be further separated from the ionized impurities by introducing undoped spacer layers in the barriers just before the well region. This mobility enhancement will play an appreciable role in controlling the turn-on and turn-off characteristics of highspeed switching applications, since the device is driven from off to on and back to off conditions very rapidly. The enhancement in the two-dimensional mobility will, of course, reduce the parasitic resistance components much as the source resistance. The heterojunction FETs go by various names, e.g., MODFET for modulation-doped field-effect transistors, SDHT for selectively doped heterojunction field-effect transistors. HEMT for high-electron-mobility transistors, and TEGFET for two-dimensional electron-gas field-effect transistors. In this article we shall refer to this device as MODFET. 11. How Modulation Doping Works
Modulation doping relies on the selective doping of a semiconductor layer adjacent to an undoped smaller-bandgapsemiconductor layer. Theoretically, as long as the donor energy is larger than the conduction-band energy of the smaller-bandgap material, the electrons diffuse into the smaller-bandgap material. The three most commonly studied systems so
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
137
Undoped L
J
Undoped
Doped
AIxGal-,As
f
Undoped Ga As
FIG. 1 . Since interface heterostructures are used for heterojunction FETs, the structure with (Al,Ga)As grown on top of GaAs, a "normal modulation-doped structure," is the one that is commonly used for FETs. The diagram on the right-hand side shows the conductionband edge with respect to distance. When the order of growth is inverted [bottom sketch, doped (A1,Ga)As first], the interface quality is not sufficiently good to provide enhanced device performance. The relatively small surface mobility of A1 is thought to lead to a degraded interface in the inverted structure.
far have been GaAs/(Al,Ga)As,' Ino.s~Gao.47As/Irb.,,Ab,,As,Z and (Al,Ga)A~/(1n,Ga)As~-~ lattice-matched systems. Systems such as III,,~~G~~.~~As/I~P have also been in~estigated.~ In this article we shall concentrate on the GaAs/(Al,Ga)As heterojunction systems, with which the vast majority of the devices and the only device with high-speed data have so far been fabricated. Furthermore, the newly developed pseudomorphic (In,Ga)As/(Al,Ga)As heterojunction systems and their superior potential for microwave and digital applications will be discussed. Figure 1 shows the schematic cross section of a single heterointerface modulation-doped structure along with its energy-band diagram. The structure is grown typically by molecular-beam epitaxy, generally at a growth rate of 1 pm/h and at a substrate temperature of 580-620°C. Following the growth of an undoped buffer layer, typically 1 pm thick, the (A1,Ga)As layer, part of which is left undoped at the heterointerface to reduce the Coulombic interaction between donors and is grown. A thin surface layer of GaAs, either doped or undoped, is optional. For structures intended for very high electron mobilities, the undoped
138
H. MORKOC AND H.UNLU
A),
layer thickness is quite large (2200 and for them to be used in FETs this layer is quite thin, about 30 A.The latter is to obtain large transconductances, which lead to improved device performance. Although predicted in the late 1960s, the first experimental realization of modulation doping came out of independent research at AT&T Bell Laboratories in the late 1970s. The improved transport obtained has quickly led to a worldwide effort to use this structure for high-speed device^.^-'^ 111. FET Fabrication
The first step in device fabrication is generally the device isolation, which in most cases is done by etching mesas down to the undoped GaAs layer or to the semi-insulatingsubstrate, or by an isolation implant. The source and drain areas are then defined in positive photoresist, and typically an AuGe/Ni/Au metallization is evaporated. Following the lift-off, the source-drain metallization is alloyed at or above 400°C for a short time (- 1 min) to obtain ohmic contacts. During this process Ge alloys down past the heterointerface, thus making contact to the sheet of electrons, as shown in Fig. 2. The gate is then defined, and a very small amount of recessing is done by either chemical etching, reactive ion etching, or ion milling. The extent of the recess is dependent upon whether depletion- or enhancement-mode devices are desired. In depletion-mode devices the remaining doped layer should be just the thickness to be depleted by the gate Schottky barrier. In
Source
Gate
Droin
ric
Passivotion
aI s
Irnplanl Isololion
FIG.2. Cross-sectional view of a commonly used MODFET. Alloyed contacts,e.g., AuGe/ Ni/Au, diffise past the interface, forming contacts to the 2DEG.
FIG.3. Top view of a fabricated device intended for microwave applicationsand having a 1 X 290 pm2 gate. For logic circuits the gate width is reduced to 20 pm to minimize the real
estate and power consumption.
140
H. MORKOC AND H. UNLU
enhancement-mode devices the remaining doped (Al,Ga)As is much thinner, and thus the Schottky barrier depletes the electron gas as well. In test circuits composed of ring oscillators, the switches are of enhancementmode FETs, which conduct current when a positive voltage is applied to the gate and the loads are of depletion type. Figure 3 shows the top view of a MODFET with a gate dimension of 1 X 300pm2 intended for microwave applications. For logic circuits, the gate width is typically 20 pm. In the long run, more advanced fabrication procedures for reducing parasitic resistances must be employed (these will be described later). In addition, the structures must be designed to allow a reproducible gate recess. This can be achieved by placing a very thin GaAs layer a specific distance away from the interface, determined by the desired device characteristics. The recessing, preferably done by dry processing, can be stopped selectively at the GaAs surface, which also makes possible the deposition of the gate on the more stable GaAs as opposed to (A1,Ga)As. IV. Principles of Heterojunction FET Operation 1. GENERAL
Since the electrons moving parallel to the heterointerface (in GaAs) encounter reduced scattering by ionized donors located in the (A1,Ga)As layer, the current-conducting channel must be parallel to the heterointerface. Modulation of the channel current is done by a third terminal, the gate, placed on the doped (Al,Ga)Aslayer. Since the transport properties of the (A1,Ga)As is much inferior even to the bulk GaAs, care should be exercised to make certain that the (A1,Ga)As layer is entirely depleted by the gate and heterointerface fields. Only then is the current carried entirely by the two-dimensional electron gas (2DEG), which has enhanced properties. In discrete devices the power consumed can be dissipated rather easily, allowing the use of normally-on devices. In this case with no external gate bias, only the (A1,Ga)As layer is depleted, and the device conducts the maximum amount of current. When a negative gate bias is applied with respect to the source, the channel (2DEG) is depleted, pinching off the device. In integrated circuits, however, the device density is very high, and the power consumption of normally-on devices prohibits their use. The doped (A1,Ga)As layer under the gate is then made thinner, such that the gate with no external bias depletes the electron gas as well. The current flows only when a positive voltage is applied to the gate. The device operation is to some extent analogous to that of the Si/Si02 metal - oxide- semiconductor field-effect transistors (MOSFETs). While
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
141
the basic principles of operation are similar, the material systems and the details of the device physics are different. The most striking difference, however, is the lack of appreciable interface states in this heterojunction system, where the gate metal and the channel are separated by only about 300-400 A.This, coupled with the large dielectric constant of (A1,Ga)As as compared to SO2, gives rise to extremely large transconductances. In addition, large electron densities, about 10l2 cm-2, and higher electron velocities and mobilities can be achieved at the interface, which lead to high current levels. The effective mass of electrons in GaAs is much smaller than in Si, and therefore the electron concentrationsunder consideration raise the Fermi level up into the conduction band, which is not the case for Si MOSFETs. It is therefore necessary to develop a new model for this device, as has been attempted by the Thomson CSF groupl5and by the team at the University of Minnesota and the University of Illinois.16 In order to calculate the current -voltage characteristics, we must first determine the 2DEG concentration. 2. TWO-DIMENSIONAL ELECTRON-GAS CONCENTRATION As indicated earlier, the electrons diffuse from the doped (A1,Ga)As to the GaAs, where they are confined by the energy bamer and form a 2DEG. This was verified by the Shubnikov- de Haas oscillations and their dependence on the angle between the magnetic field and the normal of the ~amp1e.l~ The wave vector for such a system is quantized in the direction perpendicular to but not parallel to the interface. The electric field set up by the charge separation causes a severe band bending in the GaAs layer, with a resultant triangular potential barrier where the allowed states are no longer continuous in energy, but discrete. As a result, quantized subbands are formed, and a new two-dimenstional model is needed to calculate the electron concentration. In most cases the ground subband is filled, with the first excited subband being partially empty. Since the wave function or the electron concentration is distributed slightly in the direction perpendicular to the heterointerface, we shall refer to the areal density of the electron gas from now on. To determine the electron concentration we must first relate it to the subband energies. The rigorous approach is to solve for the subband energies self-consistently, with the solution for the potential derived from the electric charge distribution. This has been done by Stem and Howard'* for the silicon-silicon dioxide system in the 1960s, and more recently by Ando and Stern and Das Sarma19for the GaAs/(A,Ga)As system. A workable approximation is to assume that the potential well is perfectly triangular, and that only the ground and first subbands need be considered. Using the experimentally obtained subband populations, adjustments in the pa-
142
H. MORKOC AND H. UNLU
rameters can be made to account for the nonconstant electric field and nonparabolicity in the conduction band. Solving Poisson’s equation in the (A,Ga)As and GaAs layers and using Gauss’s law, one can obtain another expression for the sheet electron concentration in terms of structural parameters, e.g., the doping level in (A,Ga)As, doped and undoped (A,Ga)As layer thicknesses, and the magnitude of the conduction-band energy discontinuity or the AlAs mole fraction in (A,Ga)As.I6 Analysis of the Fermi level shows’6that it is nearly a linear function of the sheet carrier concentration, n,, for n, 2 5 X 10” cm-*. Taking this into account, one can eliminate the iteration process because analytical expressions become available. Another feature that must be considered in the model is the necessity of using the Fermi-Dirac as opposed to the commonly used Maxwell- Boltzmann statistics.2oThis term is particularly important at room temperature because of larger thermal energy. In the case of Si/SiO, MOSFETs, three-dimensional analyses work quite well because the Fermi level is not as high, but they fail for this heterojunction FET. This will be explained in detail below. The interface density of the 2DEG, n,, is determined as
where E, is the bottom of the conduction band, g(E) is the density of states and 1
=
1
+ exp[(E - E,)/kT]
(2)
is the Fermi-Dirac distribution. In Si MOSFETs the integral in Eq. (1) may be evaluated using the foIlowing assumptions: ( I ) the Maxwell- Boltzmann (rather than the Fermi - Dirac) distribution function may be used; and (2) the density of states in the potential well near the interface is continuous.
Both of these assumptions are justified for Si MOSFETs, but are not justified for modulation-doped structures. This is illustrated in Fig. 4, where interface carrier densities for intrinsic Si and GaAs are plotted as functions of the Fermi level using the three-dimensional Joyce - Dixon approximation2’ (dashed line) and a more accurate “two-dimensional” formula proposed in Ref. 15. This two-dimensional model is based on considering the quantized energy levels in the potential well near the interface. It is also shown that a simple linear approximation for the n,
2. (A1,Ga)AslGaAsA N D (Al,Ga)As/InGaAs MODFETs
143
Fermi Level (Volts)
FIG.4. Variation of the electron-gas density with Fermi level as measured from the bottom of the conduction band in GaAs at 300 K. Since the conduction-band density of states in Si is very large, the F e m i level even for the largest sheet canier concentration, 2 X lo1*cm-2, is still below the conduction band, and predictions are reasonably accurate when the problem is treated as a three-dimenstional(3D) one and the quantization is neglected (---). For GaAs, however, the density of states is smaller (or the effective mass is smaller) and the quantization of the electron population at the heterointerface cannot be neglected. Models encompassing ) nature of the electron population must be utilized. the two-dimensional (2D, -
versus EF curve may be used in analytical calculations22for interface densities greater than about 5 X 10” cm-2. Due to a large effective electron mass in Si (rn*/rno = 0.3 for one ellipsoid in the [ 1001 direction) and six ellipsoids included in the density-of-states effective mass, there are many levels in the potential well, so that a “three-dimensional” theory works quite well. The Maxwell -Boltzmann distribution function may be used because the position of the Fermi level (for realistic interface densities) is several kT (about 0.1 eV at room temperature) below the bottom of the potential well. In GaAs the Fermi level is in the potential well, necessitating the use of the Fermi-Dirac distribution; therefore, the discrepancy between the three-dimensional and two-dimensional models is quite large in GaAs. The theory of modulation-doped structures should also account for the fact that the position of the Fermi level in Si is much less sensitive to the interface carrier density n, as compared to GaAs. For n, 2 5 X 10” cm-2 the linearized relationship between n, and EF (Fig. 4) can be used, which allows the following analytical expression for the maximum density of the 2DEG to be derived:20
+
+
+
n, = [N$(di Ad)2 (2&2Nd/q2)(AE, 6 -
+
- Nd(di A d ) ( 3 ) where Nd is the donor density, di is the thickness of the undoped (A1,Ga)As layer, Ad = 80 A is a constant related to the EF versus n, curve,2oc2 is the
144
H. MORKOC AND H. UNLU
dielectric permittivity of (Al,Ga)As, 4 is the electronic charge, A E, is the discontinuity in the conduction band,
+
6 = -kT[ln( 1 g’y)
+ ( 4 / N 3 In( 1 - 1/(4y))]
(4)
and y = { [( 1 - 1/4Na2
+ 4g’NJ’’’
-(1
- 1/(4N3)}/2g’
(5)
where
N i = NJN,,
g’ = g exp(EJkT)
(6)
and
y = exp(- En/kT) Here EF is the energy difference between the bottom of the conduction band and the Fermi level in (Al,Ga)As, N, is the density of states in (Al,Ga)As, g is the donor g factor, and Ed is the donor ionization energy. Comparison of the exact solution (dotted line) with our analytical expression for n, (solid line) in Fig. 5 illustrates the accuracy of the approximation.
3. CHARGE CONTROL AND I-V CHARACTERISTICS So far we have related the interface charge, which is to carry the current parallel to the heterointerface, to the structural parameters of the heterojunction system. To control and modulate this charge, and therefore the
:
H
0 lo17
10”
Doping Density (cm-3)
FIG. 5. Interface carrier density n, as a function of doping density Nd, with various undoped (A1,Ga)As layer thicknesses. ( . . * f, the exact solution; (-), the analytical expression, Eq. (3).
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
145
current, a Schottky barrier is placed on the doped (A1,Ga)As layer. As indicated earlier, the doped (A1,Ga)As is depleted at the heterointerface by electron diffusion into GaAs, but this is limited to about 100 A for an (Al,Ga)As doping level of about 1OI8~ m - ~ It .is also depleted from the surface by the Schottky barrier. To avoid conduction through (Al,Ga)As, which has inferior transport properties, and screening of the channel by the carriers in the (Al,Ga)As, parameters must be chosen such that the two depletion regions just overlap. A detailed description of the device modeling and analysis has been given elsewhere.23 In normally-on devices the depletion by the gate’s built-in voltage must be just enough to have the surface depletion extended to the interface depletion. For example, devices designed for 1OI2cm-2 electrons in the channel can be turned off at a gate bias of - 1 V by the gate located 600 A away from the interface on the (Al,Ga)As layer. This is the structure used for discrete high-speed analog applications, e.g., microwave low-noise amplifiers. In normally-off devices the thickness of the doped (A1,Ga)As under the gate is smaller and the gate built-in voltage depletes the doped (Al,Ga)As, overcomes the built-in potential at the heterointerface, and depletes the electron gas. No current flows through the device unless a positive gate voltage is applied to the gate. This type of device is used as a switch in high-speed integrated digital circuits because of the associated low power dissipation. The loads may be normally-on transistors with the gate shorted to the source, or an ungated “saturated resistor,” which has a saturating current characteristic due to the velocity saturation of the carriers. Away from the cutoff regime, it is quite reasonable to assume that the capacitance under the gate is constant and thus the charge at the interface is linearly proportional to the gate voltage minus the threshold voltage. As the threshold voltage is approached, the triangular potential well widens, and the Fermi energy of the electrons is lowered. This change in surface potential subtracts from the change in the applied gate bias, so that a lesser change in potential acts across the (A1,Ga)As layer, reducing the transconductance of the device, and causing the curvature of the transfer characteristic near threshold, as will be discussed later. This curvature is more pronounced at room temperature, due to the thermal distribution of the electrons; however, some curvature will persist down to the lowest temperatures, due to quantum-mechanical confinement energies. This has profound implications for device operation, since it precludes high-speed operation at voltages less than a few tenths of a volt. This means that ultralow-power-delayproducts, similar to those of Josephson-junction devices, which operate at only a few millivolts, would not be realized. Near complete pinchoff (defined loosely as when the free-electron gas concen-
146
H. MORKOC AND H. UNLU
tration drops to 10%of its maximum value) the effective position22of the electron gas may be 200 A away from the interface, as shown in Fig. 6. This simply implies that it will require larger gate voltages to deplete the electron gas, leading to a slow gradual pinch-off. In addition, the gate capacitance near pinch-off will show a decline as well. Away from cutof€, the charge can be assumed to be linearly proportional to the gate voltage, and in the velocity-saturated regime the current will then be linarly proportional to the gate voltage, and the transconductance will approach a constant [except when the (A1,Ga)As starts conducting]. These arguments apply to the velocity-saturated MESFET as well. For the MESFET, in constrast, the transconductanceincreases with increasinggate biases, since the depletion layer width narrows and modulation of the channel charge increases. In order to calculte the current-voltage characteristics, one must know the electron velocity as a function of electric field. Since the device dirnensions (gate) used are about 1 pm or less, high-field effects such as velocity saturation must be considered. Even though the electrons in MODFETs are located in GaAs, and the electron transport in GaAs is well known, there was some confusion in the early days as to what one should expect. There were, in fact, reports that this heterojunction structure held promise only because of the high mobilities which are measured at extremely small voltages (electric field cr: 5
Gate Voltage ( V ) FIG.6. Effective position of the 2DEG from the heterointerface, Ad versus gate bias.
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
147
V/cm). In short-channel devices the electric field can reach tens of kV/cm, making it necessary to understand the high-field transport. Using 400 pm long conventional Hall bar structures, the velocity -field characteristicshave been measured.24A dc technique below 300 V/cm and a pulsed technique up to 2 kV/cm were used to measure the current versus field characteristics. Knowing the electron concentration from the same sample by Hall measurements, the electron velocity versus electric field characteristics were deduced on many modulation-doped structures. Above 2 kV/cm the electric field was suspected to be developing nonuniformities, as determined by the voltage between equally spaced voltage wings along the sample. Depending on the low-temperature low-field mobility of the sample, hot-electron effects, even at fields as low as 5 V/cm, have been observed.25At very low fields piezoacoustic phonons and, at fields of about 100-200 V/cm, optical phonons are emitted. This leads to electron velocities smaller than predicted by simple extrapolations based on low-field mobility. The velocity versus field characteristics below 300 V/cm for a typical modulation-doped structure intended for FETs is shown in Fig. 7. Also
A
300
Electric Field (V/cm)
FIG. 7. Velocity versus electric field measured in a single-interface modulationdoped structure at 300 and 77 K. The electron-gas concentration at the heterointerface is about 7 X lOI1cm--’, and the unintentional background acceptor concentration in GaAs of the modulation-doped structure is about lOI4 ~ m - ~For . comparison, calculated velocity-field characteristicsof bulk GaAs with zero ionized impurity density (Nr= 0) at 300 and 77 K and with N, = lot5~ m at- 77~ K are also shown. It is clear that at 300 K the transport properties of the modulation-doped structure with as many electrons as needed for FETs is comparable to the pure GaAs. At 77 K it is almost comparable to pure GaAs at fields below 300 V/cm and quite comparable at 2 kV/cm and above (estimated from FET performance).
148
H. MORKOC AND H. UNLU
shown are the Monte Carlo calculations performed for lightly doped and ion-free bulk GaAs layers. The agreement between the modulation-doped structures and undoped GaAs (Ni5 lOI5 ~ m - is ~ striking. ) The agreement at low temperatures is even better at high fields, as determined from the heterojunction FXT performance. It is clear that having electrons but not the donors in concentrations of about 10l2cm-2 in modulation-doped structures does not degrade the velocity. The most important aspects of these results can be summarized as follows. (1) A quasi-saturation of electron velocities is obtained at fields of about 200 V/cm. This implies that the extremely high electron mobilities obtained at very low electric fields have only a secondary effect on device performance. (2) The higher mobilities at low fields help give the device a low saturation voltage and small on-resistance and help enhance its speed during turn-on and turn-off transients. (3) Since the properties of the pure GaAs are maintained, electron peak velocities of over 2 X lo7 and 3 X lo7 cm/s at 300 and 77 K, respectively, can be obtained. These values have already been deduced using drain current versus gate voltage characteristics of MODFETs. (4) Perhaps the velocity overshoot is more pronounced.
It can be simply concluded that modulation-doped structures provide the current transport that is needed to charge and discharge capacitances, without degrading the properties of pure GaAs. To get electrons into conventional structures, the donors have to be incorporated, which degrades the velocity. From the velocity considerations only, these devices offer about 20% improvement at 300 K and about 60% at 77 K. However, other factors, e.g., large current, large transconductance, and low source resistance, improve the performance in a real circuit far beyond that predicted by the velocity enhancement only. For small-signal operation, e.g., microwave small-signal amplifiers, the improvement in the device performance as compared to the conventional bulk FETs may actually be very close to the figures mentioned above. In the presence of a Schottky gate on (Al,Ga)As, the density of the 2DEG is approximately described by the modified charge control model:
n, = (&2/d( Vg- V,,)/(d + A 4 (7) where d = dd di, dd is the thickness of the doped (A1,Ga)As beneath the gate, VOs= - (I/q)(AE, AEm)- V,,, #, is the Schottky barrier height, A Em is a temperature-dependent parameter (0 at 300 K and 25 meV at 77 K),17*19and V,, = qNdod32&,.A similar calculation for a SiSiOz interface leads to A d = 10 A (compared to A d = 80 A for GaAs).
+
+
2. (Al,Ga)As/GaAs AND (A1,Ga)AslInGaAs MODFETs
149
This 10 A correction may not be very important in a typical MESFET but should be considered when the oxide thickness becomes less than 200 A. In Fig. 8, the exact value of interface carier density (dotted line) is plotted against the applied gate voltage. As can be seen from this figure, the analytical expression (dashed line) of Eq. (7) is quite good except near threshold. The solid line is plotted assuming no dependence of EF on n,, i.e., A d = O.I5 Using two-piece and three-piece approximations for the velocity-on field characteristics, simple analytical expressions for I - V characteristicswhich include the source resistance R, and the drain resistance were developed earlier.20*22 V. Optimization
In a normally-off MODFET, the type used for the switches in an integrated circuit, a positive gate voltage is applied to turn the device on. The maximum gate voltage is limited to the value above which the doped (A1,Ga)Aslayer begins to conduct. If exceeded, a conduction path through the (A1,Ga)Aslayer, which has much inferior properties, is created, leading to reduced performance. This parasitic MESFET effect for typical parameters becomes noticeable above a gate voltage of about +0.6 V, which determines the gate logic swing.26Using alternate methods to improve this shortcoming should be very useful.
FIG.8. Interface cyrier density versus voltage difference between gate and channel (V,, = 0.15 V and d = 400 A). (-), the simple chargeantrol model proposed in Ref. 12, where the Fermi level is assumed to be constant; ( . . .), the numerical exact solution; and (---), the analytical model of Ref. 19. Note this figure should be used only up to the value of n, which is typically about 9 X 10" cm-z.
150
H. MORKOC AND H. UNLU
4. TRANSCONDUCTANCE
Since the ultimate speed of a switching device is determined by the transconductance divided by the sum of the gate and interconnect capactiances, the larger the transconductance, the better the speed is. MODFETs already exhibit larger transconductances because of higher electron velocity, and, in addition, since the electron gas is located only about 400 A away from the gate metal, a large concentration of charge can be modulated by small gate voltages. The latter comes at the expense of a larger gate capacitance. Considering the interconnect capacitances, any increase in transconductance (even with increased gate capacitance) improves the speed. The transconductancein these devices can be optimized by reducing the (A1,Ga)As layer thickness. This must accompany increased doping in (Al,Ga)As, which in turn is limited to about lo’* cm-3 by the requirement for a nonleaky Schottky barrier. By decreasing the undoped setback layer thickness, one can not only increase the transconductance, but also the current level (through the increased electron gas concentration). There is, of course, a limit to this process as well because thinner setback layers increase the Coulombic scattering. All things considered, a setback layer thickness of about 20-30 A appears to be the best at present. Setback layers less than 20 A have led to much inferior performance. Transconductances of about 225 mS/mm (275 being the best) and 400 mS/mm gate width have been demonstrated at 300 and 77 K, respectively. The current levels of MODFETs also depend strongly on the setback layer thickness and on the doping level in (A1,Ga)As. For good switching and amplifier devices, a good saturation, low differential conductance in the current saturation region, and a low saturation voltage are needed. These are attained quite well, particularly at 77 K, as shown in Fig. 9. The increased current level at 77 K is attributed to the enhancement of electron velocity. The rise in current would have been greater if it were not for the shift in the threshold voltage, about k 0.1 V, as the device is cooled to 77 K. This may be due to the freeze-out of electrons in the (A1,Ga)As layer. The maximum gate voltage Vimiu that can be applied is the pinch-off voltage of the 2DEG:
This, together with the 2-piece model, leads to the following expression for the maximum “intrinsic” transconductance (R,= 0)26:
2. (Ai,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
151
O.SV/div
FIG.9. Drain I - Vcharacteristic of a MODFET with a 300,um gate width at 300 (---)and As indicated, the extrinsic transconductance increases from about 225 (best 77 K(-). 275 mS/mm) to 400 mS/mm as the device is cooled to 77 K. The improvement in the drain current observed at 77 K could be much larger if it were not for the positive shift in the threshold voltage. This shift is attributed to defects in (AI,Ga)As and is a subject of current research.
where gm is the intrinsic transconductance per unit gate length, p the low-field mobility, us the saturation velocity, and where VBi= $, - AEJq is the effective built-in voltage. One of the consequences of Eqs. (9) and (10) is that higher doping of the (A1,Ga)As reduces the minimum thickness of the doped (A1,Ga)As beneath the gate given by the second term on the right-hand side of Eq. (lo), leading to a higher transconductance. The results indicate that at small gate lengths the transconductance becomes nearly independent of the gate length due to the velocity saturation. In reality, an additional enhancement of the transconductance in short-gate structures is possible due to overshoot and/or ballistic effects. For very short gate lengths when
wn,(d+ Ad)/eZusL>> 1
+
(1 1)
and one finds (g,)= = E2usZ/(d Ad). This expression, together with Eq. (lo), sets an upper limit for the transconductance of short-gate heterojunction FETs. At room temperature when p is only a weak function of di,the transconductance should increase both with a decrease in di (in agreement with experimental results) and with a decrease in gate length. This reduction in di has two effects. First, it increases both the capacitance and transconduct-
152
H. MORKOC AND H. UNLU
ance. Second, it increases n,, the maximum voltage swing [Eq. (S)], the maximum drain saturation current, and (gm)- through Eq. (9). Assuming VBi = 0.7 V, Ad = 80 A, and p = 7000 cmZ/V s, which is independent of di at 300 K (Fig. 10 and Ref. 25), we calculate (gm)- as a function of dj for a 1 pm gate noqnally-off device. The results are shown in Fig. 1 1. Also shown are the values of the highest intrinsic transconductance obtained in our laboratoryz8at 300 and 77 K. The transconductance is considerably larger for small values of di,especially at higher doping levels. This result is in good agreement with the experimental data reported earlier.z9It should be noted here that the values of transconductance are somewhat overestimated for reasons not yet understood, although it could possibly be due to the uncertainties in (A1,Ga)As thickness under the gate. To some extent the current swing is even more important than the high transconductance in logic devices designed for maximum speed because the current determines the time necessary to charge the effective input capacitance. The maximum current from the 2DEG layer is given by where Z is the width of the device. Using the theory described earlierZO (I&)- as a function of difor lo'* ~ r n can - ~ be calculated. The results of this calculation are shown in Fig. 12, where they are compared with the experimental results. As can be seen from the figure, the trend in (4)variations with di seems to agree with the experimental results. However, the calculated values of the current are considerably higher than the experi-
..- 4-
n
r"
0 0
0
0
-?
0
Q
-0
c 4-
9
u W w
Doped (AI,Ga)As (300-7008,) Undoped (AI,Ga)As (di) Undoped GaAs
c
0-
I
I
I
I
I
FIG. 10. Measured (0),deduced (A), and calculated (-) two-dimensional electron mobility at 300 K versus the set-back layer thickness. Extremely good agreement is obtained when the effect of parallel conduction through (A1,Ga)As on the mobility is accounted for.
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
4
'
20 I 40 ' ' 60 ' 00 I Undoped Layer Thickness ( A )
'
153
'
100
FIG. 11. Since transconductance is inversely proportional to the gate-toelectron gas separation, the undoped (Al,Ga)As layer at the heterointerface can influence the transconduo tance substantially. Considering that the gate-toelectron gas distance is about 300 A, an undoped layer thickness of greater than 100 A can have a dramatic i n f l u p x on the transconductance. For best results an undoped layer thickness of about 20-30 A must be used. This imposes stringent requirements on the epitaxial growth process, and only molecular-beam epitaxy has so far been atle to produce such structures. (0),experimental data points; (-), theory. Below 20 A, the performance degrades.
mental values. One of the reasons for this is that Eq. (3) slightly overestimates the measured n,, perhaps due to uncertainties in the electron concentration in the (A1,Ga)As layer. This discrepancy may also be a result of the reduction of the effective mobility and/or saturation velocity at large values of n, due to the intersubband scattering. As can be seen from Eq. ( 12), the maximum drain current is determind by n, and v, independent of the series source resistance R,. Thus, more detailed studies of the maximum current may yield importat information about n, and v,. The steps to be taken to optimize the heterojunction structure for FETs for logic and microwave applications can be summarized as follow: (1) Increasing the A1 content in the (A1,Ga)As increases both the Schottky barrier height of the gate and the heterojunction interface barrier. These permit higher forward gate voltages on the device, reduce any hot carrier injection from the GaAs into the (Al,Ga)As, and permit higher electron concentrations in the channel without conduction in the (A1,Ga)As. The concentration of the A1 in the (A1,Ga)As should therefore be as high as possible consistent with obtaining low ionization energies for the donors, good ohmic contacts, and minimum traps. In present practice it varies from 25 to 30%.
154
H. MORKOC AND H. UNLU
T=300K x.0.3 vS= 2x10~crn/sec
-
,200 -
z
0
FIG. 12. Maximum drain current is also very sensitive to the undoped (Al,Ga)As layer thickness. For the desired large current levels a smaller electron-donor separation is needed to yield a large electron-gas cohcentration. The available data obtained in normally-on FETs while showing the general trends, should be augmented with more experiments. Maximum current levels of about 300 mA (per millimeter of gate width) at 300 K in normally-on FETs with a 1 pm gate length are possible. Large current levels obtainable at low voltages lead to fast switching speeds with low power dissipation.
(2) Maximum voltages on the gate, limited by Schottky diode leakage or by conduction in the (Al,Ga)As, are about 0.8 V at room temperature and about 1 V at liquid nitrogen temperature. Threshold voltages should be about 0. I V for good noise margins and tolerances. (3) To maximize the transconductance for logic applications (and dc current, since voltage swings are given), the (A1,Ga)As should be as thin as possible. Thinner (A1,Ga)As implies higher doping, to achieve the desired threshold voltage. Doping levels cannot be larger than about 1 X 10l8cm-3 because of possibly large gate leakage currents. If one goes ahead with a thin (A1,Ga)As layer without increasing the doping level, the gate built-in potential will widen the triangular potential well, and the electrons will not be confined to the heterointerface, as shown in Fig. 6. When this happens the transconductance near turn-on will be very small and nonlinear with respect to the gate voltage. (4) The setback layer should be as narrow as possible without compromising transport properties (20-30 A) since this gives the minimum total (A1,Ga)As thickness and maximum transconductance consistent with the above limits. Typical parameters for a nomally-off device to satisfy these
2. (Al,Ga)As/GaAsAND (Al,Ga)As/InGaAsMODFETs
155
criteria would be: A1 concentration of about 30%, (A1,Ga)As thickness of about 350 A, setback thickness of 30 A, and doping level of about 1 X 10I8~ m - ~ . VI. Performance in Logic Circuits
Interest in MODFETs was aroused almost immediately after the first working circuits were built by Fujitsu in 1980, with its 17 ps delay time and power -delay product of 16.4 ft at 77 K and 1.7 pm gate length, attained by ring oscillators operating at liquid nitrogen temperature.30These results may be explained on the basis of higher velocities and transconductance and lower saturation voltages of the device, as evidenced from the experimental characteristics. In the logic applications area, using 1 pm gate technology and ring oscillators (about 25 stages), Fujitsu reported a zD= 12.8 ps switching time at 77 K (power consumption not given), Thomson CSF reported 18.4 ps with a power dissipation of PD = 0.9 mW/stage at 300 K,31and AT&T Bell Laboratories reported zD = 23 ps and PD = 4 mW/stage with 1 pm gate te~hnology.'~ Rockwell reported a switching speed of 12.2 ps at 300 K with 13.6 fJ/stage power-delay product. Honeywell reported ring oscillators with switching speeds of zD = 1 1.6 ps at 1.56 mW/stage and 8.5 ps at 2.59 mW/stage at 300 and 77 K, respectively, using 1 pm self-aligned gates,33and AT&T Bell Laboratories reported delay times of zD = 10.2 ps at 1.03 mW/stage and 5.8 ps at 1.76 mW/stage at 300 and 77 K, respectively, with a 0.35 pm gate length device.34MODFETs have also been used to realize medium-scale integration (MSI) level circuits with a 4 X 4 bit multiplier circuit (1.6 ns multiplication time)35and even large-scale integration (LSI) level circuits with a 4 kb static memory (2.0 ns access time).36 Modulation-doped field-effect transistors have progressed from nonfunctional circuits, e.g., ring oscillators, to functional circuits such as frequency dividers. AT&T Bell Laboratories, using a type-D flip-flop divideby-two circuit with 1 p m gate technology, obtained frequency division at 3.7 GHz (with 2.4 mW/gate power dissipation and 38 ps/gate propagation delay) at 300 and 5.9 GHz (with 5.1 mW/gate power dissipation and 18 ps/gate propagation delay) at 77 Fujitsu has reported results on their master- slave direct-coupled flip-flop divide-by-two circuit. At 300 K and with a dc bias of 1.3 V, input signals with frequencies up to 5.5 GHz were divided by two. At 77 K the frequency of the input signal could be increased up to 8.9 GHz before the divide-by-two function was no longer possible, The power dissipation per gate was 3 mW, and the dc bias voltage was 0.96 V.38The best performance, however, is achieved with the type-D flip-flop circuit made at AT&T Bell Laboratorie~.~~ Their results are im-
156
H. MORKOC AND H. UNLU
pressive considering that the type-D flip-flop circuit is inherently slower than the master-slave circuit, and the transistor gate lengths were 1 pm as opposed to 0.7 pm for the master-slave circuit. AT&T Bell Laboratories reported frequency dividers of 6.3 GHz at 300 K and 13 GHz at 77 K with 0.7-0.8 pm gate lengths.39 All of the above circuits have used the simple direct-coupled logic circuits family using enhancement-mode drivers and depletion-mode loads, or saturation resistor loads. The delay for such a stage is proportional to the capacitances, and the voltage swing inversely proportional to the current drive. To achieve high speed one needs to develop a high current-to-voltage ratio. This requires more than just a high transconductance, which is simply the slope of the drain current versus gate voltage characteristics. The characteristics should also have a sharp knee so that little of the valuable swing is lost traversing the low-transconductance knee region. The sharp turn-on of the device maximizes speed. The maximum transconductance is mainly a function of the saturated carrier velocity, but the sharpness of the knee depends strongly on the lower-field part of the velocity versus field characteristics (as well as on the charge-control characteristics, as previously mentioned). While the device possesses good high-speed characteristicsat room temperature, these are enhanced considerably at liquid nitrogen temperature. Low voltages are the key to low-power operation, since the switching energy of the circuit is proportional to C - V 2 ;however, operation at low power-supply voltages would require a very tight control over the turn-on characteristics of the device. Good uniformity of the threshold voltage has been achieved over distances of a few centimeters, the best number being about a 10 mV standard deviation, achieved by Fujitsu,@and 14 mV over a 3 in. wafer, achieved by H~neywell.~' This control would be sufficient for enhance- deplete logic, if it could be obtained reproducibly. The experimental findings can be further improved by developing optimal inverter design rules through an accurate and practical large signal device model. Such a model should give some insight as to what properties of MODFETs make them perform exceptionally well and what parameters should be optimized for maximum speed and power dissipation as well as the maximum device parameter variations tolerable for reliable circuit operation. Ketterson et ~ 1and. Ketterson ~ ~ and M o r k o ~have ~ ~ ,made ~ attempts to develop such a model for both current-voltage and capacitance- voltage characteristics of experimental devices produced in this laboratory. This device model takes the charge response of (A1,Ga)As gate voltage swings into account. Such important high gate bias effects as free-electron generation neutralization of donors in the (A1,Ga)As were included by numerically solving Poisson's equation in the (Al,Ga)As using
2. (Al,Ga)As/GaAsAND (Al,Ga)As/InGaAs MODFETs
Threshold Voltoge
157
(V)
Threshold Voltage (V) FIG. 13. The effect of drive threshold voltage on propagation delay and minimum noise margin at (a) 300 and (b) 77 K for a supply voltage of 1 V (-) and 2 V (---). Increasing V,, decreases r, and leads to improved noise margins, expecially "3.
158
H. MORKOC AND H. UNLU
Fermi- Dirac statistics?*-* Using this model various direct-coupled FET logic (DCFL) ring oscillators with saturated resistor loads were simulated, and the agreement with the experimental r e s ~ l t sis~reasonably ~,~ good for the effect of threshold voltage, supply voltage, gate length, fan-out, and parasitic capacitance on propagation delay time and noise margin. Figure 13 shows the effect of threshold voltage V,, on propagation delay time zD at 300 and 77 K for two different supply voltages. The propagation delay time shows a steady decrease with increasing V, reaching a minimum for V, above 0.6 V. The origin of this decrease in T D with VTHis similar to that for the decrease in tDwith smaller V D D , with the exception that the logic swing and noise margin are not degraded. Figure 14 shows the supply voltage V D D dependence of propogation delay time z, showing the trend of increasing T D with increasing VDD and saturating for large V D D . Figure 14 also shows the power-dissipation dependence on the supply voltage, showing good agreement with that of Hendel et d . I 9 for the same gate width. As discussed in Ref. 42, the increase in zD for large vDD is for the most part due to the saturation of drain current and the resulting decrease in transconductance at high gate voltages. Therefore, by decreasing V,, so that V,, is at or below the point where g, degrades and where large capacitances due to (A1,Ga)As charge changes begin, Z , improves remarkably. This decrease in V D D , however,
5
.!5
I
I
1.25
I
I
I
lo
150 L75 200 225 Supply Voltage (V) FIG.14. Effect of supply voltage on propagation delay time and power dissipation at 300 K (-) and at 77 K (---). Minimum delay times at VDD= 2 V are 14.4 and 8.6 ps at 300 and 77 K, respectively. 1.00
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
159
results in a small logic swing and noise margin. Rather than reducing VDD to prevent these high gate bias effects, a large Vm can be used to effectively push the gate voltage necessary to see the transconductance degradation above VoH. Furthermore, a wide logic swing and noise margin can be maintained with a large V, . The effect of V, on noise margin NM is also illustrated in Fig. 13. Only the minimum, either NMO or NMI, is plotted, since it is the smallest noise margin that determines the overall noise margin for the circuit. The NM is much larger at 77 than at 300 K because of the smaller on resistance and sharper transfer characteristics. For small Vnr it is NMO which is the smallest and determines the overall NM. Because noise margins represent an additional voltage difference that must be tranversed before the actual switching takes place, large noise margins should result in an increase in delay time?’ However, as Fig. 13 shows, minimal delay times are possible even with large NMs by using high threshold voltages, while large NMO should not affect 7D, due to the high gate capacitance well above threshold. Nevertheless, the gain in zD of reducing high gate bias effects by raising v, is more significant than the detrimental effect of large noise margins. The effect of reduced gate lengths LG on device and circuit performance is very important.42Submicrometer devices exhibit superior current -gain cutoff frequencies and improved ring oscillator switching times due to the increased transconductance and reduced gate capacitance. The simulations made for ring oscillatorsof LGranging between 0.25 and 2 pm show that 7, is virtually the same for a given gate length even though the L,-dependent transconductancevaried from 197 to 258 mS/mm at 300 K and 348 to 393 mS/mm at 77 K. This demonstrates the real gain of short gate length MODFETs in digital circuits is in reducing the gate capacitance rather than being from the increased transconductance. In summary, the simulations indicate that a large driver threshold voltage should be used to provide ample noise margins (especially NMO) and to increase the gate voltage where high gate bias effects become noticeable. Circuits utilizing such high-threshold drivers have improved propagation delay times and large logic swings. The simulations for a minimum tolerable noise margin of 0.2 V and a threshold voltage of 0.5 V under various conditions indicate that delay times can be obtained at small supply voltages, in agreement with the majority of experimental results. This is explained as being due to the reduction in transconductance and the increase in gate capacitance at large voltages. Fan-out resistivities of 10.8 and 7.1 ps per FO at 300 and 77 K show the superiority of (Al,Ga)As/GaAs MODFETs over values predicted for Si MOSFET and GaAs MESFET circuits. The simulations further show that, at least for circuits dominated by gate capacitance loading, the
160
H. MORKOC AND H. UNLU
important effect of reducing the device gate length on delay time is in reducing the gate capacitance rather than increasing transconductance. Although these results are strictly valid only for ideal circuits (i.e., ring oscillators),they do provide an indication of how actual circuits dominated by interconnect capacitance and multiple fan-out capacitance can be optimized. VII. Microwave Performance While a great majority of heterojunction-related research has so far been directed toward the logic applications due to their distinct advantages over conventional GaAs MESFETs, with the increasing interest in industrial laboratories, impressive results have become available on the microwave low-noise performance of MODFETs. Even though this device is being considered for power applications as well, its power-handling capabilities are limited by the relatively low breakdown voltage of the gate Schottky bamer. Approaches such as the camel which utilizes a p+-n+ structure on n-(A1,Ga)As for an increased breakdown voltage, will have to be improved before the MODFETs can be a good contender in the power FET area except at extremely high frequencies. In the microwave low-noise FET area, there have been recent reports of successful operation of MODFETs with gate lengths as short as 0.35 pm.49-s8The results are very impressive as compared to conventional GaAs MESFETs. Devices with a 0.35 pm gate length operated at 35 GHz exhibited an impressive low-noise figure of only 2.7 dB at 300 K,s3 and MODFETs with relatively long gate lengths, i.e., 1.4 pm, have demonstrated current-gain frequencies as high as 25 GHz. Most recently, the General Electric groups8has reported (Al,Ga)As/GaAsMODFETs with a noise figure of 0.8 dB at 18 GHz and a current - gain cutoff frequency offT of 80 GHz for 1 pm gate length. As will be seen from the comparison, the available current-gain and noise-figure data on nonoptimized MODFETs are extremely superior to the optimized conventional GaAs MESFETs with the same gate length. The microwave MODFET models show that a 50% improvement in the power-gain cutoff frequency can be expected as 5. EQUIVALENT CIRCUITS
In understanding and modeling the high-frequency small-signal and large-signal performance, one needs to develop an equivalent circuit for the MODFET structure, such as by assuming a lumped-element circuit model.60 One can perform microwave S-parameter measurements as a function of applied bias, from which Y parameters and the equivalentcircuit parameters can be calculated. Measurements made between 2 and
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
161
FIG.15. Scatteringparameters measured as a function of frequency (data)at V, = 0 V and V, = 2.5 V for a normally-on FET. The outer circle corresponds to 1 for S,, and S,,, 5 for S,, ,and 0.2 for S,, ,respectively. The calculated S parameters using the equivalent circuit of Fig. 14 are also shown (-). The measurements for this particular figure were performed at Tektronix.
18 GHz are illustrated in Fig. 15. S parameters, which are easily measured for a two-port device, are connected to gain and admittance parameters corresponding to the equivalent-circuit model commonly used for highspeed FETs, as shown in Fig. 16. The intrinsic transistor elements are: C, and C,, the sum of which is the total gate-to-channel capacitance; and rh and R&, which relate the channel current to the voltage drop across CP. The transadmittance is characterized by a transconductance g,, and a transit time rt for electrons in the high-field region beneath the gate. In some cases a feedback resistance R, is placed in series with C,. The extrinsic circuit elements are: R, and R,, each of which is the sum of an ohmic contact resistance; the semiconductor resistance between the ohmic contact and the edge of the gate; and part of the channel resistance under the gate.61,62Rg is the meta gate resistance, and Lg, L,, and Ld are the inductances associated with wires bonded to the transistor. The current-gain cutoff and unilateral power-gain cutoff frequencies may be estimated from the equivalent circuit given by Ref. 63.
fT = gmd2nCgS
(13)
162
H. MORKOC AND H. UNLU
FIG. 16. Equivalentcircuit parameters calculated from Y parameters which in turn are calculated from S parameters.
and fmax =fTl{2[(&
+ Rg + RJRG' + 2nhRgcds]''2)
(14)
The current-gain cutoff frequency can also be found directly by measuring the current gain as a function of frequency and extrapolating the unity of gain. Table I shows the values of the small-circuit parameters reported for (Al,Ga)As/GaAs MODFETs with 0.35,52 0.50,64and 1 pm50 gate lengths with estimated and/or measured values of& and f ,,. One should keep in mind that these values are bias dependent and do not represent the device operation under identical conditions. Arnold et have studied the bias dependence of small-signal parameters of MODFETs as compared to GaAs MESFETs which were fabricated in the same manner. The values for GaAs are given in parentheses in Table I. Furthermore, the circuit elements g,, , C, ,and Rdsfor MODFETs displayed sharper pinch-off characteristics than in the GaAs MESFETs, as demonstrated in Fig. 17. The superiority of MODFETs over the conventional GaAs MESFETs for low noise can be seen by using a first-order analysis. For a short-gate MODFET operating in the saturated velocity mode, the intrinsic transconductance g,,, is given by gm =~
2 ~ sW Zl and C, is primarily due to the gate capacitance, given by
+
C, = E2LZ/(d A d )
+ c,LZ/W
(15)
(16) where E~ and dare the dielectric and the thickness of the (A1,Ga)Asbeneath the gate, Ad is the average effective displacement of the electron gas in the GaAs from the heterointerface, and L the gate length. For a MODFET, W
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
163
TABLE I SMALL-SIGNAL CIRCUIT-ELEMENT VALUES, CURRENT-GAIN CUTOFF FREQUENCIES V;) AND POWER-GAIN CUTOFFFREQUENCIES & ), FOR
MODFETs" ~
L,@m) g,,,(mS/mm) Tz (PS)
C,(pF/mm) C, (pF/mm) c d s (pF/mm) R, (a) Rin (amm) Rds(Rmm) R,,(Rmm) R, (Q mm) Rd mm) f, (GH-8 f, ( G W
0.39 230
0.78 0.23 0.17
03
1.od
235 2 1.3 0.15 0.45
140 (100) 2.4 (4.4) 1.3 ( 1 . 1 ) 0.09 (0.11) 0.16 (0.28) 4 (4) 7 (3.6) 220(120) 0 (0) 1.5 (0.93) 0.84 (0.15)
2.4
1
1.11
0.4 66 0.2
44 0
0.38 0.39 47 75
~~~~
0.8
1.0 35*/29 96
18(14)
38 (30)
The values in parentheses are for a GaAs MESFET with the same geometry as the MODFET. The starred values of f, were obtained by measuring lhz,l versusf and extrapolating to the intercept with the faxis. * Reference 52. Reference 64. * Reference 50.
and therefore C, are relatively bias independent except near the pinch-off, as shown in Fig. 18. This change may also show that C,, calculated here, merely represents an effective input capacitance. The combined capacitive effects in the input circuit is here termed and modeled as C,,.shown in Fig. 14. The large drop in C, as the gate voltage approaches pmch-off is explained by the widening of the potential well in the undoped GaAs channel near pinch-off, as shown in Fig. 17. Combining Eqs. (14) and (1 5) we have
fT = vJ2;nL an identical expression to that obtained for GaAs MESFETS.~',~~ This is confirmed in Fig. 19, which showsf, andf,, as a function of the gate bias for both devices. For the MODFETs one would have expectedf, to remain constant if the effective input capacitance C, were bias independent. MODFETs have higher effective V , at 300 K (- 1.8-2 X lo7 ~ m / s ) ~ ~ @ than GaAs MESFETs (- 1.3 X lo7 ~ m / s ) , 6which ~ gives the MODFETs a
164
H. MORKOC AND H. UNLU f [
l 051
-7
70t
I
I
I
0.25
"
0.50 0.75 1.00 IVdV, FIG.17. Small-signal transconductance calculated from Y,, at 4 GHz for a MODFET (0) and conventional MESFET (A) versus the gate bias normalized with respect to the pinch-off voltage and to 1 mm of gate width. The fact that the transconductance of the MODFET stays high for about half the gate bias range supports the theory presented. 6%
I
v)
u" 1.0-
0.8
0
I
I
0.25
0.5 vgs / v p
I 0.75
1
1.o
FIG. 18. The variation of gate capacitance with gate bias in a conventional MESFET (A, experimental) and a MODFET (0,experimental; ---, theoretical). Again, invariance away from the pinch-off is characteristic of the MODFET.
2. (AI,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
165
40
30.-/-
c
I n
0
0
2 0
c3
-
Y
-
0
0.25
0.50
0.75
1.00
IVqdVpI
FIG. 19. Current-gain cutoff frequency, fT (0, MODFET A, MESFET), and unilateral power-gain frequency,f- (A, MODFET; 0,MESFET) a (in GHz) versus gate bias normalized to pinch-off. V, = 4 V.
speed advantage of about 1.5 over MESFETs for a 1 pm gate device. At 77 K the speed advantage of MODFETs increases to 1.5 as Y , increases to 3 X lo7C ~ / S , ~ ~ while , ~ O us is independent of temperature for MESFETs. The feedback capacitance C, is mainly due to gate fringing capacitance between the gate and drain; therefore, it is almost in contact with the increase in reverse gate b i a ~ . ~At, ~lower ' drain biases C, contains parts of C,as well. As the device reaches velocity saturation C, reflects only the gate fringe capacitance. This explains the rapid decrease of C, with the increase in drain bias in the vicinity of saturation, as illustrated in Fig. 20. Similar behavior has also been reported for conventional GaAs MESFETS.~,~' -73 The drain resistance %, is another crucial parameter in deterrniningf,, and, in turn, the power gain of the MODFETs. Figure 21 illustrates how the drain resistance (output resistance) %, of the MODFET changes with V,, with the gate voltage used as a parameter. At lower drain voltages %, is in good agreement with the theory developed for conventional GaAs
-
166
H. MORKOC AND H. UNLU
FIG.20. Feedback capacitance for a MODFET versus V , with V, as a parameter ( 0 , O ; A, V,-0.6 V) in pF per mm gate width.
-0.2; 0, -0.4;
Here I d and I, are the drain and drain saturation currents, respectively. The V, dependence of R, at lower drain voltages can be immediately seen in Fig. 2 1. Figure 22 demonstrates how Rdsvaries with Id.As can be seen, the linear depenence of R, on the drain current I d does not hold over the entire operational range of the device. The observed nonlinearity can arise from parallel resistance in the substrate, as observed for GaAs MESF E T s . ~ ~The , ~ * increase of the L$d+ Ad ratio in MODFETs decreases R, and in turn increases f-. In MESFETs it is desirable to keep L J a >> 3.75 For MODFETs d A d is less than 0.05 pm, so that gate lengths can be reduced to 0.15 pm before suffering serious degradation. The final scaling consideration is that the threshold voltage of MODFETs has been found to be independent of L down to at least 0.25 This is due to the fact that the large electron density is located in the vicinity of the gate. In conventional GaAs MESFETs threshold voltage shifts of - 1 or -2
-
+
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
167
FIG.21. Output resistance for a MODFET versus Vdawith VW as a parameter (0,0; A,
-0.2; 17, -0.4; V, -0.6 V) in R mm.
V have been observed as the gate length is decreased to 0.3 One can overcome this in MESFETs by increasing the channel doping up to 1 X lo1*~ m - ~ Figure . 23 show the current-gain cutoff frequencyf, as a function of gate length L (in pm). The slope shows that fT is inversely proportional to the 1.5 power of the gate length over the entire gate length. Here we should like to point out that for submicrometer gate lengths velocity overshoot effects on fT may be i r n p ~ r t a n t . ~ ~ , ~ ~ The low-noise performance of state-of-the-art MODFETs is illustrated in Fig. 24. The noise value is plotted as a function of gate length. Goronkin and Nair78have pointed out that the noise value M = 1 - F is a useful figure of merit in comparing FETs. The method is based on the form of physical or empirical noise models which can be written as F = 1 fL.F is related to the measured noise figure by NF = 10 log,,, F. The solid and dashed lines are the best published data for MODFETs and GaAs MESFETs characterized at 8 and 18 GHz, respectively, in 1984.
+
168
H. MORKOC AND H. UNLU
FIG.22. Output resistance for a MODFET as a function of drain current in R mm.
VIII. Anomalies in the Current - Voltage Characteristics As indicated earlier, these devices have the potential for better performance at cryogenic temperatures; however, a peculiar behavior in the drain I- V characteristics of these devices upon cooling to 77 K has been ob~ e r v e d . ' ~In. ~particular, ~ when the device is cooled to 77 K without exposure to light, the drain I- I/ characteristics collapse at drain to source voltages less than about 0.5 V, while at voltages greater than about - 5 V the characteristics look normal. In addition, a marked reduction in the drain current has been reported in some devices for large drain biases:' which was attributed to carrier injection over the barrier.82 Modulation-doped FETs would be of limited value for cryogenic operation if this effect could not be understood and preferably eliminated. By careful control of the fabrication process and growth conditions it has been shown to be possible to fabricate heterojunction FETs that do not exhibit this effect at cryogenic temperature^.^^ Figure 25 shows a schematic cross section of a fabricated device and also indicates one possible mechanism by which the distortion of the drain I- V characteristics may occur. Since the collapse in the I- V characteristics of MODFETs is observed only after a drain bias greater than 1 V has been
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
169
00 7060 -
5040 N
I (3
Y
+I-30-
20 -
101
0.2
I
I
I
0.4
0.5
I
I
l
l
0.6 0.7 0.8 0.9 Gate Length ( p m ) FIG. 23. The comparison of experimental values of current gain cutoff 6 ) versus gate length for state-of-the-art(In,Ga)As/(Al,Ga)As (---) and GaAs/(Al,Ga)As (-) MODFETs. 0.3
applied, it seems likely that the mechanism responsible for the collapse is related to some charge-injection/trappingprocess. Also, since the magnitude of the electric field is largest at the drain end of the channel, the charge-injection mechanism would be expected to occur near the drain. Drummond et al.76have suggested that the mechanism responsible for the observed effect is due to charge injection and trapping in the bulk (A1,Ga)Asnear the drain end of the channel. A similar collapse in the drain current has also been observed in CdSe thin-film transistors” and in insulated gate (SiOJ FETs.85,86 In both cases the collapse was attributed to electron trapping in the gate insulator. In the MODFET the depleted
170
H. MORKOC AND H. UNLU 1s
0.
7 0.
s
v) 0
P .%
20
0
L
I
1
0.2
I
I
0.4
I
0.6
I
Gote Length (,urn) FIG. 24. Noise measure of state-of-the-art (Al,Ga)As/GaAs MODFETs as a function of gate length. (---), the performance of state-of-the-art GaAs MESFET for comparison (see Ref. 50).
Depletion
Doped AlxGo~-xAs
Undoped AIxGal-xAs 2 DEG
4
Electron Injection
lf’
FIG.25. Schematiccross section of a MODFET, indicatinga possible mechanism by which I - V collapse occurs.
2. (AI,Ga)As/GaAs AND (AI,Ga)As/InGaAs MODFETs
171
(a 1
300 K
r
;jE 60
77
F
Dorh
-
Light
32u
L
o
"
1
a5 1.5 ~5
,
0
0
5
5
Drain to Source Voltage (V)
(b)
r
77 K Dark
r
Light
*c---
_----
--
0 0.5 1.0 L5 Drain to Source Voltage (V)
/----
o
a5
LO 1.5
FIG.26. Current-voltage characteristics from two devices, (a) one With 1.0 pm long gates and (b) one with 1.6 pm long gates. The gate recess was the same length in both devices. The step size was +0.2 V for all characteristics. Some of the traces at 77 K are shown as dashed lines because of oscillations.
(AI,Ga)As, undoubtedly containing a very large concentration of defects, can trap the injected electrons. This was also observed by Kastalsky and KiehLS7 It is well known that (A1,Ga)As contains a defect center presumably induced by donors and has quite a large barrier to electron capture and emission at cryogenic temperatures.88It is this center which is believed to give rise to the persistent photoconductivity effect in ( A I , G ~ ) A sand , ~ ~it is conceivable that this particular center could capture the electrons injected into the (A1,Ga)As for a sufficient period of time so as to give rise to the observed phenomenon. Once electrons are injected into the (A1,Ga)As near the drain, the reduction of net positive space charge in the depleted
172
H. MORKOC AND H. UNLU
(A1,Ga)As calls for a smaller concentration in that region. A depleted 2DEG would restrict current flow in the channel. As the drain bias increases, however, the depleted region can be punched through, resulting in the resumption of drain current. The mechanism is supported by the experimental results shown in Fig. 26a, where the typical behavior of the collapse of I- Vcharacteristics at 77 K is plotted. The drain I- V characteristics are shown at room temperature, after cooling to 77 K in the dark, and after exposure to light at 77 K. Some traces in the I- V characteristics of these devices at low temperature are shown as dashed lines. The larger transconductances achieved at low temperatures lead to bias instabilities due to oscillations caused by parasitics of the TO- 18 headers used. Both of the devices shown in Fig. 26 were fabricated from the same epitaxial layer. Once the source and drain ohmic contacts were formed, the wafer was cut in half. On one piece, 1.6 pm long gates were fabricated, while on the other, 1.0 pm gates were fabricated. The gate recess was the same length in both pieces. The characteristics shown in Fig. 26b correspond to the device having the 1.6 pm gate length, while those in Fig. 26a correspond to the device with the 1.O pm long gate. The I- Vcharacteristics of the device in Fig. 26b demonstrate that is indeed possible to fabricate FETs whose performance improves substantially when cooled to 77 K in the dark. Furthermore, when the characteristics of Fig. 26b are compared with those of Fig. 26a, it is obvious that at least part of the mechanism resonsible for the collapse is related to the geometry and/or particular fabrication procedures used. The transconductanceof the device in Fig. 26b was 170 and 280 ms/mm at 300 and 77 K, respectively. The source resistance of this device estimated from the drain I- V Characteristics was about 1.5 51 mm at 300 K and decreased to 0.36 R mm at 77 K. The value for the source resistance measured by forward biasing the gate with respect to the source and recording the drain voltage while monitoring the gate current was slightly larger than 1 SZ mm at 300 K. On the same wafer, using the transmission line method, a specific contact resistivity of slightly less than 2 X lo-' 51 cm2 was measured for this particular structure. This resistivity should be treated with some caution, because the sheet resistivity underneath the contact may be different. The results of Fig. 26 can be understood in light of the phenomenological model outlined above. It has been shown that when the distance between two electrodes is less than about 0.4 pm, the surface depletion does not occur to any appreciable extent,w which has been confirmed in camel diode gate GaAs FETs9' as well. Since the (A1,Ga)Asis thinner in the gate recess, any surface depletion can extend closer to the 2DEG than in
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
173
regions away from the gate recess. In addition, this results in lifting the conduction-band edge in (A1,Ga)As with respect to the Fermi level. The entire thickness of the (A1,Ga)As could in fact be depleted, enhancing the charge-trapping mechanism alluded to earlier. Since little or no effective surface depletion was shown to occur for distances of less than about 0.4 pm,w this allows some variation in the positioning of the gate metallization within the recess. As long as the gap in the recess is kept below about 0.4 pm, the devices should not show any collapse. The results of Fig. 26 support this observation, since the only difference between the device of Fig. 26a as opposed to that of Fig. 26b is in the size of the gap between the gate metal and edge of the recess. Figure 27 shows the drain I- V characteristics of the device of Fig. 26b with the source and drain leads interchanged (inverted). For this particular device, the 1.6 pm gate was not placed in the center of the 2.2 p m long recess. The gap between the edge of the gate metal and gate recess on the drain side was about 0.1 pm when the device was operated in the normal configuration, while that for the inverted configuration was about 0.5 pm.The fact that collapse occurred only for inverted operation but not for normal operation further verifies the proposed mechanism. The results of Fig. 26 also indicate that the drain I- Vdistortion in MODFETs is not necessarily related to problems associated with contacting the 2DEG through (Al,Ga)As. The devices of Fig. 26 had their source and drain ohmic contacts formed at the same time, yet one exhibited collapse while the other did not. If indeed the collapse of the drain I-V characteristics at cryogenic temperatures is related to charge trapping in the (Al,Ga)As, the density of defects in the (A1,Ga)As should have a profound impact on this phenomenon. In order to determine the influence of defect concentration, FETs were fabricated from an expitaxial layer on which several monolayers of Ga had been deposited on half the surface of the substrate prior to the
+0.2V/step, 77K Dork
Drain to Source Voltage (V) FIG.27. Drain I - V characteristic from the device of Fig. 22b with source and drain leads interchanged.
174
H. MORKOC AND H. UNLU
initiation of growth to achieve a varying As vacancy-related defect concentration across the wafer. The characteristics shown in Fig. 28a correspond to devices taken from areas of wafers with no Ga predeposition, while those of Fig. 28b are associated with the area of the wafer with predeposited Ga layer. The characteristics of Fig. 28a show no collapse, while those of Fig. 28b do show collapse. This demonstrates that the presence of traps plays an important role in the drain I- Y collapse. The characteristics of Fig. 28a demonstrate another important feature having to do with the lack of performance enhancement as it is cooled to 77 K. At room temperature, the transconductance and source resistance were 150 ms/mm and 1.5 R mm, respectively. As the device was cooled to 77 K in the dark, the
(a)
77 K
300 K -a1vistep
-0.1Vlstep
Dark
a
,
0
10-
-
0
0
Light
0.5 1.0 L5 Drain to Scurce Voltage (V)
(b)
77 K
300 K 3or
-0.2 V/step
-0.2 V/step
-0.2 V/step
Light
25
2
3
I0 5-
Drain to Source Voltage
(V)
FIG.28. Drain current- voltage characteristicsfrom two devicestaken from a layer in which about one monolayer of Ga was predeposited to study the effect of traps. The characteristics shown in (a) are from a device with a smaller amount of predeposited Ga, and those in (b) are from one with a larger amount.
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
175
transconductance increased slightly to 155 mS/mm, but the source resistance increased to about 2 R mm. Upon exposure to light, however, the transconductance increased to 3 15 mS/mm, and the source resistance decreased to about 0.5 Q mm. This demonstrates the importance of eliminating defects and shows that the cryogenic performance of the device is sensitive to the molecular-beam epitaxial growth conditions. Since it is well known that ternary compounds such as (A1,Ga)Asare much more sensitive to the growth conditions used than are binaries, the defects are predominantly believed to be associated with (A1,Ga)As. A further verification of this conclusion was obtained in another experiment in which the GroupV/III ratio used during the growth of the (A1,Ga)Aswas varied from the optimum vaue (about 5 : 1) to a much lower value (2 : 1). The points to be drawn from the results of this series are that the device performance at room temperature and 77 K (dark and light) all degrade as the Group-V/III ratio decreases. This underscores the importance of obtaining the highest possible quality (A1,Ga)As. Another important point is that, although the device results degrade with lowering the Group-V/III ratio [and hence (A1,Ga)As quality], the Hall measurements do not necessarily reflect this difference. With the exception of the 77 K mobility for the layer grown with the lowest GroupV/III ratio, the Hall data do not vary appreciably from layer to layer. This demonstrates that the Hall mobility of modulation-doped layers gives a rather poor indication of device performance. Again the device with the lowest (2 : 1) GroupV/III ratio showed collapse while the others did not.
IX. Advanced Technology Requirements In the ultra-high-speed area, the technology for fabricating submicrometer gate devices with extremely small source and drain contact resistances must be developed. In order for the heterojunction FET to retain its speed advantages over self-aligned-gate GaAs FETS?~a postgrowth ion implantation and annealing process (Fig. 29) must be used. It is then imperative that after annealing the heterojunction structure maintain its electrical properties. Initial furnace annealing studies have revealed an extensive degradation, which casts doubt on the development of this important t e c h n ~ l o g yFurther .~~ investigation with furnance annealing showed that the interface sheet carrier concentrations must be made very very small to reduce d e g r a d a t i ~ n(but ~ ~ failed to eliminate it). This is contradictory to the requirements for a fast switching device, since large interface electron concentrations are needed. Using conventional single-interface modulation-doped structures, Henderson et al.95have shown that almost no degradation occurs after flash
176
H. MORKOC AND H. UNLU nf Implant
I\ I
tI
I
n
I
I
I
'
I
1-7-1 2DEG
Goas
___--
i
t
FIG. 29. Cross-sectional view of a self-aligned modulationdoped MODFET where the T-shaped gate is used as a self-aligned n+-implantation mask. The gate metal in contact with the semiconductor is a refractory material, and the cap on it may or may not be left in place depending on whether or not it tolerates the annealing process.
$1012
'
I
I
Y
b
b
c
4 c
0
w
-.a s
t -
c .-
a
r"
2-
TA ("C) FIG.30. The sheet-electron areal density and the electron mobility at 77 K of modulationdoped heterostructures flash annealed between 750 and 900°C (A, layer A; 0, layer B).
2. (Al,Ga)AsfGaAs AND (A1,Ga)AsfInGaAsMODFETs
177
annealing at 8OO"C,which is believed to be sufficient to produce about 80-90% activation in GaAs when the dose is about 10l2 cm-2. In this particular study the samples were flash annealed at TA= 750, 800, 850, and 900°C in a commercially available annealing apparatus in an inert atmosphre of 3%H2 in Ar, and in contact with an undoped GaAs wafer to minimize As desorption. A similar technique has been shown95to yield an activation efficiency of >90% in Zn-implanted bulk GaAs at TA= 800°C. The results of HalI-mobility and sheet-camer-concentration measurements are presented in Fig. 30. Note that, while in both instances annealing at 800°C preserves over 85% of the virgin 77 K mobility, n, at 77 K is reduced 25% in layer A, but remains virtually unaffected in layer B. Layer A consisted of a 1.0 ,urn GaAs buffer layer beneath a 30 A undoped (A1,Ga)As setback layer topped with a 400 A n-(A1,Ga)As layer which is then capped with 200 A n-GaAs. The thicknesses in layer B were 3.0 pm, 30 A, 600 A, and 50 A, respectively. Diffusion of Si from the doped (Al,Ga)As layer can be ruled out on two premises: (1) insufficient time at the elevated temperature and (2) the more heavily doped layer is less
I
I
8200
8300
x ti, FIG. 3 1 . Low-temperature photoluminescence spectra obtained From layer A after flash annealing between 750 and 900°C.
178
H. MORKOC AND H. UNLU Excitons
I
I
8200
8300
x (8) FIG.32. Low-temperaturephotoluminescence spectra obtained in layer B following flash annealing at temperatures between 750 and 900°C.
affected with the same undoped layer thickness. The remaining plausible mechanisms are As desorption, diffusion of impurities from the substrate, and site transfer of the amphoteric Si dopant in the doped (A1,Ga)As layer. The photoluminescence (PL) spectra reproduced in Figs. 31 and 32 exhibit trends similar to those observed in the Hall data. Both samples display an overall decrease in integrated PL intensity as TA increases, except that the control from layer B produced weak luminescence, possibly due to misalignment during measurement. Deterioration of the GaAs buffer at higher TAis evidenced by a broadening of the exciton lines and an X) increase in the ratio of intensities of peaks associated with defects96(D, with respect to the free-exciton (F, X) line. It should be pointed out, however, that recent results on high-purity samples indicate that what were previously thought to be (D, X)97lines may actually be associated with donor- acceptor pairs. The difference in the deterioration of the optical properties of the two layers is again one of degree. The overall similarity of PL spectra from
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
179
samples annealed at 750 and 800°C to their respective controls supports the conclusion that decreases in n, stem from either As depletion or site transfer of the amphoteric Si dopant in the doped (A1,Ga)As layer rather than deterioration of the GaAs itself The decrease in intensity of exciton lines is due to and correlates well with the decrease in n,: In layer A as TA increases from 750 to 800°C, An, = 16%,while A(F, X) = 28%; in layer B as TA increases from 750 to 8OO"C, An, = I%, while A(F, X) = 1.5%. The above data present some clues regarding the processes leading to the degradation observed at 900°C. The mechanisms that may be responsible are: (1) diffusion of impurities from the substrate into the epilayers, (2) lattice damage and autocompensation due to As desorption from surface layers, and (3) diffusion of Si from the n-(A1,Ga)Aslayer into the 2DEG region. The role of substrate impurity outdiffusion is addressed by a comparison of layers A and B. These structures are similar except that B is grown on a Cr-doped substrate and includes a buffer layer between the substrate and the heterointerface that is three times the thickness of the buffer layer in structure A, which was grown on an undoped substrate. Because the substrate type and buffer layer thickness do not appreciably affect the degradation of those two structures, substrate impurity outdiffusion can be eliminated as a significant mechanism. A second possible mechanism is surface desorption of As. Because the total thickness of GaAs or (A1,Ga)Asabovethe 2DEG in these structures is less than 700 A,this mechanism might be expected to play a significant role. The increased density of surface defects observed in structures annealed at TA = 900°C supports this hypothesis. If As desorbs to the extent of creating lattice vacancies near the heterointerface, a significant degradation in mobility as well as camer concentration would be expected as a result of the large density of scattering centers and traps. In addition, because PL measurements made on Si-doped epitaxial GaAs indicate increasing probability of the transfer of Si from donor to acceptor sites with increasing TA,a similar trend can be expected for the Si-doped (A1,Ga)As layers in these structures. This would tend to further decrease measured electron concentrations. Above 850°C both samples show a drastic decrease in mobility and carrier concentration. This is undoubtedly due in part to the creation of a large number of GaAs lattice defects, as seen in the PL spectra of those samples. This is consistent with extensive loss of As from the samples. The importance of a third mechanism, diffusion of Si from the doped (A1,Ga)As layer to the heterointerface, is less certain. A simple extrapolation of results available in the literature indicates that Si in GaAs may diffuse as far as - 50 A for TA= 800°C or - 200 A for TA = 900°C with an
180
H. MORKOC AND H. UNLU
annealing time of 1 s. Assuming that these lengths are similar for diffusion in (Al,Ga)As, one might expect that diffusion of Si to the heterointerface could cause a large reduction in the 2DEG mobility for TA2 800°C. Our data indicate that this is not the cause, even for annealing temperatures as high as 850°C. Whether or not significant Si diffusion takes place from 850 to 900°C is difficult to determine because the effects of As desorption on mobility will obscure any similar effects that Si diffusion might produce. PL measurements on annealed modulation-doped structures do not show the spectral lines characteristic of Si in GaAs. This indicates that, if diffusion of Si into the GaAs takes place, the difision depth is indeed fairly small. Microprobe measurements should be of some value in this regard. Despite the fact that some degradation is observed in transient annealed modulation-doped heterostructures at the higher end of the temperature range, the results of this experiment are quite promising and demonstrate that the modulation-doped structures can withstand the annealing process that makes 90% implant activation possible in GaAs and retain over 95% of the virgin mobility and sheet carrier concentration. The successful results reported here serve to remove a significant obstacle on the path to a new generation of ultrafast, MODFET-based circuits.
X. Pseudo-morphic MODFET- (In,Ga)As/(Al,Ga)As So far our focus has been on the most extensively studied MODFET structure, (Al,Ga)As/GaAs. It is shown that although the (Al,Ga)As/GaAs structure is one of the most promising and widely studied transistors, cryogenic operation of this device (where its advantages over the conventional GaAs MESFET are most apparent) is not ideal. Deep levels (DX centers) in the (Al,Ga)As with peculiar persistent properties can lead to the “collapse” of the drain I - V characteristics. More troublesome is the threshold voltage, which occurs at 77 K after the gate has been forward biased. To reduce these problems one may use (A1,Ga)As with x 5 0.20, where the DX occupation probability is significantly decreased. This, however, reduces the conduction-band discontinuity at the heterointerface, which results in less efficient electron transfer and therefore a smaller 2DEG concentration. Furthermore, the use of a low Al mole fraction in (A1,Ga)As also compounds the problems of parasitic MESFET effects. An alternative solution proposed recently, which replaces (A1,Ga)As altogether, is to use (In,Ga)As as the narrow-bandgap material and dope the larger-bandgap GaAsW A thin layer of the narrow-bandgap (In,Ga)As, which is lattice mismatched to GaAs (- 1%) is sandwiched between an undoped GaAs buffer and a doped GaAs cap layer. The (In, Ga)As is thin enough (-200 A) that the lattice strain is taken up coherently by this
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
181
quantum well, resulting in a dislocation-free “pseudo-morphic” material. More recently it was demonstrated that, by replacing the GaAs with the even higher-bandgap low mole fraction (Al,Ga)As, device performance rivaling the best reported (Al,Ga)As/GaAs MODFET results are possible. Ketterson et aL3have made the first s u m f u l fabrication of I ~ . 1 5 ~ . s s A l / Ab.lsG~.ssAs MODFET exhibiting good dc characteristics. dc transconductances are 270 mS/mm at 300 K and 360 mS/mm at 77 K for devices with 1-pm gate lengths and 3-pm source-drain spacings. Their results rival the best results obtained for (Al,Ga)As/GaAs MODFETs. A currentgain cutoff frequency of about 20 GHz was found, and no persistent photoconductivity or drain collapse was observed. Most recently, Ketterson et aL5 characterized the pseudo-morphic InyGaI-yA~/AI,,15Ga,,85A~ (0.05 I y I0.20) MODFETs grown by molecular-beam epitaxy (MBE) at dc (300 and 77 K) and rf frequencies. The pseudo-morphic single-quanturn-well (In,Ga)As/(Al,Ga)As structures studied were grown by MBE on GaAs substrates. The 1 pm unintentionally doped GaAs buffer layer was followed by either 150 or 200 A quantum well of undoped InYGal-,As, with y varing from 0.05 to 0.20. Finally, a structure with a 30 A Ab.15Gh.sy4sundoped setback layer, a 350 A n-Ab.lSG%.85As layer doped to 3 X lo1*cm-3 with Si, and a 200 A n+-GaAs cap layer to facilitate ohmic contact formation is grown. Hall measurements performed on the sample shown in Fig. 33 indicate a 300 K low-field mobility and areal concentration of 6000 cm2/V S and 1.4 X 10I2 cm-2; at 77 K these values are 29,000 cm2/V S and 1.2 X 10l2 cm-2 and showed no peristent photoconductivity effect. When the setback layer thickness was reduced to 30 A, the 2DEG concentration decreased to 3 X 10’’ cm-2, and the Hall mobility increased to 8000, 95,000 and 158,000 cmF2/VS at 300, 77, and 10 K, respectively. They represent the highest yet reported mobilities for a strained-layer MODFET structure. However, the low sheet carrier concentration makes this structure less suitable for FETs. In order to demonstrate the superior cryogenic performance of (In,Ga)As/(Al,Ga)As pseudomorphic MODFETs compared to the more conventional GaAs/(Al,Ga)As structure, Hall measurements were made down to 12 K in the light and dark. At 12 and 77 K persistent photoconductivity measurements were done by measuring the sample in the dark just following the illumination, and the results are illustrated in Fig. 34. As Fig. 34 shows, there is virtually no change in either the mobility of sheet carrier concentration between light and dark for the (In,Ga)As/(Al,Ga)As sample as compared to a conventional GaAs/(Al,Ga)Assample of a similar structure. Figures 35 and 36 show the mobility and velocity of electrons in an A~.IsG~.ssAs/I~.lsG~.ssAs pseudo-morphic structure as a function of
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200.4
n-GoAs
Al Gate
p- GaAs Undoped Undoped
- -- - - --- - -
EC
EF
FIG. 33. (a) Typical structure for MBE-grown (In,Ga)As/(AI,Ga)As pseudo-morphic MODFET and (b) the associated conduction-banddiagram. The conductingchannel forms a two-dimensionalelectron gas in the strained-layer(In,Ga)Asquantum well.
electric field at both 77 and 300 K, respectively. Although the 77 K mobility is not extremely high at low electric fields (29,000 cm2/V S), the high-field mobility is as high as or higher than typical (Al,Ga)As/GaAs structures. Furthermore, it was shown earlier that the low-field mobility does not affect the device performance of FETs. The dc characterization was made by using an HP4145 semiconductor parameter analyzer at both 300 and 77 K. The current-voltage and FET transfer characteristics are shown in Fig. 37, indicating excellent saturation and pinchoff characteristics with an output conductance of 700 p s and an on resistance of 18.4 a. As the In mole fraction increased form 15 to 20%, the peak extrinsic transconductance was increased from 270 to 310 ms/mm at 300 K. This is superior to the best reported transconductances for I pm non-self-aligned GaAs/(Al,Ga)As MODFETs. Figure 38a and b illustrates the 77 K state of the dc characteristics for the y = 0.15 device in the dark. There is no collapse observed under this condition. As the device is illuminated, no visual change in the curves was seen, and they com-
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
183
T(K)
FIG.34. Hall mobility and sheet carrier concentration as a function of temperature for the conventional Ga/As/Ab3Ga,,,As and pseudo-morphic I ~ . I s G ~ . 8 5 A ~ / A b . 1 5 GMOD~.,,As FET structure. The pseudo-morphic structure exhibits virtually no light sensitivity or PPC effects due to the lower mole fraction of (A1,Ga)As used. (0,light; A, dark; 0 persistent.)
pletely return to their original values when the source of the illumination is removed. Table I1 summarizes the In mole fraction effects on the transconductance, agreeing with expectations. An important problem with conventional GaAs/(Al,Ga)As MODFETs is the positive shift in the threshold voltage after a gate bias sufficient to
184
H. MORKOC AND H. UNLU 3.55104
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103
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FIG. 35. Electron mobility as a function of electric field for the A& In,,,,Ga,,,,As structure at both 77 and 300 K.
fully turn on the channel is applied. The bending in the conduction band, due to the large positive gate voltage, in the (A1,Ga)As is enough to allow energetic electrons to fill DX traps. The injected charge acts to decrease the 2DEG concentration and therefore shifts the transfer characteristics toward higher gate voltages. Figure 39 illustrates the transfer characteristics before and after bias stress for both a pseudomorphic 1%.IsG%,ssAs/
Electric Field (V/crn) FIG. 36. Electron velocity as a function of electric field for the Al,,.,sGa,,.ssAsl In,,,Ga,,,,As structure at both 77 and 300 K. The straight low-field lines indicate the velocity expected from a constant mobility.
2. (Al,Ga)As/GaAsAND (Al,Ga)As/InGaAsMODFETs
185
Drain Voltage Vos ( V )
a
CI
-
L
3
0
-
Gate Voltage VGS
(V)
FIG.37. (a) 300 K current-voltage characteristics and (b) transfer characteristics for 1 X 145 pm2 pseudo-morphic MODFET. Excellent pinchoff and satu,ration behavior are evident, with a peak transconductance of 270 mS/mm and a maximum current density of 290 mA/mm at V,, = 2 V.
Ab.,sG%.ssAsand a more conventional GaAs/&,,,G%,,+ MODFET with a similar threshold voltage and doping concentration. As seen clearly in Fig. 39, the pseudomorphic MODFET shows virtually no threshold shift, while the Ab.wG+.7&/GaAs MODFET shows a 0.12 V shift. This bias stability is extremely important for practical cryogenic device operation.
186
H. MORKOC AND H. UNLU 50
,
,
,
,
,
,
,
,
,
2
1
0
Drain Voltage VDS ( V 1
- 0.6
0
1.0
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FIG. 38. (a) 77 K Current-voltage characteristics and (b) transfer characteristics for a pseudo-morphic MODFET with a 3 pm source-drain spacing. The gate potential ranges from -0.3 V (bottom curve) to +0.9 V (top curve). The I - V curves remain virtually unchanged under illumination.
To demonstrate the superiority of (Al,Ga)As/(In,Ga)AsMODFETs over the more conventional (Al,Ga)As/GaAs MODFETs in large-signal logic applications, the (Al,Ga)As/(In,Ga)As MODFETs were also simulated in ring 0scillators.4~~~ Model parameters were chosen to agree with the dc current - voltage characteristics and velocity-field measurements of Ah,,5G+.ssAs/Iq,2G~.sAsMODFETs fabricated in our laboratory. The effects of the second heterobarrier of the (In,Ga)As quantum well (1 50 A wide) was neglected, and the (Al,Ga)As/GaAs expression used in Ref. 100
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
187
TABLE I1 EFFECTOF INDIUMMOLEFRACTIONON DC TRANSCONDUCTANCE OF (In,Ga)As/(M,Ga)As MODFETs
5 10 15
20
253 234 270 310
-0.24 -0.99 -0.33 -0.32
303 276 360 380
-0.13 -1.03 -0.19 -0.27
for the quasi-triangular potential solution is assumed in the model calculations. Nevertheless, a reasonable fit to the experimental device was possible. Using the (Al,Ga)As/(In,Ga)As device model for a 1 X 20 pm2 driver FET with a 10pm saturated resistor load, ring oscillators were simulated at various supply voltages. Figure 40 shows the same trend of decreasing delay time with V , , as seen for the (Al,Ga)As/GaAs MODFETs in Fig. 14.
C Gatevoltage V,
(V)
FIG. 39. Gate-bias stress measurements for conventional GaAs/A&,G%,,As (---) and (-) MODFETs at 77 K. The threshold pseudo-morphic In,,,G~~,,As/A~,,,G~~ssAs voltage shift is due to electron trapping in the (A1,Ga)As and is a measure of the quality of the material. The pseudo-morphic MODFET shows very little shift due to the reduced trapoccupation probability of low mole fraction Ab.,sGa,,ssAs.
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H. MORKOC AND H. UNLU
FIG.40. Effect of supply voltage on delay time and power dissipation at 300 K (-) and 77 K (---) for (In,Ga)As/(Al,Ga)As pseudo-morphic MODFET inverters. Delay times are 18 and 22% smaller than for conventional Ga/As/(Al,Ga)As at 300 and 77 K, respectively.
The minimum delay time and power dissipation at V,, = 0.8 V is 11.8 ps and 0.62 mW/stage at 300 K and 7.5 ps and 0.9 mW/stage at 77 K. These delay times represent a 18 and 12% improvement, respectively, over the 300 and 77 K results for conventional (Al,Ga)As/GaAs MODFETs. The power dissipation is larger, however, due to the higher oscillator frequency and current levels. The large current densities possible with the small logic swings of MODFETs and, in particular, (Al,Ga)As/(In,Ga)AsMODFETs make them perform so exceptionally well. Although these results are strictly valid only for ideal circuits (i.e., ring oscillators), they do provide an indication of how actual circuits dominated by interconnect capacitance and multiple fan-out capacitance can be optimized. In order to determine the bias dependence of equivalent-circuit parameters of these MODFETs, scattering parameter measurements were made between 2 and 18 GHz. The parameters giving the best overall fit were plotted versus gate and drain voltage. Figure 41a and b demonstrate the gm,ext, and gm,int for gate voltage dependence of tranconductances g,,,dc, V, = 0.5 and 3 V at room temperature, showing a good agreement with the expected trend. Figure 42a and b demonstrate the gate voltage dependence of C, and C, capacitances for V, = 0.5 and 3 V at 300 K and Fig. 43 shows the drain-source voltage dependence of G, and C, for VGs= 0.7 V at 300 K, both showing good agreement with the expected trend from these devices.
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
Gate Voltage
189
(V)
Gate Voltage (V) FIG. 41. The effect of gate voltage on the large-signal transconductance of (In,Ga)As/ (Al,Ga)As MODFETs for (a) 0.5 V and (b) 3 V drain-source voltages V , at 300 K.
The microwave performance of pseudomorphic MODFETs were measured from l to 26.5 GHz using a Cascade Microtech on-wafer prober and automated network analyzer. the device current gain hS, was determined from the measured S parameters. Figure 44 shows the maximum available and short-circuit current gain comparisons of I~.15G~.s5As/~.,5G~.s5 and,GaAs/Al,,mG+.7& MODFETs, which demonstrate the superiority of the former to the latter. Table I11 shows element valuesh and , f for devices with increasing In mole fractions. As seen from Table 111, the increase in In mole fraction improves the device microwave performance.
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H. MORKOC AND H. UNLU
,
0.61
I
I
I
I
- 0.4
I
I
I
I
I
0
I
0.4
0.8
Gate Voltage (V)
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FIG.42. The effect of gate voltage on the large-signal capacitancesof (In,Ga)As/(Al,Ga)As MODFETs for (a) 0.5 V and (b) 3 V at 300 K.
Henderson et al. Io1 have reported excellent dc and millimeter-wave pseudo-morphic MODFETs performance in I~.lsG~.ssAs/A~.lsG~.ssAs with 0.25 ,urn gate lengths. Extrinsic transconductances as high as 495 mS/mm at 300 K and unprecedented power performance in the 60 GHz range were observed. Although not yet optimized, excellent low-noise characteristics, 0.9 dB, with an associated gain of 10.4 dB at 18 GHz, and a
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAsMODFETs
..
"
191
0
-
L
L
-
1 x 290pm
-
-
-
0,
\
I
I
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I
I
I
FIG.43. The effect of drain voltage on the large-signalcapacitance of (In,Ga)As/(AI,Ga)As MODFETs for VGs= 0.7 V at 300 K.
noise figure of 2.4 dB, with an associated gain of 4.4 dB at 62 GHz, were obtained, This is the best noise performance ever reported for a MODFET in this frequency range. Furthermore, much better efficiency and power output levels (28% and 0.43 W/mm, respectively) than in conventional MODFETs were observed. Figure 45 illustrates the drain-voltage dependence of 62 GHz output power and power-added efficiency for a 0.25 X 50 pm2 gate pseudo-morphic device. The device is tuned for maximum output power with 3 dB gain. The superior performance of these 0.25 pm gate pseudomorphic MODFETs is attributed to the excellent carrier confinement and transport properties that the (In,Ga)As quantum well provides, in addition to a high gate-to-drain reverse breakdown voltage. These results clearly demonstrate the superiority of pseudo-morphic MODFETs in high-frequency applications. XI. Remaining Problems and Projections
Since the (Al,Ga)As/GaAs heterojunction FETs are large-current and small-voltage devices, the saturation voltage and transconductance are very sensitive to the contact resistance. In fact, the higher the transconductance the more severe the effect of the source resistance becomes. In order to fully take advantage of the device potential, it is essential that extremely low contact resistances be obtained. Not only the contact resistance but parasitic resistances such as the source and drain semiconductor access resistance must be minimized. This could be done using the gate as an ion
192
H. MORKOC AND H. UNLU
T A B L E I11
EQUIVALENT-CIRCUIT PARAMETERS FOR DEVICES WITH INCREASING INDIUM MOLEFRACTIONS OF (In,Ga)As/(Al,Ga)As MODEFTs ~~~
~
~
[Indium](mol %) 0
5
10
15
20
2.5
2.5 0.0 37.0 80.80 0.69 3.87 21.19 63.87 4.00 1.20 5.67 1.26 3.09 26.29 49.49 13.58 36.0 19.0 0.0 18
2.5 -0.4 23.0 16.72 0.61 4.38 43.94 61.13 2.51 1.16 5.79 6.66 3.53 27.97 40.28 10.45 34.5 18.5 0.0 17
2.5 0.0 45.0 71.94 0.58 4.23 32.82 62.62 2.87 1.20 1.05 6.49 3.39 22.88 35.16 10.36 37.0 21.5 0.0 19
2.5 0.0 24.0 81.01
0.0
43.0 42.15 0.54 3.26 29.28 50.29 2.51 1.04 4.82 7.80 1.95 26.57 48.41 10.05 30.5 12.0 0.013
-
0.53 3.90 35.84 62.76 2.82 1.20 7.31 6.40 3.39 22.82 34.90 10.45 40.0 24.5 0.016
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs 501
I
62 GHz
I
I
I
193
10.5
I
<
- 0.3
-
-0.2
E
g 0
n
FIG.45. Output power and power-added efficiency as a function of drain voltage for a 0.25 X 50 pm2 pseudo-morphic MODFET. The device is tuned for maximum output power with 3 dB gain.
implantation mask to increase the conductance on each side of the gate, as described in Section IX. In some prototype devices this technique has successfully been applied, but high-speed results are not yet available.'O2 Currently many laboratories, both university and industrial, are looking into the degradation mechanism occurring during the annealing step. Although preliminary, the transient annealing technique looks very promising in this regard. Modulation-doped structures also suffer from the persistent photoconductivity (PPC) efect below 100 K.Io3This is believed to be the result of donor-induced defects in (Al,Ga)As, which, once ionized, exhibit a repulsion towards capture. It has also been suggested, however, that defectrelated processes in GaAs as well can play an important role.'04As a result, increased carrier concentrations, which persist unless the sample is warmed up, are obtained. The electron mobility also increases with illumination in samples with low areal carrier density. This is tentatively attributed to the neutralizing of some defect centers in the depleted (A1,Ga)As near the heterointerface, which then do not cause as much scattering. This PPC effect has been shown to decrease when the (Al,Ga)As layer is grown at high substrate temperatures.*05 It should be pointed out that, using an Asz dimeric source, modulation-doped structures with minimal light sensitivity have been grown. For the most part the heterointerface is almost perfect in that the interface states encountered in Si MOSFETs do not occur. However, the
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H. MORKOC A N D H. UNLU
(A1,Ga)As layer contains a large concentration of traps which can give rise to threshold voltage shifts with temperature and perhaps with time. The temperature dependence of the threshold voltage in long-gate FETs has been studied in detail,'% and these studies show a positive shift as the temperature is lowered. Part of this threshold shift can be attributed to a freeze-out of electrons to relatively deep donor level in (A1,Ga)As. Since the (A1,Ga)As under the gate is thicker in normally-on devices, the voltage shift as compared to normally-off devices is much larger. The emission time from the traps is dependent exponentionally on the sum of the trap level and the barrier against emission. This barrier against emission may be obtained from transient gate capacitance measurements, as reported earlier. lo6 A highly nonexponential time response of the gate capacitance to a gate voltage pulse was found to be indicative of the time-dependent threshold voltage resulting from the change of trap occupation. In addition, from the temperature dependence of the threshold voltage, donor traps were found to be 42 meV below the conduction-band edge with a 30%A1 mole fraction. The value of the thermal activation energy for emission from the traps is estimated to be 450 meV. The density of the native traps and donor-induced traps in (A1,Ga)As can be comparable to the electron concentration, which makes the deep-level analysis by transient capacitance somewhat difficult. Alternative studies, such as the drain-current transient, can be used in FETs to deduce similar information when large trap concentrations are encountered.'07 Again, recent results obtained in our laboratory show that the threshold voltage shift when the device is cooled to 77 K is much less than 0.1 V. The drain-current response to a gate bias with varying temperature can be used to calculate the activation energies of the traps as well. With this method an activation energy of 0.47 eV was deduced for a 30%A1 mole fraction, which is in good agreement with the data obtained from transient capacitance measurements.lo6 In addition, other techniques, such as lowfrequency generation-recombination noise characteristics of MODFETs, can be used to characterize deep levels at the h e t e r o i n t e r f a ~ e . The ~~~~'~~ generation-recombination noise is a result of fluctuations in the number of electrons,'@' in this case the number of electrons at the heterointerface trapped by defects located in the forbidden band of (A1,Ga)As. Using the low-frequency noise characteristicsof FETs measured in a frequency range of 1 Hz-25 kHz and a temperature range of 100-400 K, four deep levels at 0.4,0.42, 0.54, and 0.6 eV below the conduction band were detected.'1° These energy levels are in general agreement with those deduced from deep-level transient spectroscopy performed in bulk (Al,Ga)As, which indicates the presence of deep levels. There are some discrepancies among the results of deep-level transient spectroscopy in bulk (A1,Ga)As per-
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAsMODFETs
195
formed at various laboratories as well. The details of the deep-level study is beyond the scope of this text and will not be covered here. It is obvious that the defects associated with the (A1,Ga)As must be minimized so that their influence on device performance is not noticeable. Realizing the importance of the issue, many researchers are looking into sources and causes of the traps and electronic defects in (A1,Ga)As. Modulation-doped FETs, in contrast to injection lasers, are the first devices utilizing (A1,Ga)As where charge and defect concentrations of about 10" cm-2 can give rise to unacceptable adverse effects on the device performance. There are also efforts to explore device structures that are not very sensitive to at least some of the obstacles discussed above. The questions of yield and reliability may, however, take a little longer to resolve. For yield, the processing philosophy with regard to GaAs must change. Instrumentation, care, and environment similar to that used for Si ICs must be implemented. There is also the question of epi defects either introduced by the epi process or present on substrates. Some of these are morphologicaldefects, which not only degrade the semiconductor,but also cause processing defects. The present state of the art of molecular-beam epitaxy when used with average GaAs substrates is such that only MSI circuits with some success in terms of yield may be possible. There are already encouraging results that tend to suggest that by the latter part of the decade the substrate quality, the epi morphological quality, the processing that introduces few defects, and thus functional circuits with active elements in the mid to upper thousands may be possible. Like that of MESFETs, the threshold voltage of FETs is very sensitive to the epi properties. For a normally-off device, a thickness control to about 2 monolayers (- 5 A) and doping control and AlAs mole fraction control of about 1% are needed to control the threshold voltage within about 10 mV. Controls like this have already, though occasionally, been obtained on wafers slightly less than 3 in. in diameter. The repeatability of this technology is one of the questions that is also being addressed. Perhaps the most difficultproblem is to prepare (A1Ga)As of the quality with defect and trap concentrations of 1014~ m - as ~ ,compared to the present concentration of high loL5~ m - ~ Again . more effort and time will undoubtedly result in substantial reductions in the defect concentration. Finally, it is clear that this device has many of the attributes required by high-speed devices, particularly those of the integrated circuits. Present results with moderate devices are very encouraging, and with more effort even better results are expected. In fact, MODFETs with only 1 pm gate length and 3 pm source- drain spacing have surpassed the performance of other techniques, e.g., conventional GaAs with sub-0.5 pm dimensions, as shown in Fig. 46.It should be kept in mind that the delay times shown in
196
H. MORKOC AND H. UNLU
Power Dissipation Per Gate ( p W ) FIG. 46. Gate delay versus power consumption of various technologies obtained in (Al,Ga)As/GaAs ring-oscillator ckcuits.
Fig. 46 would increase by a factor of about 3 in a real circuit with loaded gates. Nevertheless, the MODFET is capable of providing functional operations in a large system by at least a factor of 10 faster than the current state of the art. With more advanced fabrication technologies, even better performance can be expected. The motivation for obtaining high-quality 111- V materials such as those used for MODFETs on Si substrates is quite substantial. Si integrated circuits containing 111- V devices would have the capacity for higher speed for certain functions and the capability of producing optical signals. Progress and the remaining problems on this subject have been discussed by Drummond et ~ 1 . " ~ XII. Summary and Conclusions
Modulation-doped (Al,Ga)As/GaAs field-effect transistors (MODFETs), whose operation is similar but performance is superior to Si/Si02 MOSFETs, have been analyzed throughout this work. It has been shown that MODFETs are inherently superior to other FET technologies in terms of achieving higher speeds of operation, lower power dissipation, and lower noise. These advantages are due to the superior transport properties of undoped GaAs, which is used as the channel layer for the FET. The electrons transferred from the wide-gap (Al,Ga)As into the narrow-gap GaAs experience reduced interactions with the parent donors, and form a 2DEG at the heterointerface.
2. (Al,Ga)As/GaAs AND (Al,Ga)As/InGaAs MODFETs
197
The transport properties of the 2DEG are strongly dependent on the structure parameters and the interfacial properties as well. An undoped (A1,Ga)As set-back layer has been used to decrease the electron-donor interaction further. The realization of the full potential of GaAs in a FET structure has become possible with improvements in the modulation doping, which combine features of both MOSFETs and MESFETs. Although MODFETs have long been based on the (Al,Ga)As/GaAs material system for high-speed digital and high-frequency microwave applications, the more through understanding of the concept of modulation doping shows that other MODFET structures can be more attractive for high-speed digital and high-frequency applications, along with the (Al,Ga)As/GaAs system. In fact, the charge transfer and enhanced transport due to the reduction of ionized impurity scattering has been demonstrated for two other lattice-matched semiconductor combinations. In these MODFET structures the electrons are confined in a (In,Ga)As layer, which showed superiority over other I11 - V semiconductors recognized as competitors for high-speed digital and high-frequency microwave device applications. Since the first demonstration of mobility enhancement in modulationdoped heterostructures, the potential application to make high-speed devices has driven remarkable attention in this area. The rapid advances in developing the potential of these high-speed devices is due to its compatibility with the conventional GaAs FET technology, bringing a new dimension to the microelectronics industry. Despite the remarkable advances already made, both the fabrication and material preparation technologies are far from the maturity needed to obtain the expected benefits from these devices. With respect to fabrication, improvements need to be made in ohmic and Schottky contact formation. Crystal growth technology is at a stage at which one can obtain high-quality GaAs and “normal” heterointerfaces. Further work is needed in improving the quality of the alloy semiconductors, such as (Al,Ga)As, (In,Ga)As, and (Al,In)As, and in the preparation of the “inverted” heterointerfaces. To summarize, MODFETs based on strained-layer systems have demonstrated a superiority both as high-speed digital and high-frequency microwave devices. In order to use their high potential, further research is needed which may lead to new device structures and device concepts. ACKNOWLEDGMENTS This work was supported by the Air Force Office of Scientific Research. The results reported here would not have been possible without the contributionsof T. J. Drummond, R. Fischer, K. Lee, W. T. Masselink, and B. Nillson, graduates students D. Arnold, J. Klem,T.
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H. MORKOC AND H. UNLU
Henderson, and P. Pearah, A. A. Ketterson, and research associate W. Kopp. The author has benefitted greatly from discussions, collaborations, and exchange of data and ideas with M. S. Shur, L. F. Eastman, K. Heime, and with A. Y. Cho, R. Dingle, A. C. Gossard, P. M. Solomon, F. Stem, M. I. Nathan, P. J. Price, H. L. Stormer, and N. T. Linh. We also would like to thank A. A. Ketterson for providing Figs. 41,42, and 43.
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SEMICONDWTORS A N D SEMIMETALS,VOL. 24
CHAPTER 3
Two-Dimensional Electron Gas FETs: Microwave Applications Nuyen T.Linh* THOMSONCSF CENTRAL RESEARCH LABORATORY DOMAINE DE CORBEVILLE, 91401 ORSAY, FRANCE
I. Introduction During the past decade, the GaAs MESFET has dominated the world of microwave solid-state devices as a power source and a low-noise amplifier. GaAs FETs cover a large field of applications: satellite communication, radar links, CB radios, car telephones, mobile receivers, direct broadcast satellite TV (DBS-TV), cable TV converters, phased array antenna radar, receivers for radioastronomy, etc. The market demand is pushing the GaAs MESFET to its highest limit. Some examples can be presented as follows: ( 1) Spectrum congestion in telecommunication leads microwave systems to operate at higher and higher frequency; satellite communication is now reaching the 30 GHz range, while military applications are working at 94 GHz, wherq.no transistor can operate yet, the best GaAs MESFET being at 60 GHz with a gate length as short as 0.25 pm.* By reducing the gate length, one hopes to improve the cutoff frequency of the GaAs MESFET, but limitations are foreseen due to technological difficultiesand basic physical properties of the material and the devices themselves. (2) The increase of the distance covered by microwave networks necessitates higher-power sources and lower-noise amplifiers. In DBS-TV equipment, for example, the performance of the low-noise amplifier will contribute to reducing the antenna dimension and then to its cost, which is one of the major problems in DBS-TV. Therefore, the search for extremely low-noise transistors is necessary. Usually the improvement of the noise figure in GaAs MESFETs is obtained by reducing the gate length. As was stated above, however, limitations are foreseen.
* Present address: Picogiga, 5 Rue de la Rkunion, Z.A. de Courtaboeuf, 91940 Les Ulis, France. 203 Copyright 0 1987 Bell Telephone Labratorig Incorporated. AU rights of reproduction in any form reserved.
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(3) Many other examples can be cited. Let us just mention one more application for which high cutoff frequency transistors are the key element in the electronic system: the phased array antenna radar. This antenna is constituted by thousands of transistors assembled in microwave integrated circuits (MICs). One of the main problems related to this type of radar is the reduction of its weight and dimension; therefore, one has to reduce the dimension of the integrated circuit to a minimum size, i.e., to integrate all the elements of the circuits (transistors, capacitors, inductors, etc.) in the same GaAs chip as has been done for many years in digital integrated circuits. In these monolithic microwave integrated circuits (MMICs) the active element (the transistor) is often small compared to the passive elements. Then, to miniaturize the MMIC, an effort to diminish the area occupied by the passive elements (particularly the inductances) has to be made. The problem related to this dimension reduction is that the transistor cannot be perfectly adapted to the passive elements, a compromise having to be found between a small area and a good adaptation. In this
FIG. 1. Extremely high-performance FETs are needed in small-dimension monolithic microwave integrated circuits for phased-array radar antennas as well as in direct broadcasting satellite receivers. The picture shows two GaAs monolithic microwave integrated circuits having the same function but whose areas differ by a factor of 10.
3.
TWO-DIMENSIONAL ELECTRON GAS
FETS
205
condition, the transistor is not used at its optimum performance. Therefore, extremely high-performance transistors are needed. The higher the transistor performance, the smaller the circuit area. The two MMICs shown in Fig. 1 illustrate the size reduction: A factor of 10 has been obtained. The smaller circuit has an area of 0.16 mm2.2To be compatible with the phased array antenna a further reduction of area by a factor of 5 is necessary. Through the examples cited above, one can realize that the achievement of a transistor exhibiting higher performance than the GaAs MESFET will lead to applications of tremendous interest. The question remains, however, as to which way to follow in the search for a new transistor. Figure 2 briefly summarizes the transistor story. The GaAs MESFET presents over Si transistors the advantages of high mobility (5000 cm2V-' s-' versus 1000 cm2 V-' s-') and high electron velocity (1.5 X 10' cm s-' versus 10' cm s-'). GaAs MESFETs can operate up to 60 GHz, while the best Si bipolar is at 10 GHz. This comparison leads one to think that some other 111-V compound semiconductors are also good candidates for low noise and high gain amplification. (1) InP, another binary compound with high electron velocity3 (-2.5 X lo7 cm s-'), was studied, but experimental results do not show any improvement with respect to GaAs in the field of low-noise amplification." At the present state of knowledge, the InP FET is more suitable for high power amplification, where an improvement by a factor of three was ~bserved.~ (2) Ternary and quaternary alloys such as GaInAs and GaInAsP, which present high mobility (- 10.000 cm2 V-' s-') and high electron velocity, were thought to be excellent material^.^,' But due to some difficulties in the device realization, experimental data are rather poor. (3) In fact, 111-V compound semiconductors belong to a family in which heterojunctions with extremely good interface quality can be grown: Ge
1 ,Bipolar) 1 GaAs (FET) Si (MOS
InP GalnAs HETEROJUNCTIONS (FET, Bipolar) (FET, MOS) (FET , MOS) FIG.2. The transistor story started with germanium, is developingwith Si and GAS, and is entering into the world of heterojunctions.
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NUYEN T. LINH
Al,Ga,-,As on GaAs, G%.48I%,,As on InP, etc. Heterojunctions have been used in optoelectronics for many years. In the field of microwaves, the heterojunction bipolar transistor (HBT) is the first example of a heterodevice,8 but its performance has not yet reached that of the GaAs MESFET.9*10 Since the pioneering work on modulation-doped superlattices, which demonstrated a mobility enhancement with respect to the bulk crystal," interest has been focused on the utilization of the high-mobility two-dimensional electron gas (2DEG) for microwave amplifi~ation.'~,'~ The transistor made of a single modulation-doped heterostructure was labeled TEGFET (two-dimensionalelectron-gas FET)', or HEMT (high-electronmobility transistor).I3 The TEGFET has been found to be definitely the best low-noise transistor: with a gate length of 0.5 pm,a noise figure as low as 0.85 dB has been measured at 10 GHz and 1.3 dB has been measured at 17.5 G H d 4 With a similar gate length, the best GaAs MESFET is far behind with 1.3 dB at 12 GHz and 2.2 dB at 18 GHz.l5Since the TEGFET technology has not been optimized yet, further improvement is sure to be made in the near future, at which point a large field of applications will open up. In the following sections we will first describe the TEGFET structures (Section 11) and transport properties (Section 111). Device modeling and performance will be represented in Sections IV and V, respectively. Discussions are given in Section VI. 11. TEGFET Structures
The TEGFET is essentially a MESFET in which the Schottky gate can be deposited on one side of the modulation-doped heterojunction (Fig. 3). Figures 4 and 5 schematically show how the 2DEG density n, can be controlled by the gate bias. It can be noticed that the control of n, is performed through the variation of the 2DEG well position with respect to the Fermi level. This type of charge control is similar to that of a MOSFET and is quite different from the conventional MESFET, for which the Schottky gate bias induces a variation of the depletion depth. According to the diagrams shown above, the TEGFET can theoretically work with the Schottky gate on GaAs or on AlGaAs, but experimental results have shown that the latter structure is more suitable for various reasons. (1) When the ndoped AlGaAs is underneath, its thickness has to be exactly controlled to be completely depleted of electrons, otherwise parallel conduction occurs.1zThere is no way to recalibrate the thickness of this
3.
TWO-DIMENSIONAL ELECTRON GAS
G
S
,,
ETS
207
0 n- GaAs
.. , . .. . . .
n- Al Go As Buffer SI AlGaAs
m. Si Substrate
S
,
,
G
D
n- A l Ga As
., . . . _. : . . . .- . _ . _ . ,
p- Go AS
Si Substrate
FIG.3. Cross-sectional view of TEGFETs: the bottom transistor is called normal and the upper inverted; most of the TEGFETs studied have the khottky gate on AlGaAs, i.e., are normal mode.
metal AlGaAs
v,=
I
GaAs pCB
0 r n \ \ \ l S
EF
VB
FIG.4. Schematic band diagram of the normal TEGFET at equilibrium and at negative gate bias.
208
NUYEN T. LINH
2DEG
ki
m\,
CB EF
. L
FIG. 5. Schematic band diagram of the inverted TEGFET. As in Fig. 4, notice that the charge control is similar to that of a MOSFET, rather than a MESFET.
AlGaAs layer as can be done in a recessed structure when the AlGaAs layer is at the top (Fig. 6). (2) Under gate bias, hot electrons in the 2DEG channel are pushed by the gate electric field toward the heterojunction interface. Since the heterojunction barrier height is 0.3 eV (for an A1 fraction in the ternary alloy of -0.3), hot electrons with an energy above 0.3 eV can be injected into
, w o A ' \ W OO%')-rJ--.--r,.
,,
f
,.# ._
.
spacer G a A s p-
1 Pm l,
i-
FIG.6. Recessed gate TEGFET.
3.
TWO-DIMENSIONAL ELECTRON GAS
5
I/
high access resistance
FETS
209
deep depletion loyer f G 0
-1
buffer layer
7 shallow depletion layer AIGOAS
low access , resistance
I
n=6x1d7cm-3
two-dimensional electron gas
1/S
i
substrote
FIG.7. Owing to the low surface potential of AlGaAs, a normal-mode planar enhancement-mode TEGFET exhibits low parasitic source resistance.
AlGaAs, the low electron velocity material. This real-space transfer mechanism has been shown to occur in GaAs/AlGaAsquantum-well structures'6 and to a smaller extent in a TEGFET structure." This injection mechanism, which would give rise to a large output conductance, is undesirable. (3) When GaAs is grown on top of the AlGaAs n-doped layer, the modulation-doped heterojunction exhibits low mobility.l8 The origin of this poor mobility is not clearly understood. Recently, by using a superlattice structure in the spacer layer, some mobility enhancement was observed.l9 (4) We have found that the surface potential of AlGaAs is low (-0.3 eV) compared to its Schottky barrier height (- 1 eV).ZOThen the depletion layer under the free surface is thin compared to the depletion layer under the gate. This allows one to obtain a low parasitic resistance even in a planar or quasi-planar structure; that is not the case for the conventional GaAs MESFET, particularly in the enhancement-mode (or normally-off) configuration, where the active layer is thin (Fig. 7). In fact, for microwave FETs the source resistance reduction is usually obtained by recessing the gate. Because of the reasons cited above, experimental results reported so far principally concern TEGFETs having their Schottky gate on AlGaAs. In the following sections, only this TEGFET configuration, which is called the normal structure as opposed to the inverted one, will be dealt with.
210
NUYEN T. LINH
111. Transport Properties in TEGFETs 1. HIGHMOBILITY OR HIGHVELOCITY?
It is now well known that modulation-doped heterojunctions present extremely high mobilities: 9000, 140,000, and 1.5 X lo6 cmz V-’ s-’ at 300, 77, and 4 K, respectively.21*2z These high mobilities, which are in good agreement with theoretical c a l ~ u l a t i o n s ,confirm 2 ~ ~ ~ ~ the important concept of electron-impurity spatial separation and reduced Coulombic scattering, and clearly show the extremely high quality of the epitaxial layers. Figure 8 compares the mobilities of a 2DEG heterostructure and an active GaAs MESFET layer: the mobility enhancement reaches a factor of 2 at 300 K and 15 to 20 at 77 K. Therefore, can we imagine that for these reasons the TEGFET would be twice or 20 times “better” than a conventional GaAs MESFET at 300 and 77 K, respectively? First, ultrahigh mobilities are obtained with a spacer thickness of 150 to 200 A. One can see in Fig. 9, however, that an increase of the spacer thickness leads to a decrease of the sheet carrier concentration n, in the 2DEG. Then the contribution of the 2DEG to the source resistance de10
\twd.dimensional eleitron gas
9 8
I 4
0
c
-x
6
5
5 t
I
\
t
c
t
1-
GaAs FET active layer
I
I ~
0
~~
100
200
300
TEMPERATURE I K 1
FIG.8. The electron mobility in TEGFET structures is extremely high with respect to conventional GaAS MESFETs.
3.
TWO-DIMENSIONAL ELECTRON GAS
ETS
211
0 0.5 1 1.5 AlGaAs FREE CARRIER CONCENTRATION I ~ l O ' ~ c r n - ' I FIG. 9. Increasing the spacer thickness leads to a large enhancement of mobility but reduces the sheet electron concentration in the 2DEG. What value is the best compromise?
creases. Moreover, with a large spacer layer, electrons cannot be easily injected from AlGaAs to GaAs. Therefore, the contribution of AlGaAs to the source resistance is weak. From this point of view, strictly speaking, the suppression of the spacer layer would be the most favorable situation. But this configuration presents a poor electron - impurity spatial separation. Thus a compromise has to be found between high mobility (thick spacer layer) and low source resistance (no spacer). Second, in a short-gate-length FET (51 pm), the intrinsic performance of the transistor is not directly related to mobility but to electron velocity. For example, the'intrinsic cutoff frequency of a transistor is proportional to the electron velocity: fT
- g,K,
- l/z - v
where g,, is the intrinsic transconductance, C, the gate to source resistance, z the transit time under the gate, and v the mean value of electron velocity under the gate.
21 2
NUYEN T. LINH
How high is the electron velocity in a TEGFET? How is it compared to the GaAs MESFET situation? The answers to these questions are difficult because, in contrast to mobility, electron velocity cannot be easily determined. Studies on electron velocity in the 2DEG can be summarized as follows. (1) Drummond et aLZ5have deduced the electron velocity from the current - voltage characteristics of a Hall bridge sample. They have found in the modulation-doped single heterojunction values of 1.7 X lo7 and 2.24 X lo7 cm/s at 300 and 77 K, respectively, at an electric field of 2 kV/cm. These data represent the lower limit, since sample heating may be affecting the measurements. Therefore, it seems that electron velocity in the 2DEG is close to that of an undoped GaAs crystal. It is higher than that of an n-doped crystal by a factor of 30%at 300 K and 200% at 77 K. (2) In the above experiment the velocity is determined in its steadystate regime. It is well known that in an FET structure where the gate is short, nonstationary effects (overshoot) have to be taken into account. The overshoot phenomenon in a TEGFET has been studied by Cappy et u Z . ~ ~ and Mudares and F o ~ l d s It . ~ was ~ found that the overshoot effect is stronger in the TEGFET structure than in the MESFET, because of the higher mobility. Figure 10 compares the electron velocity distribution under the gate between a TEGFET and a GaAs MESFET. It can be noticed that at 300 K the peak velocity is 40% higher in the TEGFET but the mean value is twice as high. The improvement of the mean velocity reaches a factor of 4 at 77 K (see Section 3). (3) Aside from this theoretical estimation of the electron velocity in a submicrometer gate length TEGFET, experimental data are poor. As will be shown in the next section, the electron velocity in a TEGFET can be deduced from the measurement of the intrinsic transconductance of the transistor [Eq. (24)]. Values of electron velocity of 1.1 to 2 X lo7 cm s-' have been determined at 300 K by this m e t h ~ d ,but ~ ~this . ~ procedure ~ just gives a rough estimate of the velocity since the analytical model of the TEGFET is itself also approximate. Therefore, it is difficult to deduce from the transconductance measurement the exact value of the electron velocity. Nevertheless, it is perfectly clear that the enhancement of electron velocity reaches a factor of more than 2 at liquid nitrogen temperature, which is much higher than in the conventional GaAs MESFET.29Because of such observations, it is often concluded that the TEGFET is interesting only at low temperatures. We will see in Section V that the room-temperature performance of the TEGFET is also tremendously high. Discussion of this point will be given in Section VI.
3.
TWO-DIMENSIONAL ELECTRON GAS
t
--_
ETS
213
G a A s FET
GATE FIG.10. Reduced electron interaction with the lattice enhances electron-velocityovershoot in the TEGFET. According to this nonstationary electron dynamic effect, the mean electron velocity in a TEGFET is twice that in a MESFET even at 300 K (after Cappy ef U L ~ . ~ ~ ) .
2. SCREENING EFFECT Reduced Coulombic scattering in modulation-doped structures and the overshoot effect in FETs are two phenomena that were treated frequently in the literature. On the other hand, the screening effect by free electrons is often ignored in FETs. Recently Wallisw demonstrated that the screening of the effect of scattering due to ionized impurities and optical phonons by free electrons plays an important role in determiningthe electron mobility of I11- V semiconductors. In particular, for GaAs MESFETs the mobility of the electrons in the channel decreases as the pinch-off regime is a p proached (Fig. 1 l), because then there are few free electrons and therefore the screening by free electrons is reduced. Figure 12 shows that the mobility in an FET decreases as the sheet carrier concentration in the channel decreases, in good agreement with theoretical calculations based on screening effects.
214
NUYEN T. LINH
h
0
.
6
.5000 4ooo>T m
u
W
0
z
V $
0.4 4
3000
6
> k 2000 =' m
-
3
0.2
1000
0
0
I
V
-1
-2
-3 -4
-5
GATE BIAS (volts) FIG.1 1. Electron mobility in a GaAs MESFET decreases as the gate voltage is approaching pinch-off (after WallisM).
How important is this phenomenon in microwave transistors? It is known that the low-noise transistor operates at a gate bias (VJ condition where the drain current is approximately 5 to 10 mA, i.e., at of the total current, Iass( Vg= 0). One can see in Fig. 12 that in this condition ( n , 0.5 X lo'* cm-*) the electron mobility in the MESFET channel is -2700 cm2 V-' s-'. The same type of measurement has been achieved in TEGFETs. As an example, the decrease of electron mobility versus gate voltage of the low-noise TEGFET reported in Ref. 30 is shown in Fig. 13. The gate bias that gives the minimum noise figure is 1.2 V in that case.31As shown in Fig. 13, the electron mobility in the TEGFET channel is then -4000 cmz V-' s-', i.e., much higher than in the MESFET. Since the TEGFET reported above has poor mobility (38,000 cm2V-' s-' at 77 K), it is
+
FIG. 12. The decrease of electron mobility near pinch-off is attributed to the reduction of screening of ionized impurity and optical phonon scattering by free electrons. Good agreement was found between experimental data (0)and calculated results (A) (after WallisM).
.....
I - 5000 7 ; -4000
w
a 3
g
40 30
"E
.
-3000 2
*..
12000
2ol
1000
10
0
-0.4 -0.8 -1.2
-5 >
58
I
-1.6
GATE BIAS ( V ) FIG. 13. Screening is also responsible for the decrease of electron mobility near pinch-off in TEGFETs, but at the minimum noise figure operation condition the mobility in the 2DEG is higher than in the GaAs MESFET (after Wallis-'O).
3001
I
3
8
r
I
8
1
I
I
*
9
r
200 -
h
c
C
E0
z
150-
2si w
a
z
2 100? W W
a
3 0
cn
0
-0.5
0 0.5 GATE VOLTAGE ( V I
1.o
FIG.14. The sheet free-electron concentration dependence of the mobilityin TEGFET was first observed by Delagebeaudeuf et who found the relationshipp =p0(nJk,k sz 0.5- 1.5.
216
NUYEN T. LINH
thought that with better material quality mobilities as high as 6000 cm2 V-I s-' can be obtained at the minimum noise figure bias condition. By studying the gate voltage dependence of the source-to-drain resistance of a TEGFET, Delagebeaudeuf et aZ.28have found that the electron mobility under the gate varies with the sheet free-electron concentration according to the formula (Fig. 14) ,u = ,uo(rQk,
k - 0.5 - 1.5
The role of screening effects has been pointed out by Linh32to explain this result and other mobility behavior in modulation-doped structures, such as the decrease of mobility observed in certain modulation-doped layers as the spacer thickness increases.33
IV. Device Modeling Device modeling was first achieved by Delagebeaudeuf and Linh28,M*35 on normal and inverted structures. These first works led to an analytical model of the TEGFET, which is approximate but useful because it can be used in a simple way to correlate device characteristics with material parameters (doping concentration, thicknesses, etc.) More recently, VinteIj6 performed an accurate modeling of charge distribution in TEGFETs at low drain voltage, while Cappy et ~ 1proposed . ~ a~ TEGFET model which takes into account nonstationary electron-dynamic effects. 3. TEGFET ANALYTICAL MODEL
a. Sheet Density in 2DEG Figure 15 represents the schematic band diagram of the heterojunction and gives the notation used. We start from the assumption of a quasi-constant electric field F,, in the potential well, i.e., we assume a triangular potential well. The solution for the longitudinal quantized energy is then well approximated by the formula
where rnf is the longitudinal effective mass,z ! the Planck constant, and q the electron charge. For GaAs, and considering only the existence of two subbands, we obtain Eo(eV)- 1.83 X 10-6F:(,3 E,(eV)
- 3.23 X 10-6F:(,3
(2)
3.
TWO-DIMENSIONAL ELECTRON GAS E T S
217
L FIG. 15. Schematic TEGFET band diagram at equilibrium.
In GaAs, the electric field Fl obeys the Poisson equation
where el is the dielectric constant of GaAs, n(x) the free camer concentration, and N the ionized acceptor concentration (the low doped GaAs is p-type) with n(x) >> N. By solving the Poisson equation with the appropriate boundary condition (Fl = Flo)at the heterojunction interface, Fl = 0 far from the interface, we find
where n, is the sheet carrier concentration of the 2DEG. Equation (2) becomes
El - y1n,2/3 (5) with yo = 2.26 X 10l2and y, = 4 X 1OI2 in SI units. By considering the relationship between n, and the Fermi level position, it can be easily found that Eo -yon:’3,
218
NUYEN T. LINH
where D is the density of states in the 2DEG:
D = qm: /z@
(7)
At the heterojunction interface, we find the same value of the electric displacement vector (neglecting interface states): E,FlO = E2F20
- 9%
(8)
c2FzOcan be determined by assuming the total depletion approximation in
the space charge layer: E
+
~ =F (2qN2v20 ~ ~ q2N$a2)'f2 - qN2a
(9)
where N2 is the electron concentration in the AlGaAs layer, a is the thickness of the undoped AlGaAs spacer layer, and v2o=AEc-(r2- Em (10) Combining Eqs. (6), (S), (9), and (lo), n, can be determined, and the solution is obtained numerically. The variation of n, versus the electron concentration N2 in AlGaAs is represented in Fig. 9.
b. TEGFET Characteristics Figure 16 shows the band diagram of the heterostructure submitted to the influence of a Schottky gate. We suppose that for a certain gate bias VBs there is interpenetration between the Schottky depletion layer and the heterojunction space-chargelayer. Under these conditions we have E2F20
vp2
= ( & 2 / d 2 ) ( vp2- v2) = (4N2/2E2)@2
(1 1 )
- a)2
(12)
Vg-EF-AEc (13) where d2 is the total thickness of the AlGaAs layers (see Fig. 16). Thus we deduce a gate voltage dependence of the sheet concentration Q.: ~ 2 = & -
- +M - EF-I-AEc i- vB> If E Fis neglected, the following can be written: Qs = ( E 2 /d2)( vp2
voff = +M
- AEC - v p 2
(15)
ld2)( Vg - v.,~) If EFis not neglected, E , can be written as
Q, = (
EF(ns, T ) -1.21
where T is the temperature.
(16)
~ 2
x 10-17n3, + 3.3 x
(14)
i - T/280)
(17)
3.
TWO-DIMENSIONAL ELECTRON GAS E T S
219
CB
VE
I
,, I
I
FIG.16. Schematic TEGFET band diagram at negative bias on the Schottky gate, showing curves at equilibrium (- - -) and with V bias (-).
Then one obtains
Q,= (82 /4(v g - GF) with
+ +
d$= dz 80 A e; = e2 80 A A EL = A E, - 3.3 X lo-’( 1 - T/280) Similar corrections have been given by Drummond et aL38 Having obtained Q,, the drain current can be calculated as in the conventional MESFET as follows. ( 1 ) Assuming a distribution of voltage V(x) in the channel, which induces a distribution of charge
Qd4 -(&z /&A[
v* - V x ) - KffI
(19)
220
NUYEN T. LINH
(2) Calculating the current in the classical manner: 1, = QMzW
(20)
where Z is the gate width and v(x)the electron velocity at the abscissa x. It can be assumed that, above a critical field F,, the velocity is saturated at the value us at the drain side. Then we deduce the saturation current I, for a short-length transistor:
where L is the gate length. Equation (2 1) can be written as = g m ( Vg - Viff - FcL)
(22)
where g, is the transconductance:
gmois the intrinsic transconductance of the transistor. Equations (21) and
(22) are valid in gate voltage regions which are not too close to the pinch-off voltage. They clearly show a linear dependence between Ids and V,. In other words, the transconductance and the capacitance remain constant over a large variation of gate voltage. Recently, Lee et ~ 1have . given ~ ~ a more complete calculation, which takes into account the conduction in the AlGaAs top layer in the case where the gate voltage is sufficiently small. The analytical calculations contain many approximations, particularly in the description of the potential well and the subband structure. We will see in Section 5 a more accurate description of the subband structure and in Section 4 a microscopic model including velocity-overshoot effects. The main different result in the device dc characteristics concerns the charge distribution versus gate voltage; in particular, the gate capacitance, hence the transconductance, is not constant. Therefore Eq. (24) cannot be taken as an accurate determination of the electron velocity. Nevertheless, the analytical model described above is a useful tool to correlate TEGFET dc characteristics to material parameters. (1) The I- V characteristics of TEGFETs can be a priori estimated. Figure 17 shows good agreement between experimental results3' and calculated data.
3.
TWO-DIMENSIONAL ELECTRON GAS
i
I
ETS
l
221
-
a E
v
-
ul ul 0
t w a a
3 V
za a
D
GATE VOLTAGE ( V )
FIG.17. Comparison between experimental (-) curves. Good agreement is noted.
data and calculated (-
- -) Z& vs. Vg
(2) The dependence of the pinch-off on various parameters such as doping concentration and thickness of the n-doped AlGaAs layer is shown in Fig. 18. The chart shown in that figure is particularly useful for growing layers with a controlled pinch of voltage. (3) Another helpful representation of the correlation between material parameters and electrical characteristics of the transistors is shown in Fig. 19. Through a Hall measurement, the values of sheet resistance and Hall mobility give a good estimation of the pinch-off voltage. 4. NONSTATIONARY AND MICROSCOPIC MODELOF TEGFET Nonstationary electron dynamic effects in submicrometer gate length While direct experimental evidence is rare,44electron velocity overshoot effects due to nonsta-
MESFETs have been studied for many
222
NUYEN T. LINH
>
v
w
w
17
N, .3.10
err?
(r
r
I-
-1 -1.5
I
1
0
200 400
I
I
I
I
I
600 800 1000 1200 1400
(8)
AlGaAs THICKNESS FIG.18. Dependence of the pinch-off voltage V, on the AlGaAs layer thickness for various doping concentrations.
;_I
it;! 2 in
W U
6
1000
In
I
1 0
I
I
I
I
I
2000
4000
6000
8000
I
10000
J 12000
HALL MOBILITY ( ~ r n ~ V - ~ < ~ ) FIG. 19. Through a Hall measurement, the pinch-off voltage can be predicted if the doping concentration is known. The various iso-V, curves correspond to -0.15, -0.48, -0.71, -0.99, - 1.27, - 1.56, - 1.84, -2.12, -2.40, -2.68, -3.00, -3.24, and -3.52, respectively.
3.
TWO-DIMENSIONAL ELECTRON GAS
ETS
223
tionary phenomena are often taken into account to explain MESFET dc or microwave characteristics. The nonstationary regime in TEGFETs was studied in some detail by Cappy et al.26,37 The most exact treatment would certainly be a two-dimensional Monte Carlo calculation. But in the TEGFET case, the description of the heavily doped AlGaAs layer and the small potential well in which the camers are accumulated requires one to simulate a great number of particles and to use a very small mesh size. Therefore, a very long computation time would be needed. To simplify the problem Cappy et al. proposed a one-dimensional calculation which includes nonstationary effects. (1) The first step is to determine the dependence, in the stationary regime, of the electron velocity and energy on the parallel and transverse field under the gate. This is done by the simultaneous resolution of the Boltzmann equation by the Monte Carlo method and the Poisson equation, the boundary conditions being expressed in Fig. 20. Real-space transfer of electrons from the 2DEG to the AlGaAs layer is considered when the electron energy related to the transverse component of the velocity is higher than AE,. The parallel component and the total energy are then considered to be conservative. Transferred electrons remain in the corresponding valley on both sides of the heterojunction. If the electron energy is not high enough (
IY
undoped GaAs
IY’
€1 = E Schottky
n -AIGoAs FIG.20. Cross-sectional view of TEGFET showing transverse electric field Ek
224
NUYEN T. LINH
n
FIG.2 I. Electron concentration distribution under the gate of a TEGFET. Notice that at high longitudinal electric field real-space transfer of electrons occurs from GaAs to AlGaAs (after Cappy et af.").
The distribution of electrons in the potential well and its neighborhood is then determined and represented in Fig. 2 1. Notice that at high longitudinal electric field Ell(high drain voltage) more electrons are transferred into the AlGaAs layer and toward the substrate. These effects can be easily interpreted by an increase of electron energy. The mean electron velocity in the stationary regime in the structure (including electrons in the 2DEG and those in AlGaAs) is shown in Fig. 22. It can be noted that the mean electron peak velocity is equal to that of bulk GaAs. This result is not surprising since no particular assumption is made on the two-dimensional character of the electron gas.
v (cmls)
FIG.22. High transverse electric field prevents real-space transfer from occurring, then the mean electron velocity in the TEGFET structure is equal to that of undoped GaAs material TEGFET at EL = 250 kV/cm; (- - -) intrinsic GaAs. (after Cappy et ~21.~').(-)
3.
TWO-DIMENSIONAL ELECTRON GAS E T S
225
GATE
FIG. 23. Distribution of longitudinal electric field q,,electron velocity u,, and energy E under the gate (L, = 0.8 bm, V, = 2 V, ,u = 4000 cm2V-I s-’ ) (after Cappy et ~ 1 . ~ ’ ) .
(2) The second step is to determine the dc characteristicsof the transistor by taking into account electron dynamic effects. As has been done for MESFETS,~~ the TEGFET model is based on the resolution of the following equations: Jas = z(Q1 UI
+Qz~z)
(25)
GATE LENGTH (pm) FIG. 24. The intrinsic maximum available gain and cutoff frequency of the TEGFET (-) is higher than that of the MESFET (- - -) (after Cappy et ~ 1 . ~ ’ ) .
NUYEN T. LINH
226
N
5-
I
250 0
1
Y
h
G
v) \
-200 fj 3
l-
s!
v
-150
z:
lL
LL
100
3
V
z
’
e
I-
2
w U
012
Ol4
Ol6
o
dl8
<
0,
GATE LENGTH (pm) FIG.25. The high electron velocity of TEGFETs (-) leads to a high gm,,/2nC, cutoff frequency (after Cappy er a/.”). The MESFET curve is also shown (- - -).
aE E - E0 - = qEll- -
ax
VIT,
dv,
m*v,
dx
T~
0 = qE,,- m*v , -- -
where I , is the drain current; Q, and Qzthe sheet concentration in GaAs and AIGaAs, respectively; v1 and v2 the longitudinal mean electron velocity in GaAs and AlGaAs respectively; E the mean electron energy in GaAs; m* the effective mass of the electron; and ,z, z, the momentum and energy relaxation time. Figure 23 gives the distribution of electric field Ell, the mean electron velocity vl, and the E under the gate. A comparison of electron velocity between TEGFETs and MESFETs is given in Fig. 10; due to overshoot effects the mean electron velocity under the gate is twice as large in a TEGFET as in a MESFET (at 300 K). Therefore, Cappy et al. concluded that the intrinsic maximum available gain (MAG) and the intrinsic cutoff frequency of a TEGFET are much higher (Fig. 24), as is also the cutoff frequency, gm0/27cC, (Fig. 25). 5. ACCURATE SUBBANDS AND CHARGE CONTROL IN TEGFET AT LOWELECTRIC FIELD DETERMINATION
Recently Vinte$6 has given an accurate determination of the subband structure and charge distribution as a function of gate voltage at room
3.
TWO-DIMENSIONAL ELECTRON GAS E T S
227
temperature, by using a self-consistent calculation in which the electronic subband structure in both GaAs and AlGaAs is taken into account (Fig. 26) as well as the partial neutralization of donors in AlGaAs. The main results are as follows: (1) The lowest subband contains approximately 60% of the 2DEG electrons, and the second lowest subband 20%. (2) The gate capacitance is not a constant function of the gate voltage. Figure 27 shows the gate voltage dependence of the capacitance. We can consider in fact that the capacitance remains roughly constant in a small range of gate voltages (0.3-0.4), whereas the complete depletion of the 2DEG needs - 1 V. That figure also shows that there is a range of gate voltages where the gate bias has an effect upon both the 2DEG and the free electrons in AlGaAs. This effect is not described by the analytical model, and is at the origin of the fact that the gate capacitance is not constant.
Figure 28 shows the sheet carrier concentration dependence of the Fermi
FIG.26. Band structure in TEGFET according to V i ~ ~ t eSubbands r~~. are also created in AlGaAs because of band bending due to the Schottky gate. (a) Depletion-mode device; (b) enhancement-modedevice. The Fermi level is EF= 0. The energies of the donor level are shown broken for the higher voltage.
228
NUYEN T. LINH
-2
0
-1
1
VG(V)
.
FIG. 27. Gate voltage dependence of gate capacitance: The gate capacitance is not strictly constant as assumed in the analytical model. Nevertheless, the capacitance remains roughly constant in a small range of voltage. (a) All electrons in the system; (b) all free electrons; (c) all electrons in the GaAs channel. (0, 1, 2) correspond to the electrons in the three lowest subbands (after Vh~ter-’~).
100
t
88
//
POTENTIAL WELL APPROXIMATION ( 4 6 )
-100 0 1 2 3 4
5 6 7 8 9
3
N, (10’~
FIG.28. Sheet carrier concentration dependence of the Fermi level. The triangular potential well approximation gives a good fit at low values of n, (after Urien and Delagebeaudeufa).
3.
TWO-DIMENSIONAL ELECTRON GAS
FETS
229
energy EF for the two cases of a triangular potential well and more accurate situation described by Vinter. Urien and Delagebeaudeuf&have given a numerical formula which gives a good fit with the result obtained by Vinter: 8
EF=
a,(N,)' i-0
where a, = - 112.7366820, a, = 137.8963593, a, = - 102.3133367, a3 = 52.99028938, a4 = - 17.32353641, a, = 3.545414254, a6 = -0.439517149, a, = 0.030101690, and as = -8.72465 X with EFin mV and 0.25 X lOI5I N , i 7 X lo', m-,. V. TEGFET Microwave Performance
6. NOISEFIGURE AND ASSOCIATED GAIN Microwave measurements on TEGFETS were first reported by Deleswho showed a maximum available gain of 8 dB at 1 1 GHz. cluse et Low-noise performance was then published by Laviron et With a gate length of 0.8 pm, a noise figure (NF) of 2.3 dB was measured at 10 GHz, with an associated gain (G-) of 10.3 dB. These figures were humble with respect to the state of the art of GaAs MESFETs at that moment (mid- 198l), but by considering that the TEGFET technology was just emerging and the gate length was large, the reported performance was encouraging and indicated that the TEGFET is not worthy only at low temperature as many people believed. Better results were obtained by Niori et al.48They observed at 8, 11.3, and 20 GHz noise figures of 1.3, 1.7, and 3.1 dB and G, of 13, 1 1.2, and 7.5 dB, respectively. Further improvement was given by Joshin et and Linh? a noise figure of 1.4 dB was reported at 12 GHz by the former authors, whereas the latter obtained 1.26 dB at 10 GHz and 2.3 dB at 17.5 GHz. Both teams showed very interesting performance at low temperature with noise figures of -0.25-0.35 dB (-20 K), the physical temperature being - 100 K. The above results show that progress was quickly made, but the state of the art of GaAs MESFETs (in 1982-1983) was not yet reached. As a matter of fact, 0.6 pm gate length MESFETs presented at 12 GHz a NF = 1.3- 1.47 dB, G, = 9.9- 10.3 dB,5'-52 while the 0.25 pm gate length MESFET showed NF = 0.95 dB, G, = 11.5 dB at 12 GHz and NF = 1.55 dB, G, = 12.3 dB at 18 GHz.,~ Recent results obtained at T h ~ m s o n - C S F and ~ ~NEC55on 0.5 pm TEGFETs have placed that device on the highest MESFET level: NF = 1.2 dB, ~
1
.
~
~
3
~
'
:
230
N U Y E N T. LINH
C
l
- 167 - 15 -14
- 13 -12
T
U
v
z
9
- 11 Gass
- 10
-9
20
DRAIN CURRENT (mA) FIG.29. Drain current dependence of the noise figure, associated gain, and maximum gain in an ultra-low-noise TEGFET at 10 GHz.
at 12 G H z , ~and ~ NF = 1.07 dB, G, = 10.6 dB at = 7.5 dB at 17.5 G H z . ~ ~ It is only very recently that TEGFET performance has outdistanced the = 1 1.2 dB at 10 GHz and MESFET performance: NF = 0.85 dB, G, = 9.5 dB at 17.5 GHz for 0.5 pm gate length device14 NF = 1.3 dB, G, (Fig. 29). Figure 30 summarizes the progress made by 0.5 pm TEGFETs, and Fig. 3 1 compares the performance of our best 0.5 pm TEGFET to that of the best sub-half-micrometer GaAs MESFET: it is remarkable that the noise figure of a 0.5 pm TEGFET is lower than that of a 0.25 p m MESFET, particularly at high frequency. G,
= 11 dB
10 GHz, and N F = 1.9 dB, G,,
7. S PARAMETERS
S parameters were first given by Niori et aL4*(Fig. 32), who deduced an equivalent circuit similar to that of a conventional GaAs MESFET (Fig. 33). We have also used such an equivalent circuit to deduce the various
3.
TWO-DIMENSIONAL ELECTRON GAS
{:
0
TEGFET 0.5pm
FETS
231
THOMSON-CSF NFEUT :SU
h
m
-0
v
w
2-
CK
3
c3 1.5-
w
cn 0 z
’0.5 ’81
‘82
‘83
‘0L
YEAR FIG.30. The progress made on low-noise TEGFETs since its beginnings is impressive. It seems that there is still room for improvement.
parasitic elements of the tran~istor.’~ In fact, the establishment of an equivalent circuit is a difficult exercise even for the conventional MESFET, and then results have to be considered with care. Comments on the TEGFET circuit elements will be given in the next section. Chao et dS7 have realized a 0.25 pm TEGFET and deduced a cutoff frequency of 45 GHz (Fig. 34). The cutoff frequency (gm0/2nC,! of our 0.5 pm TEGFET was measured to be 30 GHz, while the intrinsic cutoff frequency is 80 GHz. It is remarkable that for an extrinsic cutoff frequency equivalent to that observed on MESFETs (see Ref l), the intrinsic cutoff frequency is much higher. 8. DISCUSSION ON THE NOISEFIGURE IN TEGFETs
How can we explain the excellent results obtained for The TEGFET noise figure? What is the physical parameter which is responsible for the high performance of the TEGFET? Is it the enhancement of mobility and velocity as was suggested in Section III? Is it an effect of the two-dimensionality of the electron gas, or is it other factors we have ignored up to now?
232
NUYEN T. LINH
i
v NEC
0.3pm
o HUGHES 0.3pm MESFET A TOSHIBA 0.25pm 0 AVANTEK 0.25pm TEGFET 0 THOMSON-CSF 0.5pm
0.25pm MESFET
/
0.51
8
1
I
I
I
I
I
10
12
1.6
16
18
20
I
22
24
FREQUENCY (GHz) FIG. 31. The comparison of 0.5 pm TEGFET performance to the state of the art of sub-half-micrometerMESFETs shows that the 0.5 pm TEGFET presents lower noise than the best 0.25 pm MESFET.
To answer these questions, let us consider the problem through two aspects: theoretical and semiempirical approaches. a. TheoreticalApproach
The theoretical treatment of the problem of noise in a TEGFET was achieved by the CHS Group of the University of Lille,58which undertook the MESFXT noise study some years Let us recall briefly the main results obtained on MESFETs. Carnez et al.59took into account the nonstationary electron dynamics to explain the dc characteristicsof FETs as well as their microwave properties. Noise was considered to be mainly due to the fluctuations of carrier velocity. Their contributions can be represented by a great number of noise
3.
TWO-DIMENSIONAL ELECTRON GAS
ETS
233
goo
FIG.32. TEGFET S parameters were first given by Niori et a/."
current sources distributed along the source -to -drain axis. These sources are supposed to be uncorrelated. The mean-square value of the noise current related to each section x is given by the relation = q2(A$)njyjZ/Ax
where (Au;) is the average quadratic fluctuation of the drifl velocity, nj the carrier density, and yj the channel thickness. It is assumed that the noise spectral density of the current source in the j t h section is related to the
234
NUYEN T. LINH
RG CDG R DG 7.5R 0.01pF 1 R
4.3R
S FIG.33. The equivalentcircuit of a TEGFET is similar to that of a MESFET (after Niori el al.,48see also Linh56).
longitudinal diffusion coefficient Dll:
3 = 4q2D,lnjyjZ/Ax
(31)
D,,is deduced from Monte Carlo calculations. From the knowledge of noise current sources in the channel, one can deduce the noise sources in the gate and at the drain, these two being correlated. The noise figure of the FET is then evaluated and the minimum noise figure calculated. 2 1. 5
10 9
8 7
; 4
3 2.5
2
1.5 1
5 6 78910 1.5 2 2 . 5 3 4 5 6 7 8 9 FREQ (GHz) FIG. 34. With a 0.25 pm TEGFET, Chao et aL5’ have obtained an extrinsic cutoff frequency of =45 GHz. 1
1.5
2 2.53
4
3.
TWO-DIMENSIONAL ELECTRON GAS
FETS
235
The main result to be obtained from the work of Carnez et ul. is that velocity overshoot can account for the noise figure experimentally observed in MESFETs. As a matter of fact, when neglecting the overshoot effect, the calculated noise figure is too high compared to experimental data. Such an approach of calculating noise figures in FETs was taken by Cappy et ul. to predict TEGFET performan~e.~~ The nonstationary electron dynamics as recalled in Section 1 were used. The first interesting result concerns the calculation of the diffusion coefficients D,,and D , . As shown in Fig. 35, the coefficient D,, in the 2DEG is approximately equal to the bulk case. D , in the 2DEG is much smaller than in bulk GaAs, but the latter diffusion coefficient intervenes in the noise figure to the second order. It was then concluded that the reduced dimensionality does not seem to contribute to lower the noise figure in a TEGFET. The reduction of noise figure seems to be totally due to the high electron mobility, which
300
I /\
-
f
I 1
m
;I
,,
\
N
E V
I
1
AT
-TEGFET
\
\
El = 280kVlcm
1
\1
\\Ol
___
MESFET
U
5
200
w
E LL
W
8 g 100 v)
3 LL
LL
0
0
I
I
I
10
20
30
3
ELECTRIC FIELD Ell (kV/cm) FIG.35. Monte Carlo calculations show that the longitudinal diffusion coefficient D, in a TEGFET is approximately equal to D, in MESFETs, but the transvem diffusion coefficient DI is lower in TEGFETs. Since the noise figure is directly correlated with D,l,and vanes in second order with Dl, then the two-dimensional character of the 2DEG does not seem to contribute to reduce the TEGFET noise figure (after Cappy et
236
NUYEN T. LINH
FREQUENCY (GHz) FIG. 36. Noise figure deduced from the nonstationary electron dynamic effect model is dependent on mobility, since overshoot is also dependent on that parameter. The associated gain indicated on the figure corresponds to R, R, = 6 R,p = 8000 cm2V-I s-l (after Cappy et ~ 1 . ~ ~ ) .
+
enhances the velocity overshoot (Fig. 36). The mobilities used in Fig. 36 are 4000 and 8000 cm2 V-' s-I , respectively. As reported in Section 2, a value between 4000 and 6000 cm2 V-I s-I seems reasonable since, at the minimum noise figure bias condition, the mobility is lower than at zero gate bias. By taking these figures, the calculated noise figure results seem to be in agreement with experimental data at 10 GHz. In conclusion, theoretical calculations predict that TEGFETs exhibit lower noise than MESFETs because of the enhancement of velocity overshoot related to the high electron mobility of this transistor.
b. Semiempirical Approach It is well known in MESFET technology that the noise figure is related to the various parameters of the equivalent circuit by the semiempirical Fukui formulam Fmin =
1 + 2xC,JK,E(&
+ RJ/gmo11'2
(32)
where F- is the minimum noise figure; C, the gate-to-source capacitance; R,,R , the source and gate resistance, respectively;,g the intrinsic transconductance; f the frequency; and KF the fitting factor. Numerous
3. TWO-DIMENSIONALELECTRON GAS FETS
237
experimental results show that the fitting factor KF is -2.5. This factor is often called the “material quality” factor, since it varies with the material quality. It is high when the transconductance at near pinch-off is low, in other terms when the camer concentration profile and/or mobility profile is not abrupt at the interface. Since the Fukui formula just takes into account the equivalent circuit elements, one can try to use it in the case of the TEGFET. In this way Niori et have found for the TEGFET K F= 1.6 and Linh, KF = 1.5.56 These values are smaller than those of GaAs MESFETs. Before commenting on this important point, let us state the question: Is the semiempirical Fukui formula valid for the TEGFET? To demonstrate the validity of the Fukui formula in the case of a TEGFET, let us recall the work undertaken by Delagebeaudeuf6I. The TEGFET operating condition is assumed to be at the knee voltage where the velocity saturation is just commencing. The calculations of noise figures are similar to those developed by Baechtold62and Brewitt-Taylof13 for the MESFET. They are long and will not be reviewed in detail in this paper. As in Section 8a, the noise figure of the device results from the contributions of thermal fluctuations of the current in the channel, obeying the Nyquist relation (Si,Z)j= 4kT AGj Af
(33)
where AGj is the conductance of the section j in the channel. This current fluctuation induces a correlated current fluctuation in the gate. Delagebeaudeuf demonstrated that the intrinsic noise of the FET is represented by the two sources of current noise:
(6ii) = 4kTgmoPAf
(34)
(did 6iZ)=jC((Sii>(Bi;))1’*
(36)
with the correlation
where P, R,and C are explicit functions of the dc characteristics of the FET. Let us note that Eqs. (34), (35), and (36) are valid for any field-effect transistor. By including the extrinsic parasitic elements and by neglecting the feedback capacitance C,, which is usually low, the schematic circuit presented in Fig. 37 can be used. By optimizing the input impedance Zi to have the
238
NUYEN T. LINH
minimum noise figure, one obtains, after long calculations:
where
K, = P
+ R - 2C(RP)‘I2
K, = RP(1 R,
=R,
- C2)/Kg
+Rg
(38)
By using the analytical model developed in Section 3, one can calculate the
parametersP,R,andC:P-1.16,C-O,R-0.15.ThenEq.(37)canbe reduced to
+
F ~ ,-, I + 2 @ ( 2 n C , f ) [ ( R , RJ/gmo]’/2 (39) which is in conformity with the Fukui formula with KF= 2 fi= 2.15. In fact, the exact value of the KF coefficient varies according to the approximation taken in the calculations stated above. We estimate that the calculated value of KF is -2 _+ 0.5 for both MESFETs and TEGFETs. Therefore, it is not possible to find a theoretical reason why the Fukui factor of the TEGFET is smaller than that of a MESFET. Still, experimental data collected in Table I show that K,(TEGFET) < KF(MESFET) for all studied samples, except for TEGFET No. 3562, which will be commented on further. The KF is calculated through the determination of the equivalent circuit elements, and it is well known, even for MESFETs, that these elements
I
tR’ INPUT I TRANS I STOR I OUTPUT FIG.37. Schematiccircuit representing noise sources in FETs.
TABLE I THEFUKUIFITTING FACTOR OF A TEGFET Is LOWERTHANTHATOF A MESFET, FOR COMPARABLE GATELENGTHTHE VALUES OF C,lg,, ARESMALLER IN TEGFETs
TEGFET
MESFET
3468 3524 3584 3588 3562 4105 4108 4164 HEMT HEMT NE673 AVTK Hughes 41 10 4155
300
140 200 200 280 75 I50 300 140
0.8 0.65 0.6 0.5 0.5 0.5 0.5 0.48 0.5 0.5 0.4 0.25 0.3 0.55 0.5
6 5.4 9.6 7.2 1.35 3.6 2.5 3.8 4.3 4 1.5 4 2.4 3 1.5
2 1.5 1.5 1.5 3.4 2.5 2.3 0.92 7.5 1
2 7 2.9 2.3 2
46 40 44 55 42 61 69 34 55 47 50 19 26 45 28
0.39 0.32 0.48 0.31 0.4 0.34 038 0.16 028 0.26 0.25 0.07 0.147 0.29 0.175
8.5 8 11 5.6 9.5 5.5 5.5 5.3 5.1 5.5 5.0 3.7 5.7 6.45 6.25
10
-
8 12 10 18 18 10 10
1.8 1.9 2.8 1.47 2.9 1.3 1.07 0.85 1.3 1.4 1.34 1.55 1.69 1.8 1.03
9.6 10.1 11.2 10.7 11
9.2 10.6 10.5 13 11
11.4 12.3 8.25 12.1
1.58 2.07 1.9 1.65 3.56 1.66 1.47 1.51 1.6 1.88 2.75 2.2 2.1 2.6 2.45
6 8 6 8 20 6.5 8.5 5.8 7.5 8
50 54 14 48 49
10
5 6.5 13.5
53 1 14
240
NUYEN T. LINH
cannot be determined with precision, in particular the parasitic resistances. We have used different techniques to determine R , and R,, including microwave methods.@The data reported in Table I are the mean values. For TEGFET No. 3562 the fitting factor is as high as 3.5 and the drain current I- corresponding to the minimum noise figure is unusually high (-20 mA). It was observed for the dc characteristics of the transistor that the transconductance drastically drops near pinch-off below 20 mA. This bad behavior occurs in some TEGFETs and has been shown to be due to processing procedures rather than to material quality, as is often the case for MESFETs. c. Discussion
Theoretical calculations predicted that the strong electron velocity overshoot in TEGFETs induces ultra-low-noise properties in this device. On the other hand, the semiempirical approach shows that the low value of the KFfactor is responsible for the high performance of the TEGFET. Which is the right explanation? High electron velocity should induce a high value of gmo/C,. Table I indicates that this is the case. An improvement by a factor of 1.4.- 1.8 is
I
I
I
I
I
I
20
24
LG = 0 . 5 p m
p = 8ooocm2v”<1 R,+ R, = 6Q
Z =300~~m gmo= 3 0 0 m S l m m
CGS 0.78 pFlmm
I
I
I
t
4
8
12
16
I
3
FREQUENCY (GHz) FIG. 38. Frequency dependence of TEGFET noise figures calculated according to the Cappy er nl. model5*(-) and by assuming the validity of a Fukui formula (- - -).
3.
TWO-DIMENSIONAL ELECTRON GAS E T S
241
noted, in good agreement with the theoretical calculation (see Fig. 25). But this simple analysis is not complete because it does not take into account rf characteristics. Let us present the noise figure of a TEGFET calculated according to the Cappy et al. model and compare it to the noise figure obtained by the Fukui formula with various KF values (see Fig. 38). The striking result is that in the Cappy et al. model the noise figure varies with frequency more rapidly than predicted by the Fukui relation, the latter being used with the same gm0/C,,R,, and R, parameters. The thinner the AlGaAs layer, the higher the deviation from the Fukui relation. Let us point out, however, that the Cappy et al. model applied to the MESFET case is in agreement with experimental data. Then it can be concluded that the microscopic model based on nonstationary electron dynamic effects has to be improved to account for experimental observations. However, the supporters of that model can argue that the determination of the source series resistance R, of a TEGFET is wrong and that probably R, diminishes as the frequency increases because the source resistance is constituted of two resistances (the 2DEG and the AlGaAs top layer) coupled with the heterojunction capacitance. Actually Fig. 39 shows that the TEGFET source resistance varies with frequency. More work also has to be done to clarify to which phenomenon the low value of Cm/gmoobserved on TEGFETs can be attributed. It was claimed above that it can be due to the velocity-overshoot effect. There are some
I
I
I
TEGFET #3588
0
5 10 FREQUENCY (GHz)
15
FIG.39. The TEGFET source resistance varies with frequency. This dependence is more pronounced near pinch-off. This result can explain the fact that the noise figure varies less with frequency in a TEGFET than in a MESFET (after Cappy et U Z . ~ ~ ) .
242
NUYEN T. LINH
observations indicating, however, that the low value of C,/g, in TEGFETs can be interpreted by surface effects. It is well known that the transconductance of a MESFET usually varies with frequency: a drop in g, is often observed at 100 to 1000 Hz. Detailed studies made by Wallis and c o - w ~ r k e r shave ~ ~ shown that this phenomenon is related to surface states existing along the source-to-gate and drain- to-gate GaAs surface. The electron trapping and detrapping by these surface states is frequency dependent, and then is responsible for the variation versus frequency of the depth of the surface depletion layer. The modulation of the depletion layer thickness by the microwave signal applied on the gate induces the existence of an effective gate length which is larger than the metallurgical gate length. The ratio between those two depends on the quality of the GaAs surface treatment, i.e., the passivation. This surface property also induces the so-called lagging effect. Suppose an FET is in its offstate. By applying a positive (or less negative) voltage on the gate, the FET is turned to its on state, but the drain current iilcrease presents some delay. This lagging effect is also surface dependent. The study of these phenomena on TEGFETs showed that g, is frequency independent (Fig. 40); and (2) a lagging effect is not observed. (1)
The effective gate length in TEGFETs is then shorter than in MESFETs.
I
I
I
104 106 lo8 FREQUENCY (Hz) FIG. 40. The transconductance of a MESFET (- - -) often varies with frequency. This
100
102
spurious phenomenon is attributed to surface effects. Notice that the TEGFET transconductance (-) remains constant (after D. Pons and F. Faucher, unpublished).
3.
TWO-DIMENSIONAL ELECTRON GAS
FETS
243
By referring to the C,/g,, values collected in Table I and by ignoring the overshoot effect, one can estimate that the effective gate length of TEGFETs is I .4 to 1.8 times shorter than that of MESFETs. With regard to the low KF value, no clear explanation can be given at present. Since in MESFETs this factor is referred to the channel material quality, it is concluded at first analysis that the TEGFET channel quality is better than that of the MESFET. Reduced Coulombic scattering and screeningeffectsby free electrons, particularly at the minimum noise figure bias condition, make the material behave like an ideal case, toward which the MESFETs are approaching. Notice that the best MESFETs reported in Table I have a lower KF value than usual. Notice also that, in the ideal case, the factor P [see Eqs. (34) and (39)] has the value of 0.666 near pinchoff.61,63966 Then KF = 2 0, KF= I .63. It is probable that this analysis is too simple since many approximations have been assumed, but it seems that it represents a reasonable hypothesis for the low value of K F . In summary, a definite explanation of the high performance of TEGFETs has not been found yet. Probably there is not one single reason but a combination of several factors that make the TEGFET superior to the MESFET.
VI. Conclusion Experimental results clearly show that the TEGFET is superior to the MESFET as a low-noise transistor; with a 0.85 dB noise figure at 10 GHz and 1.3 dB at 17.5 GHz (at 300 K) for a gate length of 0.5 pm, the TEGFET performance is higher than state of the art for 0.25pm MESFETs. Moreover, TEGFETs do not present spurious effects such as ‘‘lagging” or g , dependence on frequency. The reason@)why TEGFETs work better than MESFETs is not clearly understood yet. Whatever the explanation(s), ultra-low-noise TEGFETs will constitute a breakthrough in the 1980s. Combining TEGFETs and MM1Cs6’ is the future of microwave devices for satellite communication, DBS, phased array antenna radar millimeter wave systems, etc. ADDENDUM During the preparation of this paper more results on TEGFETs were presented at the IEEE MTT-S Symposium (San Francisco, May, 1984). In particular K. Ohata, H. Hida, H. Miyamoto, M. Ogawa, T. Baba and T. MizutaN (1984 IEEE MTT-S Digest, p. 434) showed a 0.5 ,um TEGFET with 1.85 dB noise figure at 20 GHz; also J. J. Berenz, K. Nakano, and K. P. Weller (1984 ZEEE MTT-S Digest, p. 98) presented 0.35-0.37 fim gate length devices
244
NUYEN T. LINH
3.5
43
COMMERCIAL PRODUCTS
TOSHIBA
4'"'Q
1
3.c
.
TRW0.25pm
I
2.5
GE .25pml
TOSHIBAW I
.25pm
-m 0 2s !A K
1
P
U
w
B
s
1.5 ROCKWELL 0 TRW 0.25
1.c
p
.25pm 0.5
0
A -GE 0.25 pm
10
20
30
40
FREQUENCY (GHz)
FIG.41. State of the art of low-noise and millimeter-wave operation TEGFETs (July, 1986). exhibiting 1.5 dB noise figure at 18 GHZ and 2.7 dB at 34 GHz. These new data confirm the extremely high performance of the TEGFET. Since May, 1984, more works have been published on low-noise TEGFETs. The main results are summarized in Fig. 41: low-noise and high-frequency operation have made tremendous progress both in the laboratory and in commercial products. Not included in this figure are the excellent work of M. Sholley et al. (Military Microwave, Brighton, July 1986) on HEMT millimeter-wave amplifiers, mixers, and oscillators working up to 70 GHz, and that of P. M. Smith et al. (Electron. Lett. 22( 19,780, 1986) on a 94 GHz amplifier using an HEMT. In addition to low-noise amplification, let us note that a high-efficiency power TEGFET also presents a very interesting performance (see,for example, H. Hida et al., Electron. Lett. 22(16), 780, 1986).
ACKNOWLEDGMENTS It is a great pleasure for the author to thank his co-workers and colleagues for their active contribution to this work, especially D. Delagebeaudeuf, P. Delescluse, M. Laviron, J. F. Rochette, J. Chewier, P. Jay, R. H. Wallis, D. Pons, A. Faucher, C. Rumelhard, and F. Diamand in Thomson-CSF and A. Cappy, G. Salmer, and E. Constant at the University of Lille. Many unpublished results were communicated by D. Delagebeaudeuf, A. Cappy, D.
3.
TWO-DIMENSIONAL ELECTRON GAS E T S
245
Pons, and A. Faucher. The permission of R. H. Wallis, M. Niori, A Cappy, J. C. M. Hwang, and B. Winter to reproduce figures from their publications is appreciated. The author would also like to express his gratitude to P. Aigrain and E. Spitz for their constant encouragement during this work, which is partially supported by DRET contract No. 81.34.570.
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Devices ED-28, 1031 (1981). 5 . M. Amand, D. V. Bui, J. Chevrier, and N. T. Linh, Electron. Lett. 19,.433 (1983);Rev. Tech. Thomson-CSF 16,47 (1984). 6. A. Cappy, B. Carnez, R. Fauquembergue, G. Salmer, and E. Constant, IEEE Trans. Electron Devices ED-27,2 158 ( 1980). 7. P. OConnor, T. P. Pearsall,, K. Y. Cheng, A. Y. Cho, J. C. M. Hwang, and K. Alavi, IEEE Trans. Electron Device Lett. EDL-3, 64 ( 1982). 8. H. Kroemer, Proc. IEEE 45, 13 (1982). 9. P. M. Asbeck, D. L. Miller, W. C. Petersen, and C. G. Kirkpatrick, IEEE Electron Device Lett. EDL-3, 366 ( 1 982). 10. D. Ankri, W. J. SchaE, P. Smith, and L. F. Eastman, Electron. Lett. 19, 147 (1983). 1 1 . R. Dingle, H. L. Stormer, H. L. Gossard, and W. Wiegmann, Appl. Phys. Lett. 33,665 ( 1978). 12. D. Delagebeaudeuf, P. Delescluse, P. Etienne, M. Laviron, J. Chaplart, andN. T. Linh, Electron. Lett. 16,667 (1980). 13. T. Mimura, S. Hiyamizu, T. Fuji, and K. Nanbu, Jpn. J. Appl. Phys. 19, L225 (1980). 14. M. Laviron,J. F. Rochette, P. Delescluse, P. Jay, J. Chevrier, and N. T. Linh, unpublished results. 15. M. Feng, V. K. Eu, H. Kanber, E. Watkins, J. M. Schellenberg, and H. Yamasaki, Appl. Phys. Lett. 40,802 (1982). 16. K. Hess, H. Morkoc, H. Shichijo, and B. G. Streetman, Appl. Phys. Lett. 35,469 (1979). 17. J. F. Rochette, P. Delescluse, M. Laviron, D. Delagebeaudeuf, J. Chevrier, and N. T. Linh, Conf Ser. --Inst. Phys. 65,385 (1983). 18. T. J. Drummond, H. Morkoc, S. L. Su, R. Fisher, and A. Y . Cho, Electron. Lett. 17,870 ( 198 1). 19. T. J. Drummond, J. Klem, D. Arnold, R. Fisher, R. E. Thorne, W. G. Lyons, and H. Morkoc, Appl. Phys. Lett. 42,6 15 ( 1983). 20. D. Delagebeaudeuf, M. Laviron, P. Delescluse, P. N. Tung, J. Chaplart, and N. T. Linh, Electron. Lett. 18, 103 (1982). 21. S. Hiyamizu, J. Saito, K. Nanbu, and T. Ishikawa, Jpn. J. Appl. Phys. 22, L609 (1983). 22. J. C. M. Hwang, H. Temkin, A. Kastalsky, H. L. Stormer, and V. G. Keramidas, U.S. Molecular Beam Epitary Workshop (1982). 23. G. Bastard and E. E. Mendez, private communication. 24. G. Fishman and D. Calecki, Proc. Int. Conf Phys. Semicond.. 16th, 1982 (1983). 25. T. J. Drummond, W. Kopp, H. Morkoc, and M. Keever, Appl. Phys. Lett. 41, 277 ( 1982).
246
NUYEN T. LINH
26. C. Cappy, C. Vernaeyen, A. Vanoverschelde, G. Salmer, D. Delagebeaudeuf, N. T. Linh, and M. Laviron, GaAs ICSymp., 1982 (1983). 27. M. A. R. Al-Mudares and K. W. H. Foulds, Eur. Spec. Workshop Active Microwave Semicond. Devices, 8th, I983 ( 1983). 28. D. Delagebeaudeuf, P. Delescluse, M. Laviron, P. N. Tung, J. Chaplart, J. Chewier, and N. T. Linh, Conf Ser.-Inst. Phys. 65, 393 (1983). 29. T. J. Drummond, S. L. Su, W. G. Lyons, R. Fischer, W. Kopp, H. Morkoc, K. Lee, and M. S. Shur, Electron. Lett. 18, 1057 (1982). 30. R. H. Wallis, Proc. Int. Conf Phys. Semicond., 16th, 1982, Part 11, p. 756 (1983). 3 1 . M. Laviron, D. Delagebeaudeuf, P. Delescluse, J. Chaplart, and N. T. Linh, Electron. Lett. 17, 536 (1981). 32. N. T. Linh, Festkoerperprobleme 23,227 ( 1 983). 33. P. Delescluse, M. Laviron, J. Chaplart, D. Delagebeaudeuf, and N. T. Linh, Electron. Lett. 17, 342 (1981). 34. D. Delagebeaudeuf and N. T. Linh, IEEE Trans. Electron Devices ED-28,790 (1981). 35. D. Delagebeaudeuf and N. T. Linh, IEEE Trans. Electron Devices ED-29,955 (1982). 36. B. Vinter, Appl. Phys. Lett. 44,307 (1984). 37. A. Cappy, A. Vanoverschelde, J. Zimmermann, P. Philippe, C. Versnaeyen, and G. Salmer, Rev. Phys. Appl. 18,7 19 (1983). 38. T. J. Drummond, H. Morkoc, K. Lee, and M. Shur, IEEE Electron Device Lett. EDL-3, 338 (1982). 39. K. Lee, M. Shur, T. J. Drummond, and H. Morkoc, ZEEE Trans. Electron Devices ED-31,29 (1983). 40. E. Constant, Conf Ser. -Inst. Phys. 57, 141 (198 1). 41. M. Shur, Electron. Lett. 12,615 (1976). 42. J. F. Pone, R. C. Castagne, J. P. Courat, and C. Amodo, IEEE Trans. Electron Devices ED-29, 1244 (1982). 43. B. Carnez, A. Cappy, A. Kaszynski, E. Constant, and G. Salmer, J. Appl. Phys. 51,784 (1980). 44. S. Laval, C. Bru, R. Castagne, and C. Amodo, Conf Ser. -Inst. Phys. 56, 17 1 ( 198 1). 45. A. Cappy, 36me cycle Thesis, University of Lille (1981). 46. P. Urien and D. Delagebeaudeuf, unpublished. Details of the calculations can be found in the Thomson-CSF report. DRET Contract No. 81-37-570. 47. M. Laviron, D. Delagebeaudeuf, P. Delescluse, P. Etienne, J. Chaplart, and N. T. Linh, Appl. Phys. Lett. 40,530 (1982). 48. M. Niori, T. Saito, S. Joshin, and T. Mimura, International Solid State Circuits Conference Fev. 1983 (1983). 49. K. Joshin, T. Mimura, N. Niori, Y. Yamashita, K. Kosemura, and J. Saito, IEEE MTT-s Symposium, 1983 (1983). 50. N. T. Linh, Eur. Spec. WorkshopActive Microwave Semicond. Devices, 8th, 1983 (1983). 51. M. Feng, V. K. Eu, H. Kanber, E. Watkins, J. M. Schellenberg, and H. Yamasaki, Appl. Phys. Lett. 40, 802 ( 1982). 52. M. Feng, V. K. Eu, I. K. DHaenens, and M. Braunstein,Appl. Phys. Lett. 41,633 (1982). 53. P. W. Chye and C. Huang, IEEE Electron Device Lett. EDL-3,401 (1982). 54. M. Laviron, J. F. Rochette, P. Delescluse, J. Chewier, and N. T. Linh, unpublished. 55. K. Ohata, H. Hida, H. Miyamoto, T. Baba, T. Mizutani, and M. Ogawa, Japanese Microwave Workshop, 1984 (1984). 56. N. T. Linh, in “Applications of GaAs MESFETs” (R. Soares, J. Graffeuil, and J. Obregon, eds.), p. 461. Artech House, Dedham Massachusetts, 1983.
3.
TWO-DIMENSIONAL ELECTRON GAS
FETS
247
57. P. C. Chao, T. Yu, P. M. Smith, S. Wanuga, J. C. M. Hwang, W. H. Perkins, H. Lee, L. F. Eatsman, and E. D. Wolf, Electron. Lett. 19, 894 (1983). 58. A. Cappy, A. Vanovemhelde, M. Schortgen, C. Versnayen, and G. Salmer, J. Natl. Microondes, 1984 (1984). 59. B. Camez, A. Cappy, R. Fauquembergue, E. Constant, and G. Salmer, IEEE Trans. Electron Devices EDZS, 784 (198 1). 60. H. Fukui, IEEE Trans. Electron Devices ED-26, 1032 (1979). 61. D. Delagebeaudeuf, unpublished. Part of the results were presented at the Maidenhead Workshop (see Linhso) Details of the calculations can be found in the Thomson-CSF report on TEGFET, DRET Contract No. 8 1-34-570. 62. W. Baechtold, IEEE Trans. Electron Devices ED-19,674 (1972). 63. C. R.Brewitt-Taylor, P. N. Robson, and J. E. Sitch, Proc. IEEE 127, 1 (1980). 64. F. Diamand and M. Laviron, Proc. Eur. Microwave Conf, 12th. 1982 (1982). 65. R. H. Wallis, D. Pons, A. Faucher, unpublished, A. Faucher, Thkse Centre National des Arts et Metiers, Pans (1 984). 66. A. Van Der Ziel, Proc. IRE 50, 1808 (1962). 67. M. Le B a n , P. R. Jay, C. Rumelhard,G. Rey, and P. Delescluse, G d s ICSymp.,Digest, p. 20, 1983 ( 1984).
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SEMICONDUCTORS AND SEMIMETALS, VOL. 24
CHAPTER 4
Ultra-High-speed HEMT Integrated Circuits M . Abe, T. Mimura, K. Nishiuchi, A. Shibatomi, M. Kobayashi, and T. Misugi COMPOUND SEMICONDUCTOR DEVICES LABORATORY FUJITSU LABORATORIES LTD., ATSUGI 10-1, MORINOSATO-WAKAMIYA ATSUGI 243-01, JAPAN
I. Introduction Information processing in 1990 will require ultra-high-speed computers, requiring high-speed LSI circuits with logic delays in the sub-100 ps range.' The evolution of high-speed GaAs integrated circuits (ICs) is the result of continuous technological progress utilizing the superior electronic properties of GaAs as compared with those of Si. GaAs metal-semiconductor field-effect transistor (MESFET) technology enables the demonstration of GaAs integrated circuits with high speed and low power c o n s ~ m p t i o n . ~ ~ ~ Fujitsu's self-aligned-gate MESFET technology has made it possible to develop a GaAs 4 kbit static RAM4 and a GaAs 3k-gate 16 X 16-bit parallel m~ltiplier.~ The GaAs 4 kbit static RAM has an address access time of 3.0 ns with a power dissipation of 700 mW. The GaAs 3k-gate 16 X 16-bit parallel multiplier has a multiply time of 10.5 ns with a power dissipation of 952 mW. Electron mobility in the MESFET channel with typical donor concentrations of around 1017cm-3 ranges from 4000 to 5000 cm2/V s at room temperature. The mobility in the channel at 77 K is not too much higher than at room temperature due to ionized impurity scattering. In undoped GaAs, electron mobility of 2 to 3 X lo5cm2/Vs has been obtained at 77 K. The mobility of GaAs with feasibly high electron concentrations for facilitating the fabrication of devices was found to increase through modulation-doping techniques demonstrated in GaAs/AlGaAs superlattices.6 As the first application of this electron mobility-enhanced phenomenon to the new transistor approach, a high-electron-mobility transistor (HEMT), based on modulationdoped GaAs/NGaAs single-heterojunction structures, was invented7 and was demonstrated to greatly improve the 77 K channel mobility. 249 Copyright 0 1987 Bell Telephone Laboratories,Incorpxated. Au rights of nproduction in any form reaemed.
250
M. ABE
et al.
HEMT technology has opened the door to new possibilities for ultrahigh-speed LSI/VLSI application^.'-'^ Due to the supermobility GaAs/ AlGaAs heterojunction structure, the HEMT is especially attractive for low-temperature operations at liquid nitrogen temperature. In 198 1, an HEMT ring oscillator with a gate length of 1.7 pm demonstrated a 17.1 ps switching delay with 0.96 mW power dissipation per gate at 77 K, indicating that switching delays below 10 ps will be achievable with 1 pm gate devices.8 A switching delay of 12.2 ps with 1 . 1 mW power dissipation per gate has already been obtained with a 1 pm gate HEMT even at room temperature." This is the shortest switching delay achieved so far in semiconductor devices. More complex circuits have achieved successful operation of HEMT frequency dividers with direct-coupled FET logic (DCFL) circuits, demonstrating a maximum clock frequency of 8.9 GHz at 77 K.12J3The maximum clock frequency achieved with HEMT technology is roughly two times higher than that of its GaAs MESFET counterpart with comparable geometry. Recently, frequency-divider circuits composed of selectively doped heterojunction transistors (SDHTs) with 1 pm gates were fabricated, showing a maximum clock frequency of 10.1 GHz at 77 K.I4 For more complex HEMT circuits, HEMT technology has made it possible to develop 1 kbit static RAMSwith access times of 0.87 ns, and has already jumped into the LSI/VLSI application field. This article first presents the technological advantages of HEMT. Next, we will describe an HEMT technology for VLSIs including material, device fabrication, and characteristics for device modeling. We will then review current work and recent advances in HEMT logic and memory 1Cs. Finally, we will project the future performance of HEMT VLSIs for ultrahigh-speed computer applications. 11. Technological Advantages of HEMTs
HEMT technology has new possibilities for LSI/VLSI with high speed and low power dissipation. This section describes the principles of the HEMT and its technological advantages compared with other technologies for high-speed devices. 1 . HEMT PRINCIPLES A cross-sectional view of the basic structure of a HEMT, with a selectively doped GaAs/AlGaAs heterojunction structure, is shown in Fig. 1 . An undoped GaAs layer and Si-doped n-type AlGaAs layer are successively grown on a semi-insulating GaAs substrate by molecular beam epitaxy (MBE). Because of the higher electron affinity of GaAs, free electrons in the AlGaAs layer are transferred to the undoped GaAs layer, where they
4.
ULTRA-HIGH-SPEED Swce
I
Gate
HEMT INTEGRATED CIRCUITS
251
Dmin
Somi -insulating GaAs substrate
I
FIG. 1. Cross-sectional view of the basic structure of a HEMT, with a selectively doped GaAs/MGaAsheterojunction structure.
form a two-dimensional high-mobility electron gas within 10 nm of the interface. The n-type AlGaAs layer of the HEMT is completely depleted in two depletion mechanisms: ( I ) the surface depletion results from the trapping of free electrons by surface states; and (2) the interface depletion results from the transfer of electrons into the undoped GaAs. The Fermi level of the gate metal is matched to the pinning point, which is 1.2 eV below the conduction band. With the reduced AlGaAs layer thickness, the electrons supplied by donors in the AlGaAs layer are insufficient to pin the surface Fermi level. Therefore, the space-charge region extends into the undoped GaAs layer and, as a result, band bending results in the upward direction, and the two-dimensional electron gas does not appear. When a positive voltage higher than the threshold voltage is applied to the gate, electrons accumulate at the interface and form a two-dimensionalelectron gas (2DEG). Thus, we can control the electron concentration to achieve depletion (D)-mode and enhancement (E)-mode HEMT operation.'.' Electron mo-. bility and sheet electron concentration (N,) in the heterostructure are shown as a function of temperature in Fig. 2.16 As temperature decreases, the electron mobility, which was about 8 X lo3cm2/V s at 300 K, increases dramatically and reaches 2 X lo5 cm2/V s at 77 K due to reduced phonon scattering. A further increase with a considerable gradient occurred even below 50 K, and a maximum value of 1.5 X lo6 cmZ/Vs in the dark and 2.5 X lo6 cmZ/Vs under light illumination was attained at 4.2 K. Sheet electron concentration decreases with decreasing temperature until it becomes constant below 150 K. The almost constant value of about 3.5 X 10'' cm-? below 150 K corresponds to that of 2DEG at the interface, since this value agrees well with the value of N, determined by Shubnikov-de Haas measurement at 4.2 K. Apparent excess camers above 150 K are attributed to free electrons which are thermally excited from relatively deep donors in n-type Ab.3Gh.7A~. Figure 3 shows annual 2DEG mobilities reported so far for selectively doped heterostructures and
252
M. ABE
-
'
I
et al.
-
".
nrn
d.20
-
'
1 ' " " ' 1
GWn-AIGaAs
N
;Id4
loB,
5
%x .c
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u
E
c
c
Id,
f
:Id3
0
c
2
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8
E lo4,
,Id2
t 0
iii
= f
5
Id
I
I
I I , I
I
.
I I
I
I
I
I I 1
I
I empmrure
I
I
I
Id'
Kj
FIG.2. Electron mobility and sheet electron concentration in GaAs/n-AlGaAs with a 20 nm thick undoped AlGaAs spacer layer, as a function of a temperature.
modulation-doped superlattices starting with 1978, when modulation doping in superlattices was first demonstrated. Open circles indicate mobilities in selectively doped heterostructuresat 77 K and solid circles show mobilities at 4.2- 10 K. During 1978 and 1979, mobility remained rather low. It began increasing rapidly, however, afler the first HEMT was developed.'
4"j, ; ,
~'p;Ki"A"
Id
'78
'79
'80 Year
'81
'82
FIG.3. 2DEG mobility improved in selectively doped GaAs/AlGaAs as a function of year.
4. ULTRA-HIGH-SPEED HEMT INTEGRATED CIRCUITS
253
2. COMPARISON WITH OTHERHIGH-SPEED DEVICE APPROACHES
The performance of various device approaches competing for high-speed applications are compared roughly in Fig. 4,on the basis of ring-oscillator results, except for Josephson junction devices, which use a gate chain. It is difficult to compare their optimized performances fairly. Here we have the criteria based on 1 pm device technology. HEMT and Josephson junction devices show excellent high-speed performance, The switching delay of GaAs MESFETs is two or three times longer than that of HEMTs. GaAs/AlGaAs heterojunction bipolar transistors (HBT) should achieve the same high-speed performance as the HEMT. The ultimate speed capability, limited by cutoff frequencyfT, is over 100 GHz, and the HBT also has the merit of flexible fan-out loading capability. The silicon MOSFET and bipolar transistor are excellent for both designing due to threshold voltage uniformity and controllability with no material problems, and for ease of fabrication in spite of complex processing steps. Configuration for both high-speed and large-scale integration with low-power performance, however, may be difficult for Si-based technology. The technological advantages of HEMTs are compared with various competing approaches to high-speed device design in Table I. HEMTs are very promising devices for high-speed VLSIs but require technological problems to be solved to achieve the LSI quality of GaAs/ AlGaAs material, using MBE and/or metal organic vapor phase epitaxy (MOVPE) technologies.
Power dissipation (mW)
FIG. 4. Performance territory of various device approaches competing for high-speed applications, based on 1 pm device technology.
TABLE I TECHNOLOGICAL ADVANTAGES OF HEMTs COMPARED WITH VARIOUS COMPETING HIGH-SPEED DEVICEAPPROACHES
Device approach
Performance
-
Speed
Power
Uniformity and controllability (uV,/swing ratio)
Fabricability
Material problems
Excellent Simple MBE and dry etching Excellent Simple
Good Defect and trap-free epi High throughput Good Defect-free ingot
Excellent
Unknown
HEMT L, = 0.5- 1 pm
Excellent Very good 10 ps 0.1 mW Highly geometry controllable
Excellent 10-20 mV (2%)
GaAs MESFET LG=0.5-1pm
Good Good 20-30 PS I mW Poor geometry controllable Excellent Good 10-30 PS 1 mW
Good 60 mV
(< 1%)
Complex New process required
Very poor 80 ps
Very good 0.1 mW
Excellent
Complex
Good Defect and trap-free epi High throughput Excellent
(1%)
Good 30-60
Poor 1-10mW
Complex
Excellent
(< 1%)
GaAsIAlGaAs HBT
Si MOSFET Si bipolar
PS
Total advantages
Good
(10%)
Excellent
Excellent
Difficult to high speeds Difficult to large scale
4.
HEMT INTEGRATED CIRCUITS
ULTRA-HIGH-SPEED
255
111. HEMT Technology for VLSI
The development of a high-performance VLSI requires new technological breakthroughs. This section describes state-of-the-art HEMT technology including material and the self-alignment-device fabrication technologies, and HEMT device modeling.
3. MATERIAL TECHNOLOGY To realize high-quality material grown by MBE, we optimized the buffer layer between the semi-insulating GaAs substrate and the two-dimensional electron-gas channel layer. The thickness of this layer is 0.6 pm. Figure 5 shows the electron mobility and sheet electron concentration in this optimized heterostructure as a function of temperature. As temperature decreases, electron mobility, which was 8 X lo3 cm*/V s at 300 K, increases to 1.2 X lo5cm2/V s at 77 K due to reduced phonon scattering." The surface defect problem of MBE is a serious one at the present time for fabrication of LSI-level complexity. Figure 6 shows the surface of an MBE-grown GaAs film. Many surface irregularities are found. These surface irregularities are called oval defects. Depending on growth conditions, the density of the oval defects was 500-3000 cm-'. The oval defects typically are from submicrometerto several micrometers, which is comparable to the size of devices in LSI circuits. These oval defects seriously affect the current -voltage characteristics of HEMTs.'* The effect of oval defects on drain current - voltage characteristics is demonstrated in Fig. 7. Figure 7a is an enlarged view of the HEMT with an arrow pointing to an
t
GaAs/n-AEaAs
".i
i
d3
.-b -
a 0
E
lo=
tI , , 3
I
4
I
I
I
I
10 100 Temperature ( K )
4
E
In
I l l
500
FIG. 5. Electron mobility and sheet electron concentration in the optimized GaAs/nAlGaAs heterojunction as a function of temperature.
256
M. ABE
et al.
FIG.6 . A micrograph of surface irregularities of an MBEgrown GaAs film.
oval defect under the gate metal. The I - Vcharacteristicsare shown in Fig. 7b. Clearly, drain current cannot be cut off by the gate. In Fig. 7b, curves on the left and on the right show the I - Vcharacteristics for positive and negative gate voltages, respectively. This suggests that oval defects produce
H
(a)
1v
(b)
FIG. 7. Effect of oval defects on drain current-voltage characteristics: (a) Top view of HEMT with an oval defect under the gate; and (b) drain current-voltage characteristics.
4.
ULTRA-HIGH-SPEED
I$,,
HEMT INTEGRATED CIRCUITS
,
,
,
257
,
0 1020304050607080 Distance (mm) (0)
F
p0j, ,
An/n
,
I
,
,
,
,
w 0 1020304050607080 Distance (mm) (b) FIG. 8. Uniformities of n-GaAs layer grown on the 3 in. substrate by MBE (a) layer
thickness and (b) electron concentration uniformities.
FIG.9. Photograph of MBE wafer of 3 in. diameter with HEMT structure.
258
M. ABE
et a!.
extra conductive channels between source and drain, which cannot be controlled by the gate, The prime challenges for LSIs from a material viewpoint must be centered on reduction of these oval defects on MBE wafers. We could achieve a density of less than 100 cme2. Another important problem in fabricating HEMT LSIs is epitaxial wafer growth technology with high throughput and large size. We optimized growth conditions for highly uniform epitaxial layers on 3 in. diameter semi-insulating GaAs substrate, which made high throughput and high quality possible. Figure 8 shows the uniformity of thickness and the electron concentration of an n-GaAs layer grown on the 3 in. substrate. Highly uniform performance of less than 1% in thickness and electron concentration have been achieved within the circular area of 60 mm diameter.lq Figure 9 is a photograph of an 3 in. diameter MBE wafer with HEMT structure. 4. SELF-ALIGNMENT-DEVICE FABRICATION TECHNOLOGY
Figure 10 is a cross-sectional view of a typical self-aligned structure of enhancement-mode (E) and depletion-mode (D) HEMTs forming an inverter for a DCFL circuit configuration. The basic epilayer structure consists of a 600 nm undoped GaAs layer, a 30 nm Al,,,G%,7As layer doped to 2 X 10l8cm-3 with Si, and a 70 nm GaAs top layer successively grown on a semi-insulating substrate by MBE. The low-field electron mobility was found from Hall measurements to be 7200 cm2/V s at 300 K and 38,000 cm2/V s at 77 K. The concentration of two-dimensional electron gas (2DEG) was ,1.0 X 1OI2 cmF2at 300 K and 8.4 X 10” cm-2 at 77 K. The AlAs mole fraction tentatively selected was 0.3, although it can be expected that higher AlAs mole fractions would increase the maximum achievable concentration of 2DEG, resulting in an increase in transconductance of HEMTs. Al,Ga,-,As with a high AlAs mole fraction, however, exhibits E- HEMT
Ohmic
,contact /Gate
D - HEMT Intercansct
,/-mrtot
insulator ( SiOp 1
/’ ,--insulator
( SiOp)
GaAs AlGoAs GaAs AIGoAs Semi- insulating GaAs submate
FIG.10. Cross-sectional view of a typical self-aligned structure of E/D-HEMTs forming an inverter for DCFL circuit configuration.
4.
ULTRA-HIGH-SPEED
HEMT INTEGRATED CIRCUITS
259
inferior surface morphology and an increase in deep traps, making device fabrication difficult. A thin Ab.3Ga,-,7As layer to act as a stopper against selective dry etching is embedded in the top GaAs layer to fabricate E- and D-HEMTs in the same wafer. By adopting this new device structure, we can apply the selective dry etching of GaAs to AlGaAs to achieve precise control of the gate recessing process for E- and D-HEMTs. Figure 11 indicates the process sequence for the self-alignedgate process in the fabrication of HEMT LSIs including enhancement-modeand depletion-mode HEMTs.~OFirst of all, the active region is isolated by a shallow mesa step (180 nm), which is produced by a very simple process and can be made nearly planar. The source and drain for E- and D-HEMTs are metallized with AuGe eutectic alloy and Au overlay alloying to form ohmic contacts with the electron layer. Then fine gate patterns are formed for E-HEMTs, and the top GaAs layer and thin Ab,3G%.7Asstopper are etched off by nonselective chemical etching. Using the same resist, after formation of gate patterns for D-HEMTs, selective dry etching is performed to remove the top GaAs layer for D-HEMTs and also remove the GaAs layer under the thin Ab,Gq,As stopper for E-HEMTs. Next, Schottky contacts for the E- and D-HEMT gates are provided by depositing Al, the Schottky gate contacts, and GaAs top layer for ohmic contacts being self-aligned to achieve high-speed performance. Finally, electrical connections from the interconnecting metal, composed of Ti/Pt/Au, to the device terminals are provided through contact holes etched in a crossover insulator film. As described above, a unique epistructure in combination with self-terminating selective dry recess etching makes it possible to fabricate super-uniform E- and D-HEMTs simultaneously, reflecting the uniformity of MBE-grown epitaxial film. The key technique to achieve stable fabrication of self-aligned gate HEMTs is the selective dry etching of the GaAs/AlGaAs layer as is understood from Fig. 1 1. Figure 12 shows etching characteristics in CCl,F2 He discharges by using GaAs (60 nm thick)/
+
Ohmic contact
Resis
AlGaAs -----
------
S.1 . GaAs substrate
(a 1
(b)
FIG. 1 1. Process sequence for self-aligned gate fabrication: (a) dry recessing, and (b) gate metallization.
260
et al.
M. ABE 3001
d GaAs (520 nm/min
I
CC12F2/He
I
,,'
7---
I
5 200 I
I
I
I
0 0
20
40
60
80
Etching time (sl FIG. 12. Optimized selective dry-etching characteristics in CC1,FJHe discharges by using GaAs/AIGaAs heterojunction MBE material.
Ab,Ga,,7As heterojunction material. A high selectivity ratio of more than 260 is achieved, where the etching rate of Ab,,Ga,,7As is as low as 2 nm/min and that of GaAs is about 520 nm/min at 140 V of self-generated bias voltage.21 Maps and histograms of threshold uniformities for E- and D-HEMTs are shown in Figs. 13 and 14. The standard deviations in threshold voltages, measured for 149 E-HEMTs and 148 D-HEMTs distributed over an area of 15 X 30 mm2, are 19 and 74 mV, respectively.22 Recently we have achieved 12 and 20 mV standard deviations in threshold voltages for Eand D-HEMTs typically due to more refined processing conditions. The ratio of standard deviation of threshold voltage to the logic voltage swing E-HEMT
D- HEMT 0.5
-->
s
0
- 1.0
-->
>' -2.0
(a) (b) FIG.13. Maps of threshold voltages for (a) E-HEMTs and (b) D-HEMTs, over an area of 15 X 30 mm2.
4.
ULTRA-HIGH-SPEED
HEMT INTEGRATED CIRCUITS
261
60
N
6149
0
-0.2 0 0.2 0.4 0.6
-2.0
-1.5
-1.0
Threshold voltage (V) (b)
Threshold voltage (V) (0)
FIG. 14. Histograms of threshold voltages for (a) E-HEMTs and (b) D-HEMTs, corresponding to data shown in Fig. 13.
(0.5 for DCFL) is 3.8%, indicating excellent controllability of MBE growth and the device fabrication process. This strongly supports the viability of these technologies for realizing ICs with LSI/VLSI-level complexities.
5. HEMT DEVICE MODELING To give reasonably accurate predictions of device performance and provide guidelines for LSI design, simple device modeling is carried out. Figure 15 shows the energy-band diagram under flatband conditions for device modeling. According to the gradual channel approximation,HEMT drain current, below the saturation region, can be written as23 ID = I O W
-
(1)
where I0 = /&w&v+/2L~d, v~# 0
FIG.15. Energy-band diagram under flatband conditions used for device modeling.
(2)
262
M. ABE
et al.
v = ( V S - VG+ VT)/VT
(3)
G + VT)/VT
(4)
and C=(VD-
Here, p,, WG,LG, E, and VT have their usual meanings, and d is the thickness of the Al,Ga,-,As layer. The quantities v and C: denote the normalized potential differencesat the source and drain, respectively. The threshold voltage VTis given by VT = CY, - A EJq - qnD( d - do)2/2~
(5)
where yM is the metal-semiconductor bamer potential, AEc is the difference in energy between the Al,Ga, -,As and GaAs conduction-band edges, nD is the donor concentration in the Al,Ga,-,As layer, and do is the thickness of the undoped spacer. The vertical threshold sensitivity is defined by the differential threshold voltage to the thickness of the AlGaAs layer and can be derived from Eq. (5) as follows: dVT dd
L2qnD(V/M
- A Ec/q - vT)/&11’2*
(6)
We use the piecewise linear approximation of Turner and Wilson for the velocity-field character is ti^.^^ In the approximation, saturation is assumed to occur when the field at the drain end of the gate reaches the peak field Em, and the saturation current is calculated from Eq. (1) by imposing the boundary condition that the field at the drain end of the gate be equal to Em. The experimental (-) and calculated (- - -) I- V characteristics for a 1 pm gate E-HEMT are shown in Fig. 16. Device parameters used in the calculation are inserted. Reasonable agreement between experimental
. 300K
s a w c E
4-
3-
L,j/Wo= 1/20 pm Em.2.8 KV/cm * 7000 c#/V.S = 30 nm VT= 0.13 V R ~ = R ~ =n ~ O
s
Drain voltage Vos (V) FIG.16. Drain current-voltage characteristics of E-HEMT at 300 K., showing the experi) and calculated (- - -) curves. mental (-
4.
ULTRA-HIGH-SPEED I
HEMT INTEGRATED
CIRCUITS
263
I
'
- 1.5
- 1.0
0
-0.5
0.5
VT (VI
Threshold voltage
FIG. 17. Vertical threshold sensitivity as a function of threshold voltage at the electron concentration of 2 X 10'8 cm-3.
and calculated results is found. It is also noted in Fig. 16 that a HEMT with a gate length as short as 1 pm, operating in the high average field region, exhibits empirical square-law characteristics, i.e., IDS= K( V,, - VT)'. Figure 17 shows the vertical threshold sensitivity calculated from Eq. (6) at a carrier concentration of 2 X 10l8~ m - The ~ . threshold sensitivity is 70 mV/nm at a VT of 0.13 V. As is shown in Fig. 14a, the deviation in threshold voltage over the wafer for the E-HEMT is 140 mV at VT of 0.13 V. This corresponds to a thickness deviation of only 2 nm over the wafer, indicating excellent controllability of MBE growth and the device fabrication process. The dependence of the K factor and transconductance g, of 100 -
:1000
W ~ = 2 0urn 1
- 500
50 N
?
4
E L
0 c
V
s
'0
5-
Y
I< 0. I
*
'
, ' l l B 1 '
0,5 I Gate length
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E-HEMTs on gate length were measured at both 77 and 300 K and are plotted in Fig. 18. Dashed lines indicate the LEI dependence of the K factor and g, expected from the gradual channel approximation. Below a 1 pm gate length at 300 K, the K factor and g , deviate from the LEI dependence. A velocity saturation effect and parasitic source resistances probably play a significant role in these results. The 0.5 pm gate E-HEMT at 77 K exhibits a g, of 500 mS/mm, which is the highest value ever reported for any FET device.I3No significant variation in threshold voltages with gate length was observed in the range from LG = 10 to 0.5 pm, as shown in Fig. 19. This horizontal sensitivity indicates that reducing the geometry of HEMTs is an acceptable way to increase performance with no short channel effect problems. IV. HEMT Integrated Circuits The current implementations and recent advances of HEMT logic and memory integrated circuits are reviewed in this section.
6. LOGICCIRCUITS In 1981, HEMT ring oscillators with gate lengths of 1.7 pm demonstrated a 17.1 ps switching delay with 0.96 mW power dissipation per gate at 77 K as shown in Fig. 20a, indicating that a switching delay below 10 ps will be achievable with 1 pm gate devices.8 Logic performance is also evaluated by testing the 19-stage HEMT ring oscillator. Ring oscillators with 0.5 pm gate HEMTs have achieved switchingdelays of 15 ps/gate with a power dissipation of 1.2 mW/gate and 25 psigate with 0.17 mW/gate at 300 K.13 To evaluate the high-speed capability of HEMTs in complex logic circuits, a single-clocked divide-by-two circuit based on the master- slave flip-flop consisting of eight DCFL NOR gates, one inverter, and four
4.
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FIG. 20. Logic circuits to evaluate the high-speed capability of HEMTs: (a) The first 27-stage HEMT ring-oscillator circuit demonstrated 17.1 ps switching delay in 1981, and (b) single-clocked divide-by -two circuit based on the master-slave tXp-flop consisting of eight DCFL NOR gates, one inverter, and four output buffers.
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output buffers, was fabricated. This circuit is shown in fig. 20b. The circuit has a fan-out of up to 3 and 0.5 mm long interconnects, giving a more meaningful indication of the overall performance of HEMT ICs than that obtained with a simple ring oscillator. The basic gate consists of a 0.5 X 20 pmZ gate E-HEMT and saturated resistors as loads. Direct-writing eleo tron-beam lithography and lift-off techniques were used throughout the fabrication process. Divide-by-two operation is demonstrated at up to 8.9 GHz at 77 K and up to 5.5 GHz at 300 K.13Figure 2 1 shows the operating wave forms of the freqency divider at 77 K. The input clock is 8.9 GHz at a supply voltage V, of 0.96 V. The values of 8.9 and 5.5 GHz, respectively, correspond to internal logical delays of 22 ps/gate with power dissipation of 2.8 mWfgate at 77 K, and 36 psfgate with power dissipation of 2.9 mWf gate at 300 K, with an average fan-out of about 2. Recently, a frequencydivider circuit composed of selectively doped heterojunction transistors (SDHT) with 1 pm gate lengths was also fabricated, showing a maximum clock frequency of 10.1 GHz at 77 K. The speed-power performances of ring oscillator and frequency divider circuits are summarized in Tables I1 and 111. Figure 22 compares switching delay and power dissipation of a variety of frequency dividers.11J3J4~25 The switching speed of HEMT is roughly three times as fast as that of a GaAs MESFET.
Input 7-
output 50 mV I
H Time 100 ps FIG.2 1 . Waveforms of the divide- by-two operation of a frequency divider. The upper signal shows the input clock with a frequency of 8.9 GHz. Horizontal scale is 100 ps/div.
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7. MEMORY CIRCUITS A HEMT 1k X 1b fully decoded static RAM has been successfully developed with E/D-type DCFL circuit configuration.26A microphotograph of the RAM is shown in Fig. 23. The RAM is organized into 1024 words X 1 bit, and arranged as a 32 X 32 matrix. Using a D-HEMT for load devices, E/D-type DCFL circuits were employed as the basic circuit. The memory cell is a 6-transistor cross-coupled flip-flop circuit with switching devices having gate lengths of 2.0 pm. For peripheral circuits, a 1.5 pm gate
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FIG.23. Micrograph of HEMT 1 k X 1 b static RAM, which measures 3.0 X 2.9 mm2 and contains 7244 E/DHEMTs.
4. ULTRA-HIGH-SPEED HEMT INTEGRATED CIRCUITS
BL
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switching device was chosen for performance reasons, and long-gate devices were used as load devices. As the result of RAM layout design, the chip size is 3.0 X 2.9 mm2. The RAM cell size is 55 X 39 pm2 (2145 ,urn2), as shown in Fig. 24. The RAM has total device count of 7244, including 2452 D-type load devices. As a result of the high-power design of the peripheral circuits, the total areas of the peripheral circuits is just the same as the cell array. The circuit diagram of the 1 kb RAM is shown in Fig. 25. The whole circuit is constructed with E/D-type DCFL circuitry. The RAM requires no synchronous mode operation and can operate fully statically. A bit line pull-up scheme and a differential amplifier type sensing circuit are adopted in order to fetch stored data quickly from the low-power memory cell. The cell is assumed to consume the retaining power of 150 pW. The data output circuits are designed to drive the large off-chip load of a 50 R resistor and 15 pF capacitor the same as in the ECL LSI. The output buffer has four amplifier stages with a final stage consisting of a push-pull circuit constructed of high-current E-type HEMTs. The output device has a gate
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ut
FIG.25. Circuit diagram of the HEMT I kb sRAM.
width of 800 pm. In order to obtain high-speed operation, sufficiently high operating current was assigned to peripheral circuits, especially to the address buffer, word driver, and output buffer which have high wiring capacitances. The partition of delay time, power dissipation, and also device count for each circuit stage in the RAM design are shown in Fig. 26. These results were obtained from circuit simulations performed by a SPICE-I1 circuit simulator. The word driver dissipates 47% of the total dissipation power. The entire peripheral circuit with 15% of the total device count dissipates 85% of the total power, whereas the delay time of this stage, from the opening of the transfer gate to the input of the sense amplifier, forms 3 1% of the total delay. The total dissipation power per chip is expected to be 1.0 W at 77 K in this design, and the row address
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FIG.26. Partition of delay time, power dissipation, and device count for each circuit stage in the RAM design.
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271
access time is expected to be 450 ps. The DCFL circuit with HEMT technology has a small logic swing of 0.9 V at 77 K and 0.7 V at 300 K. In order to obtain enough noise margin and guarantee stable logic operation, power lines have to be designed carefully to avoid voltage drops due to the large operating current, especially in the design of the ground (GND) lines. Line widths of the GND lines are from 50 to 200 pm, and the voltage drop is limited to less than 50 mV. A separate line was used to supply power to each circuit block; the cell array, row decoder, and colun decoder/output buffer. This resulted in 23 power pads in RAM. Basic read/write operation waveforms of the 1 kb static RAM at a temperature of 77 K are shown in Fig. 27. Read and write operations with “0” and “1” data for two different address points are performed. The measurement is done with a supply voltage of 1.20 V, and the RAM dissipates about 300 mA of total current. Amplitude for input signals: the row address, data input (D-in), and write enable (WE), are all 1.0 V. The test clock cycle is set to 10 kHz.The RAM can output a 0.55 V data output
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H 100 ys FIG.27. Read-write operation waveforms of the HEMT RAM at liquid nitrogen temperature. The top signal shows the address input; the second, data-input; the third, write -enable; and the bottom, data-output.
M. ABE et
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al,
(D-out) signal. The D-out signal is measured with a 50 R output load. Normal read/write operation was also confirmed at room temperature. Dynamic performance such as the address access time of the HEMT 1 kb RAM was evaluated both at room temperature and liquid nitrogen temperature. At room temperature, the wafer was tested with a probing machine whose probing pins on a card were connected with coaxial cables for high-frequency signals. The RAM chip was mounted in a flat package and tested at liquid nitrogen temperature by immersing the mounted chip into liquid nitrogen. The output signal from the chip was led to an oscilloscope whose input impedance was 50 R. The access time was measured by applying an address pulse having a 2 ns rise time to a row address input. At 300 K, the row address access time was 3.4 ns. The drain supply voltage was 1.30 V, and the chip dissipation power was 290 mW. The performance was greatly improved when the chip was cooled to 77 K, as shown in Fig. 28. An access time of 0.87 ns was obtained at 77 K, where the supply voltage was 1.60 V and the chip dissipation power was 360 mW. A HEMT 4 k X 1 b fully decoded static RAM has also been successfully fabricated and t e ~ t e d ~using ~ * ~the ’ technology described above. Figure 29 is a micrograph of the 4 k X 1 b sRAM. The memory cell is 55 X 39 pm2,the
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Time 500 ps FIG. 28. Oscillograph for memory address access operations. The upper signal shows X-address input and the lower signal, output waveforms. The horizontal scale is 5 0 0 psfdiv.
4. ULTRA-HIGH-SPEED HEMT INTEGRATED CIRCUITS
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FIG.29. Micrograph of HEMT 4 k X 1 b static RAM, which measures 4.76 X 4.35 mmz and contains 26864 E/J%HEMTs.
chip is 4.76 X 4.35 mm2, and 26,864 HEMTs are integrated in a 4 kbit static RAM. Normal read/write operation was confirmed both at 300 and 77 K.The minimum address access time obtained was 2.0 ns, with a chip dissipation power of 1.6 W and a supply voltage of 1.54 V. At 300 K, typical address access time was 4.4 ns with a chip dissipation of 0.86 W. Figure 30 shows the address access time and power dissipation of the sRAM, compared with SiMOS, bipolar, and GaAs MESFET sRAMs. The plot shows the performance of a HEMT 1 kb sRAM at 77 K and dotted lines show the projected performances of 1 and 4 kb S U M S .By using 1
274
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pm gate devices and 2 pm design rule technology, subnanosecond address access times can be projected for the 4 kb sRAM. V. Future HEMT VLSI Prospects Performance of HEMT VLSIs for future high-speed computers is projected and discussed, based on the results of HEMT performance described above.'*'3 Modern computer systems are constructed by stacking printed circuit boards. System delay mainly results from chip delay on the LSI chip and external wiring delay between chips. Chip delay time is the sum of intrinsic gate delay, logic layout delay on fan-out capability, and delay in the wiring on the chip.' Chip delays are calculated based on experimental data for HEMTs with a 1 pm gate length at 300 and 77 K, respectively. Here we assume that fan-out is 3, Y is 10, C,,, is 100 fF/mm, average line length is 1 mm in the chip, and heat flux for liquid cooling is 20 W/cm2.System delay is estimated from the sum of chip delay and external wiring delay of 1 ns, depending on the length of the external wiring network. Figure 31 shows chip and external wiring delays as functions of complexity, under a 0.5 pm design rule HEMT technology. At lo4 gates, the chip delays are 70 ps at 300 K, and 40 ps at 77 K. Figure 32 shows the LSI complexity dependence of system delays calculated for the 1 and 0.5 pm design rule HEMT technologies, at 300 K (solid lines) and at 77 K (broken lines). For a 0.5 pm design rule HEMT technology under liquid nitrogen cooling conditions, optimum system delay of 70 ps is achieved at around lo4 gate integration. By applying a 0.5 pm design rule 10 k-gate HEMT LSIs to a
4. ULTRA-HIGH-SPEED HEMT INTEGRATED CIRCUITS
275
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FIG.31. Chip (-) and external wiring (- - -) delays calculated as a function of LSI complexity,under 0.5 ,urn design rule HEMT technology.
large-scale computer, a system clock cycle time of 2 ns will be realized to achieve computer performance of over 120 MIPS for future large-scale computer requirements. Figure 33 shows the progress of IC complexity. The complexity of Si memory devices doubles each year, as seen from 4 to 256 K. Its progress is slowing down a little at 1 Mbit. Si logic ICs are progressing from 8 to 32 bit microprocessors,as seen from this figure. GaAs MESFET IC complexity is growing approximately threefold each year. Since 1982, self-aligned enhancement-mode MESFET approaches are making it possible to increase the complexity of 3 k gate logic and 4 k bit memory LSIs. Progressing
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and
276
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et al.
Year FIG. 33. Evolution of IC complexity for logic (0)and memory (e), based on Si, GaAs MESFET, and HEMT technologies.
fourfold each year, HEMT will catch up with GaAs integration in about 1 year, and more complex HEMT LSIs will be developed in the near future.
VI. Summary Current status and recent advances in HEMT technology for high-performance VLSI were reviewed with the focus on material, self-alignment device fabrication, and HEMT IC implementation. HEMTs are very promising devices for VLSIs, especially operating at liquid nitrogen temperatures, because of their ultra-high speed and low power dissipation. The projected HEMT performance target suitable for VLSIs is a fundamental switching delay below 10 ps with a power dissipation of about 100 pW per stage under 1 pm design rule technology. By evaluating the gate length dependence of threshold voltage and K factor of short-channel HEMTs, short-channel effects were found not to be a problem in microstructures of submicrometer dimensions. Master - slave flip-flop divide-by- two circuits achieved internal logic delays of 22 ps/gate at 77 K and 36 psfgate at 300 K, at an average fan-out of about 2, roughly three times faster than GaAs MESFET technology. HEMT technology has been shown to have desirable features for high-performance VLSI devices. A HEMT 1 kbit static RAM has been developed and has achieved an address access time of 0.87 ns to demonstrate the feasibility of high-performance VLSIs. Using the same technology, a HEMT 4 Kb sRAM has also been successfully fabricated and normal read/write operation confirmed. With the device technology of 1 pm gate devices and a 2 pm line process, a HEMT 4 kb sRAM should achieve
4.
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subnanosecond access operation. Using the experimental data on HEMT logic, we project an optimized system performance of 70 ps at 10 k gate integration with a HEMT VLSI at liquid nitrogen temperature. This system’s performance will achieve speeds higher than 120 MIPS for future large-scale computer requirements. ACKNOWLEDGMENTS The authors wish to thank Dr.M. Fukuta for encouragement and support. The authors also wish to thank their colleagues,whose many contributions have made possible the results described here. This work is supported by the Agency of Industrial Science and Technology, MITI of Japan, in the frame of National Research and Development Project “Scientific Computing Systems.”
REFERENCES 1. M. Abe, T. Mimura, N. Yokoyarna, and H. Ishikawa, ZEEE Trans. Electron Devices ED-29, 1088 (1982). 2. R. L. VanTuyl and C. A. Liechti, ZEEE J. Solid-state Circuits SC-9,269 (1 974). 3. R. C. Eden, B. M. Welch, and R. Zucca,ZEEE J. Solid-state Circuits SC-13,419 (1978). 4. N. Yokoyama, H. Onodera, T. Shinoki, H. Ohnishi, H. Nishi, and A. Shibatomi, ZSSCC Dig. Tech. Pap., p. 44 (1984). 5 . Y. Nakayama, K. Suyama, H. Shimizu, N. Yokoyama, A. Shibatomi, and H. Ishikawa, ZSSCC Dig. Tech. Pap., p. 48 (1983). 6. R. Dingle, H. L. Stormer, A. C. Gossard, and W. Wiegmann, AppZ. Phys. Lett. 33, 665 ( 1978). 7. T. Mimura, S. Hiyarnizu, T. Fujii, and K. Nanbu, Jpn. J. Appl. Phys. 19, L225 (1980). 8. T. Mimura, K. Joshin, S. Hiyamizu, K. Hikosaka, and M. Abe, Jpn. J. Appl. Phys. 20, L598 (198 I). 9. P. N. Tung, P. Delescluse, D. Delagebeaudeuf, M. Laviron, J. Chaplart, and N. T. Linh, Electron. Lett. 18, 5 17 (1982). 10. J. V. DiLorenzo, R. Dingler, M. Feuer, A. C. Gossard, R. Hendel, J. C. Hwang, A. Kastalsky, V. G. Keramidas, R. A. Kiehl, and P . O’Connor, Tech. Dig.-Znt. Electron Devices Meet., p. 578 (1982). 11. C. P. Lee, D. Hou, S. J. Lee, D. L. Miller, and R. J. Anderson, ZEEE G d s ZC Symp.. Tech. Dig., p. 162 (1983). 12. K. Nishiuchi, T. Mimura, S. Kuroda, S. Hiyamizu, H. Nishi, and M. Abe, 4lst Annu. Dev. Res. Conf. IIA-8 (1983). 13. M. Abe, T. Mimura, K. Nishiuchi, A. Shibatomi, and M. Kobayashi, ZEEE GaAs ZC Symp., Tech. Dig., p. 158 (1983). 14. S. S. Pei, R. H. Hendel, R. A. Kiehl, C. W. Tu, M. D. Feuer, and R. Dingle, 42ndAnnu. Dev. Res. ConJ, IIA-2 (1984). 15. T. Mirnura, S. Hiyamizu, K. Joshin, and K. Hikosaka, Jpn. J. Appl. Phys. 20, L317 ( 198 1). 16. S. Hiyamizu, Collect. Pap. Znt. Symp. Mol. Beam EpitMy Relat. Clean SurJ:Tech., 2nd. 1982. Pap. A-7-1 (1982). 17. S. Hiyamizu, T. Mimura, and T. Ishikawa, Jpn. J. Appl. Phys. 21, Suppl. 21-1, 161 ( 1982).
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18. T. Mimura, K. Nishiuchi, M. Abe, A. Shibatomi, and M. Kobayashi, Int. Electron. Devices Mater. Symp., B2.2, 193 (1984). 19. A. Shibatomi, J. Saito, M. Abe, T. Mimura, K. Nishiuchi, and M. Kobayashi, Tech. Dig. Int. Electron Devices Meet. (1984). 20. S. Kuroda, T. Mimura, M. Suzuki, N. Kobayashi, K. Nishiuchi, A. Shibatomi, and M. Abe, IEEE GaAs IC Symp., p. 30 (1984). 21. K. Hikosaka, T. Mimura, K. Joshin, and M. Abe, Proc.-Electrochem. SOC.,p. 163 (1982). 22. T. Mimura, K. Nishiuchi, M. Abe, A. Shibatomi, and M. Kobayashi, Tech. Dig.-Int. Electron Devices Meet., p. 99 (1983). 23. T. Mimura, S. Hiyamizu, M. Abe, and H. Ishikawa, Jpn.-USSR Electron. Symp. Proc., 8th, p. 98 (198 I). 24. J. A. Turner and B. L. H. Wilson, Con$ Ser.-Inst. Phys., p. 195 (1968). 25. R. A. Kiehl, M. D. Feuer, R. H. Hendel, J. C. M. Hwang, V. G. Keramidas, C. L. Allyn, and R. Dingle, IEEE Electron Device Lett. EDL-4,377 (1983). 26. K. Nishiuchi, N. Kobayashi, S. Kuroda, S. Notomi, T. Mimura, M. Abe, and M. Kobayashi, ISSCC Dig., Tech. Pap., p. 48 (1984). 27. M. Abe, T. Mimura, K. Nishiuchi, A. Shibatomi, and M. Kobayashi, Int. Conf Solid State Devices Mater. C-S-(l), 359 (1984).
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SEMICONDUCTORS AND SEMIMETALS, VOL. 24
CHAPTER 5
Nonlinear Optical Properties of Multiple Quantum Well Structures for Optical Signal Processing D. S. Chemla and D. A . B. Miller AT&T BELL LABORATORIES HOLMDEL, NEW lERSEY 07733
P. W. Smith BELL COMMUNICATIONS RESEARCH, INC. HOLMDEL, NEW JERSEY 07733
I. Introduction Optical processing of information has received increasing attention in recent years. It has several potential advantages over electronic processing. For example, optics should be ideal for handling large amounts of data in parallel because it does not suffer from the same interconnection problems. It is also in principle possible to make very fast optical switches, and optical signal processing has obvious advantages when the signals are already in the form of light. However, both analog and digital optical processing have suffered from a shortage of suitable nonlinear materials from which lowenergy optical switches or signal amplifiers can be made. Without large nonlinearities in convenient materials, the intriguing opportunities of optical processing cannot be exploited. It has recently been found, however, that semiconductor quantum well material exhibits several interesting nonlinear optical effects. Not only are these effects larger than comparable effects in other materials, but they are seen at room temperature, and at wavelength, power levels, and time scales compatible with laser diodes and/or electronics. Furthermore, being made with semiconductor materials, quantum well structures can be fabricated with a high degree of reproducibility, and the advanced technology of semiconductor preparation is available to assist in making devices. The nonlinear optical effects utilized by various of the methods of optical signal processing, whether analog or digital, are the variations of the refractive index (n) or of the absorption coefficient (a)induced in a mate-
279 Copyright 0 1987 BeU Telephone Laboratories,Incorporated. AU rightsof repduction in any form r e ~ e ~ e d .
280
D. S . CHEMLA eta/.
rial medium by an electromagneticperturbation. After traveling through a length z in the medium, an optical field experiences a phase shift A$ given bY
A 4 = ($n + f a) z The shifts associated with an optical excitation E, correspond to nonlinear optical effects, i.e., n + n(E,) and a + a(&). Those associated with a static electric field E, correspond to electro-optic effects, e.g., electroabsorption, a a(Eo), or field-induced changes in refractive index, n-n(E,). To be detected easily the phase shifts must be such that Re(A4) - a or Im(A4) 1. To be of interest for practical applications, the nonlinear refractive index ~ Z ( E , ,-~ n(0) ) or the nonlinear absorption coefficient - a(0)should be large, fast, and convenient to use. In that respect semiconductors, which are relatively polarizable, have received a lot of attention.1-4 In the transparency domain high-field excitation and/or long optical length are still necessary to produce substantial effects. In order to increase the magnitude of the nonlinear responses, it is possible to exploit the resonant enhancement obtained by using frequencies close to optical transitions of the m e d i ~ m . Large ~ . ~ enhancements are observed in the vicinity of steep and/or narrow electronic transitions such as those associated with correlated electron- hole systems, i.e., excitons. Because the interaction of camers with the vibrations of the crystal tends to destroy the electron - hole correlation, in bulk semiconductors excitonic effects are only seen at low temperature where the density of thermal phonons is small. Very large nonlinear optical responses have been observed and utilized in laboratory experiments in semiconductors at low temperature. However, these inconvenient conditions have so far limited the utilization of bulk semiconductors in practical applications. Recently modern techniques of crystal growth such as molecular beam epitaxy (MBE) or metal - organic chemical vapor deposition (MOCVD) have permitted the fabrication of heterojunctions which are smooth down to one atomic monolayer with perfectly controlled composition and doping concentration.6Using semiconductors exhibiting specific chemical and crystallographic compatibility, it is possible to grow alternatively very thin layers of each compound to form multiple quantum well structures7 (MQWS). Examples of these structures have been grown using a number of I11- V, I1- VI, and IV - IV compounds. Because of the very small thickness of the layers which can be achieved, quantum size effects occur that provide MQWS with unusual electronic and optical properties8 Since the two compounds do not have the same energy gap, for undoped samples the band structure exhibits a series of rectangular steps in real
-
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5.
MULTIPLE QUANTUM WELL STRUCTURES
281
space in the direction perpendicular to the layers. In certain systems, such as GaAs/AlGaAs, the minimum of the conduction band and the maximum of the valence band occur in the same compound (Fig. 1). The motion of the carriers is restricted in the direction of the normal of the interfaces, z, but exhibits quasi-two-dimensional behavior in the plane of the layers, xy. If the height and width of the potential barriers are large enough, the quantization of energy in the z direction results in a discrete spectrum and thus two-dimensional energy subbands. The large extension of the envelope wave functions in the z direction and the associated energy subbands are specific properties of MQWS that are most important for the effects considered in this chapter. At the optical transitions between valence and conduction subbands, excitonic effects are expected. Indeed they are enhanced whenever the thickness of the low-gap compound is comparable to or smaller than the electron-hole correlation distance, i.e., the Bohr radius of the exciton in the host material. The artificial reduction of the average distance between the electron and the hole corresponds to an increase of the exciton binding energy. The exciton can be sufficiently stabilized to be observable at room ternperat~re.~
Eg
I
FIG. 1. Schematic of the band structure of a multiple quantum well structure, in real space and along the normal of the layer. The dashed lines represent the carrier wave functions. The cross-hatched circle and ellipse illustrate the bulk exciton and how it is shrunk by the carrier confinement.
282
D. S. CHEMLA eta[.
Specifically, two properties of quantum wells which are relevant for nonlinear optical devices are discussed in this review. First, quantum wells show optical absorption saturation associated with the remarkably distinct room-temperature excitonic resonance. Second, they show a large electricfield dependence of the absorption, both associated with the excitonic transition and also directly with the transition between two-dimensional subbands. All these effects are seen near the optical absorption edge. The work reviewed here is primarily,from recent experiments on GaAs/AlCaAs MQWS. The chapter is organized as follows: In Section I1 we discuss the linear absorption in MQWS with a particular emphasis on room-temperature effects. In Section I11 we present the measurements of nonlinear optical effects in MQWS, including absorption saturation and four-wave mixing. Finally, in Section IV we review the experimental and theoretical studies of electroabsorption in MQWS, and we present the first example of a new category of devices that exploits both the absorption saturation and electroabsorption. 11. Linear Absorption in Multiple Quantum Well Structures
In this section we discuss the linear absorption in MQWS in the fundamental gap region. MQWS exhibit some very specific features intermediate between two-dimensional and three-dimensional systems. In addition, because the normal to the layer defines a natural quantization axis, the already complex valence-band structure of the host material is strongly modified, giving rise to new selection rules for the optical transitions. We will use descriptions of increasing complexity to approach the problem of absorption by MQWS. We first analyze the case of an ideal 2D semiconductor with a parabolic valence band and a parabolic conduction band. Then we discuss the quantization of states in the direction perpendicular to the layers and the consequences of this quantization for the band structure in the plane of the layers. Finally, we address the problem of real excitons in GaAs/AlGaAs MQWS with a special emphasis on the room-temperature properties. 1. THEORY OF LINEAR ABSORPTION AND BANDSTRUCTURE
Let us first consider a purely two-dimensional electron- hole system. The particles are bound to the x, y plane, and their dispersion is described by parabolic bands:
5.
MULTIPLE QUANTUM WELL STRUCTURES
283
The correspondingjoint density of states is a step function starting at Eo, i.e.,
+
where pi is the reduced mass in the plane pi' = rn;' rn:', and 0 is the Heaviside step function. The reduction of dimensionality transforms the well-known parabolic profile of the absorption coefficient into a step of constant height. As in the 3D problem, the Coulomb interaction between the electron and the hole gives rise to excitonic effects. It is important to note that only the motion of the particles is two dimensional; Coulomb interaction is still three dimensional with an r-l dependence. It turns out that the two-dimensional hydrogen atom problem can be solved exactly.'O The main effect of the reduced dimensionality is to remove the wave-function dependence on the azimuthal angle 0, with the consequence that the principal quantum number n becomes n - 4. For instance, the 2D Rydberg series is written EiD= -R,,/(n - +)2, where R,, = e44/e2h2is the Rydberg constant of the system. The oscillator strengths of the hydrogenic peaks decrease more rapidly in 2D than in 3D, i.e., (n - 4)-3 versus c3. The Coulomb enhancement of the absorption above the bandgap, which is described by the Sommerfeld factor F(fiw)gives in 2D a rise from the value of the steplike continuum far above the bandgap as the photon energy is
E:P
E,
c:
El
ti,
FIG.2. Schematic comparison of the absorption spectra of ideal three-dimensional and absorption spectra two-dimensional semiconductors. (---), joint density of states. (-), with Coulomb interaction.
284
D. S. CHEMLA eta[.
TABLE I CORRESPONDENCE BETWEEN THE PARAMETERS DESCRIBING EXCITONS IN PURE THREE-DIMENSIONAL A N D TWO-DIMENSIONAL
SEMICONDUCTORS~
2D
3D Density of states
3
V E - E, ‘1’ -2Z24Ry
[
Ry
Energy Oscillator strength
f ,=
2 IPJ2 -
nrn,&fio Z
Sommerfeld factors
F(W) =
n3
p/fi
sinh(z/m
~~
-
a The symbols used are defined as follows: rn, is the free-electron mass,
W is a reduced and normalized energy: W = ( E - E,)/Ry in 3D and W = ( E - E I I ) / R yin 2 D . The Sommerfeld factors F( W) give the absorption enhancement in the continuum due to electron-hole correlation.
reduced to a value twice as large at the ionization limit (i.e., the bandgap), as compared to the almost flat spectrum in 3D. The comparison between the absorption spectra in ideal 2D and 3D semiconductors is illustrated in Fig. 2, and the correspondence between the parameters describing 2D and 3D excitons is summarized in Table I. In an actual system the electrons and holes are bound to layers of finite thickness, and the energy has to be quantized both in the plane of layers and perpendicular to it. In the case of undoped semiconductors and for camers very tightly bound to the layers, one can approximate the potential in the z direction by an infinitely deep square well, which gives an infinite series of bound states for each particle:
5.
MULTIPLE QUANTUM WELL STRUCTURES
285
wherej is an integer labeling the subbands, L, is the thickness of the layers, and rn, is the mass in the direction normal to the layers. In this approximation the absorption spectra exhibit a series of steps starting at Eje- Ejh, with the selection rule A j =j , -j h = 0 imposed by the symmetry of the wave functions &.(ze) and rjh(zh).A more realistic model consists of a potential well of finite depth; then the number of bound states is limited. The corresponding wave functions have a sinusoidal dependence in the well and exponential tails in the barrier regions. For a symmetric well the bound-state energies Ej are solutions of
where ej = Ej/EY and v = V/EY are the normalized energies and potential. There is always at least one bound state. This model is indeed better than the infinitely deep well model; it, however, raises immediately two important questions related to the masses of the electrons and holes. In a quantum well structure a semiconductor layer of one compound is sandwiched between layers of another compound. Because the carriers have different effective masses in each compound, the proper boundary conditions at the interface must be that the wave function and the probability current (and not the derivative of the wave function) must be continuous at the interface,'' i.e., ((z) and rn:' ay/ez. The consequences on the electron states are not drastic; the second term of Eq. (6) is simply multiplied by (rnAL/rnBL)1/2,where mA, and rn, are the effective masses in the two compounds. For the holes the question is much more intricate. The valence-band structure of the host material involves the J = 4 upper valenceband multiplet and the J = lower valence band, which is depressed by the spin orbit splitting. In the bulk GaAs the dispersion of the J = 3 multiplet is given by12J3
+
+
+
E(k)= Ak2 -t [ B 2 P C2(k:k; + k;kz kzk;)]'/2 (7) where the inverse mass band parameters are defined according to Dresselhaus et al.I3;they are simply related to the Lutthger parameters12yl, y 2 , y3 and the free-electron mass m,. Along the 2 direction the masses to be used in the quantization of the energies are given by (A f B) = h2(yl k 2y2)/rn0. The degeneracy of the upper valence band is therefore lifted by the confinement,14 giving two separated hole bands, a heavy hole with mirL= mo/ (yl - 2,) 0.45% and a light hole with rn, = %/(yl + 2y2)- 0.08%. Note that this definition relates to the 2 direction only. In the plane of the layer the effective masses are much more complicated to determine. In the case of the infinite well depth where the wave vector along 2 is perfectly defined, the energy bands can be obtained by cutting the three-dimensional
-
286
D. S. CHEMLA et a/.
energy contour of the host material by the planes k, = f n/L,. This procedure has been used to describe Si p-channel 1a~ers.I~ For finite well depth the only rigorous approach is to solve exactly the 6 X 6 effective matrix equation. State mixing is expected as soon as kx, # 0 with highly nonparabolic bands for the motion of holes parallel to the layers; this has not been satisfactorily described even with only one type of layer. Of course, the boundary conditions are even less well described. Nevertheless, intuitively it can be expected that near k , = 0 the heavy-hole subband has mostly a I+, k 4 > character, whereas the light hole one is mostly 13, k >.16 In that case the selection rules imposed by symmetry are that the intensities of the transition to the conduction band are proportional to f and for a field parallel to the layers and, respectively, for the heavy- and the light-hole subbands, and to 0 and 1 for a field perpendicular to the layers.17
,,
2.
QuAsI-2D
+
EXCITONBINDINGENERGY
When the Coulomb interaction is accounted for, two types of excitons can be formed with the two hole subbands.I4 The natural parameter for measuring the amount of 2D nature of an exciton in a layer is LJa,, where a,is the bulk exciton diameter. This latter is built up from the electron and the isotropic part of the hole,18J9its reduced mass is p;; = my’ yl /mo, and the 3D Bohr radius is a,= &h2/dp3,. In GaAs, Ry- 4.2 meV and a,- 150 A.For very thick layers (L,/a,>> 1) the excitons are nearly three dimensional, whereas for very thin layers (L,/ao< < 1) they approach their 2D limit. The difference between the true Coulomb potential and the 2D limit is
+
+ +
AHc = (eZ/&)[(x2 y 2
+ y2)’/*]
z2)lI2- (x2
(8) AHc can be treated as a perturbation to analyze the effect of a finite well thickness. Its magnitude is of the order of R,,(L,/a,)2; it implies that the binding energy is very sensitive to the well thickness.” In the intermediate range (L,/ao - 1) the confinement reduces the average distance between the electron and the hole, thus increasing the binding energy over that of the bulk exciton. Let us note that the exciton wave function is an admixture of states belonging to a domain of the Brillouin zone definedI k l s a; l , where a, is the exciton radius (which may not be 4).Thus, strictly speaking, both excitons formed with the light and heavy hole utilize admixtures of the 14, -t$> and [$, f > states with complicated effective masses. The choice of these effective masses to determine the binding energies and the Bohr radius is critical. Up to now, approximate values16 have been used to calculate the binding energy El, by variational procedures.21*22 It is found that the enhancement is only substantial in the range
+
5.
MULTIPLE QUANTUM WELL STRUCTURES
287
+
I (LJa,,)I2, for instance, for L, - 4 then El,- -2Ry. In the two limits, 2D and 3D, it is found that a,&, = e2/2&.This relation can therefore be used to evaluate the exciton diameter from the binding energy. It is found that the whole charge distribution shrinks rather than flattening, giving ellipsoidal excitons with an aspect ratio close to unity. This can be interpreted as being due to the fact that a flattened charge distribution involves a mixture of high-lying states which are not energetically favorable. This discussion strictly applied to the case of a single quantum well. In the case of a multilayer system, as the thickness of the layers decreases, the exponential wave function tails extend more and more into the largegap compound, and their overlap eventually becomes large enough to give extended states with again a 3D character. The binding energy then decreases toward that of the 3D exciton of the Altogether it is found that in GaAs MQWS, the binding energy is typically maximum in the range a,,/2 5 L, I a,,, for which -3Ry I E l , I -2Ry.
3. LOW-TEMPERATURE EXPERIMENTAL RESULTS The low-temperature absorption spectroscopy of GaAs/Al,Ga, -As MQWS indeed shows the steplike structure of the absorption edge with a double exciton peak at each subband transition.l4rZ4 These measurements were used to determine how the bandgap discontinuity AEg = 1.247x, is shared between the valence and the conduction band. It was foundI4that approximately 60%of the discontinuity occurs in the conduction band and 4090 in the valence band, which therefore exhibit much shallower wells. However, very recent measurements on parabolic quantum wells tend to indicate that the bandgap discontinuity is more equally divided between the two bands.25The assignment of the light- and heavy-hole excitons to valence-band states primarily of I$, +_ 4 > and 13, f > character was obtained by spin polarization technique^.^^.^' Low-temperaturephotoluminescence measurements on undoped and modulation-doped waveguide samples have provided information on optical transition selection rules with the wave vector along the plane of the layers and polarization parallel and perpendicular to the layer^.^*,^^ It was found that close to lkl= 0 the hole characters are indeed mostly M, = _+ 3 and 3z 4 for the heavy and light holes, with an admixture 10- 20% of M, = k4 in the heavy hole dependent on the well thickness and therefore also on the subband splitting. However, for wave vectors parallel to the plane of the layers and of the order of n/L, the admixture of M, = k3 and states is almost com~ l e t e Measurements .~~ in wave guides show that the ratio of oscillator strength follows the trends indicated by the symmetry arguments,17but the exact values actually depart slightly from the theoretical ones. For a field
+
-
++
288
D. S. CHEMLA ef af.
parallel to the layers it is found in undoped samples that the heavy-hole to light-hole exciton oscillator strength ratio lies somewhere between 2.5 and 2, whereas it is predicted to be equal to 3. For a field perpendicular to the layer, the heavy-hole oscillator strength is indeed reduced substantially, to a value 0.03 to 0.1 times that for a field parallel to the layers.28The value of the binding energy was evaluated from the observation of the IS to 2s splitting,” confirming the large enhancement over that of the bulk compound exciton. The width of the photoluminescence peak was found to increase when the well thickness decreased. The result was interpreted as being due to an inhomogeneous broadening of the transition induced by fluctuations of the layer t h i c k n e s ~ . The ~ ~ *magnitude ~ of the inhomogeneous broadening was found to be consistent with an interface presenting an islandlike structure, the islands being approximately one atomic layer high and about 300 A in diameter. This interpretation is supported by resonant Rayleigh scattering and hole-burning experiment^.^'-^^ They show that within an inhomogeneous line there are two types of excitons: the low-energy ones which are localized and have a very narrow homogeneous width, r, - 0.02 meV, and delocalized ones with homogeneous widths, r, - 1.5 - 2 meV.
4. TEMPERATURE DEPENDENCE AND ROOM-TEMPERATURE EXCITON RESONANCES The thermal broadening of the exciton peaks arises from the interaction with thermal phonons, and in polar semiconductors it is dominated by the exciton-LO phonon interaction. The effect of the layered structure of MQWS on the exciton- LO phonon interaction has been investigated by resonant Raman s ~ a t t e r i n g .It~ was ~ , ~inferred ~ from the cross-section double-peak asymmetry that LO phonons couple excitons of different subband origin (for examplej = 2 to 3), in sharp contrast with bulk semiconductors, where only an intraband process exists. In addition, it was found that low-lying excitons ( j = 1, 2) only interact with GaAs phonons because their wave functions are well confined within these layers. It is only for high states ( J = 3 or excitons lying above the bandgap discontinuity) that interaction with Al,Ga, -As phonons are o b ~ e r v e d . ~ ~ , ~ ~ The confinement of excitons in the GaAs layers produces two specific and complementary effects. It increases the binding energy and consequently decreases the size of the exciton. It also changes the interaction with polar phonons, which are restricted to those of the GaAs layers only. This double feature is at the origin of the observation of exciton resonances at room t e m p e r a t ~ r e ~and , ~ ~abovem - ~ ~ in the MQWS absorption spectra. This is shown in Fig. 3, where the absorption coefficient at 300 K of a 3.2
5.
MULTIPLE QUANTUM WELL STRUCTURES
1.4
PHOTON ENERGY (eV) 1.5
289
1.6
050 800 WAVELENGTH (nm)
FIG.3. Room-temperature absorption spectra of a 3.2 pm thi.ck high-purity GaAs sample and of an MQW sample consisting of 71 periods of 102 A GaAs layers and 207 A Ab,,Gk,7,,,Aslayers. Insert:Temperature dependenceof the linewidth of thej = 1 heavy-hole exciton. (-), the fit discussed in the text.
pm thick high-quality GaAs sample is compared to that of a MQWS consisting of 77 periods of alternate 102 A GaAs layers and 207 A Al,,28G%.,2Aslayers. On the GaAs sample absorption coefficient one can observe a small bump just at the band edge which is the remains of the excitonic resonance. In the MQWS spectrum two clear resonances, corresponding to the heavy- and light-hole excitons, are seen followed by an almost flat continuum and the onset of the j = 2 resonances. The insert shows the variation of the j = 1 heavy-hole exciton width r for temperatures ranging between liquid helium and room temperature, as measured on the low-energy side of the resonance. It is approximatelyconstant under 150 K and increases with temperature thereafter. The solid line is a semiempirical fit involving two parameters r, and r,, corresponding to a constant inhomogeneous term and a term proportional to the density of thermal LO phonons of GaAs layers, i.e., r(T) = r,
+ rPh/[exp(fiRm/kT)- 11
(9)
290
D. S. CHEMLA et a/.
where kR, = 36 meV = 428 K is appropriate for the GaAs LO phonons. The fitted value, ro= 2 meV, is in good agreement with the low-temperature luminescence r e s ~ l t s . ~That ~ , ~of~ rph , ~= ~ 5, ~meV, somewhat smaller than that of bulk GaAs4' (rph - 7 meV), is consistent with a reduced thermal phonon broading. The line shape of the resonance is rather unusual for a Wannier exciton. In fact, the low-energy edge is too sharp to describe the resonance by a simple Lorentzian line shape. An excellent empirical fit has been foundm by using Gaussian line shapes for the excitons and a broadened two-dimensional continuum. This type of fit is very convenient to discuss data; however, one should avoid extracting precise conclusions from it, especially considering that the line shape of excitons in MQWS is not yet understood. It should be noted that other systems such as disordered molecular crystals exhibit excitonic peaks with Gaussian line shapes or more complex (Gaussian on the low-energy side and Lorentzian on the high-energy side). As an illustration of the parameters obtained by our fitting procedure, for a sample consisting of 65 periods of 96 A GaAs layers and 98 A Ak,,G%,,As layers, one finds for the two excitons a line width r - 3 meV, a separation fiQ, - fiR& 8.5 meV, a binding energy for the heavy-hole exciton E k - 10 meV, and a ratio of oscillator strengthf,/f, - 2. At this point it is important to discuss the very special situation in which excitons find themselves at room temperature. It is characterized by the inequality
-
-
where fiRLo- 36 meV, k T - 25 meV, Ek- 10 meV, and r 3 meV. First, because El, > r, the exciton peak is well resolved from the continuum. Then because kQm > E,,, any LO-phonon exciton collision results in the exciton ionization. The mean time to LO-phonon ionization AtT, can be evaluated from the uncertainty principle and the thermal contribution to the exciton width i.e., T ( T )- roin Fig. 2. It is foundmthat at room temperature AtT - 0.4 ps. This means that, once an exciton is created by a photon absorption, it lives less than half a picosecond before being dissociated into a free electron and a free hole by the thermal vibrations. Finally, because kT 2.5E,,, very few excitons are reformed from the electronhole plasma. To support this conclusion let us consider the thermodynamical equilibrium between a steady-state population of excitons (for instance, in a sample under constant optical excitation) and the electron hole plasma, i.e.,
-
5.
MULTIPLE QUANTUM WELL STRUCTURES
291
The exciton, electron, and hole areal densities at equilibrium are N,, N,, N,,. Let No be the areal density of exciton generated by the absorption of a constant flux of photon No = z q w L , I / h w , where z is the exciton lifetime, aQW the absorption coefficient in the GaAs layers, and I and hw the photon intensity and energy. The particle conservation condition is No.= N, -tNh, the neutrality condition, N, = Nh and the equilibrium condition, i.e., the Saha equation adapted to a two-dimensional system, N,Nh/Nx = (m:kT/ zh2) exp(- E,,/kT). These three relations completely determine N, and N,, giving an ionization ratio N,/No which is always very large. It varies from 1 for small No to 0.7 for large No, i.e., corresponding to a probability to find one carrier in an exciton area of 50%. Note that this simple evaluation in fact underestimates the ionization ratio since it neglects a number of other channels through which the number of nonionized excitons can be reduced.
5. SUMMARY To summarize this section, we can conclude that the confinement of excitons in narrow layers thinner than the host material exciton diameter increases the exciton binding energy by a substantial amount (2.5 to 3) and it restricts the exciton-LO phonon interaction, resulting in a slightly smaller thermal broadening (- 40%). The combination of the two effects makes it possible to observe exciton resonances at room temperature. However, once generated, excitons have a rather short lifetime, of the order of half a picosecond, before being ionized into a free electron and a free hole. 111. Excitonic Nonlinear Optical Effects in Multiple Quantum Well Structures
6 . NONLINEAR ABSORPTION A N D REFRACTION DUE TO EXCITONSIN SEMICONDUCTORS The refractive index and absorption coefficient of semiconductors exhibit large variations under optical excitation. For fields of moderate magnitude these effects are described by an expansion of the polarization density in powers of the fields. The relevant series expansion for nonlinear absorption and refraction is
+
P(w)= p ( w ) E ( w )
x‘3’(-
0; w, w’, - w’)lE(w’)l2E(w)
(12)
This is equivalent to saying that the dielectric constant is field dependent,
292
D. S . CHEMLA ef al
ie., 8=
+
1 s 4n(X(*)
(13) where the frequency indices have been dropped for simplicity. The complex wave vector is ,(3)142)
0 2n i -Ji=-n+-a! C 1 2 Using Eqs. (1 3) and (14) one finds for small values of $3)IE12 that the refractive index and absorption coefficient depend on the intensity according to
n(1)= n
+ nzz,
a(1)= a! + 0 2 1
(15)
with
The structure of the nonlinear susceptibility of semiconductors near excitonic resonances has been analyzed in Chemla and Maruani.’ It exhibits strong enhancement when the field frequencies are close to excitonic transitions, and given rise to very large nonlinear effects which have been observed and utilized at low temperature in high-purity crystal^.^^,^ Under these circumstances the transitions are no more virtual, and large changes of populations can occur. A review of the mechanisms involved in the nonlinear optical properties of excited bulk semiconductors is given in Miller et aL3 Close to the bandgap, exciton screening and bandgap renormalization are the most relevant processes to consider. At low temperature, where excitons are stable, the screening by neutral particles is rather weak:’ and it is only at high density that an effective screening of the Coulomb interaction by a charged electron -hole plasma occurs; the plasma is created through collision-induced exciton ionization. Exciton resonances then disappear, giving large changes in the absorption and refraction. Direct screening of excitons by the electron -hole plasma is observed when the carriers are directly created by excitation well above the and bandgap. In this case it is customary to introduce the parameters oeh,which characterize the variation of the refractive index and of the absorption coefficient induced by one electron - hole pair. Similarly to Eq. ( 15) one writes n ( N ) = n + n,,N, a ( N ) = (Y -k O h N (17) where N is the carrier density. (The correspondence between n2, a2and neb, oeh is discussed in Ref. 40.) Saturation of excitonic absorption in
5.
MULTIPLE QUANTUM WELL STRUCTURES
1.2
-
n=1
293
UNPUMPED ABSORPTION
1.0 0
0.8 pc
0.6
rn
*
0.4
0.2
-
n 1.51
1.54
1.57
1.60
ENERGY (eV)
FIG.4. Absorption spectra of an MQW sample consisting of 156 periods of 205 A GaAs layers and 224 A AI,,,G%,,As layers before (-) and at later times ( * ; ---) following excitation about 130 meV above the bandgap.
-
MQWS at low temperature by high excitation above the bandgap were observed in investigations of the dynamics of screening using ultrashort light p ~ l s e s . 4For ~ ~a~carrier ~ density - 5 X 10" cm-* complete bleaching of the j = 1 exciton resonance is obtained. The recovery of the absorption was used to measure the cooling rate of the camers injected about 130 meV above the gap (Fig. 4). The cooling occurs in approximately 100 ps. 7. OBSERVATIONS OF NONLINEAR EXCITONIC EFFECTS
In the case of room-temperature exciton resonances observed in MQWS, a new situation occurs. Large absorption is observed at the exciton resonances just as for the low-temperature bulk semiconductors. However, because the excitons are very quickly ionized by thermal phonons, a charged electron - hole plasma is generated just as for above-bandgapexcitation. Therefore, for excitation with long optical pulses or cw laser sources, the nonlinearities result from the conjunction of two factors, the concentration of a large oscillator strength in narrow energy bands and the saturation by the plasma. Saturation of excitonic absorption at room temperature in MQWS has been observed under moderate intensities using cw laser excitation at re~onance~,~' and above the bandgap.40 Figure 5b shows the intensity dependence of the optical absorption at the heavy-hole exciton peak of the MQWS sample whose low-intensity absorption spectrum is given in Fig. 3. For comparison, Fig. 5a shows the result of a similar measurement on the high-purity GaAs sample whose absorption spectrum is also given in Fig. 3. The solid curves are semiempirical Lorentzian saturation fits to the data.
294
D. S. CHEMLA eta1 INCIDENT POWER (mW) 0.1 1 10
0.01
v) v)
W
Z Y
2 r
0
0.2
00
t-
\
t'
i
I
I
I
I I
100
1000
10,000
100,000
EFFECTIVE AVERAGE INTENSITY (W/cm2)
FIG.5. Intensity dependence of the absorption at the peak of the exciton resonance for (a) the the GaAs and (b) MQW samples whose absorption spectra are shown in Fig. 2. (-), empirical fits discussed in the text.
For GaAs the saturation is well described by a constant absorption plus a saturable one with a saturation intensity of 4.4 kW/cm2, i.e., a1 = 0.5 1 S / (1 Z/4.4 X lo3).The fit obtained for the MQWS sample corresponds to two saturable species, one with a very low saturation intensity about one order of magnitude lower than for GaAs, I, = 580 W/cm2,and another one with a much larger saturation intensity Z, = 44 kW/cm2, i.e., a1 = 0.35/ ( 1 Z/580) 0.43/( 1 1/44 X lo3).This behavior is interpreted as being due to a low-intensity saturation of the exciton superimposed on that of transitions more difficult to saturate, most likely interband transitions, which, because of bandgap renormalization shifts below the position formerly occupied by the resonance. Confirmation of the easy saturation of MQWS exciton absorption was obtained using a picosecond tunable laser; at resonance the saturation intensity was found to be Z, = 500 W/cm2, in
+
+
+
+
+
5.
295
MULTIPLE QUANTUM WELL STRUCTURES
excellent agreement with the cw measurement. The very low value of I, indicates that cw diode lasers can be used to saturate exciton absorption in MQWS. This is demonstrated in Fig. 6, which shows absorption spectra measured with a tunable dye laser at very low power ( I - 1.6 W/cm2) without (dashed line) and with (solid line) pumping from a diode laser beam. The MQWS sample consists of 65 periods of 96 A GaAs and 98 A AL,zsG%.,zAslayers; it is more recent and of better quality than that from which Figs. 3 and 5 were obtained. The rapid oscillations are due to Fabry-Perot interferences in the thin glass disc covering the sample. In this experiment the cw laser diode intensity incident on the sample was 800 W/cm2. It was operating well above the exciton resonance, at fia = 1.492 eV. Note that in this case the photocarriers are directly generated as an electron- hole plasma. It can be seen in Fig. 6 that the exciton saturation corresponds to an increased transmission at the resonances. However, it is accompanied by broadening and/or shifts because the two curves cross
z 2
t a
.5
0
cn
m
a
I
I
1
I
1 1.490
4.440
PHOTON ENERGY
FIG. 6. Absorption spectra of an MQW sample consisting of 65' periods of 96 A GaAs and without (---) a pumping beam from a cw layers and 98 A AI,,,G%.,Aslayers With (-) laser diode operating 42 meV above the first exciton resonance. The intensity of the pumping beam is I = 800 W/cmZ.The fast oscillations in these spectra are Fabry-Perot fringes in a thin glass plate covering the sample.
296
D. S. CHEMLA eta/.
below the resonance, where a spectral region exhibits a decrease in transmission. The saturation intensity was obtained from measurements of this type at lower pump power, and it was found to be I, 300 W/cmZ.In all cases the number of free carriers which were generated was in the range 1-3 X 1010cm-2. An accurate measurement of the nonlinearities across the j = 1 absorption edge was obtained by pump-probe experiments using a tunable picosecond dye laser.50In such experiments it is possible to determine simultaneously the transmission spectrum, the nonlinear transmission spectrum, and that of the diffraction efficiency on the light-induced refractive and absorptive coherent gratings generated by the pump-probe interference.5,51,52 An example of the raw data is shown in Fig. 7a-c. The picosecond excitation also gives access to the absorption recovery time, from which one can deduce the recombination lifetime T~ and the diffusion coefficient. It was found that at room temperature Z , - 30 ns and D - 13 cm2/s. If one assumes that the diffusion is ambipolar and limited by the hole diffusion, this value corresponds to a hole mobility p h - 260 cm2/V s, in good agreement with the measurements on high-purity bulk GaAs at room temperature. The spectra of Fig, 7b and c give an independent determination of the parameters a& and a = #(27r/lr/A)ne,, (i/2)aehl.'" At the peak of the heavy-hole exciton resonance it is found that a,, - 7 X cm-2 or a2- 40 cm/W. This corresponds to a saturation intensity I, = 300 W/cmZ.The saturation intensities measured at the exciton resonance or far above the bandgap are in excellent agreement, showing that the saturation is the same whether the carriers are directly created or generated through exciton ionization by thermal phonons. Using the semiempirical fit described in Section 11, it was determined what changes in the parameters describing the two excitons and the continuum could reproduce the nonlinear transmission spectrum.40It was found that an excellent fit to the experimental data could be obtained by neglecting the variations in the continuum parameters and assuming a loss of absorption at the exciton peaks of the order of 5% with a small broadening of the heavy-exciton peak, consistent with a fairly constant area of the resonance. Both excitons experience a very small shift (- - 5 X eV). The fact that the contribution of the continuum saturation is not important shows that at the intensity at which the experiment was performed saturation of interband transitions was not yet reached. The fitting procedure used accounted for the change in absorption as the dye laser was tuned across the resonances,'" confirming again that the effect of the pump is directly proportional to the number of carriers it generates. The excellent agreement of the fit for the linear and nonlinear absorption is shown in Fig. 8a and b. A more severe test of the accuracy of this description was
-
+
5.
297
MULTIPLE QUANTUM WELL STRUCTURES
-
I
I
I
1
I
I
I
1
EXCITON RESONANCES
J
-
I 0
-0.2
1.5
0
I .45
1.46
1.47
4.48
PHOTON ENERGY (eV)
FIG.7. Raw data obtained by pumpprobe experiments on the sample described in Fig. 5. (a) Linear transmission:(b) nonlinear transmission;diffiction efficiency on the light-induced grating. For (b) and (c) the pump and probe beam powers were -300 and -60 p W , respectively.
298
D. S . CHEMLA eta[. v
io.0
-
7.5
-
0
$ 1
-zap
1.0
0
0.5
2
I 0
0
z
2-0.5 I-
-1.0
z
-? 1.0
d
0.8
2 z
0.6
$
0.4
PHOTON ENERGY ( e V )
FIG.8. Comparison of the experimental data of Fig. 6 , corrected for the small variations of the laser power, with the semiempirical fit for (a) the linear absorption and (b) nonlinear in (c ) is obtained by the Kramers-Kronig inversion discussed in the absorption. (-) text.
obtained in the following way. From the variation of the parameters describing the nonlinear absorption, the nonlinear refractive index nehwas calculated by Kramers- Kronig inversions at a fixed number of photocarriers.40The corresponding spectra of fseh and neh are shown in Fig. 9. They determine completely the nonlinear susceptibility in the fundamental bandgap region and therefore must reproduce the diffraction efficiency
5. MULTIPLE
I
5 f g
*
299
QUANTUM WELL STRUCTURES
-
.5
I
o
I
I I
I
h
-
c
b'
-.5
-1.0
L
-1.0
-.5
I
I
.5
-
1.0
-
i
FIG.9. Spectra of the imaginary and real parts of the nonlinearity of the MQW sample. , o and nd describe, respectively, the change of absorption and the change of refractive index induced by one electron-hole pair.
spectrum of Fig. 8c. In the small-signal regime it is given by
where I, = ( 1 - @')/a is the effective interaction length accounting for the linear absorption, N ( o ) is the number of photocarriers generated, and K is a constant. The excellent agreement with the experimental spectrum shown in Fig. 8c is obtained by assuming that only 70% of the photocarriers actually contribute to the diffraction. Note that the exact value of N ( o )only affects the amplitude of the &&action efficiency spectrum. This efficiency can be reduced by a number of effects, and the accuracy of 30% is well within the experimental uncertainty. Most important is the fact that
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the very highly structured spectra ceh(u)) and neh(o)very accurately reproduced the experimental diffraction efficiency spectrum. The maximum value of refractive nonlinearity occurs below the heavy-hole exciton resonance, in fact, close to the energy where the curve of ceh(w)crosses zero. The nonlinear refractive index is very large at this point: neh - 3.7 X cm3 or n2 2 X cm2/W. This corresponds for y3)to a maximum value of I&l6X esu, that is, about lo6 times that of silicon at room temperaturess3or lo4 times that of CuCl at liquid helium temperaIt is worth noting that, although the nonlinear absorption is important only in the exciton peak region, the nonlinear refraction extends far from the resonance and is still large in the transparency domain.
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8. ORIGINOF EXCITONIC SATURATION AT ROOM TEMPERATURE A detailed description of the origin of the nonlinear effect in MQWS must use many-body theory. Two-dimensional bandgap renormalization and Coulomb interaction screening by an exciton-electron hole gas must be accounted for. However, a simple, almost mechanical, model accounts rather well for the magnitude of the nonlinearities and describes the essential mechanisms responsible for saturation, i.e., Pauli exclusion. It is based on the assumptions that the thermodynamic equilibrium Eq. (1 1) describes correctly the population and that electrons or holes act as point defects which sufficiently perturb the semiconductor that excitons cannot be created within a certain area of the defect. A simplistic evaluation uses the exciton area as a good measure of the region perturbed by a point defect. At low excitation where the carriers are independent an analytical expression for the absorption coefficient can be obtained.40 It has the usual Lorentzian saturation functional form a(Z)= a( 1 Z/Z,)-I with a very simple expression for the saturation intensity:
+
where 0law is the absorption coefficient in the quantum well, z and Ax are the exciton lifetime and area, and fiw the energy of the incident photons. The physical interpretation of this expression is that the absorption is reduced by a factor two for an intensity incident on the sample such that either one electron or one hole is created in one exciton area per lifetime. The values obtained by this model for the two samples discussed above (I, - 490, - 190 W/cm2)compare well with the measured values (I, - 500 and - 300 W/cm2).As stated above, a correct theory explaining the spectra
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of the real and imaginary part of the nonlinearity (Fig. 9) would be much more elaborate.
9. APPLICATIONS MQWS have already been utilized in applicationswhere large changes of absorption or refractive index can be achieved with low optical excitation. Nonlinear optical processes such as degenerate four-wave mixing are usually observed in solids only in rather long samples and under intense excitation. Because of their huge linearities, MQWS can be used with path lengths of only a few pm. Also, because of the compatibility with laser diode wavelengths and powers, MQWS are very attractive for integrated optical circuits. In order to demonstrate this potential, degenerate fourwave mixing experiments were performeds4in a 1.25 pm MQWS using a cw laser diode operating at power varying from values as low as Pp 0.1 mW (I, - 0.8 W/cm2) to Pp - 2.5 mW (I, - 20 W/cm2). Figure 10 shows a typical power dependence of the forward DFWM signal. This curve was taken at 14.7"C sample temperature, which was the sample temperature
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30 I
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0
I
I
I
I
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'1
TEST BEAM POWER 400pW
I
PUMP
/'
I
1
/
2
3
POWER (mW)
FIG.10. Degenerate four-wave mixing signal found using a diode laser source on an MQW sample. The dashed line is a theoretical fit showing quadratric fit behavior up to -2 mW pump power with a small linear background (shown separately as -*-). Saturation of the nonlinearity at higher pump power shows up as a deviation from the theoretical curve.
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eta[.
for which the maximum signal was obtained, although a signal could be observed between 6 and 22°C. The curve has been fitted at low powers with a quadratic power dependence (as would be expected for this DFWM configuration) and a small linear background term to account for scattered light. The curve shown in Fig. 10 was extended to relatively high power levels to show the roll-off of the signal at high powers due to saturation of the nonlinearity. The quadratic behavior of the DWFM at low powers has, however, been checked down to < 100 pW pump power. The diffraction efficiency(i.e., the forward reflectivity of the test beam) in Fig. 10 rises to 5X The corrections for single surface reflections at each interface raise this to 5 X diffraction efficiency inside the crystal. The nonlinearities measured in these experiments are in excellent agreement with those deduced from the picosecond measurements. Recently, we have performed experiments in which MQW samples have been used as external saturable absorber elements to mode lock a semiconductor diode laser. Most previous attempts to mode lock diode lasers with a saturable absorber have utilized absorption produced by optical damage. By aging a laser to the point of severe degradation, pulses as short as 1.3 ps have been obtained for a short time before laser fai1u1-e.~~ Bursts of subpicosecond pulses have been obtained by proton bombarding one end facet of a laser.56 Recently, 35 ps pulses have been produced in a GaAlAs laser with nonuniform current inje~tion.~’ In order to characterize the physical properties of a material for use as a saturable absorber, it is desirable to have a configuration in which the absorber is completely independent of the laser structure. In this way one can study the physical mechanisms of absorber saturation and recovery, and tailor the absorber to act effectively to mode lock a diode laser. H a d 8 has analyzed the conditions necessary for mode locking a homogeneously broadened laser with a saturable absorber having a relaxation time longer than the pulse width. He showed that the relaxation time of the absorber must be faster than that of the gain and that one must have GAIAA OGIAG (19) where 0, and uGare the optical absorption cross sections of the absorber and gain media, respectively, and A , and AG are the cross-sectional areas of the laser beam in the absorber and gain media. For the MQW samples we have studied, saturation takes place at optical intensities about a factor of 30 lower than those required to saturate the band-to-band transition. Thus, for an MQW absorber we have 0, >> a,. The recombination time of photoexcited carriers in the MQW sample is much longer (- 30 ns) than the gain recovery time (- 2 ns). In order to
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303
FIG.1 1. Schematic of the mode-locking setup.
reduce the absorption recovery time below 2 ns, the saturating beam must be tightly focused on the absorber. The diffision of carriers out of this excited region then determines the recovery time. Our measurements of diffusion lead us to project a recovery time of -3 ns for a 2 pm spot size. Experiments were performed using the setup shown in Fig. 11. The laser was a commercial laser diode with one facet antireflection coated. The anamorphic prism served to convert the output beam to an approximately circular cross section, which allows tight focusing on the absorber. The MQW absorber consisted of 47 layers of 98 A GaAs and 100 a of G%.,,Al,,29As epoxied to a high-reflectivity mirror. The unsaturated reflectivity of the mirror-absorber combination was 25% at the exciton peak. With the beam focused to - 1 pm spot size on the absorber, stable mode locking was obtained, as shown by the autocorrelation trace in Fig. 12. The autocorrelation pulse width corresponds to 1.6 ps pulses and the pulse spacing was 1 ns, which means that there were two equally spaced pulses circulating in the laser resonator. Further improvements in this mode-
DELAY TIME FIG. 12. Autocorrelation trace of mode-locked output pulse.
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locked behavior are expected with MQW samples specially tailored for mode-locking experiments. Optical bistability has also been demonstrated using an MQW sample as the nonlinear element in a Fabry-Perot r e s o n a t ~ r Although .~~ the observed behavior reported in Gibbs et aLs9may be due in part to saturation of the band-to-band absorption, in principle the large effective dipole matrix element of the exciton absorption should permit very low switching energies for MQW devices.60 It is shown in Sm h@ ti' that, in principle, switching energies as low as J should be achievable in high-finesse resonant cavities; these are the lowest limits so far predicted for any all-optical switch, and reflect the very large absolute size of the nonlinearities in the MQWS. IV. Variation of Optical Properties Induced by a Static Field 10. INTRODUCTION
The MQW possesses interesting optical properties with an applied electric field, showing large changes in absorption near the band edge for only moderate electric fields.61-64The reason for such large effects is the presence of large envelope wave functions (e.g., - 100 A) with correspondingly small associated energies (e.g., 10 meV) which are consequently easily perturbed; this is true both for the excitonic envelope functions and the particle in a box envelope functions of the individual electrons and holes [especially in the first ( j = 1) confined state]. Clearly we should expect large changes in these wave functions when the applied potential across the wave function is comparable to the unperturbed ground-state energy. 10 mV across 100 A corresponds to lo4 V/cm or 1 V/,um. Such fields are readily obtainable in micrometer-sized ~ a m p l e s ~inl -the ~ ~laboratory with some care to prevent excessive ohmic heating. These fields are lower than those normally required to observe the Franz-Keldysh effect65seen in bulk semiconductors (e.g., > lo5V/cm) where it is necessary to perturb the wave function within a unit cell of the crystal. Since the optical consequences of the electric field are readily seen at room temperature in micrometers of optical thickness, new practical devices are possible with the MQW which were not conceivable with bulk semiconductors. The general behavior of the near-band-edge absorption with field now appears to be ~ n d e r s t o o dwith , ~ ~ distinct effects for fields parallel and perpendicular to the layers, and the first speculative devices have now been demonstrated.62,66We discuss both the physics of the effects and the devices below. To date, only the effects on absorption have been measured directly, but changes in refractive index comparable with those seen in the
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nonlinear optics experiments (see Section I11 above) are to be expected and may be calculated from the absorption changes using the Kramer - Kronig relations. 1 1. ELECTRIC FIELDPARALLEL TO THE LAYERS
With the electric field parallel to the layers, we should not expect the field to perturb the wave functions perpendicular to the layers directly (i.e., the particle-in-a-box wave functions). However, the exciton wave function of the relative motion of electrons and holes clearly should be perturbed, just as any atom would be. In principle, two consequences should follow from the application of this field63:(1) The exciton should be polarized, resulting in a decrease of the energy of the system with the exciton peak moving to lower photon energies; this would be the same as the quadratic Stark effect, although it should be noted that this Stark effect would be on the ground (i.e., 1s-like) state of the exciton rather than between two different orbital states, as is normal in atomic spectroscopy, as the exciton is being created rather than being raised to an excited state from the exciton ground state. (2) The exciton once created should be field ionized (in the presence of an electric field, there are no bound states of the system); i.e., the electron and hole should tunnel away from one another toward the electrodes, resulting in a broadening of the exciton resonance (sometimes called a Stark broadening) due to the reduction in the exciton lifetime. The first observations of electric field effects on the near-band-edge absorption showed a large Stark-like effect on the exciton peaks6'; later results with improved electrode geometrics6*showed that such effects are due to perpendicular fields (see below, Section 12). The predominant optical effect of parallel fields (at least at room temperature) is a broadening of the excitonic peaks which is ascribed to the reduction of exciton lifetime due to field ionization,63i.e., Stark broadening. Figure 13 shows experimental absorption spectra63taken for various parallel field strengths for a sample with 95 A GaAs layers. At zero field, the two exciton peaks are clearly resolved as usual. With 1.6 X lo4 V/cm (16 mV in 100 A), the peaks have broadened so much that they have merged, and by 4.8 X lo4 V/cm, the peaks have been totally destroyed, although increased absorption is seen both above and below the nominal bandgap energy. The behavior at high fields is difficult to analyze theoretically, and this has so far not been attempted. However, at low fields the field ionization rate has been calculated theoretically for a two-dimensional e ~ c i t o nThe . ~ ~rate is considerably lower than that for the 3D exciton, partly because of the difference in binding energy and partly because of geometrical factors. This calculation is in order of magnitude agreement with the measured broad-
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L
i
10000
w0 i i
Lc
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0 0
0
5000
k
a
5: m a
0 1.43
1
1.48 PHOTON ENERGY (eV)
FIG. 13. Absorption spectra for electric fields d parallel to the quantum well layers. (a) 6 = 0; (b) & = 1.6 X lo4 V/cm; (c) & = 4.8 X 104 V/cm. The insert shows figuratively the distortion of the electron-hole Coulomb potential with the applied field.
ening of the exciton at low fields with, for example, field ionization times of the order of picoseconds expected for fields lo4 V/cm.
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12. ELECTRIC FIELDPERPENDICULAR TO THE LAYERS With the electric field perpendicular to the layers, there are several possible sources of perturbation of excitonic or even simple interband absorption. Again, we can divide these into two categories. ( I ) The wave functions, both of the individual particles (electrons and holes) and of the relative motion of electrons and holes in excitons, can be polarized just as in the Stark effect, resulting in an overall decrease of the energy of the system and a shift of the absorption spectrum. (2) Electrons and holes can be field emitted from their potential wells (i.e., tunnel out of their wells) and excitons can be field ionized, resulting in lifetime reductions and consequent broadening of spectral features. In general, experiments show
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MULTIPLE QUANTUM WELL STRUCTURES
307
that the dominant effectis a shift of the spectrum, although some broadening is also observed e~perimentally.~~ Figure 14 shows measured absorption spectra for 95 A GaAs wells with electric field perpendicular to the layers.63These measurements were made with the MQW in the depletion region of a reverse-biased p- i- n diode to minimize conduction. The field is consequently not totally uniform within the MQW, but, despite this, the exciton peaks remain resolvable up to very high fields (- 1O5 V/cm), and large shifts of the absorption to lower photon energies are seen. With fields greater than lo5V/cm the exciton peaks are no longer resolvable, and there is some theoretical indication that, at these fields and above, the holes can tunnel rapidly out of the weW3 so that broadening due to shortening of carrier or exciton lifetimes may be significant. However, it is difficult in these results to separate broadening due to lifetime effects from broadening due to field inhomogeneity.
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4.48 PHOTON ENERGY (eV) FIG.14. Absorption spectra for electric fields C perpendicularto the quantum well layers. (a) 8 = 1 X LO4 V/cm; (b) C = 4.7 X 104 V/cm; (c) C = 7.3 X 104 V/cm. The insert shows figuratively the distortionsof the quantum well potentials with the applied field. 1.43
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As there are several possible contributions to the energy shifts in the spectrum, it is useful to write the Hamiltonian for the envelope wave functions, Hen,,explicitly for a given electron and hole:
where
are the kinetic operators for electrons (e)and holes (h) in the z direction perpendicular to the layers, ze and zh being the z coordinates of electron and hole, using the appropriate perpendicular effective masses me, and mu; V,(z,) and the vh(zh)are builtin rectangular quantum-well potentials for electrons and holes; are the potential energies for electron and hole, respectively, in the field & in the z direction;
is the kinetic energy of relative motion of electrons and holes [r = (x, xh)% + (ye - yh)9,where (xe,ye) and (xh,yh) are the coordinates of electron and hole in the plane of the layers] with p,,being the reduced mass in the plane; and
-e2 J(ze- zh )* + rz is the Coulomb potential energy of electron and hole. Only the center-ofmass kinetic energy of electron and hole in the plane has been neglected in Eq. (20),as negligible momentum is transferred to this motion under direct optical excitation. Solving for the eigenvalues of Eq. (20) would give the energy of an exciton peak relative to the bulk bandgap energy. Various parts of this Hamiltonian have been treated separately. HmzC+ V,(z,) (and the equivalent for holes) is just the usual particle-in-a-box Hamiltonian at zero field and is easily solved exactly.’ HKEze V,(z,) HKEsh v,(zh) HKEreh V,h(r) should give the zero-field energy of the exciton relative to the band edge and hence, by subtracting the exact particle-in-a-box energies of the appropriate electron and hole states, the zero-field exciton binding energy veh(r)=
E
+
+
+
+
+
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MULTIPLE QUANTUM WELL STRUCTURES
309
EB. This problem has been tackled using ~ a r i a t i o n a l , ~ l -perturbative,m ~~,~’
+
+
and adiabaW8 techniques. HmZ, Ve(ze) Hsz, (and the equivalent for holes) would give the energy of the electron (or hole) in the skewed potential well, which is the total potential energy in the presence of the electric field (see the insert in Fig. 14). This has been approached variationally for both finite and infinite has been solved exactly for the infinite well (giving Airy functions as the wave function^),^^ and the energy has been calculated exactly for a specific finite-well structure by looking for tunneling resonance^.^^ In the total problem, many of these effects are coupled. In particular, with an applied field, the electrons and holes tend to move to opposite sides of the wells, resulting in reduced Coulomb binding between electron and hole (i.e., contributing an increase in the energy of the system overall). One approach to solving the entire Hamiltonian of Eq. (20) has been taken using a variational technique63in which the variational wave function is separable with the z wave functions being the “exact” solutions for electrons and holes separately in the field and the radial wave function being a 1s-like orbit whose radius is variationally adjusted. This calculation at least estimates the increase of energy of the system due to the reduced electron- hole Coulomb binding. It also predicts incidentally that the exciton orbit should get larger with applied field with an associated decrease in electron - hole kinetic energy. The overall result of these calculations for the 95 A wells used in the experiments is that the dominant effect for fields of lo4 to lo5V/cm is the shift of the hole levels in the skewed quantum well. The hole levels are more easily perturbed than the electron levels for two reasons: for the heavy holes the larger effective mass gives lower zero-field energy; for heavy holes and especially light holes, the wave functions penetrate significantly into the barriers, giving larger overall wave functions which are therefore more easily perturbed. The electron levels with their higher energies are comparatively much less perturbed by the fields. It turns that the infinite-well model of the particles in a skewed well gives very good estimates of the overall shifts of the absorption, provided only that a larger effective well width is used to account for the penetration of wave functions into the barriers; the effective widths used are those which give the correct zero-field energies for electrons and holes. The resulting calculated shifts of the bands63are slightly larger than the observed shifts of the exciton resonance^:^ and this discrepancy can be reduced by including the change in electron-hole Coulomb and kinetic energy discussed above. This change in electron-hole binding energy is most important at low fields; it tends to saturate out at high fields to a constant value.63It is coincidental that the light-hole and heavy-hole subbands (and consequently the associated excitons) move at approximately the same rate with field in the
310
D. S. CHEMLA et a1
0
%
-10
Y
I-
LL
r v) Y
U
W
n
z 0 0
-20
X
w
hh
-30' 0
'
'
'
I
'
I
I
5
I
I
ELECTRIC FIELD ( 1 0 4 ~ / c m )
FIG. 15. Shift of the position of light-hole (Ih, 0)and heavy-hole (hh, X) exciton peaks with applied field perpendicular to 95 A GaAs quantum w e k 3 Points are experimental, lines are theoretical. (---), calculations including only the shift of the electron and hole subbands. (-) include the correction to the exciton binding energy due to the movement of electrons and holes to opposite sides of the wells. There are no fitted parameters in the theory.
sample measured63; different valence-band potential well depths would give different relative rates. Figure 15 shows the measured shifts of lightand heavy-hole excitons for 95 wells, compared with the tunneling resonance calculations of the shift of the exciton with and without the correction due to the change of binding of electrons and holes as they move to opposite sides to the well. The agreement is good, especially as there are no adjustable parameters in the theory. One remarkable feature of the spectra in Fig. 14 is that the exciton resonances remain well resolved even up to fields lo5 V/cm, which are well above those which gave strong field ionization for parallel fields; the rationalization of this is that the walls of the quantum wells prevent the electron and hole from escaping from one another.
A
-
5.
31 1
MULTIPLE QUANTUM WELL STRUCTURES
13. APPLICATIONS
a. Optical Modulators With absorption coefficient changes - 5000- 10,000 cm-I it is possible to alter the transmission of a light beam significantlyin only micrometer of thickness. This can be exploited to make novel high-speed electroabsorptive modulators62with very small active volumes and low power requirements. While such devices are in an early stage, they offer many attractive features: the small size possible offerslow capacitance and also avoids the difficulty of matching optical and electrical propagation velocities encountered at high frequencies with conventional electrooptk modulators. The
CONTACT
METAL
LIGHT OUT
M
s
ETCH STOP
A
u
n+
APGaAs CONTACT
L
L B
0
s
c
-
B
-w
u
F F E R
T I V E
F F E R
n+ i
i
i p'
LIGHT IN
p+ d
I
I"
V/cm
5 ~ 1 0 ~ V/c m
c
I
I
I
1
2
FIG. 16. Schematic diagram of the p- i- n multiplequantum-well modulator with the p - i - n layer thicknesses exaggerated for clarity.* Device diameter is 600 pm and total thickness -4 pm. The MQW is sandwiched between superlattice (SL) buffer layers. The insert shows the field distribution inside the device for 0 and 8 V reverse bias.
312
D. S. CHEMLA eta/.
small sizes and moderate drive voltage also imply that only very small energies may be required to drive these modulators. The structure of the first device to be tested as a high-speed modulator is shown schematically in Fig. 16. This is a p-i-n diode as used for the perpendicular field measurements discussed above. When operating at photon energies below the bandgap (e.g., 1.446 eV), it is possible to change the transmission of the sample by a factor of - 2 with 8 V applied reverse bias even in this thin sample. The response of this device was tested down to RC-time-constant-limited response times 2 ns with a 50 R load resistance. This first device was much larger in area (600 p m diameter) than is necessary for optical modulation (e.g., 10 pm diameter) and consequently had a large capacitance (-20 pF). Smaller devices should exhibit much faster response. Indeed, fast modulation using a smaller device (95 pm diameter) was recently demonstrated. The device was driven with a comb generator, which provides 8 V electrical pulses with a measured full width at half-maximum of 120 ps, to modulate the output of a cw single-mode GaAs diode laser. Light pulses of 170 ps width were observed. The slight broadening can be easily explained by the nonlinearity of the detection system, as well as by that of the modulator itself. One important consequence of the observation that the primary shifting effect is due to band states rather than being purely an excitonic effect is that other quantumwell systems that have too low a material quality to display exciton resonances may still be useful as modulators of this type. This is particularly important for longer wavelengths.
-
b. Self-Electrooptic-Eflect Devices It is observed experimentally that the p-i-n diode is also an efficient photodete~tor.6~ Above-2 V reverse bias when the depletion region extends all the way through the MQW region, within experimental error approximately one photocarrier is collected for every photon absorbed, regardless of photon energy or reverse bias. This is interesting in itself as a detector. However, it also opens up interesting possibilities in which the same material functions simultaneously as both optical modulator and detector. When connected in an external electrical circuit, the detected light will give a photocurrent, which in turn, through the electrical circuit, will change the voltage across the device and hence its absorption, hence changing the detected light and so on in a feedback loop. Such devices using MQW material have been called self-Electrooptic-effect devices (SEEDS).~~ The first such device to be demonstratedMconnects the p-i-n diode through a series resistor to a constant voltage reverse bias supply. The
5.
MULTIPLE QUANTUM WELL STRUCTURES
313
wavelength of the incident light is chosen near to the position of the larger (heavy-hole) exciton resonance at zero bias. With no light incident, all the bias voltage is dropped across the diode, shifting the exciton peaks to lower energy and reducing the absorption at the operating wavelength. With increasing incident light power, photocurrent is generated by the remaining absorption, resulting in a voltage drop across the resistor and reducing the voltage across the diode. As the power is increased further, the exciton peaks start to move back over the operating wavelength, giving increased absorption and a further increase in photocurrent. Under the right condit ~ n ~this , ~ process ~ , becomes ~ ~ regenerative, leading to switching into a high-absorption, low-reverse-bias state. This results in a bistable optical input-output characteristic, as shown in Fig. 17. This ingut/output characteristic can be modeled theoreticalIf6 from the measured voltage dependence of the responsivity and transmission of the diode, with good agreement with experiment, as shown in Fig. 17. Optically bistable (OB) devices have received considerable attention as possible ways of implementing optical logic s y ~ t e m s .Optical ~ ~ . ~ ~logic in principle has some advantages over electronic logic, especially in the area
3o
EXPERIMENT
n
0
20
40
60
80
100
INPUT POWER, P ( p W )
FIG.17. Theory and experiment for a SEED optically bistable device for a 1 MR load resistor.
314
D. S. CHEMLA ef a/.
of parallel processing, but few optical devices have demonstrated switching energies low enough to make them of practical interest7*;those that have low switching energies have required Fabry - Perot resonators to achieve low switching energie~,’~ thus complicating fabrication and operation. (The nonlinear optical properties of the MQW discussed above in Section I11 actually already make it one of the most attractive materials for such Fabry-Perot switches.) The OB SEED, however, is an example of a recently discovered class of optical bistability which does not require cavitie^,'^,^' and the intrinsic optical switching energy of this device even without cavities is >30 times lower per unit area (-4fJ/pmZ) than any previously demonstrated OB device at a comparable wavelength. The first device is, however, much larger than required (600 pm) and only has a moderately low switching energy (- 1 nJ). Importantly for practical applications, it also operates over a wide range of conditions depending on resistance and wavelength from 670 nW switching power at 1.5 ms switching time to 400 ns at 3.7 mW (with this switching speed limited only by power). The absence of cavities makes this device relatively simple to fabricate and operate, and smaller devices should have proportionately lower switching energies and probably much faster switching speeds. This also is the subject of current research. However, the SEED certainly offers some interesting new prospects for optical switching and signal processing devices, and has significantly reduced switching energy limits for optical devices. As high operating energies have been a major problem for both analog and digital optical processing devices, the MQW electric field effects may be able to make a significant contribution to optical processing. V. Conclusion
The work reviewed in this article demonstrates that the quantum well materials show a variety of large nonlinear optical effects. Importantly for practical applications, these effects can be seen at room temperature and are eminently compatible with convenient light sources such as laser diodes and also, in the case of the electroabsorptiveeffects, with semiconductor electronics. In all cases, the effects result directly from the quantum mechanical confinement within the quantum well layers. The existence of room-temperature exciton resonances, without which room-temperature excitonic nonlinear absorption, nonlinear refraction, and parallel field Stark broadening would be impractical, results primarily from the increase in exciton binding energy with confinement. The bandgap shrinkage with perpendicular fields, which is, of all these effects, the only one truly unique to
5.
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315
quantum wells at any temperature, results primarily from the perturbation of the large envelope functions of the particle-in-a-box states. The effects discussed here are also large enough to have significant implications for nonlinear optical processing- a field which has been especially inhibited by the shortage of materials with suitable nonlinearities. For example, the absolute size of the excitonic nonlinear absorption and nonlinear refraction is so large that they give the lowest scaled operating energies of any room-temperature all-optical bistable switch. While such scalings do not represent any current practical devices, they do give a good relative measure of the sizes of nonlinearities in different materials. The electroabsorptive effects are so large that it is possible to make highspeed optical modulators with dimensions on a scale of micrometers. This is a totally new opportunity in optical modulation, as it reduces the limiting operating energy of an optical modulator by orders of magnitude. This low modulation energy is one of the principal reasons for the low-energy operation of the SEED hybrid bistable optical switch, which has an operating energy per unit area six times smaller than any other bistable optical switch at a comparable wavelength despite the fact that it uses no resonant cavity. It will be apparent to the reader that the work reviewed here is very recent at the time of writing. This makes it difficult to obtain an accurate historical perspective on the significance of the nonlinear effects and the devices which can be made using them. The device opportunities which present themselves are particularly difficult to judge at this stage, because for the most part the quantum wells offer completely new classes of optical devices. This uncertainty in itself makes these effects particularly exciting for possible practical applications, and there can be little doubt that some genuinely new opportunities are now available for optical devices. We believe that this is a very appropriate time to write this review, because the basic physics of the effects is apparently understood and the first speculative devices have been demonstrated. Finally, it is worth noting that the work reviewed here represents only one small comer of the total field of physical systems that can be made using the current advanced semiconductor fabrication techniques. The experiments reported here, for example, are mostly performed on G A S / GaAlAs quantum wells with - 100 A layers. Other material systems, layer dimensions, and layer sequences remain to be explored for their nonlinear optical properties, and doubtless many new physical phenomena remain to be investigated and applied. The limited experience with even the GaAs/ GaAlAs system is, however, very encouraging and hoped to have ramifications well beyond the field of semiconductor physics.
316
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REFERENCES I. D. S. Chemla and J. Jerphagnon, in “Handbook of Semiconductors,” Vol. 2. North-Holland Publ., Amsterdam, 1980. 2. D. S. Chemla, Rep. Prog. Phys. 43, 1191 (1980). 3. A. Miller, D. A. B. Miller, and S. D. Smith, Adv. Phys. 30,697 (1981). 4. R. K. Jain and M. B. Klein, in “Phase Conjugation” (R. A. Fisher, ed.). Academic Press, New York, 1983. 5. D. S. Chemla and A. Maruani, Prog. Quantum Electron. 8, 1 (1982). 6. A. C. Gossard, in “Thin Films Preparation and Properties” (K. N. Tu and R. Rosenberg, eds.). Academic Press, New York, 1983. 7. R. Dingle, Festkiierperprobleme 15,21 (1975). 8. See other chapter of the present volume. 9. D. A. B. Miller, D. S. Chemla, P. W. Smith, A. C. Gossard, and W. T. Tsang, Appl. Phys. [Pard B B28,96 (1982). 10. M. Shinada and S. Sugano,J. Phys. SOC.Jpn. 21, 1936 (1966). 11. G. Bastard, Phys. Rev. B: Condens. Matter [3] 25,7584 (1982). 12. J. H. Luttinger and W. Kohn, Phys. Rev. 97,869 (1955). 13. G. Dresselhaus, A. F. Kip, and C. K. Hel, Phys. Rev. 98,368 (1955). 14. R. Dingle, W. Wiegmann, and C. H. Henry, Phys. Rev. Lett. 33,827 (1974). 15. T. Ando, A. B. Fowler, and F. Stem, Rev. Mod. Phys. 54 (1982). 16. J. C. Hensel and G. Feker, Phys. Rev. 129, 1041 (1963). 17. D. D. Sell,S. E. Stokowski, R. Dingle, and J. V. DiLorenzo, Phys. Rev. B: Solid State [3] 7,4568 (1973). 18. A. Baladereschi and N. C. Lipari, Phys. Rev. B: Solid State [3] 3,439 (1971). 19. E. Kane, Phys. Rev. B: Solidstate [3] 11, 3850 (1975). 20. Y. C. Lee and D. L. Lin, Phys. Rev. Br Condens. Matter [3] 19, 1983 (1979). 21. R. C. Miller, D. A. Kleinman, W. T. Tsang, and A. C. Gossard, Phys. Rev. B: Condens. Matter [3] 24, 1134 (1981). 22. G. Bastard, E. E. Mendez, L. L. Chang, and L. Ezaki, Phys. Rev. B: Condens. Matter [3] 26, 1974 (1982). 23. R. L. Greene and K. K. Bajaj, Solid State Commun. 45,831 (1983). 24. R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975). 25. R. C. Miller, A. C. Gossard, D. A. Kleinman, and 0.Munteanu, to be published. 26. R. C. Miller, D. A. Kleinman, W. A. Norland, and A. C. Gossard, Phys. Rev. B: Condens. Matter [3] 22,863 (1980). 27. C. Wiesbuch, R. C. Miller, R. Dingle, and A. C. Gossard, Solid State Commun. 37,219 (1 98 1). 28. A. Pinczuk, D. S. Chemla, D. A. B. Miller, and A. C. Gossard, to be published. 29. R. Sooryahumar,D. S. Chemla, A. Pinczuk, and A. C. Gossard, to be published. 30. C. Weisbuch, R. Dingle, A. C. Gossard, and W. Wiegmann, J. Vac. Sci. Technol. 17, 1128 (1980). 31. J. Hegarty, M. D. Sturge, C. Weisbuch, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 49, 930 ( 1982). 32. J. Hegarty, Phys. Rev. B: Condens. Matter [3] 25,4324 (1982). 33. J. Hegarty, M. D. Sturge, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 40, 132 (1982). 34. J. Hegarty, M. D. Sturge, L. Goldner, A. C. Gossard, and W. Wiegmann, to be published. 35. J. Zucker, A. Pinczuk, D. S. Chemla, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 51, 1293 (1983).
5. MULTIPLE QUANTUM WELL STRUCTURES
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36. J. E. Zucker, A. Pinczuk, D. S. Chemla, A. C. Gossard, and W. Wiegmann, to be published. 37. D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. T. Tsang, Appl. Phys. Lett. 41,679 (1982). 38. Y. Ishibashi, S. Tarucha, and H. Okamoto, Conf: Ser.-Znst. Phys. 63,587 (1982). 39. S. W. Kirchoefer, N. Holonyak, K. Hess, D. A. Gulino, H. G. Drickamer, J. J. Coleman, and P. D. Dapkus, Appl. Phys. Lett. 40,82 1 ( I 982). 40. D. S. Chemla, D. A. B. Miller, P. W. Smith, A. C. Gossard, and W. Wiegmann, J. Quantum Electron. (to be published). 41. V. I. Alperowich, V. M. Zalekin, A. F. Kranchinko, and A. S. Terekhev, Phys. Stutus Solidi. B 77,466 (1976). 42. J. Klafter and J. Fortner, J. Chem Phys. 68, 1513 (1978). 43. I. Abram and R. M. Hockstrasser, J. Chem. Phys. 72,3617 (1980). 44. K. Lindenberg and B. J. West, Phys. Rev. Lett. 51, 1370 (1983). 45. J. Shah, R. F. Leheny, and W. Wiegmann, Phys. Rev. B: Solid State [3] 16, 1577 (1977). 46. D. S. Chemla, A. Maruani, and F. Bonnouvier, Phys. Rev. A 26,3026 (1982). 47. G. W. Fehrenback, W. Schafer, J. Tzeusch, and R. G. Ulbrich, Phys. Rev. Lett. 49,1281 (1982). 48. C. V. Shank, R. F. Fork, B. 1. Greene, and C. Weisbuch, Surf: Sci. 113, 108 (1982). 49. C. V. Shank, R. L. Fork, R. Yen, J. Shah, B. I. Greene, A. C. Gossard, andC. Weisbuch, Solid State Commun. 47,98 1 (1983). 50. D. A. B. Miller, D. S. Chemla, D. J. Eilenberger, P. W. Smith, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 42,925 (1983). 51. J. P. Woerdman, Phillips Res. Rep. Suppl. 7 (1981). 52. H. J. Eichler and F. Massmann, J. Appl. Phys. 53,3237 (1982). 53. R. K. Jain and M. B. Klein, Appl. Phys. Lett. 35,454 (1979). 54. D. A. B. Miller, D. S. Chemla, P. W. Smith, A. C. Gossard, and W. Wiegmann, Opt. Lett. 8,477 (1983). 55. E. P. Ippen, D. J. Eilenberger, and R. W. Dixon, in “Picosecond Phenomena II” (R. Hochstrasser, W. Kaiser, and C. V. Shank, eds.), pp. 21-25. Springer, New York, 1980. 56. J. P. van der Ziel, W. T. Tsang, R. A. Logan, R. M. Mikulyak, and W. M. Augustyniak, Appl. Phys. Lett. 39,525 (1981). 57. C. Harder, J. S. Smith, K. Y. Lau, and A. Yariv, Appl. Phys. Lett. 42,772 (1983). 58. H. A. Haus, J. Quantum Electron. QE-l1,736 (1975). 59. See, for example, H. M. Gibbs, S. S. Tamg, J. L. Jewell, D. A. Weinberger, K. Tai, A. C. Gossard, S. L. McCall, A. Passner, and W. Wiegmann, Appl. Phys. Lett. 41,221 (1982). 60. P. W. Smith in “Proceedings of ELECTRO ’83 Conference,” Sess. Rec. 1 1/1. IEEE, New York, 1983. 61. D. S.Chemla, T. C. Damen, D. A. B. Miller, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 42, 864 (1983). 62. T. H. Wood, C. A. Burms, D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, and W. Wiegmann, Appl. Phys. Lett. 44, 16 (1984). 63. D. A. B. Miller, D. S. Chemla, T. C. Damen, A. G. Gossard, W. Wiegmann, T. H. Wood, and C. A. Bums, to be published. 64. Luminescence measurements with applied field have also been made at low temperatures. See E. E. Mendez, G. Bastard, L. L. Chang, L. Esaki, H. Morkoc, and R. Fisher, Phys. Rev. B: Condens. Mutter [3] 26,7101 (1982);R. C. Miller and A. C. Gossard, Appl. Phys. Lett. 43,954 (1983). 65. See, for example, G. E. Stillman and C. M. Wolfe, in “Semiconductors and SemimetaIs”
318
66. 67. 68. 69. 70. 71. 72. 73.
D. S . CHEMLA eta[.
(R. K. Willardson and A. C. Beer, eds.), Chapter 5, pp. 380-86. Academic Press, New York, 1977. D. A. B. Miller, D. S. Chemla, T. C. Damen, A. C. Gossard, W. Wiegmann, T. H. Wood, and C. A. Burms, Appl. Phys. Lett. (submitted for publication). Y. E. Lozovik and V. N. Nishanov, Sov. Phys.-Solid State (Engl. Transl.) 18, 1905 (1976). [Fig. Tverd. Tela (Leningrad) 18,3267 (1976)l. N. N. Kolychev, G. G. Tarasov, A. M. Yaremko, and V. I. Sheka, Phys. Status. Solidi. B 98,527 (1980). G. Bastard, E. E. Mendez, L. L. Chang, and L. Esaki, Phys. Rev.B: Condens. Matter [ 3 ] 28,3241 (1983). D. A. B. Miller, A. C. Gossard, and W. Wiegmann, Opt. Lett. (to be published). D. A. B. Miller, J. Opt. SOC.Am. B (submitted for publication). See, for example, D. A. B. Miller, Laser Focus Fiberopt. Technol. 18, No. 4,79 (1982). P. W. Smith, Bell Syst. Tech. J. 61, 1975 (1982).
SEMICONDUCTORS AND SEMIMETALS, VOL. 24
CHAPTER 6
Graded-Gap and Superlattice Devices by Bandgap Engineering Federico Capasso AT&T BELL LABORATORIES MURRAY HILL, NEW JERSEY 07974
I. Introduction: Bandgap Engineering The advent of molecular beam epitaxy (MBE) has made possible the development of a new class of materials and heterojunctions with unique electronic and optical properties. Most notable among these are: heterojunction and doping superlattices, modulation-doped superlattices, and strained layer and variable gap superlattices. The investigation of the novel physical phenomena made possible by such structures has proceeded in parallel with their exploitation in novel devices. As a result a new approach or philosophy to designing heterojunction semiconductordevices, bandgap engineering, has gradually emerged (F. Capasso, 1983a,b; 1984a). The starting point of bandgap engineering is the realization of the extremely large number of combinations made possible by the above-mentioned superlattices and heterojunction structures. This allows one to design a large variety of new energy band diagrams. In particular, through the use of bandgap grading, one can obtain, starting from a basic energy band diagram, practically arbitrary and continuous variation of this diagram. Thus the transport and optical properties of a semiconductor structure can be modified and tailored to a specific device application. One of the most powerful consequences of bandgap engineering is the ability of independently tuning the transport properties of electron and holes, using quasielectric fields in graded gap materials and the difference between conduction- and valence-band discontinuitiesin a given heterojunction. An excellent example of bandgap engineering is the development of the concept of the staircase solid-state photomultiplier, starting from a quantum well reverse-biased diode, as discussed in Section IV. Band discontinuities also play a key role in other devices, such as realspace transfer structures and heterojunction bipolar transistors, by confining camers to certain layers (Sections I1 and VI). 319 Copyright 0 1987 Bell Telephone Laboratories, Incorporated.
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FEDERICO CAPASSO
Some of the most intriguing applications of bandgap engineering are graded-gap superlattices, discussed in Section IV, and the design of threedimensional energy band diagrams (Section 111). Resonant tunneling transistors and tunneling phenomena in superlatticesare discussed in Sections VI and VII. The last section of this chapter discusses a technique to modify barrier heights and band discontinuities, recently introduced by the author. This method should prove extremely useful in the deisgn of novel devices. 11. Real-Space Transfer Structures Real-space transfer is a new phenomenon first proposed and discovered by Hess and co-workers (1979; Keever et al., 1981) at the University of Illinois. Real-space transfer can be defined as the transfer, by thermionic emission, of hot electrons between layers of different semiconductors. In equilibrium the electrons reside in the layers with the lowest conductionband energy. Within picoseconds the electrons can be heated by electric fields parallel to the heterolayers to energies exceeding the band-edge discontinuities and subsequently transferred to a different semiconductor layer. Storage and switching effects can be achieved in this way. In addition, if the layers into which electrons transfer are of lower mobility than the layers in which they originally reside, negative differential resistance will occur (Hess et al., 1979). The latter effect was first demonstrated in AlGaAs/GaAs modulation-doped heterostructures (Keever et al., 198I). Figure 1 shows the doping, mobility, and conduction band edge energy profile of modulation-doped material. The AlGaAs layers are intentionally doped with donors to densities in the lOI7/cm3 range, while the GaAs layers contain only low concentrations of background impurities (5lo1’/ cm3);the layer thicknesses are typically a few 100 A. At thermal equilibrium electrons reside at the bottom of the GaAs wells, spatially separated from their parent donors in the AlGaAs. The spatial separation greatly enhances the mobility of electrons parallel to the layers, especially at low temperatures (577 K), since it reduces Coulomb scattering from the ionized impurities (Dingle et al., 1978). Although to observe real-space transfer effects modulation doping is not essential, from a conceptual and tutorial point of view it is simple to explain the effect as the reverse process of modulation doping. One may ask in fact whether electrons in modulation-doped layers can be made to reunite with their parent donors. This can be done by applying a high electric field (> lo3 V/cm) parallel to the interfaces of the structure of Fig. 2. The band-bending due to the charge present in the layers is not shown for simplicity. Electrons in the GaAs layer will be heated by the field and gain energy. When their energy exceeds the barrier height given by the
6.
GRADED-GAP AND SUPERLATTICE DEVICES
321
conduction-band discontinuity of =0.2 eV, electrons are no more confined in the GaAs layers and may diffuse into the neighboring AlGaAs layers. Since these layers have much lower mobility than the GaAs layer, the real-space transfer effect may be used to produce negative differential resistance. It is important to note that this effect represents the real-space analog of the transferred electron effect (or the Hilsum - Ridley -Watkins mechanism) which is the basis of Gunn devices. Negative differential resistance in these devices is due to transfer of hot electrons from the high-mobility central valley in direct-gap I11- V materials, to higher-lying low-mobility satellite valleys. Here the T - L valley separation controls the threshold for negative differential resistance, while in real-space transfer devices it is controlled by the conduction-band discontinuity at the interface between the AlGaAs and the GaAs layers. By changing the A1 concentrations in the AlGaAs layers the discontinuity can be varied. In addition, the magnitude of the negative resistance effect can be increased by incorporating more
U
7 GAS
FIG.1. Illustration of modulationdoping technique.
322
FEDERICO CAPASSO LOW ELECTRIC FIELD
HIGH ELECTRIC FIELD
x
A5
FIG.2. Pictorial representation of real-space transfer. For low parallel electric fields electrons are confined to the GaAs layers; at high fields they transfer into the AlGaAs layers by themionic emission (courtesyof K. Hess).
donors in the AlGaAs layers. These features provide greater flexibility in device design with respect to Gunn devices. Negative differential resistance in real-space transfer devices has recently been used to implement an oscillator (Coleman et al., 1982). Another interesting application of real-space transfer involves fast switching and storage of electrons in GaAs/Al,Ga, -,As heterojunction layers (Keever et al., 1982). Electrons are initially in a “high-field” GaAs layer and are subsequently heated by the field and transfer by thermonic emission to the AlGaAs neighboring layers. Suppose now that on the other side of the AlGaAs layers there are “low-field GaAs layers.” The electrons scattered into the AlGaAs from the “high-field” layer may transfer into the “low-field layers” and lose their energy via phonon emission. Thus they become trapped as long as the temperature is not high enough for them to escape by thermionic emission. Therefore, the selective application of a high field to a group of layers allows switching and storage of electrons in the heterostructure; switching times d 1 ns have been observed (Keever et al., 1982). One of the attractive features of the real-space transfer mechanism is its high speed. There are typically four time constants involved: The time
6.
GRADED-GAP AND SUPERLATTICE DEVICES
323
required for heating of the electrons to the temperature T,, the transfer time to the wide gap layers, the transfer time of the electrons back into the starting layers, and the cooling time. The heating and the cooling time are typically of the order of the energy relaxation time in GaAs for electron energies of the order of A E,, (0.1 -0.2 eV), i.e., a few picoseconds. The two transfer times can be estimated from the formula, obtained from thermionic emission theory (Hess et al., 1979),
where A* is the Richardson constant, m* the electron effective mass, mo the free-electron mass, (b the barrier height to overcome in the transfer process, N , the density of states in the layer from which electrons transfer, and L the thickness of this layer or the electron mean-free path, whichever is smaller. For T, in GaAs = 500 K, 4 = 0.2 eV, and L = 400 A, the transfer time into the AlGaAs is = 35 ps. In the case of transfer back into the GaAs layer, the barrier is due to the space-charge potential of the ionized donors in the AIGaAs. The electron temperature in the AlGaAs is close to the lattice temperature TL because of the low mobility. For 4 / k T = 4 and T, = 300 K one has t , = 0.3 ns for the transfer time back into GaAs. In the real-space transfer negative differential resistance devices of the type discussed by Hess ( 1983), the relevant time is therefore the sum of the four previously estimated time constants, since electrons have to cycle back and forth constantly between the high- and low-mobility layers. It is clear that the dominant of the four time constants is the time to transfer back into the GaAs. An analytical model of real-space transfer valid for thick layers has been recently given by Shichijo et al. (1980). The Boltzmann equation was solved, assuming a position-dependent electron temperature and a quasiFermi level in the Al,Ga, -,As and a position-independent electron temperature and quasi-Fermi level in the GaAs layer. Conduction of hot electrons from the GaAs into the Al,Ga,-,As and the reverse flow of cold carriers back into the GaAs layer were taken into account. The variation of the fraction of electrons in the AlGaAs layer with increased field is clearly illustrated in Fig. 3 with the mobility of the AlGaAs layer as a parameter. The thickness of the GaAs is 0.1 pm and that of the AlGaAs, 1 pm, while the bqrrier height is 0.25 eV. The threshold field for transfer is = 3 X lo3 V/cm and the transfer is smaller for higher mobility. This is physically intuitive; if the mobility in the AlGaAs is higher, electrons in this material will be heated to higher temperatures by electronic heat conduction due to
324
FEDERICO CAPASSO 1
1
1
1
)
1
1
1
1
~
1
l
*
-
o t p m , L~ = i p m A E = 25omeV
L, =
tV
w
0
IL I
0
I
I
I
I
5
I
I
I
I
I
I
I
10
FIG. 3. Fraction of electrons in the GaAs layer as a function of electric field for various mobilities in the Al,Ga,_,As. L, and L, are, respectively, the GaAs and AlGaAs layer thicknesses (from Shichijo ef al., 1980).
hot electrons injected from the GaAs and by the electric field. This in turn will increase the hot electron diffusion coefficient in the AlGaAs, which in turn increases the “backflow” of electrons into the GaAs, thus reducing the percentage of electrons transferred into the A1,Ga -,As. The analog with the Gunn effect here is again evident. Calculations of the current-voltage characteristic for this case (Fig. 4) show that the highest peak- to-valley ratio is obtained for the lowest AlGaAs mobility (50 cm2/s V); it becomes unity instead for p = 500 cm2/s V. The above analytical model does not take into account the band bending caused by the ionized donors. These donors create an electric field which would tend to attract the hot electrons into the AlGaAs layer, thus increasing the transfer efficiency. 1 . NEGATIVE DIFFERENTIAL RESISTANCE FIELD-EFFECT
TRANSISTOR Recently Kastalsky and Luryi ( 1983) proposed and demonstrated a new class of charge injection devices, utilizing real-space transfer. In these structures this effect gives rise to charge injection between the conducting layers, isolated by a potential barrier and separately contacted. Figure 5 illustrates a typical charge injection device and its band diagram. It consists essentially of a three-terminal device with the electrodes
6.
325
GRADED-GAP AND SUPERLATTICE DEVICES
L 1 = 0.1p m Lp=ipm
A E = 250 meV
1
1
1
1
l
1
1
1
1
1
1
5 10 ELECTRIC FIELD, F (kv/cm)
1
1
1
5
FIG. 4. Current-voltage characteristics for a Hess-type real-space transfer device with various values of electron mobility in the AlGaAs layer (from Shichijo et al., 1980).
labeled source (S), drain (D), and substrate (SUB). The source-to-drain voltage creates a parallel electric field which heats the electrons in the channel, causing charge injection by real-space transfer into the heavily doped conducting substrate, across the graded-gap AlGaAs barrier. Although the graded barrier may slightly increase the transfer speed, its use is not essential for most applications, and a rectangular barrier may do just as well. The gate electrode serves the purpose of concentrating the source-todrain field in a 1 pm region under the gate notch. However, most of these devices can also be operated without gate metallization or at zero gate voltage. One use of the above structure is as a negative differential resistance field-effect transistor (NERFET) (Kastalsky et al., 1984a,b). In this device hot-electron injection from the channel across the gradedgap barrier is accompanied by a pronounced negative differential resist-
326
FEDERICO CAPASSO
I-
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ELECTRON ENERGY
4
7
'Pm n-GoAs S'
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vc 4
i
!
VD I
--
---
1
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FIG. 5. Device structure and band diagram of the negative-differential-resistancefield-effect transistor (NERFET) (from Kastalski et al. 1984a).
ance (NDR) in the source- to-drain circuit, observed both at 300 K and at low temperature. This is illustrated in Fig. 6 for several substrate biases. The NDR in the NERFET is due to two interdependent mechanisms: (1) the diversion of current from the channel into the substrate following hot-electron injection, and (2) the channel depletion due the negative dynamically stored charge in the graded-gap barrier, which screens the channel from the positive substrate bias. To understand the latter effect, consider that the hot electrons emitted over the barrier drift in the gradedgap layer (at the saturated drift velocity) toward the substrate. They constitute therefore a negative space charge dynamically stored in the barrier, which repels the electrons in the channels towards the source, thus depleting the channel. This further lowers the drain current, thus increasing the peak-to-valley ratio. The strong dependence of the NDR on the substrate voltage (Fig. 6) is strong evidence of the charge storage effect. Note that NDR is obtained only when VsuB 5 2; this is because electrons, dynamically stored in the barrier, create a space-charge potential Acy which opposes further charge injection; to overcome this additional barrier the substrate bias must exceed Acy, which is =2 V for the devices of Fig. 6. The increase in the peak current in Fig. 6 is a manifestation of the increased electron density in the channel. As the substrate bias is increased, the NDR becomes more pronounced and the FET drain current saturates at a lower value (Fig. 6). This is due to the charge dynamically stored in the bamer, which increases with substrate bias, and increasingly depletes the channel.
6.
GRADED-GAP AND SUPERLATTICE DEVICES
327
FIG.6. Current - voltage characteristic(drain current versus source-to-drain voltage) in the NEFWET for different substrate bias voltages. The gate voltage is zero and the temperature, 77 K (from Kastalski et ul., 1984a).
As pointed out by Luryi and Kastalsky (1985), “we can regard A y as a threshold shift in a field-effect transistor in which V,, plays the role of a gate bias. Due to the dynamically stored charge, the FET drain current saturates at a lower value corresponding to Vslb - Ay.” The rising portion of the I- V curves at high drain voltage after saturation in Fig. 6 is due to the onset of thermionic emission of cold electrons from the substrate into the channel. Figure 7 shows the drain and substrate current versus drain voltage for a NERFET. The device has no gate. The substrate current in general increases with drain voltage in the NDR region except for a small drop right at the onset of NDR. The latter effect appears to be associated with the formation of a high-field domain in the channel near the drain. In the saturation region also the substrate current saturates as expected, while at higher voltages it decreases in correspondence to the emission of cold electrons from the substrate back into the channel.
328
FEDERICO CAPASSO
' '.:s;.h y I q " D
...... : : : : : 'S'UB:::::::.:. ...... ..._...........
DRAIN VOLTAGE
FIG.7. Substrate (Isnb) and drain (ID)currents in a NERFET versus the source-to-drain = 4 V) (from Luryi et al., 1984a). voltage at a fixed substrate bias ( Vaub
The substrate current in a NERFET is concentrated in a narrow strip running along a direction perpendicular to the channel and located in the high-field domain. From simple current continuity considerations it is easy to show that the charge density in the high-field domain decays exponentially from the beginning of the domain towards the drain with a characteristic length determined by the barrier height v/ and the hot-electron temperature T,. For ty = 0.3 eV and T, = I500 K, this length is = 1000 A (Kastalsky et al., 1984a; Luryi and Kastalsky, 1985). 2. CHARGE-INJECTION TRANSISTOR
The same structure previously discussed can also be operated as a new kind of transistor (charge-injection transistor or CHINT) (Luryi et al., 1984a). The CHINT principle is illustrated in Fig. 8 by analogy with a vacuum diode. The channel plays the role of a cathode whose hot-electron temperature is controlled by the source-to-drain voltage. The second conducting layer, separated from the first by a potential barrier, acts like an anode. The anode (substrate) current is varied by changing the substrate bias and saturates at a level determined by the cathode temperature. At 77 K power gain was demonstrated experimentally in a CHINT utilizing the structure of Fig. 5. The speed limitations of the CHINT and the NERTET are due to: (1) to transit time through the high-field domain in the channel and the gradedgap barrier, amounting to a total of a few picoseconds, (2) the energy
6.
GRADED-GAP AND SUPERLATTICE DEVICES
329
SECOND CONDUCTING LAYER (ANODE)
t CHANNEL
F
S
CHANNEL (CATHODE)
Y ANODE VOLTAGE
FIG. 8. Illustration of the charge-injection transistor principle. The channel serves as a cathode whose effective electron temperature is controlled by the source-todrain field. The second conducting layer, separated by a potential barrier, as an anode and is biased positively. The anode current as a function of the anode voltage saturates at a value determined by the cathode temperature (from Luryi et al., 1984a).
relaxation time, = 1 ps, required to establish the electron temperature. Thus these devices are intrinsically faster than the original ones proposed by Hess, since the speed-limiting process of transfering back into the GaAs channel is eliminated. 3. MEMORY DEVICES If the substrate in a NERFET (or CHINT) is kept floating, the hot-electron injection produces a negative charge in the substrate and raises its potential. In steady state the emission of cold electrons from the substrate will balance hot-electron injection from the channel. If the source-to-drain voltage is not removed, charge will be retained in the substrate for a time controlled by the cold thermionic emission. This memory effect was demonstrated recently (Luryi et al., 1984b) and can be the basis of a hot-electron erasable programmable random access memory (Luryi and Kastalsky,
330
FEDERICO CAPASSO
r
(cClAS
UNDOPED
CHANNEL AkOAS
BARRIER
INSULATING
SUBSTRATE
FIG.9. Structure and energy diagram of the hot-electron erasable programmable randomaccess memory. The thickness (- 1000 A) and doping level in the second conducting layer are chosen so that this layer can be depleted by the guard-gate field (from Luryi and Kastalsky, 1985).
1985), shown in Fig. 9. The device is grown on a semi-insulatingsubstrate and contains a main GaAs channel (modulation doped as in a NERFET) and a second GaAs conducting channel contacted by a Schottky guardgate G, as in a MESFET, and by a deep drain D,, which is positively biased with respect to the source. When the guard gate voltage is negative, the second conducting channel can be charged up via hot-electron transfer by applying a voltage between D, and S (Write). This will deplete the main channel. Information is read by measuring the channel resistance. The information can be erased by applying a positive voltage to G, this discharges the second conducting layer through D,. All logic operations of this device have high speed. 4. TUNNELING REAL-SPACE TRANSFER DEVICES
Real-space transfer between different layers can also be achieved by tunneling. This was recently demonstrated by Kirchoefer et al. (1984), using the bilevel quantum-well structure of Fig. 10, The deep well is of GaAs, while the shallow one is of A10.06Ga0.94A~. Because the energy separation between the first quantum states of the two wells is larger than the linewidth of these states, the wavefunctions of the ground states of the wells have negligible overlap and in equilibrium electrons will reside in the GaAs well. When a parallel field is applied, electrons can gain sufficient energy to tunnel into the first subband of the A1,,Gao.94As well. Since this layer has lower mobility than GaAs, negative differential resistance should be observed. This was experimentally demonstrated using a superlattice of bilevel quantum wells like the ones of Fig. 10, with bamer and well thicknesses of 80 A.
6.
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331
FIG.10. Band diagram of a bilevel quantum well structure used in the tunneling real-space transfer device. Electrons transfer by tunneling from state v, to state y2 after application of an electric field parallel to the layer (from Kirchoefer et al., 1984).
111. Channeling Diodes
The real-space transfer effect previously discussed relies on the parallel transport of electrons and their thermionic emission over heterojunction barriers into layers of different bandgaps. In this section we examine another class of devices (channeling diodes) in which a high electric field is applied parallel to a set of n and p layers (in general, but not necessarily, of different bandgap) to achieve real-space transfer of one type of carrier (electrons or holes) and therefore spatial separation of electron and holes. The devices are now based on a new method of achieving total depletion of a large volume of semiconductor material and have many interesting applications (Capasso, 1982a,b; Capasso et al., 1982a,d). Some of these include ultra-low-capacitance p - i- n detectors, photocapacitive devices, and a new avalanche photodiode with ultra-low avalanche noise. 5. ULTRA-LOW CAPACITANCE p - i - n PHOTODIODES AND PHOTOCAPACITIVE DETECTORS Figure 1 1 shows a cross-section of the channeling diode. This consists of several abrupt p - n junctions. The layers are lattice matched to a semi-insulating substrate. The p + and n+ regions, which extend perpendicular to the layers, can be obtained by ion implantation, or by etching and epitaxial regrowth techniques. The voltage source supplying the reverse bias is connected between the p+ and the n+ regions. We assume, for simplicity,
332
FEDERJCO CAPASSO
r
r
SEMI- INSULATING SUBSTRATE
L
t
FIG. 11. Cross section of channeling diode under different reverse-bias conditions. The shaded portions of the layers represent the depletion region, while the white ones the undepleted portions. V,, is the voltage at which the layers are completely depleted. Any further increase in reverse bias adds a constant field E parallel to the device length. Quai doping levels (p = n) have been assumed (from Capasso, 1982b).
equal doping levels (n = p = N) for the n and p layers. The three center layers have thickness d, while the topmost and bottommost p layers have thickness d/2;the layer length is L (<< d), and the sensitive area is assumed rectangular with dimensions L and L'. For zero bias the p and n layers are in general only partially depleted, as shown in Fig. 1 la. The shaded areas denote the space-charge regions. The undepleted portions of the p and n layers (white areas) are at the same potential of the p + and n+ end regions, respectively, so that the structure appears as a single
6.
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interdigitated p- n junction. Because of this geometry, when a reverse bias is applied between the p + and n+ regions, this potential difference will appear across every p- n junction, thus increasing the space-charge region thickness (Fig. I 1b). The bias is then further increased until all the p and n layers are completely depleted at a voltage V = V,, (Fig. 1lc). At this point, analogously to ap+-i-n+ diode, any further increase in the reverse bias will only add a constant electric field E parallel to the length L of the layers. The capacitance of this novel structure has an interesting dependence on the applied voltage. For voltages < Ve, the capacitance is essentially that of the four p-n junction capacitors in parallel (since L >> d). Thus
C'
= 4&,LL'/
w
(2)
where W is the depletion layer width of each p-n junction (the dielectric constants of the n and p layers of different bandgap have been assumed equal). As the reverse voltage is increased and approaches V,,, the capacitance decreases toward the value C,, = 4&,LL'/d
(3)
At the punch-through voltage Vpa the capacitance drops abruptly from this relatively large value to
since the layers have been completely depleted and the residual capacitance is that of the "p+-i-n+ diode" formed by the p + and n+ regions. The capacitance is thus reduced by a factor (LID)', which for typical dimensions is 2 100. For L = L ' = 50 pm CV,", 5 pF. Note that in the general case of different acceptor and donor concentrations N,, N A , the thicknesses of the center p layer and of the NN layers (d,, and dp)should be in the ratio d?l/dp = NAIND
(5)
and the topmost and bottommost p layer should have a thickness dJ2, to ensure complete depletion of all the p and n layers. For a three-layer structure the p + and n+ regions may be obtained via ion implantation of relatively light ions such as Be, Mg, or Si. Schemes using etching and epitaxial regrowth techniques are also possible, as described below. The above novel steplike C- V characteristics has been experimentally verified on three-layer p- n -p structuresgrown by LPE and MBE (Capasso et al., 1982a). The center layer is of n-GaAs doped to 5 X 10'5/cm3and is = 1.5 pm thick. The two p layes are of A10.,,Gao.55Asdoped to = 5 X
334
FEDERICO CAPASSO
I
0
1
2
3
4 6 6 7 8 9 1 0 REVERSE BIAS (VOLTS) FIG. 12. Experimental C- Vcharacteristic of channeling diodes, measured at 1 MHz, from LPE (A) and MBE (B)-grown wafers (from Capasso et a[., 1982a).
cm3 and of half the thickness of the center layer. The area between the p + and n+ regions is typicaly cm2.The transverse p + and n+ regions were obtained by etching and LPE regrowth of A10.,Gao~80As.Figure 12 shows the typical experimental C- Y curves measured at 1 MHz for three-layer structures grown by LPE (A) and MBE (B). The C-V characteristics display three distinct regions: first, a decrease in capacitance with increasing reverse bias described by a linear 1/C2 versus V plot, typical of an abrupt p - n junction diode. This is followed by a large decrease in capacitance (= 1 pF) over a small voltage range (= 0.15 V) and by a region of ultrasmall nearly constant capacitance (0.05 pF). The punch-through voltage and the overall C- Y curve is consistent with the above doping and layer thicknesses and reproduces the expected behavior very well. It is important to point out that the ultrasmall capacitance of this structure above punch-through is largely independent of the detector area between the p + and n+ regions, which can be very large, and of the doping of the layers. It is also worth noting that for a large change in capacitance complete depletion of all the layers is not required. Assume, for example, that for the device in Fig. 11 the donor concentration is greater than the acceptor concentration; in this situation only the p layers will be completely depleted. The residual capacitance is that formed by the two unde-
6.
GRADED-GAP AND SUPERLATTICE DEVICES
335
pleted sections of the II layers with the p+ region. This capacitance is still much smaller than that before punch-through. The channeling diode can be used as an ultrasensitive photocapacitive detector (Capasso et al., 1982d). The capacitance of these devices was measured at 1 MHz as a function of reverse bias for different incident intensities (Fig. 13). A 2 mW He-Ne laser attenuated with neutral density filters was used as the light source. The top Alo.45Gao.55As layer is transparent to the 1 = 6328 A radiation. The other C- Vcurves were obtained by varying the incident laser power over four orders of magnitude from 20 pW to 200 nW. Note the increase of the punch-through voltage with increasing power and the larger variations in capacitance (0.6 - 1.O pF) (with respect to the “dark capacitance”) produced by the low optical power levels used. It is clear that this device can be used as an ultrahigh-sensitivity photocapacitance detector. The essential features of this novel photocapacitance phenomenon can be easily interpreted with the aid of Fig. 1 1. Assume that the device is biased at or slightly above the punch-through voltage. When light shines on the device, the photogenerated electrons and holes are spatially separated and collected in the depleted n and p layers, respectively, thereby partially neutralizing the ionized donor and acceptor space charge. The net effect is that the width of the depletion layer is reduced (Fig. 1 1b); this produces a large increase in capacitance. An addi-
I
0.0
0.5
1.0
I 1.5
I 2.0
I 25
I
I
I
3.0
3.5
4.0
I 4.5
REVERSE BIAS (VOLTS) FIG. 13. C- V characteristics of the channeling photodiode, for optical power levels in the range from 20 pW to 200 nW (from Capasso ef al., 1982b).
336
FEDERICO CAPASSO
tional voltage is thus required to deplete the layers. This explains the shift of the punch-through voltage with increasing optical power. The ultrahigh sensitivity of the structure is due to two factors. First, even a small (510%) reduction of the depletion layer width produced by a very low incident power is sufficient to cause a large change in capacitance. Second, the spatial separation of optically generated electrons and holes greatly increases their recombination lifetime so that substantial quasi-stable excess densities of electrons and holes are present in the layers to compensate the ionized space charge. The increase in lifetime due to the spatial separation of the electrons and holes was first discussed by Dohler in the context of n-i-p-i superlattices (1972). It is important to stress that to operate this detector as an ultra-low-capacitance, low-punch-through voltage p - i-n photodiode, it must be biased at voltages such that the capacitance is not affected by the incident light. It is worth emphasizing the unique features and important differences of this structure with respect to conventional p - i - n diodes. The novel interdigitated p - n junction scheme allows one to achieve an ultrasmall capacitance, largely independent of the detector area between the p+ and n+ regions of the layers’ doping. Thus the sensitive area can be maintained reasonably large and the doping moderately high. Note that conventionalp - i- n’s require very low doping levels and small areas. The p and n layers can also have the same bandgap for the above applications. Horikoshi et al. (1984) have reported on a GaAs channeling p - i - n diode with low capacitance responding at wavelengths longer than the bandgap of GaAs. This is due to the extremely thin highly doped n and p layers, which create a very high electric field normal to the layers. This gives rise to a large electroabsorption at sub-bandgap photon energies; the Franz-Keldysh effect is also enhanced by band tailing due to statistical fluctuations of the doping in the highly doped layers.
6. CHANNELING AVALANCHE PHOTODIODES To operate the device as a new avalanche photodiode with high alp ratio, the p layers should have a wider gap than the n ones. The ionization rates for electrons (a)and holes (p)must be very different from each other to minimize the avalanche noise (McIntyre, 1966). To understand the avalanche photodetector operation consider the three-dimensional picture of the APD band diagram. This is illustrated in Fig. 14 for voltages > VP*. The parallel field E is assumed high enough to cause ionization. Suppose that radiation of suitable wavelength is absorbed in the lowergap layer, thus creating electron- hole pairs. The p - n heterojunctions serve to confine electrons to the narrow-bandgap layers while sweeping
6.
GRADED-GAP AND SUPERLATTICE DEVICES
337
FIG. 14. Band diagram of a channelingavalanche photodiode (from Capasm 1982a).
holes out into the contiguous wider-bandgap p layers where they are confined by the potential. The parallel electric field E causes electrons confined to the narrow-bandgap layers to impact ionize. Holes generated in this way are swept out in the surrounding higher-gap layers before undergoing ionizing collisions in the narrower-gap layers since the layer thickness is made much smaller than the hole ionization distance 1//3. In conclusion, electrons and hole impact ionize in spatially separated regions of different bandgap. The holes in the wider-gap layers impact ionize at a much smaller rate compared to electrons in the relatively low-gap material, due to the exponential dependence of a,/3 on the bandgap, so that a//3can be made extremely large. In order for the device to operate in the described mode, several conditions should be met. First, the potential well confining the electrons in the narrow-gap layer should be equal to or greater than the electron ionization energy, so that electrons do not escape the potential wells before impact ionizing. A similar condition is required for holes in the wider-gap layers. It is also necessary that holes created by incident photons or by electron impact ionization are swept out of the lower-gap layers in a time short with respect to the average time between ionizing collisions in these layers, so that they will not avalanche in the low-gap material. Finally, it is necessary that when the layers are completely depleted the maximum field perpendicular to the plane of the layers be smaller than the avalanche threshold field E ~ so, that no multiplication occurs perpendicu-
338
FEDERICO CAPASSO
lar to the layers. Detail design considerations for this structure have been given by Capasso (1982b); recently Brennan (1985) presented an optimization of the channeling APD based on a Monte Carlo simulation.
IV. Low-Noise Muhilayer Avalanche Photodiodes and Solid-state Photomultipliers A comprehensive theory of avalanche noise in APDs was developed by McIntyre in 1966. The noise of an APD per unit bandwidth can be described by the formula
where e is the electron charge, z p h is the unmultiplied photocurrent (signal), (M) is the average avalanche gain, and F is the excess noise factor. For simplicity, the device’s dark current is neglected. In p - n and p - i- n reverse-biased photodiodes withoug gain, (M) = 1, F = 1, and the well-known shot-noise formula will indicate the device’s noise performance. In the avalanche process, if every injected photocarrier underwent the same gain M, the factor F would be unity, and the resulting noise power would only be the input shot noise due to the random amval of signal photons, multiplied by the gain squared. The avalanche process is, instead, intrinsically statistical, so that individual camers generally have different avalanche gains characterized by a distribution with an average (M). This causes additional noise, called avalanche excess noise, which is conveniently expressed by the F factor in Eq. (6). As briefly mentioned in the previous section, it can be shown that F is related to the ratio of ionization coefficients (a//?) for electrons and holes and to the gain (M). The ionization coefficient is defined as the average number of secondary pairs created per unit length by impact ionization by an electron or hole along the direction of the electric field. a and /? stand, respectively, for electron and hole ionization rates, and the ratio of ionization coefficients k = a/P. The ionization rates in all known semiconductorsincrease exponentially with increasing applied electric field and with decreasing ionization threshold energy Ei. This is the minimum energy required by an electron or a hole to create an electron- hole pair by impact ionization, and is in general greater than the bandgap of the semiconductor. For a detailed discussion of ionization rates, of threshold energies, and in general of the physics of avalanche photodiodes, the reader is referred to a recent review by the author (Capasso, 1985). The excess noise factor F is strongly dependent on a//?. To achieve a low
6.
GRADED-GAP AND SUPERLATTICE DEVICES
339
F, not only must a and p be as different as possible, but also the avalanche process must be initiated by the carrier with the higher ionization coefficient. To understand this, consider electron- hole pairs created when a! = p. If the pair is created by an electron, then there will be another pair generated by the newly created hole traveling in the opposite direction. This strong feedback effect amplifies noise fluctuations, and the situation is unfavorable for low noise detection. The feedback effect can be eliminated if only one type of carrier can impact ionize. With such a limitation, the excess noise factor can be shown to attain a constant value of 2 at gains greater than 10. Thus, even in an ideal APD where only one type of carrier can ionize, one cannot reduce the avalanche noise below such a limit if a reasonable gain (M 5 10) is to be maintained. The designer, therefore, faces the challenge of finding suitable materials with a large difference between a! and p. Silicon, for example, has a much greater than /3, and is widely used to make low-noise APDs for systems operating at wavelengthsbelow 1.1 pm.Si APDs with k values greater than 100 have been reported. The advent of communication systems that exploit the low-dispersion and low-loss regions of present fibers has stimulated considerable work to develop low-noise APDs for the 1.3 and 1.55 pm regions. Although impressive results have been reported, especially as far as the dark current and the gain are concerned, no APD capable of matching the noise performance of Si APDs has yet been demonstrated. Most materials currently available in the longer-wavelength regions do not have the desirable large difference between a and p. For example, InP, currently used as the avalanche layer in heterojunction APDs operating at 1.3 to 1.6 pm, has k = 0.3-0.4; germanium APDs have k = 2 (Capasso, 1985). The lack of suitable materials raises the question: Is it possible to artificially enhance the ratio of ionization coefficients using bulk constituents having comparable CLI and p? A second question is whether it is possible to think of the solid-state equivalent of a PMT, which has the characteristic, not available in an APD, of having virtually noise-free gain up to the highest multiplications (approximately 106). Several schemes have either been proposed or demonstrated that either enhance or inhibit ionization from one type of carrier. These proposed techniques include developing the following structures: graded-gap APDs, which exploit the difference between the ionization energies and the quasielectric fields for electrons and holes in graded-gap materials; multiquantum well and staircase APDs, which have a large difference between the ionization energies for electrons and holes created by the asymmetry be-
340
FEDERICO CAPASSO
tween the conduction- and valence-band-edge discontinuities in many 111- V heterojunctions; and channeling APDs, which use the spatial separation of electrons and holes in materials of different bandgap via n - p - n - p structures to enhance the orlp ratio. The latter devices have been discussed in the previous section. All these new schemes seek to reduce avalanche noise by modifying the conventional energy-band diagram of a reverse-biased p- n junction. Such modifications affect the electron and hole ionization rates, and they are carried out by tailoring both the composition and doping of the semiconductor alloys that constitute the avalanche region. 7. GRADED-GAP AVALANCHE PHOTODIODES
This device is probably the simplest scheme for enhancing the a!/p ratio (Capasso et al., 1982~).It is essentially a reverse-biased p- i-n diode. The high-field i region, a few hundred nanometers in width, consists of a variable-gap material and is sandwiched between heavily doped p + and n+ regions. The mechanism that enhances a!//? is illustrated by the energy-band diagram of the graded high-field layer in Fig. 15. Assume that avalanche is initiated by an electron- hole pair (1 - 1’) excited, for example, by a photon inside the graded region. Electron 1 is accelerated by the electric field toward regions of lower bandgap and generates an electron-hole pair (2- 2’) by impact ionization, after an average distance 1/a.Hole 1’, on the other hand, drifts in the opposite direction toward a higher-bandgap region and creates an electron - hole pair (3- 3’) after a distance 1/p. Thus, the effective ionization threshold energy for electrons is smaller than that for holes. Every electron- hole pair generated by impact ionization will create additional electron- hole pairs. Since the ionization rates increase exponentially with decreasing ionization energy, the ionization rates ratio crlp is expected to be enhanced. To achieve minimum excess noise, the avalanche should always be initiated by the carrier with the highest ionization coefficient. In this case, to achieve pure electron injection, the p + region, which is in contact with the high-gap portion of the i layer, should serve as the absorption layer. Since there is no electric field in this region, this layer could be graded also; this would allow injection in the high-field layer by drift rather than diffusion without sacrificing speed. For an experimental graded-gap device, the i region was graded from Alo.4sGao.ssAs to GaAs over a distance of 0.4 pm. Note that in the AlGaAs and GaAs bulk constituents of this graded region, a! is comparable to p.
6.
GRADED-GAP AND SUPERLATTICE DEVICES
341
FIG. 15. Band diagram of graded-bandgap avalanche photodiode showing impact ionization by the initial electron-hole pair 1 - I’ (from Capasso et al.. 1982~).
The p + and n+ layers were, respectively, Alo.45Gao.55As and GaAs. By measuring the multiplication under condigons of pure electron and hole injection, respectively, an effective ratio Z//3approximately equal to seven at a gain of five was estimated, where C and are suitable spatial averages of the ionization rates. It was found that C/p decreases with increasing graded layer width. A limitation of this structure is, however, that a large a//3can be obtained only at relatively modest gains.
a
8. MULTI-QUANTUM-WELL AVALANCHE PHOTODIODE Capasso et al. (198 1, 1982b)first demonstrated experimentallythat in an AlGaAs/GaAs multi-quantum well structure the effective impact ionization rates for electrons and holes are very different (a//3= 8), although they are comparable in the basic bulk materials (a = 2/3 in GaAs). This effect is attributed to the difference between the conduction- and valence-bandedge discontinuities at the Alo.,5Gao.55As/GaAsinterface, a feature common to several lattice-matched heterojunctions of use in long-wavelength
342
FEDERICO CAPASSO
FIG. 16. Band diagram of multiquantum well avalanche photodiode.
1.3 pm detectors (A10~481no,52As/Gao~4,1no~,3As; AlSb/GaSb, HgCdTe/ CdTe). This makes possible the development of low-noise APDs in these materials. The enhancement of a/j3 in a quantum well structure was first predicted by Chin er al. (1980). To understand the multi-quantum well APD, consider (Fig. 16) the energy-band diagram of the structure implemented by Capasso et al. (1981). Because of the very low doped material, the field is constant across the 2.5 pm long depletion layer. This consists of alternating GaAs (450 A) and Alo,45Gao.55As (550 A) layers, for a total of 25 periods; for illustrative purposes, assume a field F = 2.7 X lo5V/cm; at this value a sizeable multiplication was observed. For F > 1O5 V/cm, electrons gain between collisions an energy greater than the average energy lost per phonon scattering event (E P )(= 2 1 meV). Thus carriers are strongly heated by the field and can gain the ionization energy. In Al,Ga, -,As/GaAs heterojunctions the conduction-band discontinuity A E , amounts to = 60%of the bandgap difference (Miller et al., 1984a,b; Wang et al., 1984; Arnold et al., 1984). Consider now a hot electron accelerating in an AlGaAs barrier layer. When it enters the GaAs well, it abruptly gains an energy equal to the
6.
GRADED-GAP AND SUPERLATTICE DEVICES
343
conduction-band-edge discontiniuty A E, = 0.38 eV. It is important to stress the ballistic nature of this energy gain. Given the abruptness of the heterointerface, it occurs over a distance much smaller than the phonon scattering mean free path so that the average distance required to reach the ionization threshold is greatly reduced (aenhanced). Since A E, > A E, = 0.23 eV, electrons enter the GaAs well with higher kinetic energy than do holes and are therefore much more likely to produce electron- hole pairs in the GaAs (impact ionization assisted by band-edge discontinuities). Therefore, a steplike band structure as shown in Fig. 16, where the discontinuity in the conduction band is greater than that in the valence band, will enhance the ionization rate ratio a/j?considerably over the value in bulk GaAs, since Orlp increases roughly exponentially with A EJA E,. Electrons that have impact ionized in the GaAs easily get out of the well; the voltage drop across the well is > 1 V. In addition, at fields 2 lo5 V/cm in GaAs, the average electron energy is 20.6 eV (Shichijo and Hess, 1981) so that electron-trapping effects in the wells are negligible. The effective ionization rates are plotted in Fig. 17 versus reciprocal electric field l / in ~ the range (2.1-2.7) X lo5 V/cm. For example, at E =
I/E cm/v I FIG.17. Measured ionization rates for electrons (a) and holes (p) versus reciprocal electric field in a quantum well APD (from Capasso et al., 1982b).
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FEDERICO CAPASSO
2.5 X lo5 V/cm, C Y / = ~ 8 and the measured electron initiated gain is M e = 10. This gives, according to MacIntyre’s theory, an excess-noise factor F = 2.9 for electron-initiated multiplication, which represents nearly a factor of 3 reduction in the excess noise factor F with respect to a GaAs APD (a= 2p) operating at the same gain. The superlattice APD concept may be extended to other heterojunctions with an asymmetry between the conduction- and valence-band discontinuities such as AlGaAsSb/GaSb or AlIn GaAs/InGaAs (lattice matched to InP). Detectors made of such materials would be sensitive to wavelengths in the low-dispersion, low-loss window ( 1.3- 1.6 pm) of optical fibers. Other authors have observed an enhancement of the alp ratio in AlGaAs/GaAs superlattices (Juang et al., 1985); Osaka el al. (1986) have reported an enhancement of p by a factor of four over the bulk value in G ~ J ~ . , 3 A sin InP/G%.471%.,,As superlattices due to the large AEv(=0.4 eV). Monte Carlo simulations have confirmed the electron ionization rate enhancement in AlGaAs/GaAs superlattices (Brennan et al., 1985); calculations by Ridley (1986) of CY and 3/ give excellent agreement with the experimental values of Capasso et al. (198 I). One potential problem in the superlattice APD is the trapping of electrons in the well. For very large band-edge discontintuities. This problem can be avoided by inserting a compositionally graded layer between the well and the barrier layers. The resulting bandgap grading smoothens out the barrier (Fig. 18). 9. STAIRCASE SOLID-STATE PHOTOMULTIPLIERS This recently disclosed device (Capasso et al., 1982e, 1983a; Capasso and Tsang, 1982; Williams et al., 1982) is also based on the physical concept of impact ionization assisted by band discontinuity. In the staircase APD, however, the entire ionization energy may be acquired at the conduction-band steps. This device offers two other features that distinguish it from any other APD; the low operating voltage, on the order of 5 to 10 V, and the noise-free gain. Figure 19a shows the energy-band diagram of the graded-gap multilayer material (assumed of very low background doping or intrinsic) at zero applied field. Each stage is linearly graded in composition from a low (EJ to a high (E,) bandgap, with an abrupt step back to low-bandgap material. The conduction-band discontinuity shown accounts for most of the bandgap difference. The materials are chosen for a conduction-band discontinuity equal to or greater than the electron ionization energy Eiein the low-gap material immediately following the step. The band structure of the complete staircase detector under bias is
6.
GRADED-GAP AND SUPERLATTICE DEVICES
\
345
"+
FIG. 18. Band diagram of a superlattice avalanche photodiode with graded regions to eliminateelectron trapping (from Capasso et al., 1983a).
shown in Fig. 19b. Consider a photoelectron generated next to the p + contact. Under the combination of the bias field E and the grading field A EJt, it drifts towards the first conduction-band step. The combined field E - ( A E J j ) is small enough so that the electron does not impact ionize before it reaches the step. After the step, since A E , = E,, the electron impact ionizes; this ballistic ionization process is repeated in each stage. Ideally, the avalanche gain per stage is exactly two; each electron impact ionizes once after each conduction-band step. In practice, the gain is 2 - 8, where 6is the fraction of electrons that do not impact ionize. The total gain of the structure is then (M) = (2 - qN,where N is the number of stages. Any residual hole-initiated ionization can only be caused by the applied electric field E, which is chosen so that holes cannot ionize; the valenceband steps are of the wrong sign to assist ionization. However, for electron
346
FEDERICO CAPASSO
FIG. 19. Band diagram of staircase solid-state photomultiplier.The arrows in the valence band simply indicate that holes do not impact ionize (from Capasso el al.. 1983a).
transport across the graded region, this bias field E must be sufficiently high to cancel the AEJI conduction-band quasi-electric field and provide a small extra component to assure drift, rather than diffusion transport. In conclusion, only electrons impact ionize in this structure. Note the low operating voltage of this device. From Fig. 19b it can be seen that the reverse-bias voltage is equal to the energy separation, expressed in volts, between the Fermi levels (dashed horizontal lines) in the p + and n+ region. This separation is approximately equal to A E , times the number of stages. For AEc = 1 eV and N = 5, this corresponds to a bias slightly in excess of 5 V. The gain at this voltage under optimal operating conditions would be approximately equal to 32. This bias voltage is significantly smaller than in conventional APDs, which have bias voltages from 50 up to several hundred volts. The physical reason for this low operating voltage is that the ionization energy is delivered abruptly to the electrons by the conduction-band steps,
6.
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347
rather than gradually via the applied field, as in a conventional APD. Thus, the competing energy losses by phonon emission are much smaller, and most of the applied voltage is used to create electron-hole pairs. This low-voltage operation also minimizes the device leakage current. The staircase APDs excess noise factor can be calculated and is plotted in Fig. 20 as a function of gain per stage for a different number of stages (Capasso, 1983c; Capasso et al., 1983a). The plot shows that if most of the carriers ionize at each step (multiplication per stage =2), the excess noise factor is nearly unity, and independent of the number of stages. In this limit, the residual excess avalanche noise is caused by the small fraction of electrons at each stage that do not impact ionize. This implies that the multiplication process is virtually noise-free even at high gain, similar to the performance of a photomultiplier (PMT). The avalanche process in the staircase structure is much less random than in a conventional APD. This explains why the staircase APD intrinsically has a much lower noise. The multiplication occurs only at well-defined positions in space (at the conduction-band steps), and if the device is properly designed, most carriers ionize at each step. Thus, the statistical
STAIRCASE APD
100
1.25
1.50
1.75
2.0
MULTIPLICATION PER STAGE
FIG.20. Excess noise factor of the staircase solid-state photomultiplieras a function of the gain per stage for different number of stages. Note that if most electrons ionize at each stage (gain per stage =2), the multiplicationprocess is noise-free even at high gain (from Capasso et al., 1983a).
348
FEDERICO CAPASSO
variations of the gain are very small, causing the excess noise factor to be near unity. In a conventional APD, carriers can ionize essentially anywhere within the avalanche region, resulting in more random avalanche, larger gain fluctuations, and higher excess noise. The staircase structure’s noise performance, therefore, is not described by the earlier theory of McIntyre ( 1966). The staircase APD and the PMT have some structural similarities, and they also have the common feature of nearly noise-free multiplication at large gain. However, the formulas for the excess noise factor for the staircase and PMT are different. In the staircase APD, an electron usually creates no more than one electron-hole pair per dynode so that the excess noise factor is only due to those camers that cannot ionize, In a practical PMT, the number of electron-hole pairs created per dynode is always greater than two. The most interesting and potentially useful material to fabricate the staircase APD is HgxCd,-xTe. In this material system, the bandgap can be vaned from 0 eV (in HgTe, a semimetal)to 1.6 eV (in CdTe), and is always direct. Heterojunctions of this material system have essentially all the bandgap difference in the conduction band. These material features permit an extra degree of freedom in designing a staircase detector. By making optimum use of the large conduction-band difference at the step, a multiplication per stage greater than two can be achieved with the Hg,Cdl -,Te material system. Gains as high as lo’ as possible, similar to those in many PMTs. A final consideration in designing staircase APDs is that electrons do not necessarily impact ionize as soon as they reach the conduction-band step. In fact, electrons will impact ionize after a distance on the order of Ai, 1
FIG.2 1 . Band diagram of a staircase detector with ungraded sections to increase ionization probability after the steps (from Capasso ef al., 1983a).
6.
GRADED-GAP AND SUPERLATTICE DEVICES
349
which is the finite ionization mean free path. Typically liis 5- 10 nm in most 111- V semiconductors. Thus, it is advisable to insert ungraded layers having a thickness corresponding to a few libetween the steps and the graded regions, as shown in Fig. 21. This will maximize the ionization probability. Although the staircase APD represents a conceptual breakthrough, the practical implementation will require extremely good control of some of the material parameters. The interfaces must not only be abrupt over an atomic scale, but they must also be defect-free. In addition, it is absolutely essential that the background doping be as low as possible to achieve a constant electric field in the graded region.These material characteristics are difficult to achieve in HgCdTe. 10.
IMPACT IONIZATION ACROSS THE BANDDISCONTINUITY: A NEWSOLID-STATE PHOTOMULTIPLIER
Recently we have observed a new avalanche phenomenon in superlattices, namely, the impact ionization across band discontinuities of camers confined in the wells (Capasso et al., 1986). This phenomenon, independently predicted by Chwang and Hess (1986), could lead to a new type of solid-state PMT. This effect is illustrated in Fig. 22a. Consider a multiple quantum well structure with n-type doped wells and undoped bamers. Electrons from the parent donors can remain confined in the wells even in the presence of a relatively strong electric field, provided the barriers are thick enough to minimize tunneling. Consider now a hot electron in a barrier layer. When it enters the well with sufficient energy, it can impact ionize one of the bound carriers out of the well. In this ionization effect only one type of carrier is created so that the positive feedback of impact HOT ELECTRON
-
&
(a)
(b)
FIG.22. Impact ionization across the band discontinuity.(a) Quantum wells are doped. (b) Wells are undoped; shown is the ionization across band discontinuities of carriers dynamically stored in the wells. These carriers originate from thermal generation processes via deep levels.
350
FEDERICO CAPASSO
ionizing holes is eliminated, leading to the possibility of a quiet avalanche with small excess noise. Of course, in this case, one must constantly supply the electrons in the wells by applying suitable selective contacts to the well regions. From a conceptual point of view, this effect has some similarities with the impact ionization of deep levels in the sense that the well may be treated as an artificial trap. It is important to point out that doped wells are not required for observation of the effect. Due to the thermal generation of electrons and holes in the well layers (which gives rise to bulk dark current), relatively large electron and hole densities can be dynamically stored in the wells if the band discontinuities are appreciably larger than the average energies of the carriers in the wells. This situation may occur in the high field region of certain quantum well p - i - n photodiodes such as the ones investigated by Capasso et al. (1 986c), and is illustrated in Fig. 22b.
HOT HOLE
FIG.23. (a) Band structure of a multiple graded well photomultiplier (the graded regions are shaded). (b) Mechanism of hole multiplication by impact ionization across the valence band discontinuity.
6.
GRADED-GAP AND SUPERLATTICEDEVICES
-
-2
3.0
- 30
R407 2 5 - T = 90K f = 200HZ
- 25
2.0
I
t
-
-
: -
w z
-
g 1.5-
HOLE INJECTION
--- ELECTRON INJECTION
- 20 -
2
-
0
-
-15
3
-
-
0
-
-
o I
-
-
u t-
a
351
5
3a I-
1 3
I
1.0 -
u
0.5 -
0
- 40
-5
_ _ _ - -1~ ~
FIG.24. Reverse bias photocurrent and corresponding multiplication factors under conditions of pure electron and pure hole injection for the multiple graded well device of Fig. 23.
These structurescontain an Ab.4,1~.~2As/Gao.4,1%.53As superlattice in the i region, with barriers and well thicknesses in the 100-500 A range. A large ratio of the multiplications for holes and electrons was observed @&/Me), implying that holes ionize at a significantly higher rate than the electrons in these structures. A similar effect has been found in p- i- n diodes containing AlSb/GaSb (Capasso et al., 1986~).A systematic study of the temperature and chopping frequency dependence of the multiplication showed conclusively that the observed effect is not a band-to-band process. Also, deep-level ionization could be ruled out, since it would require unrealistically large densities of such centers ( Z 10'' ~ m - ~By ) . appropriately grading the interface of the wells, the storage of electrons can be eliminated, while holes are still confined (Fig. 23). This should maximize the ionization rate ratio, by minimizing electron-initiated multiplication. This structure was grown by MBE in the AlInAs/GaInAs system (Allam et al., 1987). The structures consisted of three period superlattices, placed in the i region of a p - i - n photodiode, with 50 1 A Ab.481%.52As barriers, 292 A Gao.471n,,53As wells, and 1022 A graded regions. The graded regions were grown linearly
352
FEDERICO CAPASSO
graded and lattice matched to InP by computer-controlled MBE. For hole-initiated multiplication, avalanche gain occurs at a reverse bias of 7 V and reaches -20 at - 12 V, at a temperature of - 100 K. For electrons, the multiplication is less than 1.4, resulting in an ionization rate ratio p/ cu(=M,, - l/M, - 1) in excess of 50 (Fig. 24). This is the highest value measured in a III-V material. The low breakdown field and the observed frequency and temperature dependence of this effect show that band-to-band ionization is not responsible for the gain. The multiplication is caused by ionization across the band-edge discontinuity of carriers dynamically stored in the wells. The siored carriers arise from thermally generated dark currents which provide a “reservoir” of carriers at the well-to-bamer interface, which can be ionized out of the well by carriers heated by the electric field in the barrier layers (Fig. 23). Beyond a reverse bias of - 5 V, there is no confinement of electrons in the wells, due to the grading, and thus there is no multiplication of electrons by this process. The onset of electron multiplication occurs at a field of - 1.5 X lo5V cm-*, where bulk ionization occurs. Thus we have observed near single-carrier-type multiplication of holes by ionization over the discontinuity with feedback provided by band-to-band ionization of electrons. V. Other Device Applications of Staircase Band Diagrams and Variable-Gap Superlattices
Staircase potentials such as the solid-state PMT represent some of the most exciting applications of bandgap engineering. In this section another device application of this concept to ultra-high-speed devices will be presented. Sawtooth structures in general are very interesting from both a physics and a device point of view because of their lack of reflection symmetry, which leads to polarization effects useful, for example, in novel displacement current photodetectors. The growth of such graded-gap structures or superlattices, often of very short period, represents a real challenge for the MBE crystal grower. Not only is a computer-controlled MBE system a must, but also new techniques to achieve such short-distance compositional grading are necessary. One such technique is the recently introduced pulsed-beam method (Kawabe et al., 1982), by which, e.g., a variable-gap alloy is grown by alternatively opening (and closing) the aluminum and gallium ovens with the shutters. The result is an AlAs/GaAs superlattice with ultrathin constant period (-20 A) but varying ratio of AlAs to GaAs layer thicknesses. The local bandgap is therefore that of the alloy corresponding to the local average composition determined by the thicknesses of the AlAs and the
6.
GRADED-GAP AND SUPERLATTICE DEVICES
353
GaAs. Since the period of the superlattice is much smaller than the de Broglie wavelengths of the carriers, the material behaves basically like a variable-gap ordered alloy. Such techniques have been used recently to grow parabolic quantum wells (Miller et al., 1984a,b). In subsections 13 and 14 we shall discuss applications of variable-gap ordered alloys to negative differential resistance devices and high-speed photodetectgrs. 11. REPEATED VELOCITY OVERSHOOT DEVICES
Other interesting applicationsof staircase potentials have been proposed. We shall discuss here the repeated velocity overshoot device. This structure offers the potential for achieving average drift velocities (-2-3 X lo7/ cm s) in excess of the maximum steady-state velocity over distances greater than 1 pm (Cooper et al., 1982; Capasso, 1983a). Figure 25 shows a general type of staircase potential structure. The corresponding electric field, shown in Fig. 25b, consist of a series of highfield regions of value E, and width d superimposed upon a background field E,. The background field E, is chosen so that the steady-state electron-energy distribution is not excessively broadened beyond its thermal equilibrium value, but at the same time the average drift velocity is still relatively high. Electrons, upon entering the high-field regions, rapidly gain energy and momentum, so that the drift velocity overshoots the steadystate value. The energy and momentum are then allowed to relax in the subsequent low-field region. With the staircase structure repeated velocity
9
I ‘-7 I
X ,
FIG. 25. Principle of repeated velocity overshoot. (a) Staircase potential and (b) the corresponding electric field. (c) The ensemble velocity as a function of position is also illustrated schematically (from Cooper ef al., 1982).
354
FEDERICO CAPASSO
,-Ga As
/
f
A'0.2 Ga 0.8As
FIG.26. Band diagram of a graded-gap repeated velocity overshoot device (from Cooper et al., 1982).
overshoot can be achieved (Fig. 2%). A graded-gap AlGaAs structure with energy steps AW= 0.2 eV could be used for this purpose (Fig. 26). Alternatively one can use a biased n- i - p - i structure. 12. ELECTRICAL POLARIZATION EFFECTSIN SAWTOOTH SUPERLATTICES The rectifying properties of graded-gap single and multiple triangular barriers were first reported by Allyn et al. (1 980). Sawtooth superlattices also have other intriguing physical properties, such as the possibility of generating a transient macroscopic electrical polarization extending over many periods of the superlattice (Capasso et al., 1983a). This effect is a direct consequence of the lack of reflection symmetry. The energy-band diagram of a sawtooth p-type superlattice is sketched in Fig. 27a. The layer thicknesses are typically a few hundred angstroms, and a suitable material is graded-gap Al,Ga,_,As. The superlattice is sandwiched between two high-doped p + contact regions. Let us assume that electron- hole pairs are excited by a very short pulse, as shown in Fig. 27a. Electrons experience a substantially higher quasi-electric field (typically = 1O5 V/cm) than holes. Therefore, electrons separate from holes and reach the low-gap side in a subpicosecond time (< s). This sets up an electrical polarization in the sawtooth structure, which results in the appearance of a voltage across the device terminals (Fig. 27b). This macroscopic dipole moment and its associated voltage subsequently decay in time by a combination of (1)
6.
GRADED-GAP AND SUPERLATTICE DEVICES
0
0
0
0
355
0
FIG. 27. Formation and decay of the macroscopic electrical polarization in a sawtooth superlattice (from Capasso et a/.. 1983~).
FIG. 28. Pulse response of a sawtooth superlattice device to a 4ps laser pulse (from Capasso et a/., 1983~).
356
FEDERICO CAPASSO
dielectric relation and (2) hole drift under the action of the internal electric field produced by the separation of electrons and holes (Fig. 1 lc). This polarization phenomenon has recently been observed in AlGaAs sawtooth superlattices. The transient photovoltage showed a decay time of = 150 ps (Fig. 28) (Capasso et al., 1983~).
DEVICES 13. CHIRPSUPERLATTICE This new negative differential resistance structure was proposed by Nakagawa et al.( 1983). Figure 29 shows the chirp device. The superlattice is composed of alternating thin layers (550 A) of two different semiconductors with gradually changing periodicity, to give a variable gap. The emitter and collector are heavily doped n-type. Figure 30 shows the band diagram of the ground-state conduction miniband of the chirp superlattice without and with bias field EB.With a small voltage applied the electrons can be transmitted through the superlattice by resonant tunneling through the first allowed miniband and the current is relatively high. As the bias is increased, at some point the first minigap lies horizontally, and perfect reflection is achieved since there is no allowed state in the superlattice for the impinging electrons. At this bias field EB the current reaches a minimum and then rises again if the applied voltage is further increased. Thus negative differential resistance should be possible. The above considerations assume that all the electrons propagate ballisticaly; anelastic phonon scattering processes may, however, contribute to current leakage, decreasing the peak-to-valley ratio. Negative differential resistance in a chirp device has recently been observed by Nakagawa et al. (1 985).
OHMIC CONTACTS
COLLECTOR
CHIRP SUPERLATTICE
1
i
1
1
1
J .
/ I \
\
n+-a
\
b o b a b ab
Q V
b a n+-b /
N OR NON-DOPED
FIG.29. Schematicsof chirp superlattice device (from Nakagawa et af.,1983).
6.
GRADED-GAP AND SUPERLATTICE DEVICES
357
/MINI-GAP
-
___)
X
X
(0) (b) FIG.30. Band diagram of a CHIRP device (a) without the bias field and (b) with the bias field (from Nakagawa et a[., 1983).
14. PSEUDO-QUATERNARY SEMICONDUCTORS: APPLICATIONS TO HIGH-SPEED DETECTORS Recently Capasso et al. (1984b) have demonstrated a new superlattice (pseudo-quaternary GaInAsP) capable of conveniently replacing conven-
tional GaInAsP semiconductors in a variety of device applications. It is important to recall that these quaternary materials play a key role in optoelectronic devices for the 1.3- 1.6 pm low-loss, low-disperson window of silica fibers. The concept of a psuedo-quaternary GaInAsP semiconductor is easily explained. Consider a multilayer structure of alternated Ga,,,lnO,,,As and InP. If the layer thicknesses are sufficiently thin (typically a few tens of angstroms), one is in the superlattice regime. One of the consequences is that this novel material now has its own bandgap, intermediate between that of Gao.4,1no.s3As and InP. In the limit of layer thicknesses of the order of a few monolayers, the energy bandgap can be approximated by the expression
+ E,(InP)UInP) EB= Eg(Gao.471n0.,3As)L(Ga~.471n~.,3As)
(7) L(Ga0.47In0.53As) + L(InP) where the L’s are the layer thicknesses. These superlattices can be regarded as novel pseudo-quaternary GaInASPsemiconductors. In fact, similarly to Ga -,In,As -,P. alloys, they are grown lattice matched to InP and their bandgap can be vaned between that of InP and that of Gao,71no.,3As. The latter is done by adjusting the ratio of the Gao,471no,,3,As and InP layer thicknesses. Pseudoquaternary GaInAsP is particularly suited to replace variable-gap Ga -,In,Asl -yPy. Such alloys are very difficult to grow since the mole fraction x (or y ) must be continuously varied while maintaining lattice matching to InP.
,
358
FEDERICO CAPASSO
Figure 31a shows a schematic of the energy-band diagram of undoped (nominally intrinsic) graded-gap pseudo-quaternary GaInAsP. The structure consists of alternated ultrathin layers of InP and Gap.471no.53As and was grown by a new vapor-phase epitaxial growth technique (levitation epitaxy) (Cox, 1984). Other techniques such as molecular-beam epitaxy or metal -organic chemical vapor deposition may also be suitable to grow such superlattices. From Fig. 3 la it is clear that the duty factor of the InP and Gao.471no,,3As layer is gradually varied, while keeping constant the period of the superlattice. As a result the average composition and bandgap (dashed lines in Fig. 3 la) of the material are also spatially graded between the two extreme points (InP and Gao,71no.5,As). In our structure both ten and twenty periods (1 period = 60 A) were used. The InP layer thickness was linearly decreased with distance from = 50 A to = 5 A, while correspondingly increasing the Gao.471no.s3As thickness to keep the superlattice period constant (= 60 A). The graded-gap superlattice was incorporated in a long-wavelength InP/ Ga0.q71n0.53As avalanche photodiode, as shown in Fig. 3 1b. This device is - -
-
- - _
f
Ga0.471n0.53AS
P+
0
I
(a)
n+ i
DISTANCE
FIG.3 1 . (a) Band diagram of a pseudo-quaternary graded-gapsemiconductor.(---) represents the average bandgap seen by the carriers. (b) and (c) are a schematic and the electricfield profile of a high-low avalanche photodiode using the pseudoquaternary layer to achieve high speed (from Capasso ef a/., 1984b).
6.
GRADED-GAP AND SUPERLATTICE DEVICES
359
basically a photodetector with separate absorption (Ga,,In0~,,As) and multiplication (InP) layers and a high - low electric-field profile (Hi - LO SAM APD). This profile (Fig. 3 lc) is achieved by a thin doping spike in the ultralow-doped InP layer and considerably improves the device perfonnance compared to conventional SAM APDs (Capasso et al., 1984a). The Gao.471no.s3As absorption layer is undoped (n = 1 X 1015/cm3)and 2.5 pm thick. The n+ doping spike thickness and camer concentration were varied in the 500-200 A and 5 X 1017- 1017/cm3ranges, respectively (depending on the wafer), while maintaining the same camer sheet density ( ~ 2 . X 5 1012/cm2). The n+ spike was separated from the superlattice by an undoped 700- 1000 A thick InP spacer layer. The pi region was defined by Zn diffusion in the 3 pm thick low camer density (n- = 10L4/cm3) InP layer. The junction depth was varied from 0.8 to 2.5 pm. Similar devices, but without the superlattice region, were also grown. Previous pulse response studies of conventional SAM APDs with abrupt InP/Gao,471no,,,Asheterojunctions found a long (>10 ns) tail in the fall time of the detector due to the pileup of holes at the heterointerface (Forrest et al., 1982). This is caused by the large valence-band discontinuity (e0.45 eV). It has been proposed that this problem can be eliminated by inserting between the InP and Gao.471no.53As region a Ga, -xInxAs,-yPy layer of intermediate bandgap (Campbell et al., 1983). This quaternary layer is replaced in our structure, by the InP/Gao,471no.53As variable-gap superlattice. This not only offers the advantage of avoiding the growth of the critical, independently lattice-matched GaInAsP quaternary layer, but also may lead to an optimum “smoothing out” of the valence-band barrier for reproducible high-speed operation. This feature is essential for HI- LO SAM APDs s i q e the heterointerface electric field is lower than in conventional SAM devices. For the pulse response measurement we used a 1.55 pm GaInAsP driven by a pulse pattern generator. Figure 32 shows the response to a 2 ns laser pulse of a HI-LO SAM APD with (a) and without (b) a 1300 A thick superlattice. Both devices had similar doping profiles and breakdown voltages (=80 V) and were biased at - 65.5 V. At this voItage the ternary layer was completely depleted in both devices and the measured external quantum efficiency ~ 7 0 % The . results of Fig. 32 were reproduced in many devices on several wafers. The long tail in Fig. 32b is due to the pileup effect of holes associated with the abruptness of the heterointerface. In the devices with the graded-gap superlattice (Fig. 32a) there is no long tail. In this case the height of the barrier seen by the holes is no more the valence-band discontinuity A Ev but A E = AE,
- eeiL
(8)
360
FEDERICO CAPASSO
FIG. 32. Pulse response of a high-low SAM avalanche detector (a) with graded gap superlattice and (b) without to a = 2 ns, L = 1.55 pm laser pulse. The bias voltage is -65.5 V for both devices. Time scale 2 ns/div (from Capasso et a/., 1984b).
where ei is value of the electric field at the InP/superlattice interface and L the thickness of the pseudo-quaternary layer. The devices are biased at a voltage such that ei > AEv/eL so that A E = 0 and no trapping occurs. In the devices with no superlattice, A E = A Ev for every ei so that long tails in the pulse response are observed at all voltages.
6.
GRADED-GAP AND SUPERLATTICE DEVICES
361
VI. New Heterojunction Bipolar Transistors The essential feature of the heterojunction bipolar transistor (HBT) relies upon a wide-bandgap emitter wherein part of the energy bandgap differencebetween the emitter and base is used to suppress hole injection. This allows the base to be more heavily doped than the emitter, leading to a low base resistance and emitter-base capacitance, both of which are necessary for high-frequency operation, while still maintaining a high emitter injection efficiency (Kroemer, 1982). Bandgap engineering can be used to design new HBTs. The bandgap can be graded in the base to achieve a significant improvement in the speed. This device, along with electron velocity measurements in graded-gap p+-AlGaAs, is discussed in this section. A quantum well, or a superlattice in the base layer, on the other hand, makes possible an entire new class of negative differential resistance devices, based on resonant tunneling. These functional devices can have interesting applications in multiple-state logic, and in other signal processing applications. Tailoring the bandgap in the emitter of a bipolar, on the other hand, may help maximize the injection efficiency and minimize the collector emitter offset voltage. 15. THEGRADED-GAP BASETRANSISTOR
Kroemer (1957) first proposed the use of a graded-gap p-type layer for the base of a bipolar transistor, to reduce the minority camer (electron) transit time in the base (Fig. 33a). When the base-emitter and basecollector junctions are, respectively, forward and revem biased, electrons are injected from the emitter into the base and move over to the collector layer. With no grading in the heavily doped pbase, minority carriers (electrons) are transported by diffusion, a relatively slow process. In addition, a fraction of the injected electrons recombine with holes in the base, thus reducing the base transport factor. The presence of bandgap grading in the base creates a quasi-electric field acting on the electrons. These therefore move predominantly by drift (if the field is sufficiently strong), leading to a higher velocity in the base. It can easily be shown that the ratio of the base transit times for an ungraded bipolar and a bipolar with a graded base is (Hayes el al., 1983a)
where Eg2- E,, is the bandgap difference across the base, T the lattice
362
FEDERICO CAPASSO
........................
(b)
1
FIG.33. Band-diagram of graded-gap base bipolar transistor (a) with graded emitter-base electron interface, and (b) with ballistic launching ramp for even higher velocity in the base.
temperature, and q,and z; the base transit times for the transistor without and with grading in the base, respectively. Although Eq. (9) is rigorous only in the limit E,, - Eg2= kT, it can be employed as a useful “rule of thumb” in cases where E,, - E , is several times kT.Thus the bandgap difference must be made as large as possible, without exceeding the intervalley energy separation (AErL)of the material with gap E,, which would result in a strong reduction of the electron velocity and in the nonvalidity of Eq. (9). Using E,, - Eg2= 0.2 eV, the transit time is reduced by a factor of -4 at 300 K over a bipolar with an ungraded base of the same thickness. This allows a precious tradeoff against the base resistance (&), making possible an increase of the base thickness and a consequent reduction of Rb, while still keeping a reasonable base transit time. This will increase the maximum oscillation frequency of the transistor,f , . The bandgap E,,,on the emitter side of the base, should be smaller than the gap of the emitter to avoid back-injection of holes into the emitter and reduction in the current gain. Thus the structures of Fig. 30 combine the advantages of the wide-gap emitter bipolar (Kroemer, 1982) with that of a graded base. The emitter base junction in this case may be either graded (Fig. 33a) or abrupt (Fig. 33b). The latter structure in fact may allow even higher velocities in the base compared to that of Fig. 33a (Capasso et al., 1983b). With the abrupt emitter, electrons can be launched ballistically in the base with initial velocities 2 5 X lo7 cm/s; the quasi-field in the base maintains the velocity high, thus giving a shorter base transit time than the graded emitter bipolar (Fig. 33a).
6.
GRADED-GAP AND SUPERLATTICE DEVICES
363
Recently Capasso et al. (1983b) demonstrated that the quasi-field in the base strongly reduces slow diffusion effects. They measured the response time of a phototransistor with a 0.45 pm wide graded-gap base to a short picosecond laser pulse absorbed in the base layer. The symmetric ultrafast (FWHM = 40 ps) scope limited pulse response gives evidence that transport in the base is drift limited and is not broadened by diffusion. Operation of a graded-base three-terminal bipolar transistor was also demonstrated (Hayes et al., 1983a; Miller et al., 1983). The latter authors reported a cutoff frequencyf, = 16 GHz in an 800 A base device. More recently, high-current-gain graded-base bipolars with good highfrequency performance have been reported (Malik et al., 1985). The base layer was linearly graded over 1800 A from x = 0 to 0.1, resulting in a quasi-electric field of 5.6 kV/cm and was doped with Be to p = 5 X lot8~ m - The ~ . emitter- base junction was graded over 500 A from x = 0.1 to 0.25 to enhance hole confinement in the base. The 0.2pm thick A1,,,Ga,,,As emitter and the 0.5 pm thick collector were doped n-type at 2 X lo', and 2 X 10l6~ m - ~respectively. , The Al,Ga,-,As layers were grown at a substrate temperature of 700"C. It was found that this high growth temperature resulted in better Al,Ga,-,As quality, as determined by photoluminescence. However, it is known that significant Be diffusion
FIG.34. Common-emitter characteristics of a graded-base (= 2000 A) bipolar transistor (from Malik ef al., 1985).
364
FEDERICO CAPASSO
occurs during MBE growth at high substrate temperature and at high doping levels ( p > 10l8cm-9. SIMS data also indicated a misplacement of the p-n junction into the wide-bandgap emitter at 700" C substrate growth temperatures. Therefore, it was determined empirically that the insertion of an undoped setback layer of 200 - 500 A between the base and emitter to compensate for the Be diffusion resulted in significantly increased current gains. Zn diffusion was used to contact the base and provided a low base contact resistance. The common emitter I- V characteristics of a test transistor with an emitter area of 7.5 X cm2 is shown in Fig. 34. It is seen that the current gain increases with higher current levels and that the collector current exhibits flat output characteristics. The maximum differential dc current gain is 1 150, obtained at a collector current density of J, = 1.1 X lo3 A cm-*, which is the highest yet reported for graded-bandgap base HBTs. The slight negative resistance effects at high collector currents is due to thermal heating. A small offset voltage of about 0.2 V is also evident.
FREQUENCY ( G H z )
FIG. 35. Current gain and maximum available power gain versus frequency of a gradedbase bipolar transistor (from Malik et aZ., 1985).
6.
GRADED-GAP AND SUPERLATTICE DEVICES
365
The maximum V, for these devices before collector breakdown was about 8 V. These high gains were obtained with a dopant setback layer in the base of 300 .k and can be compared with previous work which consistently resulted in current gains of < 100 in HBTs without the setback layer (Hayes et al., 1983a; Miller et al., 1983). Several transistor wafers were processed with undoped setback layers in the base of 200-500 A, and all exhibited gain enhancement. High-frequency graded-bandgap base HBTs were fabricated using the Zn diffusion process. A single 5 p m wide emitter stripe contact with dual adjacent base contacts was used. The areas of the emitter and collector junctions were approximately 2.3 X and 1.8 X lo-’ cm2,respectively. The transistors were wire bonded in a microwave package and automated s-parameter measurements were made with an HP 8409 network analyzer. The frequency dependence of the small-signal current gain and power gain (for a transistor biased at I , = 20 mA, and V , = 3 V) are shown in Fig. 35. The transistor has a current gain cutoff frequencyf, = 5 GHz and a maximum oscillation frequency off- = 2.5 GHz. Large-signal pulse measurements resulted in rise times of 7r- 150 ps and pulsed collector currents of I , > 100 mA, which is useful for high-current laser drivers. 16. ELECTRON VELOCITY MEASUREMENTS IN VARIABLE-GAP AlGaAs
Recently Levine et al. (1982, 1983), using an all-optical method, measured for the first time the electron velocity in a heavily p+-doped compositionally graded Al,Ga,-,As layer, similar to the base of the bipolar transistor illustrated in Fig. 30. The energy-band diagram of the sample is sketched in Fig. 36, along with the principle of the experimental method. The measurement technique is a “pump and probe” scheme. The pump laser beam, transmitted through one of the AlGaAs window layers, is absorbed in the first few thousand angstroms of the graded layer. Optically generated electrons, under the action of the quasi-electric field, drifi towards the right in Fig. 36 and accumulate at the end of the graded layer. This produces a refractive index change at the interface with the second window layer. This refractive index variation produces a reflectivity change that can be probed with the counter propagating probe laser beam. This reflectivity change is measured as a function of the delay between pump and probe beam using phase-sensitive detection techniques. The reflectivity data are shown in Fig. 37 for a sample with a 1 pm thick transport layer, graded from A10.,Gao.9Asto GaAs and doped to p = 2 X 101*/cm3.This corresponds to a quasi-field of 1.2 kV/cm. The laser pulse width was 15 ps, and the time 0 in Fig. 37
366
FEDERICO CAPASSO
WNDOW LAYER
WINDOW LAYER Al0.4Ga0.6As
A10.4Ga0.6As
--
EV
GRADEDTRANSPORTLAYER
-
6aAs
Alo.,Gao.,As
'P FIG. 36. Band diagram of sample used for electron velocity measurements.
- 20
20
40
60 80 100 120 TIME DELAY ( p s e c ) FIG. 37. Normalized experimental results for pumpinduced reflectivity change versus time delay obtained in 1 pm thick graded-gap p+-AlGaAs at a quasielectric field F = I .2 kV/ cm (From Levine eta/., 1982). 0
6.
367
GRADED-GAP AND SUPERLATTICE DEVICES
represents the center of the pump pulse as determined by two-photon absorption in a GaP crystal cemented near the sample. The transit time is approximately given by the shift of the half-height of the reflectivity curve from zero, which is 7 = 33 ps. Taking as the drift length the graded-layer thickness minus the absorption length of the pump beam ( l / a= 2500 A), one estimates a minority carrier velocity 2.3 X lo6cm/s. In these relatively thick samples diffusion effects are important and cause a spread in the arrival time of electrons at the end of the sample, given roughly by the 10-90% rise time of the reflectivity curve, i.e., 63 ps. It is interesting to note that the drift mobility obtained from the measurement is pLd= v,/F = 1900 cm2/V s, which is comparable to the usual electron mobility of 2200 cmZ/Vs in GaAs at the doping level of the graded layer in GaAs. Electron velocity measurements were also made in a 0.42pm thick strongly graded (F= 8.8 kV/cm) highly doped (p = 4 X 10l8cm-9 Al,Ga,-,As layer, graded from A10.3Ga0.7Asto GaAs. A transit time of only 1.7 ps was measured, more than an order of magnitude shorter than that for F = 1.2 kV/cm, as shown in Fig. 38, corresponding roughly to a velocity v = 2.5 X lo7cm/sec. The velocity can be obtained rigorously and 1.2
t ~
1.0-
t. + 0.80
w
LL
w
IX
0.6 0.6-
n w
tJ! 0.4Q
2
cr
0
z 0.20
I
-4
-2
0
2
4
6
8
I0
T I M E DELAY (Psec)
FIG. 38. Normalized experimental reflectivity change Venus time delay measured in a 0.4pm thick graded-gap p+-AlGaAs layer at a quasi-electric field F = 8.8 kV/cm (from Levine ef al., 1983).
368
FEDERICO CAPASSO
accurately (*lo% error) from the reflectivity data by solving the drift diffusion equation and taking into account the effects of the pump absorption length (especially important in the thin sample), and the partial penetration of the probe beam in the graded material (Levine et al., 1983). Including all these effects, one finds that the reflectivity data can be fitted using only one adjustable parameter, the electron drift velocity. This velocity is u = 2.8 X lo6 cm/s for F = 1.2 kV/cm and p = 2 X 1018/cm3;and v = 1.8 X lo7cm/s for I:= 8.8 kV/cm and p = 4 X 1018/cm3. We see that when we increased the quasi-field from 1.2 to 8.8 kV/cm (a factor of 7.3) the velocity increased from 2.8 X lo6 to 1.8 X lo7 cm/s (a factor of 6.5). That is, we observed the approximate validity of the relation v = p F. In fact, using p = 1700 cm2/V s (for p = 4 X lot8cm-9 we calculate u = 1.5 X lo7 cm/s for F = 8.8 kV/cm, in reasonable agreement with the experiment. It is worth noting that this measured velocity of 1.8 X lo7 cm/s (in the quasifield)) is significantly larger than that for pure undoped GaAs, where u = 1.2 X lo7 cm/s for an ordinary electric field of 8.8 kV/ cm. In fact, the measured high velocity is comparable to the peak velocity reached in GaAs for F = 3.5 kV/cm before the intervalley transfer occurs from the central to the L valley. It is noteworthy that our measured velocity is also comparable to the maximum possible phonon-limited velocity in the central valley of GaAs. This is given by V,, = [(E,/m*) tanh(EP/2kT)]'/*= 2.3 X lo7 cm/s, where E , = 35 meV is the optical phonon energy and the effective mass m* = 0.067 mo. This high velocity can be understood without reference to transient effects since the transit time is much larger than the momentum relaxation time of 0.3 ps. The large velocity results from the fact that the electrons spend most of their time in the high-velocity central valley rather than in the low-velocity L valley. This may result from the injected electron density being so much less than the hole doping density that the strong hole scattering can rapidly cool the electrons without excessively heating the holes. Furthermore, the electrons remain in the central valley throughout their transit across the graded layer since the total conduction-band-edge drop ( A E c= 0.37 eV) is comparable to the GaAs T - L separation = 0.33 eV) and therefore they do not have sufficient excess energy for significant transfer to the L valley. 17. EMITTER GRADING IN HETEROJUNCTION BIPOLAR TRANSISTORS In this section we discuss in detail the emitter grading problem in HBTs. biThe performances of recently developed A10.481no.s2As/Gao,,71no~,3As polars with graded and ungraded emitters are compared, and the optimum way to grade the emitter is discussed.
6.
GRADED-GAP AND SUPERLATTICE DEVICES
369
Most of the work on MBE-grown heterojunction bipolar transistors has concentrated on the AlGaAslGaAs system. Recently the first vertical np - n Alo.4aIno.52As/Gao.471no.53As heterojunction bipolar transistors grown by MBE with high current gain have been reported (Malik et al., 1983). The (Al,In)As/(Ga,In)As layers were grown by molecular-beam epitaxy (MBE) lattice matched to a Fe-doped semi-insulating InP substrate. Two HBT structures were grown: the first with an abrupt emitter of A10.4aIn0.52As on a Gao.471n,,,,As base, and a second with a graded emitter comprising a quaternary layer of AlInGaAs of width 600 A, linearly graded
(a) ABRUPT EMITTER
(b)
GRADED EMITTER
HETEROJUNCTION BIPOLAR TRANSISTOR
FIG. 39. Band diagrams under equilibrium of heterojunction bipolar with (a) abrupt emitter and (b) graded emitter. Note the elimination of the conduction-bandnotch through the use of a graded emitter and the increase of the emitter-base valence-band barrier (from Malik et al.. 1983).
370
FEDERICO CAPASSO
between the two ternary layers, and an A10.481n0.52A~ layer. Grading from Gao,471no.53As to Alo,481no,,,Aswas achieved by slmultaneously lowering the Ga and raising the A1 oven temperatures in such a manner as to keep the total group-I11 flux constant during the transition. It should be noted that this is the first use of a graded quaternary alloy in a device structure. The energy-band diagram for the abrupt and graded emitter transistors are shown in Fig. 39a and b, respectively. It is seen that the effect of the grading is to eliminate the conduction-band notch in the emitter junction. This in turn leads to a larger emitter-base valence-band barrier under forward-bias injection. The following material parameters were used in both types of transistors. The A10.481n0.52A~ emitter and Ga0.471n0.53A~ collector were doped n-type with Sn at levels of 5 X lOI7 and 5 X 10l6~ m - Recent ~. experimental determination of the band-edge discontinuities in the Alo,4,1no~5,As/Gao,471no~53As heterojunction indicates that A E , = 0.50 eV and AE, = 0.20 eV (People el al., 1983).This value of A E , is large enough to allow the use of an abrupt A10.481n0.52A~/ Ga0.471n0.53As emitter at 300 K. Nevertheless, a current gain increase by a factor of 2 is achieved through the use of the graded-gap emitter, which is attributed to a larger valence-band difference between the emitter and base under forward-bias injection. This increase is clearly shown in Fig. 40. It is apparent from Fig. 40a and b that there is a relatively large collectoremitter offset voltage. This voltage is equal to the difference between the builtin potential for the emitter- base p - n junction and that of the base collector p - n junction. No such offset is therefore present in homojunction bipolars. We have recently shown that by appropriately grading the emitter near the interface with the base, such offset can be reduced and even totally
FIG. 40. Common-emitter characteristics of the A1,,Ino,,,As/Gao,,In,,,As heterojunction bipolar transitors with (a) abrupt emitter and (b) graded emitter at 300 K (from Malik et aL, 1983).
6.
371
GRADED-GAP AND SUPERLATTICE DEVICES
eliminated (Hayes et al., 1983b). The other advantage of grading the emitter is of course that the potential spike in the conduction band can be reduced, thus increasing the injection efficiency. The conduction-band potential is the result of the sum of two potentials: the electrostatic potential (b, equal to Vbi (the built-in potential)- V, (the base-emitter voltage), which vanes parabolically with distance (if the doping is uniform) and the grading potential (bf. If linear grading is used, there is always unwanted structure in the conduction band (spikes or notches, Fig. 41). It now becomes obvious that any structure can be eliminated by grading with the complementary function of the electrostatic potential in the emitter region
(a) VsE = 1.35V
-0.1 -
-
1.2v
-0.2
-> W
w
-0.3 - LINEAR GRADING WIDTH = 150 1 -0.4 0.1 -
1.05V
BASE
(b)
EMITTER
a
2
0.4-
z
0
0
0.1
-
(C 1
EMITTER
BASE
0 -
VBE= 1.35V
-0.1 -0.2
-
1.2v
-o,3 - PARABOLIC GRADING
-0.4 - WIDTH = 500 I
I
1.05V
i I
I
I
I
1
10
FIG.41. Conduction-bandedge versus distance from the p+-n base-emitter junction for three different linear grading widths at different base-emitter forward bias voltages (from Hayes ef a/.. 1983b).
372
FEDERICO CAPASSO
-
w 0.1
-
(3
? I
I EMITTER
BASE
0-
2 -0.1 U
VBE= 1.35V
m -0.2-
1.2v
z
0 -0.3I$ -0.4 D
-0.5
1.05V I
I
I
I
I
I
I
U13 I A N G C PMUM p - l l J U N L I I U N I A J
FIG.42. Conductio?-band edge versus distance from the p+-njunction, using a parabolic allv mded laver 500 A wide at different forward bias voltages (from Haves et ul.,1983b).
(1 - &J over the depletion layer width corresponding to a forward-bias equivalent to the base bandgap (Fig. 42). Note that in this case if the base-emitter junction is forward biased at 1.42 eV, the two potentials (grading and electrostatic) cancel each other out and one attains the flatband condition with a built-in voltage for the base-emitter equivalent to the bandgap in the base (= 1.42 eV). An HBT with such a parabolic grading has been fabricated, using MBE, with a Ga,,Al,,As emitter and a GaAs base and collector (Hayes et al., 1983b). A schematic diagram of the transistor structure and the commonemitter characteristic is shown in Fig. 43a and b, respectively. The emitter-base junction was graded from x = 0 to 0.3 on the emitter side over a distance of 600 A, the parabolic grading function being approximated by linear grading over nine regions. It can be seen from the characteristics shown in Fig. 43b that the offset is very small, about 0.03. Virtually identical characteristics with offsets 50.03 V were obtained for all devices on the wafer.
18. RESONANTTUNNELING TRANSISTORS WITH QUANTUMWELLBASE In this section, we discuss a new class of resonant tunneling heterojunction devices which consist of heterojunction bipolar transistors, with a quantum well and a double barrier, or a superlattice in the base region (F. Capasso and R. A. Kiehl, 1985). Pioneering work on resonant tunneling through a heterostructure quantum well was first done by Tsu and Esaki (1973) and Chang et al. (1974). Miniband conduction in a superlattice, a particular case of resonant tunneling, was investigated by Esaki and Chang (1974) and Tsu et al. (1979). More recently, Sollner et al. demonstrated resonant tunneling in AlGaAs/
a
r
n = 2 x iof?cm-3
I
AU-Sn
x *O
300A
X = 0.30-0
500;
x.o.30
Au
- Sn1
x=o x=o
I 3000i
n = 2 x10'8crn-3
PARABOLIC GRADING
. n = 3 x-6101
x=o
FIG.43. (a) Schematic diagram of AIGaAs/GaAs bipolar transistor that has a parabolic grading width of 600 A at the base-emitter junction; (b) common-emitter characteristics of the transistor shown in (a). Note the negligible offset voltage (from Hayes et al.. 1983b).
314
FEDERICO CAPASSO
GaAs double bamers at terahertz frequencies (1983) and a quantum well oscillator operating at 18 GHz (1984). In the above experiments, resonant tunneling was obtained by applying a voltage to the double barrier, to achieve matching between the Fermi level in the cathode and the resonant states of the well. Recently, however, Ricco and Azbel (1984) have pointed out that in order to achieve unity transmission at all the resonance peaks, the transmission of the left and right barrier must be equal at all the quasi-eigenstate energies. This is physically identical to that which occurs in a Fabry-Perot resonator, with which resonant tunneling structures share profound analogies. If the transmissions of the two mirrors are made sufficiently different, the transmission at the resonant frequencies decreases significantly below unity. Application of an electric field to a symmetric double barrier introduces a difference between the transmission of the two barriers, thus decreasing below unity the overall transmission at the resonance peaks. Unity transmission can be achieved only if the two barriers have different thicknesses; however, using this procedure one can only optimize the transmission of one of the resonance peaks. Recently, F. Capasso and R. A. Kiehl(l985) have proposed a new class of structures where resonant tunneling through a symmetric double barrier is achieved for the first time, not by application of an electric field, but by high-energy injection. This method does not alter the transmission of the two barriers and therefore should lead to unity transmission at all resonance peaks. Figure 44a shows the equilibrium band diagram of one of the devices. The structure is a heterojunction bipolar transistor with a degenerately doped abrupt emitter and a symmetric double barrier in the base. The base - emitter and base - collector junctions are then, respectively, forward and reverse biased. As the base-emitter voltage is increased, the energy difference between the Fermi level in the emitter and the first resonant state of the quantum well decreases. When these two levels are matched, electrons tunneling from the emitter region are injected in the first state of the well and undergo resonant tunneling through the double barrier with near-unity transmission probability. Off resonance the transmission probability is typically << 1 and equal to the product of the transmission coefficients of the two barriers without the quantum well (Ricco and Azbel, 1984). The collector current as a function of the base-emitter voltage VBE exhibits a series of peaks corresponding to the various quasi-stationary states of the well. Multiple negative conductancein the collector circuit can therefore be achieved. As a result of symmetry of the double barrier under operating conditions, current densities, negative conductance, and peakto-valley ratios much larger than in conventional resonant tunneling struc-
6.
-a
GRADED-GAP AND SUPERLATTICE DEVICES
375
L
-------
FIG.44. Band diagram of the resonant tunneling transistor (RTT) with tunneling emitter under different bias conditions: (a) in equilibrium; (b) resonant tunneling through the first level in the well; (c) resonant tunneling through the second level (not in scale) (from F. Capasso and R. A. Kiehl, 1985).
tures should be possible. Thus, RTTs have potential for high-performance oscillators. The device of Fig. 44 can be implemented with AlGaAs/GaAs grown by molecular-beam epitaxy. The wide-gap degeneratedly doped emitter ( N Ek 1 X lO”/cm3) would consist typically of Al,Ga,-,As (x 2 0.3) to ensure sufficient hole confinement in the base and high injection efficiency. The double barrier in the center of the base region consists of a GaAs well (typical thickness range 30-60 A) and of Al,Ga,-,As (0.3 5 x 5 1) rectangular barriers of equal thickness (typically 15 -50 A). For example, for two rectangular barriers, height 0.24 eV (corresponding to A130Ga0.70As), width 20 A and separation 30 A, the first level has an energy El = 1 19 meV. The double barrier and the well should be undoped with ultra-low carrier concentration to minimize scattering and recombination. This is important since for high transmission wave-function coherence must be maintained. Alloy disorder in the barrier may contribute to scattering of the injected electrons. This can be minimized by the use of AlAs barriers. The rest of the base layer outside the barrier region is instead heavily doped ( 10l8Ip I5 X 10’8/cm3)and its total thickness should be typically 2000 A to provide the required low base resistance. The thickness of the
376
FEDERICO CAPASSO
base region between the double barrier and the emitter should be smaller than the scattering mean free path of the electron injected from the emitter but greater than the zero-bias depletion width on the p side. A good choice for this thickness is = 500- 1000 A, which also minimizes quantum-size effects in this region. To achieve high current at resonance, the width of the resonant peak should be of order of or less than the width of the energy distribution of the electrons in the emitter, which at 77 K is comparable to the degeneracy (EF- E,). Assuming a GaAs well width = 30 A and A10.30Ga0.,0As bamers 20 A thick, one obtains A El = 50 meV for the energy width of the first resonance. If the emitter doping level is 2 X 1018/cm3,the degeneracy EF- Ec is = 50 meV A E l so that most electrons leaving the emitter will resonantly tunnel through the well. With a tunneling emitter doped to 2 1OI8/cm3,collector current densities at resonance 2 lo4 A/cm2 are estimated for the case of negligible electron recombination in the base. To minimize thermionic emission of the electrons from the emitter and scattering effects in the base, these devices are preferably operating at 77 K. The peak-to-valley ratio, i.e., the ratio of the currents at and off resonance, is given by i= l / T i , where TB is the barrier transmission coefficient (Ricco and Azbel, 1984). For the previously considered example, this gives a peak-to-valley ratio of ~ 3 6 . An alternative injection method is the nearly abrupt emitter, which can be used to bailistically launch electrons into the resonant state with high momentum coherence. The grading (= 150 A) of the A10.3Gao,,Asemitter is essential in order to be able to vary the energy of the injected electrons over a significant range as the base-emitter voltage VB, is increased, especially if the base doping is much higher than the emitter doping. As VBEis increased, the top of the launching ramp eventually reaches the same energy of the resonant state so that electrons can be ballistically launched into the resonant state (Fig. 45a). If several peaks in the collector current versus emitter-base voltage are desired, which is important for several device applications, the well should be relatively thick (100- 200 A) and the barriers should have a high A1 concentration. To achieve equally spaced resonances in the collector current, the rectangular quantum well in the base should be replaced by a parabolic one (Fig. 45b). Parabolic quantum wells have been recently realized in the AlGaAs system (Miller et al., 1984a).Assuming the depth of the parabolic well to be 0.43 eV (corresponding to grading from AlAs to GaAs) and its width 400 A,one finds that the first state is at an energy of 11 meV from the bottom of the well and that the resonant states are separated by = 33.4 meV. This gives a total of 12 states in the well. 3
6.
-
GRADED-GAP A N D SUPERLATTICE DEVICES
377
L
FIG. 45. (a) Band diagram of RTT with graded emitter (at resonance). Electrons are ballistically launched into the first quasi-eigenstate of the well. (b) RTT with parabolic quantum well in the base and tunneling emitter. (c) RTT with superlattice base (from Capasso and Kiehl, 1985).
Finally, in Fig. 4% we illustrate another application, that of studying high-energy injection and transport in the minibands of a superlattice, using ballistic launching. Minibands are formed when the barriers are sufficiently thin that the quasi-eigenstatesof the wells are strongly coupled. For example, in an AlAs/GaAs superlattice with 40 A barriers and wells, the first miniband E, is 0.1 eV from the bottom of the wells and the second one and 0.36 eV. The widths of these two minibands are, respectively, = 10 and = 50 meV. To achieve ballistic launching into the first excited state of the superlattice, it is necessary to choose an emitter composition such that AE, = E2 = 0.36 eV. Alo.sGao.sAswould be the choice. It is important to mention the time dependence of resonant tunneling. This feature has been stressed clearly only recently by Ricco and Azbel (1984). To achieve resonant tunneling the electron probability density Iv12 must be peaked in the well. Therefore, if initially there are no electrons in the double barrier and the carriers are made to tunnel by applying a positive base-emitter voltage, it takes a certain time constant to build up the probability density in the well, via multiple reflections, and to achieve high transmission at resonance. An identical situation is present in an optical Fabry-Perot. The above time constant z, is of the order of h/A E,
378
FEDERICO CAPASSO
a
ANALOG INPUT
BINARY OUTPUT
C
“i
FIG.46. (a) Schematic of multiple-valued voltage transfer characteristics of the RTT and corresponding circuit diagram; (b) parity generator circuit; (c) analog-to-digital converter circuit (from Capasso and Kiehl, 1985).
where A E is the width of the resonant state. For A E = 50 meV (as for one of the structures previously discussed) z, = 1 X s. Note, however, that z, increases exponentially with barrier thickness. After a time of the order of a few tohas elapsed, a quasi steady state has been reached whereby electrons continuously enter in the well and exit from it to maintain a constant electron density in the well. The “traversal time” of an electron through the barriers (Buttiker and Landauer, 1982) is significantly shorter than zo for the range of thicknesses and barrier heights of interest here.
6.
379
GRADED-GAP AND SUPERLATTICE DEVICES
A particularly interesting heterojunction for RTTs is A1,~,,Ino~,,As/ Gao,,,Ino.,,As because of the large conduction-band discontinuity (A E, = 0.55 eV) and the low Gao,,Ino.,,As electron effective mass (m*= 0.041). The multiple resonant characteristic of the RTT is of interest for a variety of signal processing and logic applications. One class of applications takes advantage of the ability to achieve a multiple-valued voltage transfer characteristic with an RTT, as shown in Fig. 46a. In this configuration, the output voltage Votakes on one of two values in accordance with the level of the input voltage q.Thus the device provides a binary digital output for an analog input, or a multiple-valued digital input. This function, which is that of a threshold (or voting) logic gate, is useful for a variety of signal processing applications. For example, a single device of this sort can be used to provide a parity generator as used in error detection circuits. In this application, the binary bits of a digital word are added in a resistive network at the input of the RTT, as depicted in Fig. 46b. This produces a binary output having a value that depends on whether the total number of ones in the input word is odd or even. The advantage of this approach over conventional circuits is that the RTT implementation should be extremely fast since it uses a single high-speed switching device. Conventional transistor implementations require complex circuitry involving many logic gates with a consequent reduction in speed. By combining a number of RTTs in a parallel array, an A-to-D converter could also be realized. In
I
RL2
L
"in "CE
FIG.47. Current-voltage characteristic with multiple-valued negative resistance. R,, and R,, indicate load lines. Stable states are denoted by circles (from Capasso and Kiehl, 1985).
380
FEDERICO CAPASSO
this application, the analog input is simultaneously applied to an array of RTTs having different voltage scaling networks, as shown in Fig. 46c, thereby producing an interlaced pattern of harmonically related transfer characteristics. Similar to the functioning of conventional successive-approximation A- to - D converters, the outputs of the RTT array constitute a binary code representing the quantized analog input level. Again, the circuitry involved in this approach is simple and should be very fast. A second class of applications takes advantage of the ability to achieve a multiple-valued negative resistance characteristic. This type of characteristic is achieved at the emitter- collector terminals by holding the base collector junction at a fixed bias V,, as shown in Fig. 47. With V , fixed, variations in V, produce variations in V,, which cause the collector current to peak as V, crosses a tunneling resonance. When connected to a resistive load as shown by R,, in Fig. 47, a device having N stable states is produced, where N is the number of resonant peaks. The state of this latch can be set by momentarily applying a voltage Vk to the circuit, forcing the operating point to that of the open circle in Fig. 47. When the input line is opened, it can easily be shown that the operating point moves along the indicated trajectory, finally latching at State 2. Thus, the RTT in this configuration can serve as an N-state memory element, providing the possibility of extremely high-density data 'storage. Memories of this sort and other circuits based on a multiple-valued negative resistance, such as counters, multipliers, and dividers, have been of interest for some time (Rine, 1977). However, since no physical device exhibiting multiple-valued negative resistance previously existed, such circuits were possible only with combinations of binary devices, such as conventional tunnel diodes. In order to achieve N states, N two-state devices were connected, resulting in a complex configuration with reduced density and speed. Because it allows multiple-valued negative resistance to be achieved in a single physical element, the RTT could offer significant advantages. Shortly after this initial proposal, Yokoyama et al. (1985) reported the low-temperature operation (70 K) of a unipolar resonant tunneling (RT) hot-electron transistor (RHET). This structure contains a double barrier in the emitter. Recently, Capasso et al. (1 986b) demonstrated the room-temperature operation of the first RT bipolar transistor (RTBT). The band diagram of the transistor under operating conditions is sketched in Fig. 48, along with the schematics of the composition and doping profile of the structure (bottom). The double barrier consists of a 74 A undoped GaAs quantum well sandwiched between the two undoped 2 1.5 A AlAs barriers, and the AlGaAs graded emitter is doped to 3 X lOI7 ~ m - The ~ . portion of the base (Ab.07G%.93A~) adjacent to the emitter was anodically etched off, while the
6.
GRADED-GAP AND SUPERLATTICE DEVICES
381
rest of the base was contacted using AuBe. These base processing steps are essential for the operation of the device. The emitter area is -2 X 1 0 - ~cm2. There is an essential difference with respect to the previously discussed RT transistors which rely on quasi-ballistic or hot-electron transport through the base. These schemes place stringent constraints on the design and make it difficult to achieve room-temperature operation due to the small electron mean free path (5500 A at 300 K), since electrons that have suffered a few phonon collisions cannot reach the collector. The key to the present structure is that electrons are thermally injected into and transported through the base, thus making the device operation much less critical. This new approach has allowed us to achieve for the first time RT transistor action at room temperature. Thermal injection is achieved by adjusting the alloy composition of the portion of the base adjacent to the emitter in such a way that the conduction band in this region lines up with the bottom of the ground-state subband of the quantum well (Fig. 48a). For a 74 A well and 2 1.5 A AlAs barriers, the first quantized energy level is El = 65 meV. Thus the A1 mole fraction was chosen to be x = 0.07 (corresponding to E, = 1.52 1 eV), so that A E, = El. This equality need not be rigorously satisfied for the device to operate in the desired mode, as long as El does not exceed AE, by more than a few kT. The quantum well is undoped; nevertheless it is easy to show that there is a high concentration (=7 X 10" cm-*) two-dimensional hole gas in the well. These holes have transferred from the nearby Al,-,0,G%.93A~region, by tunneling through the AlAs barrier, in order to achieve Fermi-level line-up in the base. Consider a common emitter bias configuration. Initially the collector- emitter voltage VcEand the base current IB are chosen in such a way that the base-emitter and the base- collector junctions are respectively forward and reversed biased. If V, is kept constant and the base current IB is increased, the base-emitter potential also increases until a flat-band condition in the emitter region is reached [Fig. 48b (left)]. In going from the band configuration of Fig. 48a to that of Fig. 48b, the device behaves like a conventional transistor with the collector current increasing with the base current [Figs. 48a,b (right)]. The slope of this curve is, of course, the current gain /3 of the device. In this region of operation electrons in the emitter overcome, by thermionic injection, the barrier of the base-emitter junction and undergo RT through the double barrier. If the base current is now further increased above the value IBmcorresponding to the flat-band condition, the additional potential difference drops primarily across the first semi-insulating AlAs barrier (Fig. 48c), between the contacted and uncontacted portions of the base, since the highly doped emitter is now fully conducting. This pushes the conduction band edge in the A&,07G%.93A~ above the first
382
FEDERICO CAPASSO
I EMITTER I
I
COLLECTOR
I
0
X dopant
BASE
n
FIG.48. Energ{-band diagrams of the resonant tunneling bipolar transistor (RTBT) and corresponding schematics of collector current Zc for different base currents ZB at a fixed collector emitter voltage VCE (not to scale). As ZB is increased the device first behaves as a conventional bipolar transistor with current gain (a), until near flat-band conditions in the emitter are achieved (b). For ZB> ZeTH, a potential difference develops across the AlAs barrier between the contacted and uncontacted regions of the base. This raises the conduction-band edge in the emitter above the first resonance of the well, thus quenching resonant tunneling and the collector current ZE. Shown also is the composition and doping profile of the structure; u stands for unintentionally doped.
6.
GRADED-GAP AND SUPERLATTICE DEVICES
383
energy level of the well, thus quenching the RT. The net effect is that the base transport factor and the current gain are greatly reduced. This causes an abrupt drop of the collector current as the base current exceeds a certain threshold value ZBm [Fig. 48c (right)]. The devices were biased in a common emitter configuration at 300 K and the I - V characteristics were displayed on a curve tracer. For base currents 52.5 mA, the transistor exhibits normal characteristics, while for IBz 2.5 mA, the behavior previously discussed was observed. Figure 49 shows the collector current versus base current at V, = 12 V, as obtained from the common emitter
B A S E CURRENT, mA FIG.49. Collector current versus base current of the RTBT in the common-emitterconfiguration at room temperature with the collector-emitter voltage held constant. The line connectingthe data points is drawn only to guide the eye.
384
FEDERICO CAPASSO
characteristics. The collector current increases with the base current and there is clear evidence of current gain (/I = 7 for lc > 4 mA). As the base current exceeds 2.5 mA, there is a drop in Ic, because the current gain is quenched by the suppression of RT. Single-frequency oscillations (at 25 MHz, limited by the probe stage) have been observed in these devices when biased in the negative conductance region of the characteristics. VII. Sequential Resonant Tunneling and Effective Mass Filtering in Superlattices
In a strong electric field in a superlattice the miniband picture breaks down when the potential drop across the superlattice period exceeds the miniband width. When this condition is satisfied the quantum states become localized in the individual wells. In this limit an enhanced electron current will flow at sharply defined values of the external field, when ground state in the nth well is degenerate with the first or second excited state in the (n 1)th well, as illustrated in Fig. 50a. Under such conditions, the current is due to electron tunneling between the adjacent wells with a subsequent de-excitation in the (n 1)th well, by emission of
+
+
b
FIG. 50. (a) Band diagram of sequential resonant tunneling. (b) Band diagram of new infrared laser band on sequential resonant tunneling.
6.
GRADED-GAP AND SUPERLATTICE DEVICES
385
3
$ 2 [L [L
3 0
0 !-
0 I O-1
OO
2
4 6 8 10 REVERSE EIAS,(V) FIG.5 1. Photocurrent-voltage characteristic at 1 = 0.6.328 pm (pure electron injection) for a superlattice of ~,,,Ilb,,,As/G~,,I~.,~Aswith 138 A thick wells and barriers and 35 periods. The arrows indicate that the peaks correspond to resonant tunneling between the ground state of the nth well and the first two excited stages of the (n 1)th weU.
+
phonons. In other words, electron propagation through the entire superlattice involves sequential RT. Experimental difficulties in studying this phenomenon are usually associated with the nonuniformity of the electric field across the superlattice and the instabilities generated by negative differential conductivity. To ensure a strictly controlled and spatially uniform electric field, Capasso et af.(1986a) placed the superlattice in the n- (5lOI4 ~ m - ~region ) of a reverse-bias p + - n-- n+ junction. This structure allowed for the first time to observe the sequential RT. Two NDR peaks observed in the photocurrent characteristics, Fig. 5 I , correspond to the resonances shown schematically in Fig. 50. For the sequential RT regime, there is the possibility of a laser action at the intersubband transition frequency-an effect not yet observed experimentally in superlattice (Fig. 50b). Capasso et al. (1985b), recently reported the observation of a new extremely large photocurrent amplification phenomenon at very low voltage in a superlattice of Ab,4gIno.52As/Gao.471no.~3As in the quantum coupling regime (35 A wells, 35 A barriers). Room-temperature responsivities at A = 1.3 pm are typically 2 X lo3 and 300 A/W, at 0.3 V and 0.08 V bias,
386
FEDERICO CAPASSO
a)
FIG.52. Band diagram showing effective mass filtering effect in the case of (a) phonon-assisted tunneling; (b) miniband conduction.
respectively, while the highest measured value is 4 X lo3A/W, corresponding to a current gain of 2 X lo4. This effect, which represents a new quantum-type photoconductivity, is caused by the extremely large difference in the tunneling rates of electrons and heavy holes through the superlattice layers (efective muss filtering; Fig. 5 1). When thickness and compositional fluctuations cause fluctuations in the subband energies of the order of or greater than the miniband width AE, miniband conduction cannot be sustained and hence conduction will proceed by phonon-assisted tunneling between adjacent wells (hopping conduction). Since electrons have a much smaller mass than heavy holes, their tunneling rate between adjacent wells is much larger (efective mass Jiltering). Photogenerated holes therefore remain relatively localized in the wells (their hopping probability is negligible), while electrons propagate through the superlattice (Fig. 52). This effective mass filtering effect produces a photocurrent gain, given by the ratio of the lifetime to the electron transit time. The gain barrier layer thickness and strongly decreased with increasing A&.481h.52A~ becomes unity when this exceeds 100 This confirms effective mass filtering as the origin of the large gain, since, as the barriers are made
A.
6.
GRADED-GAP AND SUPERLATTICE DEVICES
387
thicker, electrons also eventually tend to become localized, thus decreasing the tunneling probability and increasing the recombination rate. The temperature dependence of the responsivity conclusively confirmed hopping conduction. For superlattices made of the same two materials with wider electron minibands (achieved by using thinner barriers), the electron transport is by miniband conduction, while holes are still localized (Fig. 52b). Such superlattice effective mass filters will have a much greater gain - bandwidth product than the other kind (Fig. 52a) due to the much shorter electron transit time. VIII. Doping Interface Dipoles: Tunable Heterojunction Barrier Heights and Band-Edge Discontinuities It is clear from the material presented in the previous sections that band discontinuities and, in general, barrier heights, play a central role in the design of novel heterojunction devices. For example, their knowledge is absolutely essential for devices such as NERFETs, multilayer APDs, and heterojunction bipolar transistors. If a technique were available to artificially and controllably vary band offsets and barrier heights at abrupt heterojunctions, this would give the device physicist tremendous flexibility in device design, as well as many novel opportunities. Recently Capasso et al. (1985a) demonstrated for the first time that barrier heights and band discontinuitiesat an abrupt, intrinsic heterojunction can be tuned via the use of a doping interface dipole (DID) grown by MBE. This concept is illustrated in Fig. 53. Figure 53a represents the band diagram of an abrupt heterojunction. The material is assumed to be undoped (ideally intrinsic) so that we can neglect band-bending effects over the short distance (a few hundred A) shown here. We next assume to introduce in situ, during the growth of a second identical heterojunction, one sheet of acceptors and one sheet of donors, of identical doping concentrations, at the same distance d/2 (5100 A) from the interface (Fig. 53b). The doping density Nis in the 1 X 10”- 1 X lOI9/ cm3 range, while the sheets’ thickness t is kept small enough so that both are depleted of carriers (t 5 100 A). The DID is therefore a microscopic capacitor. The electric field between the plates is G / E , where CT = eNt. There is a potential difference A @ = (a/&)dbetween the two plates of the capacitor. Thus the DID produces abrupt potential variations across a heterojunction interface by shifting the relative positions of the valence and conduction bands in the two semiconductors outside the dipole region (Fig. 53c). This is done without changing the electric field outside the DID.
388
FEDERICO CAPASSO
FIG.53. (A) Band diagram of an intrinsic heterojunction. (B) Schematics of doping interface dipole. c is the sheet charge density and A @ the dipole potential difference. (C) Band diagram of an intrinsic heterojunction with doping interface dipole. For simplicity of illustrais assumed small compared to A@ tion, the potential drop across each charge sheet [)(u/e)f] (from Capasso ef al., 1985).
The valence-band barrier height at the heterojunction is increased by the DID to a value A E, e A CD ( n / / ~ )If t .A CD is dropped over a distance of a few atomic layers and the total potential drop across the charge sheets [ = ( a / ~ )ist ]small compared to A@, the valence-band discontinuity has effectively been increased by e A CD. The DID reduces the energy difference between the conduction-band edges on both sides of the heterointerface to A E, - e ACD. On the low-gap side of the heterojunction, a triangular quantum well is formed. Since typically the electric field in this region is 2 lo5 V/cm and e ACD = 0.1 0.2 eV, the bottom of the first quantum subband E, lies near the top of the well. Therefore the thermal activation barrier seen by an electron on the low-gap side of the heterojunction is = A E, - e ACD/2. Electrons can also tunnel through the thin (5100 A) triangular bamer, and this further reduces the effective barrier height. In the limit of a DID a few atomic layers thick and of potential A@, the triangular barrier is totally transparent and the conduction-band discontinuity is lowered to A E, e ACD. By inverting the position of the donor and acceptor sheets, one can instead increase the conduction-band discontinuity and decrease the valence-band one. Note that experimental evidence suggests that “natural” dipoles may
+
+
6.
GRADED-GAP AND SUPERLATTICE DEVICES
389
occur at polar heterojunction interfaces, causing the orientation dependence of band discontinuities (Grant et al., 1983). Interface defects may also produce dipoles capable of altering band discontinuities (Zur et a)., 1983). To verify the barrier lowering due to the DID, we have grown by MBE heterojunction AlGaAs/GaAs p- i- n diodes on ptype (100) GaAs substrates. Two types of structureswere grown: one with and the other without dipole. The one with dipole consists of four GaAs layers first, p+ > 10l8/ cm3(5000 A),undoped (5000 A), p+ = 5 X 1017/cm3(100 A), forming the negatively charged sheet of the dipole, and undoped (100 A), followed by four A10.26Ga0.74A~ layers, undoped (100 A), n+ = 5 X 10i7/cm3(100 A), forming the positively charged sheet of the dipole, undoped (5000 A), and n+ 2 1018/cm3(5000 A). The second type of structure is identical, with the exception that it does not have DID. They were grown consecutively in the MBE chamber without breaking the vacuum to ensure virtually identical growth conditions. Beryllium was used for the p-type dopant and silicon for the n-type. The substrate temperature was held at 590" C during growth. The background doping of the undoped layers is 5 loi4~ m - It ~ is . important to note that the charged sheets were introduced by controlling the aperture of the shutters of the ovens, without interrupting the growth of the GaAs and AlGaAs layers. This minimizes the formation of defects in the interface region. The solid and dashed lines in Fig. 54a are, respectively, the band diagram of the diodes at zero applied bias, with and without dipole (not to scale). In the structure with the DID the electric field inside the dipole layer is strongly increased while it is slightly (= 10%)decreased outside the dipole (compared to the structure without dipole) since the potential drop across the depleted i layer is identical to that of the diodes without dipole. Figure 54b gives the conduction-band diagram (to scale) near the interfaces for the cases with and without dipole. The potential of the dipole is A@ = 0.14 V; for A E, we have used the value 0.2 eV, followingthe new band line-ups for AIGaAs/GaAs. The bamer height EBis about a factor of 2 smaller than in the case without dipole (=AE,). We have measured the photocollection efficiency of the two structures; light chopped at 1 KHz and incident on the AlGaAs side of the diode was used and the short-circuit photocurrent was measured with a lock-in. Absolute efficiencydata were obtained by comparing the photoresponse to that of a calibrated Si photodiode. In Fig. 55 we have plotted the external quantum, efficiency q as a function of wavelength for devices with and without dipole. In the ones without dipole q is very small (52%) for 2 2 7100 A; this wavelength corresponds to the bandgap of the A10.26Ga0,74A~ layer as determined by photoluminescence measurements. At wavelengths longer than this and shorter than -8500 A,photons are
I
AE,= 0.2ev
I
-300
I
I
-100
I
I
0
100
1
I
1
300
DISTANCE t i ,
FIG. 54. (a) (-) and (---) represent, respectively, the band diagram of the p-i-n diodes with and without interface dipole (not in scale). (b) Band diagram of the conduction band near the heterointerface of the diodes with dipole and without (in scale) (from Capasso ef al., 1985).
0.3 -
> V
z
2- 0.2 L LL
w
5I-
2 0.1 4 1 0
-
-
FIG. 55. External quantum efficiency of the heterojunctions with (-) and without (---) dipole at zero bias versus photon energy; T = 300 K, zero bias voltage. Illumination is from the wide-gap side of the heterojunction (from Capasso ef al., 1985).
6.
39 1
GRADED-GAP AND SUPERLATTICE DEVICES
absorbed partly in the GaAs electric field region and partly in the p+-GaAs layer within a diffusion length from the depletion layer. Thus most of the photoinjected electrons reach the heterojunction interface and have to surmount the heterobanier of height A E, = 0.2 eV to give rise to a photocurrent. Thermionic emission limits therefore the collection efficiency, which is proportional to exp(-AE,)/kT (Te Velde, 1973; Shik and Shmartsev, 1984). This explains the low efficiency for 1 > 7 100 A,since A E, is significantly greater than kT. For 1< 7 100 A the light is increasingly absorbed in the AlGaAs as the photon energy increases, and the quantum efficiency becomes much larger than for 1 > 7100 A since most of the photocarriers do not have to surmount the heterojunction barrier to be collected, For 1 < 6250 A the quantum efficiency decreases, since losses due to recombination of photogenerated holes in the n+-AlGaAslayer and to surface recombination start to dominate (Womac and Rediker, 1972). The above behavior of the efficiency is consistent with predictions for abrupt AlGaAslGaAs heterojunctions without interface charges. The solid curve in Fig. 55 is the photoresponse in the presence of the DID. A striking difference is noted compared to the case with no dipole.
. .
I
0
0
0
0
0
1 *.
.
0 0
0
0
0.2
0.4
0-6
0.0
1.0
FORWARD B I A S ( V O L T S )
FIG. 56. External quantum efficiency at 1 = 8000 A versus forward bias voltage of the diodes (0)with and @) without dipoles. T = 300 K (from Capasso el a[., 1985).
392
FEDERICO CAPASSO
While the quantum efficiencies for A 5 7 100 A are comparable, at longer wavelengths it is enhanced by a factor as high as one order of magnitude in the structures with dipoles. This effect was reproduced in four sets of samples. The physical interpretation is simple. The barrier height E , has been lowered by =87 meV (Fig. 55b), which enhances thermionic emission across the barrier. Tunneling through the thin triangular barrier and hotelectron effects due to the smaller reflection coefficient also contribute to the enhanced collection efficiency. Figure 56 shows the quantum efficiency versus forward vias at IE = 8000 A for the two structures. It decreases first gradually and then rapidly above a certain cutoff voltage. This behavior, due to band flattening, is well known and has been observed previously. The efficiency rapidly increased with reverse voltage in both structures and then saturated. Above 10 V the quantum efficiency in the energy range 1.5 - 1.7 eV was identical in both structures and -60%. This is expected, since at fields > lo5 V/cm the electrons acquire so much energy that the barrier height is no more a significant limiting factor to the efficiency. The smallest size of the DID depends on the diffusion coefficient of the dopants, which depends on the doping density, the substrate temperature, and the growth time. For Si and Be in the AlGaAs/GaAs system, one should be able to place the doping sheets as close as 10 A,for substrate temperatures 5 600" C and growth times of < f h without significant interdiffusion. The DID concept may have important implications for the design of new heterojunction devices. For example, dipoles at the band steps of superlattice or staircase avalanche photodiodes may be used to further increase electron-impact ionization and to eliminate hole pile-up at the interfaces. Other areas of applications include heterojunction lasers and real-space transfer devices. ACKNOWLEDGMENTS The author is indebted to S. Luryi, A. Kastalski, K. Hess, S. Forrest, B. Ridley, R. J. Malik, and R. A. Kiehl for many fruitful discussions.
REFERENCES Allam, J., Capasso, F., Alaui, K., and Cho, A. Y. (1987). IEEE Electron Dev. Lett. EDG8,4. Allyn, C. L., Gossard, A. C., and Wiegmann, W. (1980). Appl. Phys. Lett 36, 373. Arnold, D. A., Ketterson, T., Henderson, T., Klem, J., and Morkoc, H. (1984). Appl. Phys. Left.45, 1237. Brennan, K. (1985). IEEE Trans. Electron Dev. ED-32,2467. Brennan, K., Wang, T., and Hess, K. (1985). IEEE Electron Dev.Lett. EDL-6, 199. Buttiker, M., and Landauer, R. (1982). Phys. Rev. Lett. 43, 1739. Campbell, J., Dentai, A. G., Holden, W. S., and Kasper, B. L. (1983). Electron. Lett. 19,818.
6.
GRADED-GAP AND SUPERLATTICE DEVICES
393
Capasso, F.(1982a). Electron. Lett. 18, 12. Capasso, F.(1982b). IEEE Trans. Electron Devices ED-29, 1388. Capasso, F. (1983a). J. Vac. Sci. Technol. B [2]1,457. Capasso, F. (1983b). SuI$ Sci. 132, 527. Capasso, F. (1983~).IEEE Trans. Nucl. Sci. NS30,424. Capasso,F. (1984a). SuI$ Sci. 142, 513. Capasso, F. (1984b). Laser Focus Fiberopt. Technol.,July issue. Capasso, F.(1985). In “Semiconductors and Semimetals” (R. K. Willardson and A. C. Beer, eds.),Vol. 22, part D,1. Academic Press, New York. Capasso, F., and Kiehl, R. A. ( 1 985). J. Appl. Phys. 58, 1366. Capasso, F.,and Tsang, W. T. ( 1 982). Tech. Dig. Init. Electron Devices Meet., 1982, p. 334. Capasso, F., Tsang, W. T., Hutchinson, A. L., and Williams, G. F.(1981). Tech. Dig.-Int. Electron Devices Meet., 1981, p. 284. Capasso,F., Logan,R. A., and Tsang, W. T. (1982a). Electron. Lett. 18, 760. Capasso, F., Tsang, W. T., Hutchinson, A. L., and Williams, G. F. (1982b). Appl. Phys. Lett. 40,36. Capasso, F., Tsang, W. T., Hutchinson, A. L., and Foy, P. W. (1982~).Con$ Ser. -Inst. Phys. 63,473. Capasso, F. Logan, R. A., Tsang, W. T., and Hayes, J. R. (1982d). Appl. Phys. Lett. 41,344. Capasso, F., Tsang, W. T., and Williams, G. F. (1982e). Proc. SPIE 340,50. Capasso, F., Tsang, W. T., and Williams, G. F. (1983a). IEEE Trans. Electron Devices ED-30,38 1. Capasso, F., Tsang, W. T., Bethea, C. G., Hutchinson, A. L., and Levine, B. F. (1983b). Appl. Phys. Lett. 42,93. Capasso, F., Luryi, S.,Tsang, W. T., Bethea, C. G., and Levine, B. F. (1983~).Phys. Rev. Lett. 51,23 18. Capasso, F., Cho, A. Y., and Foy,P. W. (1984a). Electron. Lett. 20, 635. Capasso, F.,Cox, H. M., Hutchinson, A. L., Olsson, N. A., and Hummel, S. G. (1984b). Appl. Phys. Lett. 45, 1 193. Capasso, F., Cho, A. Y., Mohammed, K., and Foy, P. W. (1985a). Appl. Phys. Lett. 46,664. Capasso, F., Mohammed, K., Cho, A. Y., Hull, R., and Hutchinson, A. L. (1985b). Appl. Phys. Lett. 47,420. Capasso, F., Mohammed, K., and Cho, A. Y. (1986a). Appl. Phys. Lett. 48,478. Capasso, F., Sen, S., Gossard, A. C., Hutchinson, A. L., and English, J. E. (1986b). IEEE Electron Dev. Lett. EDL-7,573. Capasso, F., Allam, J., Cho, A. Y., Mohammed, K., Malik, R. J., Hutchinson, A. L., and Sivco, D. (1986~).Appl. Phys. Lett. 48, 1294. Chang, L. L., Esaki, L., and Tsu, R. (1974). Appl. Phys. Lett. 24,593. Chin, R.,Holonyak, N., Stillman, G. E., Tang, S. Y., and Hess, K. (1980). Electron. Lett. 16, 467. Chwang, S . L., and H a , K. (1986). J. Appl. Phys. 48,2885. Coleman, P. D., Freeman, J., Markoc, H., Hess, K., Streetman, B., and Keever, M. (1982). Appl Phys. Lett. 40,493. Cooper, J. A., Capasso, F., and Thornber, K. V. (1982). IEEE Electron Device Lett. EDL-3, 497. Cox, H. M. (1 984). J. Cryst. Growth 69,64 1 . Dingle, R.,Stormer, H. L., Gossard, A. C., and Wiegmann, W. (1978). Appl. Phys. Lett. 33, 665. Esaki, L., and Chang, L. L. ( 1 974). Phys. Rev. Lett. 33,495 1. Forrest, S. R., Kim, 0. K., and Smith, R. G. (1982). Appl. Phys. Lett. 41,95. Grant, R. W., Waldrop, J. R., and Krawt, E. A. (1983). Php. Rev. Lett. 40,6.
-
394
FEDERICO CAPASSO
Hayes, J. R., Capasso, F., Gossard, A. C., Malik, R. J., and Wiegmann, W. (1983a). Electron. Lett. 13, 410. Hayes, J. R., Capasso, F., Malik, R. J., Gossard, A. C., and Wiegmann, W. (1983b). Appl. Phys. Lett. 42, 410. Hess, K., Morkoc, H., Shichijo, H., and Streetman, B. G. (1979). Appl. Phys. Lett. 35,469. Horikoshi, Y., Fisher, A., and Ploog, K. (1984). Appl. Phys. Lett. 45,919. Juang, F. Y., Das U., Nashimoto, Y., and Bhattacharya, P. K. (1985). Appl. Phys. Lett. 47, 972. Kastalsky, A., and Luryi, S. (1983). IEEE Electron Device Lett. E D M , 334. Kastalsky, A., Luryi, S., Gossard, A. C., and Hendel, R. (1984a). IEEE Electron Device Lett. EDL-5, 57. Kastalsky, A., Kiehl, R. A., Luryi, S., Gossard, A. C., and Hendel, R. ( 1 984b). ZEEE Electron Device Lett. EDL-5, 32 1 . Kawabe, M., Matsuuza, N., and Inuzuka, H. (1982). Jpn. J. Appl. Phys. 21, L447. Keever, M., Shichijo, H., Hess, K., Baneerjee, Witkowski, L., Morkoc, H., and Streetman, B. G. (1981). Appl. Phys. Lett. 38, 36. Keever, M., Hess, K., and Ludawise, M. (1982). IEEE Electron Device Lett. EDL-3, 297. Kirchoefer, S. W., Magno, R., and Comas, J. (1984). Appl. Phys. Lett. 44, 1054. Kroemer, H. (1957). RCA Rev. 18,332. Kroemer, H. (1982). Proc. IEEE 70, 13. Koremer, H., Wu, Y. C., C a y , H. C., and Cho, A. Y. (1978). Appl. Phys. Lett. 33,749. Levine, B. F., Tsang, W. T., Bethea, C. G., and Capasso, F. (1982). Appl. Phys. Lett. 41,470. Levine, B. F., Fkthea, C. G., Tsang, W. T., Capasso, F., Thornber, K. K., Fulton, R. C., and Kleinmann, D. A. (1983). Appl. Phys. Lett. 42, 769. Luryi, S., and Kastalsky,A. (1985). Superlattices and Microstruct. 1, 389. Luryi, S., Kastalsky, A., Gossard, A. C., and Hendel, R. (1984a). IEEE Trans. Electron Devices ED31,832. Luryi, S., Kastalsky, A., Gossard, A. C., and Hendel, R. (1984b). Appl. Phys. Lett. 45, 1294. Mclntyre, R. J. (1966). IEEE Trans. Electron Devices ED-13, 164. Malik, R. J., Hayes, J. R., Capasso, F., Alavi, K., and Cho, A. Y. (1983). IEEE Electron Device Lett. EDIA, 383. Malik, R. J., Capasso, F., Stall, R. A., Kiehl, R. A., Wunder, R., and Bethea, C. G. (1985). Appl. Phys. Lett. 46, 600. Miller, D. L. Asbeck, P. M., Anderson, R. J., and Eisen, F. H. (1983). Electron. Lett. 19, 367. Miller, R. C., Kleinmann, D. A,, and Gossard, A. C. (1 984a). Phys. Rev. B: Condens. Matter [3] 29,7085. Miller, R. C., Kleinmann, D. A., Gossard, A. C., and Munteanu, 0. (1984b). Phys. Rev. B: Condens. Matter [3] 29,3740. Nakagawa, T., Kaway, N. J., Ohta, K., and Kawashima, M. (1983). Electron. Lett. 19, 822. Nakagawa, T., Imamoto, H., Sakamoto, T., Ohta, K., and Kawai, N. J. (1985). Electron. Lett. 21, 882. Osaka, I., Mikawa, T., and Wada, 0. (1986). IEEE J. Quantum Electron. QE-22. People, R., Wecht, K. W., Alavi, K., and Cho, A. Y. (1983). Appl. Phys. Lett. 43, 1 18. Ricco, B., and Azbel, M. Ya. (1984). Phys. Rev. B: Condens. Matter [3] 29, 1970. Ridley, B. K. (1987). Semicond. Sci. Technol. 2, 1 16. Rine, C., ed. (1977). “Computer Science and Multiple-Valued Logic.” North-Holland Publ., Amsterdam. Shichijo, H., and Hess, K. (1981). Phys. Rev. B: Condens. Matter B 23,4197. Shichijo, H., Hess, K., and Streetman, B. G. (1980). Solid StateElectron. 23, 817. Shik, A. Ya., and Shmartsev, Y. V. (1984). Sov. Phys.-Semicond. (Engl. Transl.) 15,799.
6.
GRADED-GAP AND SUPERLATTICE DEVICES
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Sollner, T. C. L. G., Goodhue, W. D., Tannenwald, P. E., Parker, C. D., and Peck, D. D. (1983). Appl. Phys. Lett. 43,588. Sollner, T. C. L. G., Tannenwald, P. E., Peck, D. C., and Goodhue, W. D. (1984).Appl. Phys. Lett. 45, 1319. Te Velde, T. S. (1973). Solid-state Electron. 16, 1305. Tsu, R., and Esaki, L. (1973). Appl. Phys. Lett. 22,562. Tsu, R., Chang, L. L., Sai-Halasz, G. A., and Esaki, L. (1979). Phys. Rev.Lett. 34, 1509. Wan& W. I., Mendez, E. E., and Stem, F. (1984). Appl. Phys. Lett. 45, 1984. Williams, G. F., Capasso, F., and Tsang, W. T. (1982). IEEE Electron Device Lett. EDL-3, 71. Womac, S. F., and Rediker, R. H. (1972). J. Appl. Phys. 43,4129. Yokoyama, N., Imamura, K., Muto, S., Hiyamizu, S., and Nishi, H. (1985). Jpn. J. Appl. Phys. 24, L583. Zur, A., McGill, T. C., and Smith, D. C. (1983). Surlf:Sci.132, 456.
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SEMICONDUflORS AND SEMIMETAU, VOL. 24
CHAPTER 7
Quantum Confinement Heterostructure Semiconductor Lasers W. T. Tsang AT&T BELL LABORATORIES HOLMDEL, NEW JERSEY 07733
I. Introduction The two-dimensional nature of electron motion in quantum well heterostructures produces several unique and important features in semiconductor lasers. For instance, these quantum-size effects shorten the emission due to the radiative transition between confined states and significantly reduce the threshold current densityg-l3and its temperature dependence14-l9 (when properly designed) as a result of the modification in the density-of-states function of the electrons. This modification is brought about by the decreased dimensionaIity of the free-electron motion from three dimensional to two dimensional.
11. Theory of Quantum Confinement Heterosbucture Lasers: Quantum Well, Quantum Wire, and Quantum Bubble Lasers 1 . DENSITY-OF-STATE FUNCTIONS
Quantum confinement of electrons or holes (charge cariers) arises from a potential well in the band edges when the well width L, is of the order of the de Broglie wavelength & of the camers. Figure la shows a schematic diagram of a conventional double-heterostructure (DH) laser, in which the active layer has all three dimensions larger than ;1, of the carriers. The corresponding density-of-state function due to electron motion in the x, y, and z directions is schematically shown in Fig. Ib and expressed as follows:
where m: is the electron effective mass, E is the energy measured from the conduction-band edge E,, and A is Planck's constant. pi3)(E)is a parabolic function. By reducing the active layer thickness I,, to the order of ;1, as
397 Copyright 0 1987 Bell Telephone Laboratones,Inmmratcd. Au rights of reproduction in any form m c m d .
398
W. T. TSANG
t
(0)
DH
tg (el Q W i
p
(b)
ENERGY
(f)
?
oooooooo
(g) QB (h) FIG. 1. (a) Schematic diagram of a conventional DH laser in which the active layer has all three dimensions larger than the de Broglie wavelength A, of the camers. (b) The corresponding density-of-state function. (c) A two-dimensional quantum well laser structure. (d) The corresponding density-of-state function. (e) A quantum Wire laser structure. (f) The corresponding density-of-state function. (g) A quantum bubble laser structure. (h) The corresponding density-of-state function.
shown in Fig. lc, a two-dimensional (2D) quantum well (QW) heterostructure laser is realized. The corresponding density-of-state function due to confined electron motion in the z direct is shown schematically in Fig. Id and is given by'
where H ( E ) is a unit step function with H ( E 2 EiZ)= 1 and H(E < E&) = 0. E L denote the quantized energy levels with quantum
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
10
100 V,(h2/2mL:)
1000
399
OD
4
FIG.2. The calculated energy level of a particle in a symmetrical rectangular potential well of depth V,.
number n. In this case, the pi2)(E)becomes a step-type function. In the case of a symmetrical rectangular potential well of depth Voand width L,, E & is given by
where F takes account of the finite depth V,. Figure 2 shows the calculated energy level of a particle in a symmetrical rectangular potential well of depth VO. For all positive values of Vothere will be at least one bound state. When Vo-+ 03, F i n Eq. (3) equals 1. The electron energies due to motion in the x and y directions remain the same as in the bulk. Thus, the electron bound energy states in the conduction band are given by E'= E&
+(k: + k;) 2m: fi2
(4)
400
W. T. TSANG
where ki = nn/ai and ai is the lattice constant in the i direction. Since there are heavy and light holes in the valence band, the hole bound energy states are given by
x2
where E z and EE are given by similar equations to Eq. (3) with parameter values for holes, m& and m;"hare effective masses of heavy and light holes, respectively. For QWs with a parabolic shape, Miller et a1.*O generated these parabolic compositional profiles by alternate deposition of thin undoped layers of GaAs and Al,Gal-,As of varying thickness. Computer control was employed in the deposition. The relative thicknesses of the Al,Ga,-,As layers increased quadratically with distance from the well centers, while that of the GaAs layers decreased. An example is shown in Fig. 3a. With parabolic wells
E& = (n - +)Awm
(7)
where again n = 1, 2, 3, etc., and w, = a e / m : with k, equal to the curvature of the parabolic well. Defining the curvature k, by the potential height of the finite parabolic well at z = k LJ2, namely, Q,AE,, where AE, is the total energy-gap discontinuity between the GaAs at the bottom of the wells and the Al,Ga,-,As at the top of the wells and Q,is the fraction of A E, for the ith particle well, Eq. (7) becomes
Similar equations can be obtained for heavy and light holes. Figure 3b shows the 5 K excitation spectrum from such a parabolic quantum well. The various exciton transition peaks are indicated by Figs. 3a and c. With such parabolic wells, Miller et aL2' also show that the energy-gap discontinuity between GaAs and AlGaAs layers is evenly split between the electron and valence-band wells instead of the previously observed value of 85%15% split.2 Therefore, as a result of quantization of the particle motion normal to the film, discrete bound states will emerge, and the energy of the lowest state will be higher than the band edge of the bulk materials and increase as L, is decreased.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
401
0.1 5
-300
-200
-100
0
100
200
300
ANGSTROM
FIG.3. (a) Parabolic compositional profiles generated by alternate deposition of thin layers of GaAs and AI,Ga,-,As of varying thickness. The relative thickness of the AI,Ga,-,As layers increased quadratically with distance from the well center. The quantum levels for an electron are also shown in the parabolic well. (b) The 5 K excitation spectrum from a parabolic quantum well. (c) The quantum levels of heavy and light holes in a parabolic well.
Similarly, one can further limit the motion of the carriers in the L,, direction, as shown by the quantum wire (QWi) laser depicted in Fig. le. In this case the density-of-state function is given byI9
and shown schematically in Fig. 1f.
402
W. T. TSANG
In this case, the p f ) ( E )becomes almost like discrete spikes beginning to resemble the discrete levels in conventional gas and solid-state lasers. Such semiconductor quantum-wire lasers are expected to resemble more closely, in particular, the spectral linewidth of gas and solid-state lasers than the conventional DH and QW lasers. Finally, if one further limits the carrier motion in the L, direction, as shown by the quantum bubble (QB) laser in Fig. Ig, the density-of-state function will be given byI9 PiO’(E)=
c
1
n,l,k ( L z LY
d(E - Egz - E$ - EL),
(10)
LX)
where 6 ( E ) is a delta function, Egz, E f ; , and E L denote the quantized energy levels with quantum numbers n, 1, and k, and are given by the form of Eq. (3). In this case, the density-of-state function is truly discrete, as shown in Fig. I h, and so the QB laser should behave similarly to conventional gas and solid-state lasers, even more so than QWi lasers. 2. GAINSPECTRA OF QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
Theoretically, because of the modification of density of states from the parabolic distribution in bulk material, as in conventional DH lasers, to the staircase distribution in the QW heterostructure (Fig. 4a), the injected carrier distribution, and hence the gain spectra,22-26will be different in both cases, as depicted in Figs. 4b and c, respectively. For the laser to lase, the overall cavity losses are about the same in both the DH and QW lasers; the modification of the density of states in the QW lasers should require that fewer carriers be injected for the laser to reach threshold. This means that the threshold current for the QW laser should be lower than the conventional DH laser. Further, the spectral gain profile should be narrower. In fact, if one uses the “no k-selection rule,” the gain coefficient g ( E ) of the i-dimensional quantum confined laser for photon energy E can be formally expressed as
X [f,(E’)--f,(E’ - E ) ]dE”
where n, is the refractive index, c the velocity of light, Eg the energy gap, and Mi)a constant representing the probability of dipole transitions.f , ( E ) andf,(E) are the distribution functions of electrons and holes, respectively.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
403
30
I
I
El
E2’4E4
-E
n=z.ox~0‘*cm-3(BULK) n=1.4x10”%m-3 (OUANTUM WELL)
W
2g cz
NEEDED TO REACH THE SAME PEAK GAIN IN ( C )
LoW A C 3
urn Iu
E c1
E
0
ELECTRON ENERGY
FIG.4. (a) Schematic diagrams of the density of states for bulk material and QW heterostructures. (b) The distribution of inected carriers in bulk and QW structures needed to achieve the same peak gain spectra as shown in (c).
The corresponding electron distributions and gain spectra for QWi and QB lasers are also schematically shown in Fig. 5a-d. It is seen that the threshold current should decrease with increasing degree of confinement of carrier motion provided other threshold-affecting factors were maintained the same. More importantly, the gain spectrum becomes a discrete level (&function-like) in the case of QB lasers. This indeed approached the discrete-level nature of conventional gas and solid-state lasers. also calculated the gain spectrum sensitivity to camer density in QW lasers as a function of well thickness.
F t u) 2 W
n
z 0
a
c
0
w -I w
ENERGY, E
4
ENERGY, E (a)
ENERGY, E
t
ENERGY,
E
(bl
FIG.5. The electron density distributions and gain spectra for (a) QWi and (b) QB lasers are shown schematically.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
405
In Fig. 6 the derivative of the peak gain with respect to electron density, dg/dn, evaluated at threshold, is plotted against well width. The value obtained for the bulk case is also shown for comparison. It is seen that the gain is much more sensitive to changes in electron density in the quantum well case than in the bulk. Indeed, the sensitivity in the extreme quantum limit (L, --* 0) is almost an order of magnitude greater than that in the bulk. This means that good optical confinement is not as critical for optimizing the threshold current for narrow quantum wells as for conventional DH structures. It is worthwhile appreciating how the difference in sensitivity of peak gain to electron concentration between quantum wells and bulk structures comes about. This difference is related to the difference dependence of the density of states on photon energy in the quantum well and the bulk. The sensitivity of the gain to electron density is determined by the relative size of the density in energy of those states contributing to the gain and the average of the density of states over the thermal electron and hole distributions. For the bulk this ratio is much smaller than in a narrow quantum well. Support for this interpretation is seen in the calculated curves. As the quantum well width increases, the fraction of carriers occupying higher quantum well subbands and hence regions of higher density of states increases, and hence the sensitivity of the peak gain to changes in electron density decreases. No such calculation has been made for QWi and QB lasers yet. But it is easy to see that dg/dn will be drastically larger in these two cases.
0
200
100 WELL WIDTH,
i
FIG.6 . The derivative of the peak gain with respect to electron density, dgldn, evaluated at threshold, is plotted against well thickness.
406
W. T. TSANG
3. TEMPERATURE DEPENDENCE OF THRESHOLD CURRENT The temperature dependence of the threshold currents of conventional DH, quantum well,'* quantum wire, and quantum bubble lasers have been theoretically calculated. Arakawa and SakakiI9 have found theoretically that the threshold current density Jthof a quatum well laser is proportional to Tln(T/const) near room temperature, whereas J* for a QB laser is independent of T. Figure 7 shows their calculated results for T near room temperature for all four types of semiconductor lasers. It is seen clearly that the temperature dependence of Jthdepends drastically on the degree of confinement of the carrier motion. If the results are expressed in terms of Jth = J, exp(T/To), To values for conventional DH, quantum well, quantum wire, and quantum bubble lasers are 104, 285, 481, and 00, respectively. Again, the quantum bubble semiconductor laser behaves like conventional gas and solid-state lasers.
1.5
I= Iz w (L (L
3 0
n
-I 0
0
I u)
w
a
I t-
n w
N-I a
I U 0
z
0.5
-40
-20 0 20 TEMPERATURE ("C)
40
60
FIG.7. Numerical example of J, calculated for (a) DH laser, To= 104"C, (b) QW laser, To= 285"C, (c)QWi laser, To= 48 I T , and (d) QB laser, To= m. J, is normalized by J, at 0°C.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
407
The reason for such a dramatic increase in To can be understood as follows: for a conventional DH laser, the intrinsic (those not due to a leakage over the barrier and Auger processes) temperature dependence of Jth is ascribed to the thermal spreading of the injected carriers over a wider energy range of states, which leads to decreases of the maximum gain g(E,,) at a given injection level. Consequently, in quantum well lasers, where pi2)(E)and pL2)(E)are steplike, the effect of such thermal spreading is expected to be smaller. In the case of quantum wire lasers, one expects a further suppression of the temperature effect because pil)(E)has a spikelike structure and is a decreasing function of E. In quantum bubble lasers, the thermal spreading of carriers should vanish because the state density is d-function-like. Hence, the temperature dependence of Jth will totally disappear, as long as the electron population in the higher subbands remains regligibly small. 4.
QUANTUM NOISE AND DYNAMICS IN QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
Both the broad-band modulation and low-noise characteristics of semiconductor lasers are desirable features for use as optical communication sources. Two important parameters which have determined, at least in the past, such properties are the relaxation oscillation frequency,f,,which sets the useful direct modulation bandwidth, and the linewidth enhancement factor a,which determines the relation of AM to FM modulation indices as well as the degree to which spectral purity is degraded by amplitude phase coupling. The expressions forf, and a are given by27,28
where P,o,and z are the photon density, frequency, and passive cavity lifetime of the lasing mode; is the nonresonant value of the refractive index; n is the carrier density; and XR(n) and XI@)are the real and imaginary parts of the complex susceptibilities of the active medium. Their derivatives with respect to the carrier density are given, re~pectively,?~ by
408
W. T. TSANG
where El and T2are the lasing photon energy and the collisional broadening time due to carrier-carrier and carrier-phonon interaction, and gi(E)is the gain envelope function, which is given by Eq. (1 1). Figure 8 shows the calculated results for a andf, as a function of well width L, in GaAs. In this calculation, the maximum internal gain that is necessary for laser oscillation is assumed to be 100 cm-I. The broken lines gives the values for a conventional DH laser. In the calculation off, we have assumed z = 2.6 ps, T2= 0.2 ps, and P = 3.8 X 1013~ m - As ~ . shown in the figure, it should be possible to doublef, in a quantum well over its value in conventional DH lasers using L, < 80 A.For the range of L,, a is also reduced. This latter result was also found by Burt,= who, however, did not estimate the value of a at El. It should be noted that a also contains a free-carrier plasma dispersion contribution which was neglected in this calculation by Arakawa et a/.23 Figure 9 shows the calculated values off, and a as a function of L, (= LJ for a quantum wire laser. These results indicate thatf, can be made about three times larger than that of a DH laser and a can be substantially reduced. Thus, the calculated results suggest that a quantum wire structure should prove effective for improving quantum noise characteristics and dynamics. Of the three types of quantum confinement heterostructuresemiconductor lasers discussed above, only the QW lasers have been demonstrated so
l5
c 3.0
-
0 t-
I W
c)
2
*
b
5
u
a
c H
t-
10
z
W
2
I
W
3
w
W
2.0 z
U
0
a I z
a LL
z 0 l-
W
5
I t-
a X
0
J
w
U
3
W
I J
K I
50
4 00
I
J
1.0
200
WELL WIDTH d) FIG.8. The calculated results for a and f, as a function of well width L, in GaAs for a quantum well laser.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
409
1.0 50
100 WELL WIDTH t i )
200
FIG.9. The calculated values forf, and a as a function of L, (L,,= L,) for a quantum wire laser.
far. This is because at present the fabrication of the other two types of lasers is still technologically difficult, even with the most advanced epitaxial growth and device processing technologies. However, the quantum wire and quantum bubble semiconductor laser structures can be effectively achieved if one places conventional DH lasers or quantum well lasers in a strong magnetic field, in which the electron motion is confined in two dimensions. However, such lasers will only be useful for investigation purposes but not as practical devices. Therefore, in the following, only quantum well heterostructure lasers, which have been most successfully prepared by molecular beam epitaxy (MBE)29 and organometallic vapor phase epitaxy, (OM-VPE)30will be presented. Up to now, most of the studies have been on the threshold current density reduction, the temperature dependence of &, the achievement of visible emission, and reliability. Rather few studies have been made on the dynamic and spectral purity properties. 111. Short-Wavelength (- 0.68 -0.85 pm) Quantum Well Heterostructure Lasers
The emission wavelength of a quantum well heterostructure laser can be varied by the well thickness, as discussed above. A calculated example is
400
-
I
I
I
I
t
I
AlAs-GaAs ENERGY BANDS
300
-
200
-
100
-
c
2
r W (z
Y w
O
400
200
0
40
4 20
80
WELL SIZE. Lz
160
ci,
FIG. 10. The lowest (n = 1 ) confined-particle energy bands for electron (e), heavy holes (hh), and light holes (Ih) as a function of well thickness L, for GaAs wells coupled by AlAs bamers of thickness L, = 20 A.
I
m
800 c
:
v
r I-
(3
z w
>
-
750
20 WELLS
clcl
s
0 0 CALCULATION
rn
IL W v)
4 700 -
650
1
0
5 z 0 . 4 5
I
I
I
I
I
I
I
10
20
30
40
50
60
70
WELL WIDTH, L z ,
I
(i)
FIG. 1 1 . Plot of emission wavelength as a function of well width. The crosses are the calculated wavelengths of n = 1 (e-hh) transitions for each sample.
7. QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
411
shown in Fig. 10 for the lowest ( n = 1) confined-particle energy bands (“minibands”) for electrons, heavy holes, and light holes as a function of well thickness L, for GaAs quantum wells coupled by AlAs barriers of size LB= 20 A.31Indeed, Woodridge et al.,32by using MBE, have prepared current-injection lasers with multiple quantum wells as thin as 13 A and obtained lasing at the shortest wavelength of 7040 A (300 K). Their results of lasing wavelength as a function of well thickness are given in Fig. 11 for multiquantum well lasers with 80 A AL.MG~.MAs barrier and cladding layers. Also plotted is the calculated wavelength for n = l(e-hh) transitions for each sample. It can be seen that for a given well width the measured emission wavelength is longer than the calculated transition
-
1-
L
A
FIG. 12. Schematic diagram showing the layer structure and doping levels of the MQW lasers. The multilayers were unintentionallydoped. The SEM photograph is of the cleaved cross-sectional view of the actualMQW laser structure at high magnification. There are 14 GaAs quantum wells each - 136 A thick and I3 Ab.2,G%,7fisbarriers each 130 A thick.
-
412
W. T. TSANG
-
wavelength by 150-200 A over the whole range of well widths studied and that this cannot be accounted for by uncertainty in the well thickness. This difference is larger than the exciton binding energy and varies from sample to sample, so that Woodridge et al.32concluded that the participation of LO phonons in the emission process is also unlikely. Reabsorption in the cavity or effective gap shrinkage at high injection may account for these observations.
3.0 I
N
E
a
x
f
7
>
tgj 2
2.0
w
n n _J
3
0
I 0
W [r
I kI
k-
2
2
1.0
[r
3 0
0
I
I
0.1
0.2
T O T A L G a A s ACTIVE M A T E R I A L ( p m ) FIG.13. Summary of the distribution, as represented by the shaded region, of the Ja’s of all the MQW wafers grown by MBE during a period of about one and half years. The (0)and (A) represent Ja’s of two systematic consecutiveseries of MQW wafers. (-) represents the best average J,,, of standard DH lasers having A&,,,G%.,As cladding layers grown also by MBE.
7.
413
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
5. DEVICE CHARACTERISTICS OF CONVENTIONAL MULTIQUANTUM WELLHETEROSTRUCTURE LASERS The ability of MBE to prepare ultrathin (5200 A) GaAs and Al,Ga,-,As layers with the latter free of alloy clusters33resulted in the preparation of high-quality multi-quantum well (MQW) heterostructure lasers3-9,14,34,35(Fig. 12). In these conventional MQW laser^,^,'^*'^ the barriers and the cladding layers have the same AlAs composition, x 2 0.3. With these MQW lasers, an extensive study has been made on the device characteristic^.^ Wafers with different numbers of wells and different well and barrier thicknesses have been investigated. These results showed that threshold current densities Jthas low as the lowest J, (800 A cm-2) obtained from standard DH lasers36with approximately the same AlAs composition in the cladding layers were obtained despite the reduced optical confinement factor r and the increased number of interfaces (Fig. 13). Significant beam-width reduction in the direction perpendicular to the junction plane was obtained. Half-power full widths as narow as 15 were measured for some MQW wafer^.^ Extensive studiesNon spectral behavior have been carried out by Holonyak and co-workers on MQW lasers prepared by OM-VPE by Dupuise and Dapkins. O
6 . DEVICE CHARACTERISTICS OF MODIFIED MULTIQUANTUM WELLHETEROSTRUCTURE LASERS
a. Laser Layer Structure and Threshold Current Density Theoretically, because of the modification of the density of states in quantum well lasers, the Jtbof quantum well lasers should be lower than that obtainable with DH lasers. However, the experimental results shown in Fig. 13 do not reflect such impr~vement.~ This has been found by Tsang to be related to the injection efficiency of the carriers over the various barriers in MQW lasers! In order to determine the optimal barrier height of the Al,Ga,-,As barrier layers for obtaining low Jtb,a series of eight-well MQW laser wafers with Al,Ga,-,As (0.3 5 y 5 0.35) was grown. In this series, all the layer structures were maintained approximately the same, whereas only the AlAs composition x in the Al,Ga,-,As barrier layers was varied (Fig. 14). It is seen that indeed the averaged Jthdoes vary with the barrier height of the Al,Ga,-,As barrier layers, as shown in Fig, 15, in which the average Jth of each wafer is plotted against the AlAs composition x (and the barrier height) of the Al,Ga,-,As barrier layers of that wafer. As the AlAs composition x increases from 0.08, the Jthdecreases first significantly to a minimum at about x = 0.2 (the cross over point of the two dashed lines) and then increases with increasing x for x greater than -0.2. Such behavior can
414
W. T. TSANG
A t , Ga +x As EARRIERS
J ' -
\\! t
GaAs WELLS FIG.14. Schematicenergy-band diagram of the modified MQW laser.
be understood in the following manner. The Jthdecreases with increasing x at first because of two possible reasons: ( 1 ) As the barrier height of the Al,Ga,-,As barrier layers increases, the modification of the density of states becomes increasingly significant. Specifically, the density of states increases with increasing depth of the wells. This increased density of states leads to a corresponding lowering of the threshold needed for achieving population inversion. This effect is expected to continue for all x but gradually saturates for large x. (2) As observed by P e t r ~ f f in , ~ ~contrast to regular DH, the MQW structure shows that the dislocations are not behaving as nonradiative BARRIER H E I G H T OF T H E GaAS/A!2xGa1-xAS M U L T I L A Y E R S (mev)
-
0
2.0
100
2 00
300
400
N I
E
0
a
1.0
5 0.8 I
-k
0.6
D
w
$
0.4
(L
w
> 4
0.2
0
I 0.1
I 0.2
I
0.3
AeAS M O L E F R A C T I O N X I N AJ2,Gal-,As BARRIER LAYERS
FIG. 15. Shows the variation of the average J,,, of several wafers as a function of their respective AlAs composition x (and barrier height) in the AI,Ga, -,As barrier layers.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
415
centers. This effect is believed to be related to the two-dimensional nature of the carrier ~onfinement.~~ As the well depth is increased by increasing x, the increased two dimensionality due to camer confinement decreases the effectiveness of any dislocations present as nonradiative centers. The resultant lowering of Jtbwith increasing x will be faster, will gradually slow down after a certain x, but will continue to decrease. However, the present results show a turnover at x of about 0.2. The increase of Jtbwith increasing x beyond 0.2 can be understood as follows, As the barrier height becomes too high, it becomes increasingly difficult for the carriers to pass over the barrier and be injected into the next well. This decreasing carrier-injection efficiency with increasing x results in increasing Jtb.It is interesting and important to note that the turnover point occurs at x 0.2, a lower limit of AlAs composition in the cladding layers above which serious carrier leakage over the barrier into the cladding layer is avoided in regular DH lasers when operating near room temperature. This observation provides strong support for the previously described model.
-
By further optimizing the barrier and well thicknesses and increasing the AlAs mole fraction in the cladding layer to y - 0.45, an extremely low Jtb of 250 A cmb2 (average value) for broad-area Fabry-Perot diodes of 200 X 380 pm2 was 0btained.4'~ Such an extremely low Jtbis to be compared with - 800 A cm2 for the previous conventional MQW lasers3 and for otherwise similar-geometry DH lasers.36 Gain-guided proton-bombarded stripe-geometry lasers fabricated from these MMQW wafers have a cw threshold current of - 30 mA instead of 80 mA,38 compared with typical conventional MQW and nonoptimized DH laser wafers also prepared by MBE.3,36Such a cw threshold still represents a very significant reduction even when compared with the median value of 70 mA of the best LPE and MBE DH laser wafers.39Since these lasers are shallow protonbombarded gain-guided stripe-geometry lasers, the component of threshold current due to lateral current spreading in the cladding layers and camer out-diffusion in the active layers is expected to be about the same in all three types of lasers. This constant component makes the threshold reduction appear smaller in stripe-geometry lasers than in broad-area lasers. The net optical gain and carrier lifetime at threshold as a function of injection current and temperature are also measured for single-quantum well (SQW) and modified MQW (MMQW) heterostructure lasersm Figure 16 shows such an example. It is seen that the rates of change of net gain with respect to injection current are significantly enhanced in QW heterostructure lasers (10 cm-' mA-' for MMQW lasers and 3.8 cm-I mA-' for SQW lasers) compared to the DH laser, which is -2 cm-' mA-'. These
-
416
W. T. TSANG
O
-E
-20 -
I
MMQW
+--
V
Z a
C
-40-
0
+ W 2
-60 -
-80 20
30
iocm-1 mA-f
DH LASER
40
50
60
70
CURRENT (mA)
FIG. 16. The net optical gain as a function of injection current for a single quantum well and a modified multiquantum well heterostructure laser.
p - G a As P-GaQ55A'0.45 A s n -GaAs p -G a0.55A I 0.45As MQW LAYER n-Gaa55 A 1 o . d ~ n-GaAs SUB
"&-I
51 nm
(
0 0.2 0.45 FIG. 17. A schematic diagram of an MMQW guided-index GaAs/AlGaAs visible (7800 A) laser.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
- 4 0 -20
0
20
417
40
0 50400450 -40 -20 0 20 4 0 ( 0 1 CURRENT(mA) (b) ANGLE(DEG) FIG. 18. (a) cw light-current characteristics for a visible MMQW laser. (b) Far-field patterns for the same laser.
enhanced rates of QW lasers over DH lasers are consistent with the reduced threshold currents of the former and theoretical calculations of BurtzZas depicted in Fig. 6. High-quality MMQW lasers have also been prepared by OM-VPE for visible and high-power ~ p e r a t i o n .Figure ~ ~ ~ ~17 ' shows a schematic diagram of an MMQW index-guided visible (7800 A) GaAsIAlGaAslaser prepared by OM-VPE. Low threshold current (35 mA) and high output power (up to 40 mW cw) in the fundamental transverse mode as shown in Fig. 18 were ~btained.'~
W ACTIVE LAYER
PROTON IMPLANT
\ h-0% 6A'0.4AS b-GoAs
FIG.19. Schematic of a coupled multiple-stripe MMQW laser.
418
W. T. TSANG
Because of the excellent material uniformity prepared by OM-VPE and MBE, coupled multiple-stripe MMQW lasers were fabricated by Scifres et uL4' from OM-VPE-grown wafers, as shown in Fig. 19 for very high output power operations. Cw output power up to 400 mW and pulsed (75 ns) output power of 2.1 W from an uncoated mirror facet have been obtained. The threshold currents are of the order of -300-330 mA pulsed and 320 - 350 mA cw, corresponding to Jtb of 1.2- 1.3 kA/cm2 over a 250 100,urn area.
-
2 .o
:::I 0.3 0
,
I
20
80
60
40
,
,
100
120
HEAT-SINK TEMPERATURE ( " C )
4.0
(b)
-f ?- 3.0
-
-
b)
W
9 r
0
2.0 I 0
-
n
-
0
I
r u
To= 2 2 OK
I
I
I
I
20
40 TEMPERATURE ( " C )
60
80
FIG.20. Threshold-temperature dependence of a proton-bombarded stripe-geometry laser (0,MQW 18 18-2; A, MQW 18 18-4) and (b) (0)buried MQW laser and (0)buried DH laser under cw operation.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
419
b. TemperatureDependence of Threshold Current With A1,Ga,~,As/A1,Gal~,As QW heterostructure lasers it is also generally observed that the threshold temperature dependence is less sensitive than in conventional DH lasers in both broad-area and stripe-geometry A Toin the range 1 7 0 - 2 5 0 K is quite typical. Figure 20 shows the threshold temperature dependence of a porton-bombarded stripegeomtry laser,16 a buried MQW laser,42and a buried DH laser under cw operation. The increased Toobserved in broad-area diodes is also preserved regardless of the stripe geometry used. Such an improvement in Toappears to be consistent with expectations from the modification of the density-ofstate function for quantum well lasers. However, a To as low as 80 K has also been observed in otherwise low-threshold MQW laser wafers. It appears that the value of To may depend to some extent on the layer structures, as suggested by theoretical treatment^.^^ In fact, it has been found experimentally that Todid indeed depend on the well thickness.
c. Reliability of Quantum Well Lasers Preliminary cw accelerated aging results (in dry nitrogen, 70°Cambient, at a constant power output of 3 mW per mirror) on 5 pm shallow protonbombarded uncoated stripe-geometry lasers fabricated from conventional MQW wafers prepared with GaAs wells are shown in Fig. 2 1. Even though the MQW lasers have pure GaAs wells and more interfaces, a median lasing lifetime of - 5000 h at 70°C was obtained.29 Proton-bombarded stripe-geometry GaAs/AslGaAs laser diodes with MMQW active layers grown by OM-VPE have been operated continuously at 5 mW/facet for over 1 8 0 0 h at 70°C with an average degradation rate of 3.5Yo/kh and over 1 1 0 0 h at 100°Cwith an average degradation rate of - 13%/kh.44Uomi et all3 also reported more than 1100 h of constant power operation of 20 mW at 70°C with their index-guided MMQW lasers prepared by OM-VPE emitting at 7800 A. 7. DEVICE CHARACTERISTICS OF GRADED-INDEX
WAVEGUIDE SEPARATE CONFINEMENT HETEROSTRUCTURE QUANTUM WELLLASERS
As was first demonstrated by T ~ a n g , ~the . ~ability , ~ ~ to profile the AlAs composition of the epilayers by MBE also made possible the preparation of a heterostructure semiconducting laser with graded-index waveguide and separate carrier and optical confinements (GRIN-SCH),S*6,39 as shown in Fig. 22. Such structure not only provides separate confinement of light and carriers to provide further optimization possibilities for Jth, but also an arbitrarily graded-index profile outside the camer-confinement region.
420
W. T. TSANG
POWER LIFETIME OF-0.87,um LASERS
W
z
t
w
100---A10.08G00.92 A s ACTIVE LAYER
!k
A K W
(MBE) A FAILED CONVENTIONAL
v)
a
A ALIVE
-I
10 -
L
I
I 1
,
l
l
I
,
J
,
]
MOWLASERS
(MBE)
, , ,, 39
FIG. 21. Log-normal plot of 70°C cw aging results of MBE-grown conventional MQW lasers.
The combination of graded index and the use of very thin camer-confinement regions, even into the quantum-well regime, produces the GRINSCH structure and permits the .I tot continue h decreasing with decreasing d even for d 5 700 A.It has been shown that the threshold current density Jth of broad-area Fabry-Perot DH lasers can be described by
with the gain -current relation assuming the linear form
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
U N2 Nz-yPx py-2
421
N i.PiwN ,P- AL;Ga;-iAS
pz
n , p a n,p- GOAS
A"B (b) FIG.22. Energy-band diagram of a GRIN-SCH laser: Nipi stands for N,P - Al,Ga,-iAs; n,p stands for n,pGaAs; CB, conduction band; VB,valence band.
In Eqs. (16) and (17), d is the active layer thickness in micrometers, q the internal quantum efficiency at threshold; aiincludes all the internal optical losses; r is the optical confinement factor; L the cavity length; R the power reflectance of the mirror (assumed identical for both end mirrors);,,g the gain coefficient; p the gain factor; Jnomthe nominal current density for a 1 pm thick active layer and unity quantum efficiency; and Jo the value of Jnomat which g,, is linearly extrapolated to zero. It is interesting to compare the optical confinement factor achieved in these GRIN-SCH single-quantum well laser structures with conventional single-quantum well structures. For a parabolically graded-index profile, the optical field is approximately Gaussian. With proper normalization factors, the confinement factor can be shown to be
r=
(d/w,)
(18)
where d is the active region width and Wois the Gaussian beam radius. For d = 100 A and W, = 2336 A,we find = 0.034. By comparison, the confinement factor in a conventional single quantum well is given by
r = 100 X d2/G
(19)
422
W. T. TSANG
where x is the composition of the barrier AlxGa,-,As and A,, is the lasing frequency. In that case, we find for d = 100 A, r = 0.0026. Thus, the use of the graded-index separate confinement structure represents a 13-times improvement in optical confinement over the conventional single-quantum well structure. From Eq. (1 6) it is seen that the Jth of a laser is due to three different contributions. The first term is the intrinsic term. The second term is the internal loss term, with aigiven by
+ +
ai= Tar, + (1 - T)afc,, a, a, 7 X 10-'*p arc (cm-') = 3 X lo-%
+
(20) (21)
In Eqs. (20) and (21), arcis the free-carrier absorption loss in the active layer and at threshold is 10 cm-'; arc,,is the free-carrier loss in the adjacent AlxGa,-,As cladding layers and for the usual doping concentrations (- lo1*~ m - ~is )- 10 crn-'; asis the optical scattering loss due to irregularities at the heterointerfaces or within the waveguide region (measured losses of - 12 cm-l can be accounted for by a roughness amplitude of only 100 A in conventional LPE-grown DH lasers); and a, is the coupling loss when the optical field spreads beyond the AlxGa1-,As cladding layers and is usually negligible when the AlXGal-,As cladding layers are thick (-2 pm). Thus, the measured ol, so far is typically 10-20 cm-' in LPE-grown lasers. The third term is the mirror loss term, which is -30 cm-' for L = 380 pm and R = 0.32. The values of Jo/q and l / $ ?that best fit the experimental results, especially when d 2 1000 A, are 4500 and 20, respectively, as suggested by case^.^' Using these values and an aiof 10 cm-', the relative importance of the three terms in Eq. ( 16) is shown in Fig. 23 by the solid curves as a function of GaAs active layer thickness d for DH lasers with Ab,3Gh.,A~cladding layers. It is seen that for the usually used active layer thickness of k 1000 A, the main contribution to Julcomes from the intrinsic linear term. Both the internal loss and mirror loss terms remain relatively unchanged and unimportant in this regime. However, for d 5 700 A,the contribution to Jth due to the mirror loss and internal loss terms becomes dominant and increases rapidly with decreasing d as a result of decreasing optical confinement I'. The effect on nonradiative recombination velocity at the interfaces are neglected.* Included in Fig. 23, as shown by the dashed curves, are the mirror loss and the internal loss terms calculated for a graded-index waveguide separate confinement heterostructure (GRIN-SCH) laser with the minimum beam width W, = pm. The inset depicts the energy-band diagram of the GRIN-SCH laser. The same parameter values as those used in the previous DH laser except r, which is calculated for the parabolically graded waveguide, are used in obtaining the previous curves. The intrinsic term remains the same. For
-
-
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
423
ACTIVE LAYER THICKNESS d ( p )
FIG.23. Relative importance of the contributions to Jta by the intrinsic, the internal loss, and the mirror loss terms: (-), calculated for regular DH lasers; (- - -), GRIN-SCH lasers. Both use previously determined parameter values as described in the text. ( * . * ), calculated for GRIN-SCH lasers using the parameter values determined in this experiment. The symbols I, A, and M refer to the first (intrinsic), second (internal loss), and third (mirror loss) terms, respectively, of Eq. (16).
d 5 700 A, although both the mirror loss and internal loss terms remain dominant over the intrinsic term, they are significantly reduced from those of DH lasers and stay almost constant with decreasing d. These result from an increased r and the feature that d/T remains almost constant in the GRIN-SCH lasers in the very thin d regime. By comparison of the DH
424
W. T. TSANG
lasers, the present calculation shows that (1) a reduction in Jthis obtained only when d is thinner than certain value d, depending on the W, of the GRIN-SCH laser; and (2) for the same W, the Jtbof the GRIN-SCH lasers should continue to decrease with decreasing d even when d 5 d,. Indeed, both features have been confirmed by experimental results. Had a superlinear gain - current been assumed in Eqs. ( 16) and ( 17), the decrease of Jth with decreasing d would have been even more drastic. In fact, the presence of the quantum size effect in such a thin active layer regime, as discussed in Section 11, will significantly increase the gain coefficient G. Kasemset et aL4’ have calculated the gain-current relation for QW heterostructure lasers. For a typical well width of 100 ,h, the gaincurrent relation is shown in Fig. 24. Presented also in the figure for comparison is the gain - current relation for normal heterostructure lasers (i.e., one which does not show quantum size effects). It can be seen that the use of the quantum well structure results in a significant enhancement of the optical gain at any particular injection level. This is due primarily to the increased density of states at the lasing energy achieved by quantization, as discussed in Section 11. Such an effect has not been included in the calculation of the intrinsic term in Fig. 23. The effect of an enhanced carrier transport to, and confinement in, the quantum well, due to electrons “funneling” by the graded composition
’
/’ 0 Zx?03
I
4x?O3 (,,,J
6x403
8x1~1~
404
A /cm2.pm)
FIG.24. Optical gain as a function of normalized current density in a 100 A quantum well calculated values for the laser and that of the normal double heterostructure laser. (-), 100 A quantum well; (- - -), calculated values for normal DH lasers, both of which assume parabolic bands with the k selection rule.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
425
regions has also not been taken into account in calculating the curves in Fig. 23. This effect may actually be important in reducing the threshold current and increasing the Toof the GRIN-SCH quantum well lasers. Further reduction in the Jthof the GRIN-SCH lasers can also come from optimization of the mirror loss and/or the internal loss terms. The mirror loss term can be reduced by having long L and reflective mirror coatings; however, these are only external structural variations. Extremely lowthreshold GRIN-SCH lasers with single and double active layers (layer thickness 200-400 A) were prepared first by MBE.6 As a result of an increased optical confinement, a significant reduction in the internal loss cui is 5 3 cm-I, and the gain constant p is 0.08-0.12. The internal loss a; is reduced by having the p and n-Al,Ga,-,As cladding layers doped to - 10'' ~ r n -and ~ both the active layer and the graded-index waveguide layer undoped (- 1014- 1015 ~ m - ~ The ) . quantity d/T is increased due to the use of GRIN structure. The present measured values for j? of 0.8 -0.12 is also larger than those estimated by Stem (1973) in his calculations. This can be the combined result of the quantum size effect and increased efficiency in utilizing the injected carriers due to the built-in graded bandgap layers on both sides of the active layer, which essentially funnels the camers into the active layer. The increased p as discussed earlier in relation to Eq. (16) and Fig. 23 is particularly advantageous for lasers with very thin active layers. Plotted in Fig. 23, as given by the dotted curves, is the relative importance of three different contributionsto J& using the various parameter values determined experimentally and to Jthusing the various parameter values determined experimentally and with L = 380 pm. As a result of reduced ai and increased gain constant p, the internal loss term is negligible. Although the mirror loss term remains dominant, its magnitude is also significantly reduced due to increased /I. Threshold current densities similar to those obtained here, -250 Acm-2, have also been obtained by Yamakoshi et U I . ' ~with ~ MBE, as shown in Fig. 25, in which the threshold current density of GRIN-SCH lasers is plotted as a function of L, for different AlGaAs cladding layers. Similar results have also been obtained by several groups with OMVPE.'0,47,4s The results obtained by Hersee et a1.'O are shown in Fig. 26. It is also shown that there is a significant reduction in Jthof GRIN-SCH lasers over the conventional SCH structure. They also show that the temperature dependence of Jtb,To decreases with decreasng L, and is substantially higher for GRIN-SCH than for conventional SCH lasers, as shown in Fig. 27. Recently Fuji et ~ 1by. incorporating ~ a superlattice buffer layer below the GRIN-SCH layers, as shown in Fig. 28, have obtained an average Jthof 190 A/cm2 for broad-area Fabry-Perot lasers with a cavity length of 450
426
W. T. TSANG
r
1500 r
II
67 67 E \
-2
a > f 1000 z
-
L
2 W
0
500
--v--EC vEV
W
a a
1
-
n II-
3
1
II
0 -1
0 I v)
w
a
I II
-
--IF-
L= 4 0 0 p m
I
0’
I
1
pm and an internal quantum efficiency of - 95%. Such a Jtbrepresents the lowest value obtained thus far in any laser structure. Figure 29 gives the variation of Jtbwith cavity length for DH, conventional multi-quantum well, GRIN-SCH, and superlattice-buffer-layered GRIN-SCH AlGaAs/ GaAs lasers.
::I ; 0 10
,
20
I
40 60 100
,
,
, , ,~
200 4000600
QUANTUM WELL THICKNESS L, (A)
FIG.26. Threshold current density as a function of well width in GRIN-SCHe(X,x = 0.4; A,x = 0.6)and SCH (0,x = 0.4) lasers prepared by OM-VPE. L = 4000-5000 A; x, = 0.18.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
427
200
150
Y
1
c
100
50 B xc'0.4 I 100
I 200
GRIN-SCH
I
I
300
400
L,d)
'
FIG.27. Variation of Towith well width for SCH and GRIN-SCH lasers.
Since the layer thicknesses, materials, and heterointerface qualities of MBE-grown lasers are so uniform, lasing-power distribution across the entire width of the broad-area lasers is always extremely uniform and not of isolated filaments. As a result, a very high output power (pulsed) of 4 W per facet from 20 X 380 fim2 broad-area lasers has been obtained with
-
-GaAs (0.5pm. 1 x l 0 ~ ~ c m - ~ ) -AIxGal-,As ( 1 . 3 p m , Ix101ecm-3) -AIGaAs (0.2p m , 3 x 1 0 1 7 c m - 3 ) NDOPED GaAs (Lz : 6 n m ) -AIGaAs -AI,Ga+,As
(0.2prn, 3 X 1 0 1 7 c m - 3 ) (1.3 p m , 1 X 1 0 1 8 c m - 3 )
a A s (15 n m l - A I G a A s ( 1 5 n m ) ( 5 + 5 ) -GaAs (3.0pm, 2 ~ l O ~ ~ c m - ~ )
AI,Gal-,As
( X -0.7)
A '0.lBGaO.Bz As GaAs
FIG. 28. A schematic diagram of a GaAs/AIGaAs GRIN-SCH laser structure with a superlattice buffer layer and the corresponding energy-band diagram.
428
W. T. TSANG
300
I
GRIN-SCH
2 2 00
W P
I-
Z W
a a: 2
“
4
00
0 TSANG
A HERSEE e t . a l . 0
0
10
20
MO-CVD
30
I / L ( l n l / R ) cm-’ FIG.29. The variation of J& with cavity length. An average Ju, is plotted as a function of (l/L)ln( I/R). GRIN-SCH lasers with superlattice (0)and without a superlattice buffer layer Heme et a/.lo (A). (O),
a),
SCH lasers.49What is more important is that all this power is concentrated in a single spot of 30” (0,) by 5” (el,),as shown in Fig. 30. Although high output powers are obtained in array lasers of both gain- and index-guided type^,^,^^ the far-field patterns tend to be double lobed. For gain-guided array lasers under high injection levels for high output powers, the lateral current spreading and carrier out-diffusion processesSZare expected to smear out the individual laser stripes, resulting, in actuality, in a broadarea laser. Thus, for the purpose of high output powers, one can employ MBE- or OM-VPE-grown laser wafers and simply form broad-area (250 pm) stripe lasers. This will yield a single-spotted far-field pattern with narrow beam divergence. The GRIN-SCH laser wafers just described were also processed into cw stripe-geometry lasers. The cw electrooptical characteristics of proton-deheated stripe-geometry lasers fabricated from GRIN-SCH wafers were studied39and compared with similar geometry lasers fabricated from LPEand MBE-grown DH wafers with standard composition. Lasers were fabricated into a 5 pm wide stripe geometry using shallow proton irradiation and 380 pm long optical cavities. The same processing and characterizing procedures were employed for both GRIN-SCH and DH lasers. The cw light-current (L-I) characteristic from each mirror was linear up to 10 mW per mirror. In addition, the “tracking” of the light outputs from both
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
I
A
2.65 A
429
(c)
0.63
x
1.0
0.55 0.52 0.47
L
-20"
0"
20'
I
-40'
1 -ZOO
I
Oo
I
20
I
40"
FIG. 30. Far-field intensity distributions at current levels near and above lasing in (a), (b) parallel and (c), (d) perpendicular directions to the junction plane of a typical symmetric SCH laser 200 prn wide and 380 p m long prepared by MBE; the threshold current was 0.42 A.
mirrors up to 10 mW cw per mirror were significantly better than that of similar geometry LPE- or MBE-grown regular DH lasers fabricated during the same time period. In Fig. 3 1, the cw threshold distribution at 30°C of GRIN-SCH protonbombarded stripe-geometry lasers is compared with those of the lowest threshold wafers of MBE- and LPE-grown DH lasers. It is seen from Fig. 3 1
110
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1
0
g 100 (u
70-
z w
(L
5
60-
V
n J
0
50-
I
cn
40I-
I
0.01
I
1 2
I
I
I
I
I
I
I
I
I
1
1
1
1
5 10 203040506070 80 90 95 9899 CUMULATIVE ('10)
FIG.31. The cw threshold distribution at 30°C of GRIN-SCH proton-bombardedstripgeometry lasers compared with those of the lowest threshold wafers of MBE- and LPE-grown DH lasers.
UNDOPED
FIG. 32. Schematic diagram of a GRIN-SCH DH laser prepared by hybrid-crystal growth.
J
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
431
*
PULSED CURRENT ( m a )
2c
20
30
s
c
A
-3E
I
& 0
n 0 a
n
u_
u fa
z
\
n W
g 1c
w
z
a
a
I3
n
b-
3
a
2
b-
I-
0 0
2 0
w
3
v, -1
3
a
d c CURRENT (mA)
FIG. 33. Light-current characteristics of a GRIN-SCH BH laser under cw and pulsed conditions.
that there is a significant reduction, by - 25%,to 52 mA dc in the median cw threshold current of the GRIN-SCH stripe lasers. This occurs because the current components (carrier out-diffusion and lateral current spreading) not contributing directly to the lasing are to first order fixed by the structure. The percentage reduction in the current component producing the lasing threshold should be even larger for the GRIN-SCH wafers. This larger reduction can be made more readily apparent by fabricating buriedGRIN-SCH lasers, as is shown next. In addition, these lower thresolds were achieved without compromising the improved distribution (a = 3.4 mA dc) available using MBE growth procedures. That is, even though the active layers are 200-400 A and the structure far more complex, the similarity of the 0’s indicates that the same degree of control and reproducibility in material quality and layer thickness uniformity is still well main-
432
W. T. TSANG
P
LASER
_K Ql
'G2
UNDOPED GaAs
1
'n+-GaAs
S.I.GoAs SUB.
/ RIDGE WAVEGUIDE
n-AIGaAs n+-GaAs
-x 0 0.18 0.45
--+
FIG.34. (a) Equivalent circuit of a monolithic laser driver. (b) Cross-sectional structure of GRIN-SCH laser/MESFET driver circuit monolithically integrated on a semi-insulating substrate.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
433
tained. As mentioned before, wafer uniformity can be further improved by having continuous substrate rotation during growth. Buried-heterostructure (BH) lasers operating in the fundamental transverse mode fabricated from GRIN-SCH wafers have cw thresholds as low as 2.5 mA (Fig. 32). This is the lowest ever reported for any laser. The active stripe is zs 3 pm in width and 250 pm long. An external differential quantum efficiency as high as 80% was obtained. Compared with regular three-layer BH the threshold currents have been significantly reduced, and more important, output powers of 20 mW per mirror (Fig. 33) have been obtained, a considerable increase over regular three-layer BH lasers of similar stripe widths ( 5 mW per mirror). The output powers are similar to those obtained from large optical waveguide BH laser^,^^-^^ but the threshold current of the present GRIN-SCH BH lasers is significantly lower. (Threshold is - 10 mA for W = 1 pm; 30 mA for W = 3 pm;
a-
-
300K
-
7-
t-
6-
W
u
i? \ z
2
5 -
E
4-
t3 0
5(3
3-
J
2-
-
1-
-
0 -3
-2
-1
0
1
2
3
values of FIG.35. Relationship of light output as a function of input voltage V,, at various values ofL VG3.The bias current of the laser is set at onehalf of the threshold current by Q, .
I
434
W. T. TSANG
L = 300 pm for four-layer buried optical waveguide (BOG) lasers,54and -23 mA for five-layer BOG laserP with W- 2 ,um and L = 380 pm.) Because of the extremely low threshold currents achievable with GRINSCH lasers, this laser structure proves to be particularly suitable for integration with other optoelectronic devices. As a demonstration, Sanada et have achieved monolithic integration of a GRIN-SCH laser with a driver circuit on a GaAs substrate. Figure 34a shows the equivalent circuit of the monolithic laser driver, while Fig. 34b shows the cross-sectional structure of GRIN-SCH laser/MESFET driver circuit monolithically integrated on a semi-insulating substrate. The measured relationship of light output is a function of input voltage V,, at various values of V,, .The bias current of the laser is set at one-half of the threshold current by Q1and is shown in Fig. 35. Using the GRIN-SCH single-quantum well (6 nm thick) structure, the integrated laser has exhibited room-temperature cw operation characteristics with an extremely low threshold current of 15 mA as well as a high quantum efficiency of 50%. Measurements have also shown the conversion ratio of laser output power to input gate voltage of 4.3 mW/V, and the turn-on and turn-off time of the light output of 400 and 900 ps, respectively, demonstrating high-sensitivity and fast-response performance of the present monolithic laser driver. IV. Long-Wavelength ( A - 1.3 - 1.6 pm) Quantum Well Heterostructure Lasers 8. IQ.,,G%.~~As-I~P QUANTUM WELLS Quantum well structures of I Q . ~ , G % . ~ ~ A s with / I ~ PL, as small as 25 A have been prepared by Razeghi and Hirtz5*using low-pressure OM-VPE Figure 36 shows the Auger spectrum of a chemically etched level which cuts all four 1~.~,Ga,,,~As layers of the four IQ,,G%,~,As well separated by InP barrier layers. The 2 K photoluminescence spectrum of the sample from the four different wells as excited with a Nd-YAg laser at 1 170 meV is shown in Fig. 37. Such quantum wells have also been prepared by hydride vapor phase e p i t a ~ y . ~ ~ Tsanga has also prepared current-injection IQ.~~G%.~, As/InP quantum well lasers using MBE. A scanning electron micrograph (SEM) of a G%,47 As/InP MQW heterostructure (layers chemically delineated to enhance the heterointerfaces) is shown in Fig. 38. The Ga,-,4,1q.s3As wells are - 250 A,and the InP barriers are - 330 A for this wafer. Lasers with such well thicknesses did not show any significant upward energy shift at room temperature. Thus, results from a wafer with a well thickness of
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
435
K
w
(3
3
a
0
100
200
300
400
500
(pm) FIG. 36. Auger spectrum of a chemically etched level which cuts all four Ga,,In,,,As layers of the four-well G%.47 In,,,,As/InP sample.
- 70 A are presented in the following discussion. Although the layers of the wafer shown in Fig. 38 are too thick to produce an observable quantum size effect, it is seen that they are very smooth and uniform is thickness. Figure 39a shows the L-I curves of a laser diode at various heat-sink temperatures fabricated from a MQW laser wafer having four Gq,, Iq,s3Aswells of 70 A and InP bamers of - 150 A.These thickness were estimated from growth-rate measurements. The room-temperature (24°C)threshold is about 2.7 kA cm-*, which is about 15%lower than that of AlGaInAs/InP DH lased1emitting at 1.5 pm also prepared by MBE. In the temperature range 10- 75 "C, the threshold temperature dependence can be described very closely by a single dependence with T0-45 K, as shown in Fig. 39b. The usually observed breaking point in threshold temperature dependence?' i.e., different Tofor the low- and high-tempera-
-
436
W. T. TSANG I
I
I
I
I
I
Ga0471n0.53As-1np
T= 2K C 598
700
800 900 PHOTON ENERGY ( m e V )
1000
FIG.37. Photoluminescence spectrum of a Ga.4,1n,,s3As/InP sample measured at 2 K with excitation at 1170 meV (Nd:YAG laser, 20 mW focused beam).Well assignments are indicated above each peak. Note that the peaks assoCiated with the 25 and 50 I\ wells are clearly multicomponent i? nature. The full width at half-maximum of the lowest energy peak (associated with the 200 A well) is 8.3 meV. The inset schematically illustrates the sample structure.
FIG. 38. An SEM p h o t o p p h of a G~,4,1n,,s,N/InP MQW heterostructure with four Ga,,471q,53A~ wells of 250 A and three InP barriers of 330 A. The cladding layers are InP the structure is grown by MBE.
-
-
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
a
437
7
10
-
2 1 G00.47In0,~3As(~70%)/InP(-1 5 0 % )
; i 7D
-I
0
6 - MQW LASER 5 -
x 4 -
v)
w
P
I
3-
I-
&w
2-
P
[L
3 V
1
I
I
I
I
I
I
I
ture regions, was not observed or at least was not as obvious in these MQW lasers. However, the Tomeasured is not higher than for AlGaInAs/InP DH where To- 40 K for temperatures between 10 and 45"C, also prepared by MBE and emitting at 1.5 pm. It has been suggested from theoretical s t ~ d i e sthat ~ ~ the , ~ ~Toof the 1.3 - 1.5 pm MQW lasers should be
438
W. T. TSANG
significantly increased as a result of the reduced-phase space for Auger recombination processes. However, the present initial results with G%.471q,53As/InPMQW lasers do not show such improvement. One obvious reason is that the present MQW lasers, as indicated by the still high-threshold current density (2.7 kA cm-2), instead of less than 1 kA cm-2, is still not perfect enough; another reason is that the layer structures are not of the right design to reveal such predicted improvement. Theoretical studies by S ~ g i m u r indicate a ~ ~ that the Auger component of the threshold current and its temperature dependence strongly depend on QW structure. The other explanation comes from a theoretical investigation by B ~ r twhose , ~ preliminary prediction indicates that the ratio of the Auger recombination rates in bulk (DH lasers) to that in two-dimensional confined structures (QW lasers) may actually be proportional to (EJkT)'/2, where E, is the activation energy of the Auger process involved in the bulk, k the Boltzmann constant, and T the temperature. It is seen that if E, is comparable to kT (-24 meV at room temperature), then (EJkT)'/2is approximately unity, and no significant improvement in To can be expected for QW lasers. The question of To in 1.3- 1.6 pm QW lasers is therefore still quite complex and unclear both theoretically and experimentally. Current injection InGaAsP/InP quantum well lasers have also been prepared by LPE recently.65(See Appendix for further discussion.) 9. II~o.~~G~,,,,AsIn,,,2Ab,4,As QUANTUMWELLS
Temkin et ~ 1 investigated . ~ ~ the properties of MBE-grown I Q . ~ ~ GA %s./~I ~ , ~ ~ A ~ , ,multi-quantum As well lasers. These devices, operating at room temperature in the 1.5- 1.6 pm range, have well thick-
p -1nGoAs p-In P
p - I n AlAs n-InAIAs n -1nP
n - I n P SUB
FIG.40. Structure of MBE-grown InGaAs/InGaA1As/InAls/InPMMQW laser.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
439
nesses as low as 80-90 A and barrier thicknesses as low as 30 A. In the broad-area devices with a total active layer thickness of 0.14 pm, they have observed threshold current density as low as 2.4 kA/cm2. Kawamura et al.,67using MBE, also obtained current-injectionInGaAs/ InGaAlAs/InAlAs modified multi-quantum well lasers operating at 1.57pm. This MMQW laser is composed of InGaAs wells. The InGaAlAs quaternary barriers and InAlAs and InP cladding layers are as shown in Fig. 40. Photoluminescence (4 K) as short as 9668 A,equivalent to 0.474 eV above the band gap of I%.53Ga,,.47As,has been obtained by Welch et a1.68 from a single quantum well of 15 A prepared by MBE with I%.szAb,48As cladding layers. In addition to InGaAs/InAlAs quantum wells, optically pumped GaSb/ Al,,6Ga,,4As MQW lasers operating at - 1.5 pm have also been obtained by Temkin and T ~ a n gusing ~ ~ MBE. Superlattices of GaSb/AlSb have also been investigated.
-
V. Very-Long-Wavelength ( A 2.5-30 pm) Quantum Well Heterostructure Lasers
Lead-salt diode lasers provide tunable laser sources in the 2.5 - 30 pm wavelength range. The entire wavelength range can be covered with PbSnSe, PbSSe, and PbCdS diodes. Alternatively, PbSnTe/PbSnYbTe can be used for wavelengths in the 6- 30 pm range, and a new material, PbEuSeTe, can be used to cover the 2.6-6.6pm wavelength range.70*71 Molecular beam epitaxial (MBE) growth of Pb,-,EuxSeYTel-, lattices matched to PbTe substrates has been used to fabricate double-heterojunction diode lasers with - 1.5 pm wide active regions operating up to 147 K cw (180 K pulsed). This is the highest cw operating temperature ever achieved with lead-salt diode lasers.'l Recently, Partin7* prepared single-quantum well lead-chalcogenide lasers by MBE. The dopant and composition (x) profiles of a Pbl-,Eu, Se,,Te,-,, diode laser are shown in Figs. 41a and b, respectively. The selenium concentration was adjusted to obtain lattice matching between the PbEuSeTe layers and the PbTe substrate. This laser structure has a PbTe single-quantum-well active region of thickness I,= 300 A. The Pb1-,EuxSeyTe,-, confinement layers have x = 0.018 near the active region, yielding an increase in energy band gap of 99 meV at 80 K. The europium concentration was increased further from the active region to form a separate optical cavity structure, since the index of refraction of PbEuSeTe decreases with increasing europium concentration. Mesa stripegeometry diode lasers were fabricated as previously reported using an
440
W. T. TSANG
0 (a)
5 DEPTH ( p m ) I
(b)
10
I
DEPTH ( pm)
FIG. 41. (a) Dopant profde an4 (b) europium concentrations versus depth for a single quantum well laser with L, = 300 A.
anodic oxide for electrical in~ulation.’~ The stripe widths for these lasers were 16-22 pm, and the cleaved cavity lengths were 325-450 pm. The threshold current for a 300 A quantum well active region is shown as a function of temperature in Fig. 42. Pulsed ( 1 ps, 1 kHz) and cw data are shown for transitions between the n = 1 states in the conduction and valence bands. Below about 130 K, a mode with much higher photon energy (corresponding to transitions between n = 2 states) was observed at a higher “threshold” current. This n = 2 threshold current decreases with increasing temperature until it becomes approximately equal to the n = 1 threshold at 140 K (pulsed). Above this temperature, the n = 2 threshold current increased rapidly, and the n = 1 transition was not observed. The PbTe quantum well width L, was varied in the sequence 300,600, 1200, 2500 A in a series of otherwise similar growths. The high-temperature performance improved up to L, = 1200 A. The threshold of a laser with this value of L, is shown as a function of temperature in Fig. 43. The
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
441
10'
Lz = 300A E p , PULSED
--a I0
0
W
K K 3 V
D
10'
0
I00
200
TEMPERATURE (K)
FIG.42. Threshold current versus temperature for a laser with L, = 300 A.
cw and pulsed curves have a kink at about 100 K, apparently caused by a switch from laser operation between n = 1 states at low temperature to operation between n = 2 states at high temperature. However, laser operation was observed from 13 K (at 6.45 pm wavelength) up to 174 K cw (at 4.4 1 pm) and to 24 I K pulsed (at 4.0 1 pm). These are the highest operating temperatures ever observed for lead-salt diode lasers. The operating temperature of the 300 A quantum well lasers was probably limited by leakage current out of the well. Pulsed operation at temperatures as high as 235 K has also been obtained with a GRIN-SCH Pb,-,Eu,Se/Pb,-,Sn,Se laser, as shown in Fig. 44, by Norton et ~21.'~These lasers were prepared by MBE. The quantum well is 1000 A.
-
442
W. T. TSANG I
I
0
I
I
I
I
100
I
I
200
I
I
300
TEMPERATURE ( K ) FIG.43. Threshold current versus temperature for a laser with L, = 1200 A.
A
'"8/;;;....
Pb+xEu,Se
Pb+,Eu,Se
EgPb4-ySnySe
FIG.44. A GRIN-SCH Pb,-,Eu,Se/Pb,-,Sn,Se
EFFECTS
laser prepared by MBE.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
443
VI. Summary The theoretical analyses of the density-of-state functions, the spectral gain, the temperature dependence of threshold currents, the quantum noise and dynamic characteristics of quantum confinement heterostructure lasers, as well as quantum well, quantum wire, and quantum bubble lasers, were reviewed. Because of the decreased dimensionality of the carrier motion from three dimensional to two dimensional (quantum well laser) to one dimensional (quantum wire laser) to zero dimensional (quantum bubble laser), successive significant modifications result in the density-of-state function as the dimensionality decreases. This modification results in a shortened emission wavelength due to radiative between confined states, and significantly reduces the width of the gain spectrum, the threshold current density, and its temperature dependence, and improves the quantum noise and dynamic characteristics. In the case of quantum bubble laser, the performance characteristics should almost completely resemble those of a conventional gas or solid-state laser due to the fact that the density-of-state function becomes 6-function-like. Experimental results from short-wavelength (A 0.68-0.85 pm), longwavelength (A - 1.3- 1.6 pm), and very-long-wavelength (A 2.5 -30 pm) quantum well heterostructure lasers were reivewed. These include quantum well lasers from AlGaAs/GaAs, InGaAs/InP, InGaAs/InAlAs, AlGaSb/GaSb, PbEuSeTe/PbTe, and PbEuSe/PbSnSe heterostructures. Of the various types of quantum well laser structures, the modified multiquantum well heterostructure and in particular the GRIN-SCH quantumwell laser have been established widely to give the lowest threshold current density ever achieved with semiconductor lasers. For instance, J,,, of 190 A/cmZhas been obtained in broad-area Fabry - Perot GRIN-SCH diodes of AlGaAs/GaAs with a cavity length of 450 pm, and a threshold current as low as 2.5 mA has been obtained with GRIN-SCH buried heterostructure lasers. An internal quantum efficiency as high as 95% has been obtained with GRIN-SCH AlGaAs/GaAs lasers. Though theoretical analysis shows that further significant improvement in laser performance can be expected with quantum wire and quantum bubble semiconductor lasers, at present no such laser structures have been constructed due to a lack of suitable material preparation and fabrication technologies.
-
-
Appendix
Very recently, high-quality Ga,,47 Iq,,,As/InP quantum wells have also been prepared by a new epitaxial technique, chemical-beam epitaxy (CBE).* Results obtained on Gao.47I%,,As/InP current-injection lasers showed that there was a definite improvement in To.
444
W. T. TSANG
In all kinds of chemical vapor deposition (CVD), because the pressure inside the reactor is typically greater than torr and up to atmospheric, the flow of the gaseous reactants is viscous. If, however, the pressure is sufficiently reduced (down to < tom) so that the mean-free paths between molecular collisions becomes longer than the source inlet and substrate distance, the gas transport becomes a molecular beam. Such thin-film deposition process is called chemical beam deposition or chemical-beam epitaxy7’ if the thin film is an epitaxial layer. Thus, CBE is the newest development in epitaxial growth technology. It combines many important advantages of molecular beam epitaxy (MBE)76and organomet a l k chemical vapor deposition (OM-CVD),’7 both of which were first developed in 1968. And, therefore, it promises to advance the epitaxial technology beyond both techniques. In CBE, unlike MBE, which employs atomic beams (e.g., Al, Ga, and In) evaporated at high temperature from elemental sources, all the sources are gaseous at room temperature. They can be organometallic or inorganometallic compounds. For 111-V semiconductors the Al, Ga, and In are derived by the pyrolysis of their organometallic compounds, e.g., trimethylaluminum, triethylgallium, and trimethylindium, at the heated substrate surface. The As2 and P2 are obtained by thermal decomposition of their hydrides passing through a heated baffled cell. The use of hydrides was first introduced into the MBE process in 1974 by Moms and Fukui7*and later applied to the growth of GaAs and InGaAsP by Calawa and P a n i ~ h . ~ ~ Unlike OM-CVD, in which the chemicals reach the substrate surface by diffusing through a stagnant gas boundary layer above the substrate, the chemicals in CBE are admitted into the high-vacuum growth chamber in the form of a beam. Further, in OM-CVD, most of the pyrolysis of the organometallics is believed to occur in the gas phase, while in CBE there is no gas-phase reaction. Therefore, comparing with MBE, the main advantages include: (1) the use of room-temperature gaseous group-I11 organometallic sources, which simplifies multiwafer scale-up; (2) semi-infinite source supply and precision electronic flow control with instant flux response (which is suitable for the production environment); (3) a single goup-I11 beam that guarantees material composition uniformity; (4) no oval defects even at high growth rates (important for integrated-circuit applications); and ( 5 ) high growth rates if desired. Comparing with OMCVD, these include: (1) no flow pattern problem encountered in multiwafer scale-up; (2) the beam nature produces very abrupt heterointerfaces and ultrathin layers conveniently; (3) clean growth environment; (4) easy implementation of in situ diagnostic instrumentation, e.g., RHEED and RGA; (5) compatible with other high-vacuum thin-film processing techniques, e.g., metal evaporation, ion-beam milling, and ion implantation.
-
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
445
A gas-handling system75 similar to that employed in organometallic chemical-vapor deposition (OM-CVD) with precision electronic mass flow controllers was used for controlling the flow rates of the various gases admitted into the growth chamber, as shown in Fig. 45. Hydrogen was used as the carrier gas for transporting the low-vapor-pressure group-111 alkyls. Separate gas inlets were used for group111 organometallics and groupV hydrides. A low-pressure arsine (ASH,) and phosphine (PH,) ~ r a c k e rwith ~ ~ a, ~reduced ~ input pressure of - 200 torr (in fact, - 40 torr is sufficient) maintained on the high-pressure side of the electronic mass flow controller was used. The cracking temperature was 920°C. Complete decomposition of arsine and phosphine into arsenic, phosphorous, and hydrogen was routinely achieved, as observed by the absence of arsine and phosphine peaks inside the growth chamber with an in situ residue gas analyzer. Triethylgallium (Et,Ga) maintained at 30 "C, triemethylindium (MetJn) at 3 7 T , and trimethylaluminum (Met,Al) at 25°C were used. The Et,Ga, Met31n,and Met,Al flows were combined to form a single emerging beam,
-
RHEED GUN LIQUID NITROGEN COOLED SHROUDS
/ +H2
CONVENTIONAL MBE OVEN
0PRECISION ELECTRONIC MASS @
FLOW METER VALVE
RHEEDSCREEN
I
RESIDUAL GAS ANALYZER
FIG.45. Gas-handling system and growth chamber with in situ surface diagnostic capabilities incorporated into a CBE system and atomic beams of Be and Sn for p and n-type dopings, respectively.
446
W. T. TSANG
impinging by line of sight onto the heated substrate surface. This automatically guarantees composition uniformity.80The typical growth rates were 3.65 pm/h for GaInAs and 1.5-2.5 pm/h for InP, although even higher rates have been achieved. Such growth rates are higher than those typically used in MBE. The growth temperatures were usually - 550- 580°C. Elemental Be and Sn were used as the p- and n-type dopant, respectively. Note that the use of CBE allows the use of evaporated atomic beams as dopants. Gas source dopants can also be used. Continuous growth was employed at the interfaces by switching out and in the appropriate gas components. Before the growth of GaInAs/InP double-heterostructure and quantum well laser wafers, the technique was first studied by investigating its ability to grow high-quality InP and G%.47In,,,,As epilayedl and quantum well structures**lattice matched to InP substrates. Excellent material quality and heterointerfaces were obtained. Typical 2 pm thick undoped InP layers were n-type - 5 X lo1,- 1 X 10l6 cm-, with a 300 K mobility of -4500 cm2/V s and a 77 K mobility of - 30,000 cm2 V-’ s-l. Typical 2 - 5 pm thick G%.47I%,,As epilayers with no two-dimensional electron-gas effect have mobilities of 10,000- 12,000 and 40,000- 57,000 cm2/V s at 300 and 77 K with n = 5 X 1014-5 X lo1, cmb3. Bulk G%.471%,,,Asepilayers also show a very intence efficient luminescence exciton peak with linewidths as narrow as 1.2 meV, which is equivalent to the calculated intrinsic (full width at half-maximum, FWHM) alloy broadening for G%.471q,53A~ (1.3 meV). Such a linewidth is the narrowest ever measured for any alloy semiconductors, including AIxGal-,As with x > 0.1. One extreme way of testing the technique is to evaluate the quality of the quantum well (QW) heterostructures grown by it. High-quality QWs should have smooth and abrupt (“squareness” of the QW) interfaces, few background impurities, and a high PL efficiency. In order to facilitate the study of more than one quantum well simultaneously, multilayer structures consisting of a 0.5pm InP buffer, a 0.2pm G%.471~.,3As control layer, plus G%.47III,,~~ASquantum wells of different thicknesses alternated with 700 A InP barriers were grown on InP(Fe) substrates. The 0.2 pm thick G%.47I%.,,As control layer (behaving as a bulk material) served as a reference wavelength in the photoluminescence (PL) spectrum from which the upshifts of the quantum wells can be calculated precisely. The quantum well thicknss was determined from transmission electron microscopy (TEM) measurements and from the steady-state growth rate. In CBE, this latter process was found to be very reliable and reproducible from run to run. Photoluminescence measurements were made at 2 K using the 647.1 nm line of an Kr ion laser as the optical pump. The pumping power used typically ranged from 0.1 to 10 pW over a pumping area of 50 pm in diameter. Very sharp intense efficient luminescence peaks due to excitonic
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS I
I .I
1.2
I
I
I
447
I
Ga47InwAs/InP QUANTUMWELLS T =2K
1.3 1.4 1.5 WAVELENGTH X (pm)
1.6
FIG. 46. A typical photoluminesce.nce spectrum from a stack of quantum wells with different thicknesses separated by 700 A InP barriers at 2 K. The pumping power is 1 pW and the pumping area is of - 50 Grn diameter (Ref. 82).
transitions in the quantum wells were obtained, as shown by a typical example in Fig. 46.With the 10 A well, the emission peak has been shifted from 1.57 pm (the bulk Ga,-,471n,-,,3As reference) to 1.17 pm. Note that, with a growth rate of 3.6 pm/h employed in the present experiment, the 10 A well required a growth time of only 1 s, yet extremely sharp and intense luminescence was obtained, indicating that the heterointerfaces were extremely uniform and smooth. Separate samples with 6 A wells have also been successful grown with similar luminescence quality and with an emission peak at 1.09 pm. Further, such results have been reproduced from run to run. It should be pointed out that all the CBE-grown wells exhibited a single sharp peak due to excitonic transitions. No extrinsic transition peaks were observed, indicating the purity of the material. Figure 47 represents a comparison of PL linewidths (FWHM) as a function of quantum well thickness for the best published G~.4,1~.5,As/InP83-87 quantum wells grown by either OM-CVD or MBE. It is clear that the present quantum wells have significantly narrower PL linewidths than any of the previous G%.471q,3As quantum wells ever reported at all well thicknesses. Such high-quality quantum wells achieved with CBE indisputably demonstrated the superiority of this technique over OM-CVD and MBE in producing highquality G%.47 I%.,,As/InP quantum wells. The low-temperature PL linewidth for Ga,,47 1q,53Assingle quantum wells are
448
W. T. TSANG 100
a0
60
40
v
2i
F
20
: z w
J
w
0
z
w
0
m w
z = 3
10
a
1
6
4
l
l
~
l
1
1
1
~
1
1
1
(
1
1
1
1
1
'
CBE (Tsang et. al.)
W
MBE (Marsh et. al.) GSMBE (Panish et. al.) 0 LPOMCVD (Rezeghi et. al.) A Atm-OMCVD (Miller et. al.)
A
-
?&
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A': : :
N
.
--D
4 : ' O
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m
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BROADENING DUE TO BAND-FILLING
M
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A
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** . . **** , .
-
CBE GalnAsllnP
.<.
-
/
D-.2
1
0
-
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l
T=2-4K
-
0 $
-
-
BROADENING DUE TO A~,=a,/2
/.
INTRINSIC ALLOY BROADENING\
--.-.----
0. 9.
I
1
0
I
I
I
50
I
I
I
I
I
I 100
I
I
I
t 0 * L l
L
150
WIDTH OF QUANTUM WELL L,
I
I
I
200
(A)
FIG. 47. Represents a compilation of PL linewidths (FWHM) as a function of weU thickness for the best published Gq,,,In,,,,As/InP quantum wells grown by OM-CVD and MBE together with present results grown by CBE. (- - -), calculated broadening due to band filling impurities. A sheet carrier density of 2 X 10" cm-2 was used. (. * .), calculated broadening due to "effective" interface roughness, L,, of 4 2 , assuming finite-height barriers (Ref. 82).
determined by three major contributions,i.e., alloy broadening,88broadening due to geometric well-width fluctuation^,*^^^^-^^ and broadening due to an equilibrium concentration of carriers (band-filling effects) and defects associated with the heterointerfaces.88In G~.471q,,,As/InP quantum wells, alloy broadening dominates for well thicknesses of L, Z 50 A and
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
449
amounts to about 1.3 meV. Alloy broadening becomes reduced though not negligible in narrow wells because the electronic wave function spreads more into the InP cladding layer. We have measured a value of 1.3 meV in extremely high-quality G%.47 I Q . ~ ~ Abulk S layers.8* The dashed curve shown in Fig. 47 was calculated by Welch et dg8 for broadening due to an equilibrium number of carriers produced from impurities in the well and/ or the barrier layers. A sheet carrier density of 2 X 10” cm-2 was used. The dotted curve was the calculated linewidth broadening, A E, due to a total (both heterointerfaces) geometric well-width fluctuation, L,, of one monolayer (a,,/2 = 2.93 A) using the relationship AE = [d(AE,,)/dL,]AL,. Elh is the energy upshift due to quantum size effect in wells with finite-height barriers. For narrow wells (550 A) broadening due to well-width fluctuation becomes very severe and is the dominant contribution to PL linewidths. Our linewidths are far significantly narrower than the calculated broadening. Neglecting contributions due to alloy broadening in narrow wells, we estimated an “effective” interface roughness of 0.124, which can be interpreted as showing that the quantum well was largely consisting of a big domain of the same thickness L, perforated with a small fraction of small domains of (L, ~ , / 2 ) From . the above discussions, we conclude that our quantum wells have extremely flat heterointerfaces, no band filling due to background impurities, and a minimal alloy broadening of 1.3 meV for wells 250 A. This is further supported by the results of excitation ~pectroscopy~~ obtained from GaAs/AlGaAs quantum wells also grown with the same technique, which show an abrupt heterointerface smooth to within one monolayer. In Fig. 48, we show the measured PL energy upshifts, A Elh, of five different Ga,,471%.,3As/InP quantum well samples, each having stacks of quantum wells of different thicknesses as thin as 6 A as a function of well thicknesses. The three solid curves were calculated with different ratios of conduction-band edge differences to valance-band edge differences,A EJ A E,. For the first time, experimental values agree well with the theoretical curves. Further, the extreme consistency and well-behaved nature of the various different samples prove the reliability of the data and the reproducibility of the growth technique. Based on the present data alone, it is difficult to determine the actual A EJA E, ratio. From the above results on bulk G%.471~,,As and InP epilayers and Ga,,471%.,3As/InP quantum wells, we are confident that CBE is capable of producing extremely high-quality materials and heterointerfaces. We shall next present our results on G%.47I%,,As/InP DH and QW lasers. Quantum-well lasers, although more complex than DH lasers, are of great interest because they offer emission wavelength tunability through well thickness adjustment^:^ lower threshold:’ reduced threshold-temperature dependen~e,9~.~~ narrower gain spectra,99and an enhanced rate of
+
-
450
W. T. TSANG
-z
350
300
W
Q
t-
i
250
W
B2O0 w V
z $150
w
z
5
3 100
P
ea
50
0
0
50
100
150
200
250
THICKNESS OF QUANTUM WELL L,
300
(2)
FIG.48. The measured PL energy upshifts of five different Ga,,,,Iq,,,As/InP quantumwell samples (symbols) each having stacks of quantum wells of different thicknesses as a function of well thickness. The three (-) were calculated with different ratios of A E J A E,. The G ~ o .III,,~~AS ~, dispersion relation was taken to be of parabolic shape (Ref. 82).
change of peak gain with respect to changes in injected carrier density.lOO.'O1 a1 these unique properties have been well studied and confirmed to a large extent in GaAs/Al,Ga,-,As MQW laser^.^^-^^* On the other or InP hand, MQW lasers with Gao.471q,,3Aswells having Ab.481~,52As barriers prepared by MBE,'08-111atmospheric or low-pressure OMCVD, * L 2 ~ 1 1and 3 hydride vapor-phase epitaxy114have scarcely been studied. Their threshold current densities were typically at least two to three times larger than in DH lasers, and no improvement in threshold-temperature dependence has been reported to date except in those prepared by liquidphase epitaxy (LPE).1'5*'16 Note that in these LPE MQW lasers, GaInAsP quantum wells were employed in order to avoid melt-back problems during the growth of the InP barriers. Here, we shall show that low-threshold
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
451
G%.471%.53As/InP MQW lasers can be grown by CBE and that there is a definite improvement in the threshold-temperature dependence when compared with similarly grown G%.47I%,53As/InPDH lasers. Both DH and MQW laser wafers were grown under similar conditions. For DH wafers, the materials employed were four-layer epitaxial structures of 2.0 pm n-InP (- 1 X loL8~ r n - ~ confinement , layer), - 0.1 -- 0.3 pm undoped G%.471%,53As (- 1 X loi5 ~ m - ~ active , layer), -3.0 pm p-InP (-8 X lOI7 ~ m - ~confinement , layer) and -0.1 pm P + - G % , ~ ~ I % . ~ ~ A S (-5 X loi8~ m - cap ~ , layer). For MQW laser wafers, the layer structures were the same except the active layer was replaced with an active region comprising typically 4 - 8 G%.47IQ,,~As wells of 70 - 150 A thick separated by InP barriers of 150 A.Note that, unlike LPE, no anti-melt-back GaInAsP layer was needed above the G%.471%,53As active layer. Figure 49 shows the Auger depth profile of a DN sample. It is clear that the composition switching at the heterointerfaceswas abrupt to within the resolution of the Auger profiling limit and there was no composition transients. The dc drift in signal is due to a drifl in the ion current collecting system. For current threshold density (&) evaluation, broad-area lasers were fabricated from each wafer. The area of the diodes was 375 X 200pm2with two cleaved mirrors and two scribed sidewalls in order to avoid internal circulating modes. The current pulses were - 100 ns- 1 p s and lo3 pulses per second. Figure 50a shows the light output versus pulsed current amplitude for a typical DH laser at different heat-sink temperatures. A plot of the threshold
-
-
0
7 14 21 28 35 42 49 56 SPUTTER TIME (min ), 750%/min
63 70
FIG.49. The Auger depth profile of a DH laser wafer. The slight dc drift in signal is due to
a drift in the ion current collecting system.
452
W. T. TSANG 1
2' 612518 25 '
TEMP.('C)=
I
I
1 2 PULSED CURRENT (AMP)
0
4.O
3
I
I
I
I
I
I
I
I
I
25.0
37.5
50.0
62.5
I - ]
3.0
2Y 2 2.0
0-
1.0
5 0.5 (b) 0.1
0
12.5
I
75.0
HEAT-SINK TEMPERATURE ("C)
FIG.50. (a) The light output versus pulsed current amplitude for a typical Ga.4,1n,,53As/ InP DH laser at different heat-sink temperatures. (b) A plot of the threshold current versus heat-sink temperature.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS
4
453
MOW LASER
-t
s3 m
-a a I-
=)
a I3 2 0
t
r
(3
-I
4
n 0
'
0.1
0
I 2 PULSED CURRENT (AMP)
I
I
I
I
3
I
I
1
12.5 25.0 37.5 50.0 62.5 75.0 HEAT-SINK TEMPERATURE ("C)
FIG.51. (a) The light output versus pulsed current amplitude for a typical Ga,,,.,,Iq,53As! InP MQW laser at different heat-sink temperatures. This wafer has 8 quantum wells of 70 A separated by 150 A InP barriers. (b) A plot of the threshold current as a function of heat-sink temperature. (- - - ), replotted from Fig. 5 I b for a DH laser.
454
W. T. TSANG
current, I,, versus heat-sink temperature is given in Fig. 50b. It is seen that in the temperature range of 2 -6O"C, the threshold-temperature dependence can be exactly described by a single dependence relation, a exp(T/To), with To= 45 K. For G%.471q,,3As/InP DH laser wafers, the emission wavelengths ranged from 1.68 to 1.72 ,urn, depending on the degree of lattice matching. The best wafers have an average threshold current density Jthof 1.3 kA/cm2 at 25 "C for an active layer thickness of - 0.3 pm. We believe that this Jth is the lowest value reported thus far for G%.47I%.,,As/InP DH lasers. The differential quantum efficiency was as high as 18%per facet. Multi-quantum well laser wafers were evaluated in the same manner. Figure 51a shows the light output versus pulsed current amplitude for a typical MQW laser at different heat-sink temperatures. This laser wafer has 8 quantum wells of 70 A separated by 150 A barriers resulting in an emission of 1.47 pm.This represents an energy upshift of 100 meV due to quantum size effects, in reasonable agreement with low-temperature photoluminescence measurements on single quantum wells.82A plot of the threshold-temperaturedependence is given in Fig. 5 1b. In the temperature range of 2-80°C, a To of 80 K was obtained. The dashed curve is the threshold-temperature dependence of a G%,4,1n0.,~As/InPDH laser replotted for convenience of comparison. At 25"C, the averaged J, was as low as 1.5 kA/cm2, and the differential quantum efficiency was - 18% per facet, Again, we believe this J, to be the lowest reported for Gh,47 I%,,,As/ InP MQW lasers grown by any technique. Table I lists some of our better DH and MQW laser wafers grown during this period. It is quite evident that the T,'s for MQW lasers are in general about 1.5 -2 times higher than for DH lasers. This represent the first conclusive comparison performed.
-
-
TABLE I SOMEOF THE BETTER DH AND MQW LASERWAFERS GROWN BY CBE AND THEIR PERFORMANCE CHARACTERISTICS~ Laser type
DH DH MQW MQW MQW MQW
No. of QWs
QW or active thickness(&
Jth (kA/cm2)
To (K)
Lasing (pm)
-
1000
1.6
37
1.68
1.3
8
3000 70
45 80
5 6 4
80 100 150
1.72 1.47 1.50 1.54 1.60
-
1.5 2. I 1.75 1.6
75 75 65 -
a
The InP barriers are 150 A in all NQW laser wafers.
7.
QUANTUM CONFINEMENT HETEROSTRUCTURE LASERS r
I
I
.
U
I
I
455
r
Ga0,471n0,53 As/InP
-t
LASERS
MOW LASER I- 4 . 2 X I ~ n
z a
m
-a* P
Irr v) z
w
+f t
3
n I-
3 0
17,160
17,250
17,330
WAVELENGTH t i )
J
14,650 14
'50
WAVELENGTH t i )
FIG.52. A comparison of the lasing spectra at room temperature under pulsed operation at
- 50 Zthfor (a) DH and (b) MQW lasers.
The Jth's are also substantially lower than for previously reported MQW lasers. Previously, by MBE growth, Tsang1Ioobtained a Jthof -2.7 ka/cm2, Temkin et al.'O* obtained -2.4 kA/cm2, Asahi et all@'obtained -3.5 kA/cm2, and Panish et al."' obtained 3.5 kA/cm2. By OM-CVD, Nelson et al.lI3reported a Jth of 7.5 kA/cm2. A comparison of the lasing spectra at room temperature under pulsed operation at 1.2 Ithis given in Fig. 52a and b for a DH and a MQW laser, respectively. Such spectra were rather characteristic for each type. As previously observed,110.'13 it is seen that the MQW laser spectrum (broadarea diode) shows a substantial reduction in spectral envelope width when compared with that of the DH laser. This narrowing of emission spectral envelope is believed to be related to the gain narrowing due to the modification of the density of state in quantum well structures.
-
-
REFERENCES 1. R. Dingle, Festkoerperprobleme, 15,21 (1975). 2. R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev.Lett. 34, 1327 (1975). 3. W. T. Tsang, Appl. Phys. Lett. 38,204 (1981). 4. W. T. Tsang, Appl. Phys. Lett. 39,786 (1981).
456
W. T. TSANG
5. W. T. Tsang, Appl. Phys. Lett. 39, 134 (1981). 6. W. T. Tsang,Appl. Phys. Lett. 40,217 (1982). 7 . T. Fuji, S. Yamakoshi, K. Nanbu, 0. Wada, and S. Hiyamizu, J. Vac. Sci. Technol. B [2] 2,259 (1984). 8. S. Yamakoshi, T. Fuji, 0. Wada, and T. Sakurai, ZEEE Intern. Semicond. Laser Conf, 9th, p. 24 (1984). 9. T. Fuji, S. Yamakoshi, I. Wada, and S. Hiyamizu, Intern. Conf Solid State Devices Mater. 16th, Paper C-3-1,p. 145 (1984). 10. S. D. Hersee, B. de Cremoux, and J. P. Duchemin, Appl. Phys. Lett. 44,476 (1984). 11. R. D. Bumham, W. Stnefer, D. R. Scifres, N. Holonyak, Jr., K. Hess, and M. D. Camras, J. Appl Phys. 54,26 18 ( I 983). 12. M. D. Camras, N. Holonyak, Jr., M. A. Nixon, R. D. Bumham, W. Stnefer, D. R. Scifres, T. L. Paoli, and C. Lindstrom, Appl. Phys. Lett. 42,761 (1983). 13. K. Uomi, S. Nakatsuka, T. Ohtoshi, Y. Ono, N. Chissone, and T. Kajimura, Appl. Phys. Lett. 45, 8 I8 ( 1984). 14. W. T. Tsang, C. Weisbuch, R. C. Miller, and R. Dingle, Appl. Phys. Lett. 35,673 (1979). 15. R. Chin, N. Holonyak, Jr., B. A. Vojak, K. Hess, R. D. Dupuis, and P. D. Dapkins, Apply. Phys. Lett. 36, 19 ( 1 980). 16. W. T. Tsang and R. L. Hartman Appl. Phys. Lett. 38,502 (1981). 17. N. K. Dutta, R. L. Hartman, and W. T. TsanG IEEE J. Quantum Electron. QE-19, 1243 (1983). 18. K. Hess, B. A. Vojak, N. Holonyak, Jr., R. Chin, and P. P. Dapkins, Solid State Electron. 23, 585 ( 1980). 19. A. Arakawa and H. Sakaki, Appl. Phys. Lett. 40,939 (1982). 20. R. C. Miller, A. C. Gossard, D. A. Kleinmaw, and 0.Munteauc, Phys. Rev. B: Condens. Matter [ 3 ] 29,3740 (1984). 21. R. C. Miller, D. A. Kleinman, and A. C. Gossard, Phys. Rev.B: Condens. Matter [3] 29, 7085 (1984). 22. M. G. Burt, Electron. Lett. 19,210 (1982). 23. Y. Arakawa, K. Vahala, and A. Yariv, Appl. Phys. Lett. 45,950 (1984). 24. M. Asada and Y. Suematsu, IEEE Znt. Semicond. Laser Conf.’,9th, p. 28 (1984). 25. M. Yamada, S. Ogita, M. Yamagishi, T. Tabata, and N. Nakaya, Znt. Semicond. Laser Conf. 9th, p. 30 (1984). 26. A. Sugimura, Appl. Phys. Lett. 43,728 (1984). 27. K. Y. Lau, N. Barchaim, I. Ung, C. Harder, and A. Yariv, Appl. Phys. Lett. 43,1(1983). 28. K. Vahala and A. Yariv, ZEEE J. Quantum Electron. QE18, 1101 (1982). 29. W. T. Tsang, IEEE J. Quantum Electron. QE20, 1 119 (1984). 30. N. Holonyak, Jr., R. M. Kolbas, R. D. Dupuise, and P. D. Dapkins, ZEEE J. Quantum Electron. QE-16, 170 (1980). 31. B. A. Vojak, W. D. Laidig, N. Holonyak, Jr., M. D. Camras, J. J. Coleman, and P. D. Dapkins, J. Appl. Phys. 52,621 (1981). 32. K. Woodbridge, P. Blood, E. D. Fletcher, and P. J. Hulyer, Appl. Phys. Lett. 45, 167 (1984). 33. R. C. Miller and W. T. Tsang, AppI. Phys. Lett. 39, 334 (198 1 ). 34. S. Yamaskoshi, T. Sanada, 0.Wada, T. Fuji, and T. Sakurai, Proc. Znt. Conf Integrated Opt. Opt. Fiber Commun., Ith, 27B3-1 (1983). 35. H. Iwamura, T. Saku, H. Kabayashi, and Y. Horikoshi, J. Appl. Phys. 54,2692 (1983). 36. W. T. Tsang, F. K. Rienhart, and J. A. Ditzenkrger, Appl. Phys. Lett. 39,683 (1980). 37. P. M. Petroff, (1984). in “Defects in Semiconductors” (J. Narayan and T. Tau, eds.), p. 457. North-Holland Publ., Amsterdam.
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38. W. T. Tsang and R. L. Hartman, Appl. Phys. Lett. 38,502 (1981). 39. W. T. Tsang and R. L. Hartman, Appl. Phys. Lett. 42,551 (1983). 40. N. K. Dutta, R. L. Hartman, and W. T. Tsang, IEEE J. Quantum Electron. QE-19, 1243 (1983). 41. D. R. Scifres, R. D. Burnham, and W. Streifer, Appl. Phys. Lett. 41,118 (1982). 42. W. T. Tsang, N. A. Olsson, and R. A. Longan, unpublished (1984). 43. A. Sugimura, IEEE J. Quantum Electron. QE9,290 (1983). 44. C. Lindstrom, T. L. Paoli, R. D. Burnham, P. R. Scifres, and W. Strief, Appl. Phys. Lett. 43,278 (1983). 45. H. C. Casey, J. Appl. Phys. 49,3684 (1978). 46. W. T. Tsang, Appl. Phys. Lett. 33, 1022 (1978). 47. D. Kasement, C. S. Hong, N. B. Patel, and P. D. Dapkins, Appl. Phys. Lett. 41, 912 (1982). 48. R. D. Dupuis, R. L. Hartman, and F. R. Nash, Electron. Lett. EDL4,286 (1982). 49. W. T. Tsang, Electron. Lett. 16,939 (1980). 50. W. T. Tsang, R. A. Logan, and R. P. Salathe, Appl. Phys. Lett. 34, 162 (1979). 5 1. D. R. Scifres, R. D. Burnham, C. Lindstrom, W. Strief, and T. L. Paoli, Appl. Phys. Lett. 42,645 (1983). 52. W. T. Tsang, J. Appl. Phys. 49, 1031 (1978). 53. W. T. Tsang, R. A. Logan, and J. A. Ditzenberger, Electron. Lett. 18,845 (1982). 54. K. Saito and R. Ito, IEEE J. Quantum Electron. QE-16,205 (1980). 55. W. T. Tsang and R. A. Logan, Appl. Phys. Lett. 36,730 (1980). 56. C. H. Henry, R. A, Logan, and F. R. Memt, IEEE J. Quantum Electron. QE17,2196 (1982). 57. T. Sanada, S. Yamakoshi, H. Hamaguchi, 0. Wada, T. Fuji, T. Horimatsu, and T. Sakurai, Appl. Phys. Lett. (1985). 58. M. Razegh~and J. P. Hirtz, Appl. Phys. Lett. 43,585 (1983). 59. M. A. DiGiuseppe, H. Temkin, L. Peticolas, and W. A. Bonner, Apply. Phys. Lett. 43, 906 (1983). 60. W. T. Tsang, Appl. Phys. Lett. 44,288 (1984). 61. W. T. Tsang and N. A. Olsson, Appl. Phys. Lett. 42,922 (1983). 62. W. T. Tsang, F. K. Reinhart, and J. A. Ditzenberger, Appl. Phys. Lett. 41, 1094 (1982). 63. N. K. Dutta, J. Appl. Phys. 54, 1236 (1983). 64. Burt, private communication (1983). 65. N. K. Dutta et al. 66. H. Temkin, K. Alavi, W. R. Wagner, T. P. Pearsall, and A. Y. Cho, Appl. Phys. Lett. 42, 845 (1983). 67. Y. Kawamura, H. Asahi, and K. Wakita, Electron. Lett. 20,459 (1984). 68. D. F. Welch, G. W. Wicks, and L. F. Eastman, Appl. Phys. Lett. 43,762 (1983). 69. H. Temkin and W. T. Tsang, J.Appl. Phys. 55, 1413 (1984). 70. D. L. Partin, Appl. Phys. Lett. 43,996 (1983). 71. D. L. Partin and C. M. Thurst, Appl. Phys. Lett. 45, 193 (1984). 72. D. L. Partin, Appl. Phys. Lett. 45,486 (1984). 73. D. L. Partin, R. F. Majkowski, and C. M. Thrust, J. Appl. Phys. 55,678 (1984). 74. P. Norton, G. Knoll, and K.-H. Bacheni, Int. Conf: MBE, 3rd (1984). 75. W. T. Tsang, Internatl. Conf: MBE, paper K2, San Francisco, 1984; also W. T. Tsang, Appl. Phys. Lett. 45, 1234 (1984). 76. J. R. Arthur, J. Appl. Phys. 39,4032 (1968). 77. H. M. Manasevit, Appf. Phys. Lett. 12(1 & 6) (1968). 78. F. J. Moms and H. Fukiu, J. Vac. Sci. Technol. 11,506 (1974).
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79. A. R. Calawa, Appl. Phys. Lett. 38, 701 (1981); and M. P. Panish, J. Electrochem. SOC. 127,2729 (1980). 80. W. T. Tsang, J. Appl. Phys. 58, 1415 (1985). 81. W. T. Tsang, A. H. Dayem, T. H. Chiu, J. E. Cunningham, E. F. Schubert, J. A. Ditzenberger, and J. Shan, Appl. Phys. Lett. 49, 170 (1 986). 82. W. T. Tsang and E. F. Schubert, Appl. Phys. Lett. 49,220 (1986). 83. M. S. Skolnik, P. R. Tapster, S. J. Bass, N. Apsley, A. D. Pitt, N. G. Chew, A. G. Cullis, S. P. Aldred,and C. A. Warwick, Appl. Phys. Lett. 48, 1457 (1986). 84. M. Razeghi and J. P. Duchemin, J. Cryst. Growth 70, 145 (1984). 85. J. H. Marsh, J. S. Roberts, and P. A. Claxton, Appl. Phys. Lett. 46, 1 161 (1985). 86. P. M. Panish, H. Temkin, R. A. Hamn, and S. N. G. Chu, Appl. Phys. Lett. (in press). 87. B. I. Miller, E. F. Schubert, A. H. Dayem, A. Ounnazd, and R. J. Capik (unpublished). 88. D. F. Welch, G. W. Wicks, and L. F. Eastman, Appl. Phys. Lett. 46,991 (1985). 89. C. Weisbuch, R. Dingle, A. C. Gossard, and W. Wiegmann, Solid State Commun. 38, 709 (1981). 90. M. Tanaka, H. Sakaki, J. Yoshino, and T. Furuta, Proc. 2nd Internati. Con$ Modulated Semicond. Structures, p. 310, Kyoto (Sept. 1985). 91. T. Hayakawa, T. Suyama, K. Takahashi, M. Kondo, S. Yamamoto, S. Yano, and T. Hijikata, Proc. 2nd Internati. Conf: Modulated Semicond. Structures, p. 322, Kyoto (Sept. 1985). 92. L. Goldstein, Y. Horikoshi, S. Tarucha, and H. Okamoto, Jpn. J. Appl. Phys. 22, 1489. 93. B. Devaud, J. Y. Emery, A. Chomette, B. Lambert, and M. Dandel, Appl. Phys. Lett. 45, 1078 (1984). 94. W. T. Tsang and R. C. Miller, Appl. Phys. Lett. (May 12, 1986). 95. R. Dingle, W. Wiegmann, and C. H. Henry, Phys. Rev.Lett. 33,827 ( 1 974). 96. W. T. Tsang, Appl. Phys. Lett. 39,786 (1981). 97. W. T. Tsang, C. Weisbuch, R. C. Miller, and R. Dingle, Appl. Phys. Letr. 35,673 (1979). 98. N. Holonyak, Jr., R. M. Kolbas, R. D. Dupuis, and P. D. Dapkus, ZEEE J. Quantum Electron. QE-16, 170 (1980). 99. R. Dingle and C. H. Henry, U.S. Patent No. 3,982,207 (September 21, 1976). 100. M. G. Burt, Electron. Lett. 19,210(1983). 101. N. K. Dutta, R. L. Hartman, and W. T. Tsang, IEEE J. Quantum Electron. QE-19, 1243 (1983). 102. H. Temkin, K. Alavi, W. R. Wagner, T. P. Pearsall, and A. Y. Cho, Appl. Phys. Lett. 42, 845 (1983). 103. H. Asahi, Y. Kawamura, and K. Wakita,, 9th Internat. Semicond. Laser Conf: Proc. 82, Rio de Janeiro, Brazil (Aug. 1984). 104. W. T. Tsang, Appl. Phys. Lett. 44,288 (1984). 105. M. B. Panish, H. Temkin, and S. Sumski, J. Vac. Sci. Technol. B3,657 (1985). 106. M. Razeghi and J. P. Duchermin, 4th Internat. Conf: Integrated Optics and Optical Fiber Commun., paper 27B4-1, Tokyo, Japan (June, 1983). 107. A. W. Nelson, R.H. Moss, J. C. Regnault, P. C. Spurdens, and S. Wong, Electron. Lett. 21,329 (1985). 108. T. Yanase, Y. Kato, I. Mito, M. Yamoykuchi, K. Nishi, K. Kobayashi, and R. Lang, Electron. Lett. 14, 700 (1983). 109. E. A. Rezek, N. Holonyak, Jr., and B. K. Fuller, J. Appl. Phys. 51,2402 (1980). 110. N. K. Dutta, S. G. Napholtz, R. Yen, T. Wessel, and N. A. Olsson, AppZ. Phys. Lett. 46, 525 (1985).
SEMICONDUCTORS AND SEMIMETALS, VOL. 24
CHAPTER 8
Principles and Applications of Semiconductor Strained-Layer Superlattices G. C.Osbourn, P. L. Gourley, I. J. Fritz, R. M . Biefild, L. R. Dawson, and T. E. Zipperian SANDIA NATIONAL LABORATORIES
ALBUQUERQUE, NEW MEXICO
81 185
I. Introduction The advent of techniques for growing semiconductor multilayer structures with layer thicknesses approaching atomic dimensions has provided new systems for both basic physics studies and device applications. Most of the work involving these thin-layered structures, called quantum wells or superlattices, has been restricted to layer materials with lattice constants that are equal to within about a tenth of a percent (e.g., GaAs/AlGaAs). However, it is now recognized that interesting and useful quantum well/ superlattice structures can also be grown from a much larger set of materials that have lattice-constant mismatches in the percent range. This broader class of new semiconductor structures, called strained-layersuperlattices (SLSs), offers a wealth of properties with which to study materials physics or design devices. This article reviews recent developments in the SLS field and consists of: a background section on the structural features and crystal growth of SLSs; a section discussing some general features of the electronic properties of SLSs along with specific examples from the GaAsP, InGaAs, and InAsSb systems; and a section dealing with device concepts and experimental results on several SLS prototype devices. The large body of work on the closely lattice-matched systems is covered elsewhere in this volume and will not be discussed in detail in this article. 11. Background 1. ELASTICSTRAINACCOMMODATION
Early work on mismatched semiconductor multilayers with layer thicknesses 2 1000 A and lattice mismatches 20.1%indicated that the difference in lattice constants is accommodated primarily by misfit dislocations 459 Copyright 0 1987 Bell Telephone Laboratories, Incorporatad. All rights of reproduction in any fonn reserved.
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at the layer interfaces.' These misfit dislocations are defects which severely degrade the electronic properties of the multilayers and render them useless for many applications. As a result, significant lattice mismatch usually cannot be tolerated in multilayers with layer thicknesses 2 1000 A.Work on mismatched multilayer structures with layer thicknesses less than a few hundred angstroms revealed a different behavior. For layer thicknesses less than a certain critical thickness h,, it is energetically favorable to totally accommodate the lattice mismatch with uniform elastic ~ t r a i n s . ~ This .~ behavior is illustrated schematically in Fig. 1. Since no misfit defects are generated in this case, these strained-layer structures can have good crystalline qualitf and exhibit interesting electronic proper tie^.^ Values of h, as a function of lattice mismatch have. been estimated using simple model^,^,^ and the result of Matthews and Blakeslee is shown as the solid line in Fig. 2. Recent experimental determinations of crystalline quality as a function of layer thickness and strain have been carried out in the InGaAsIGaAs system: and Fig. 2 also summarizes those results. Here we see the InGaAs layer thickness plotted versus the InGaAs layer strain for a variety of SLS samples. Samples with high crystalline quality, as deter-
j -1
Thin, mismatched layers
I
I
I
, u p e r l a t t i c e : rnisma t c h accommodated by strain
2 1
1
5
Graded layer
Substrate
FIG. 1. Schematic illustration of tetragonally strained layers in an SLS structure. Also shown are the graded layer and substrate on which the SLS is grown.
8.
STRAINED-LAYER SUPERLATTICES
InGaA./QaA. SLS.
461
.
a W
lo1
I
10-2
10-1
STRAIN
FIG.2. Plot of InGaAs layer thickness versus strain for InXGa,-,As/GaAs SLSs. Solid plotting symbols: highquality material. Open plotting symbols samples with structural defects. (-), theoretical expression proposed by Matthews and Blakesee.
mined by electrical or optical techniques, are shown with solid plotting symbols, whereas poorquality samples are indicated by open symbols. We see that this expression is in excellent agreement with the data, as the theoretical line exactly separates the high- and low-quality samples. However, experimental values in the SiGe system' appear to be an order of magnitude larger than these theoretical estimates. This is probably indicative of metastable SLS growth resulting from the lower growth temperatures used in SiGe epitaxy and the large energy barriers against dislocation formation? 2. CRYSTAL GROWTH
The techniques of metal-organic chemical vapor deposition (MOCVD) and molecular-beam epitaxy (MBE) have been used to prepare strainedlayer superlattices (SLSs) because of their demonstrated capability for growing very thin (<50 A) Both techniques are capable of producing abrupt interfaces and layers which are uniform in composition and thickness and contain very few defects.l0 Abrupt doping profiles can a!so be obtained using these techniques. MOCVD was used to prepare SLSs in the GaAs/GaP system. Although the detailed growth procedures have been published elsewhere, I certain
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aspects of the SLS preparation will be emphasized here. The epitaxial layers were grown on GaP or GaAs substrates by decomposing a mixture of trimethylgallium, phosphine, and arsine in a hydrogen carrier gas. Usually the SLSs were grown on either a graded alloy layer whose composition was varied continuously from that of the substrate to the average composition of the SLS (as indicated schematically in Fig. 1) or on top of a buffer layer of a fixed composition approximately equal to the average of the SLS. The graded or buffer layer was used to minimize the misfit dislocations due to lattice mismatch between the SLS and the substrate. The SLS would then be grown on top of the graded or buffer layer by growing a thin layer of either GaAs or Gap, purging the reaction chamber with a mixture of ASH3 and PH3, and then growing a thin GaAs,P,-, layer. The growth of the layers was controlled by turning off and on the trimethylgallium source. The layer thicknesses were varied by changing the growth time. The growth rate of the superlattice was found to be very similar to the growth rate of the graded and/or buffer layers. The conditions used for the preparation of SLSs in which layers of Gap or GaAs were alternated with GaAs, -xPx were very similar except for the growth temperature. A temperature of 800825 C was used for the GaP SLS, while a temperature of 700- 750"C was used for GaAs. If GaAs was present as one layer in the SLS, disordered layers were obtained at 2 800" C. Growth of SLSs containing layers of GaP at temperatures 5 750" C resulted in poor surface morphology and hillock formation due to preferential nucleation or island growth.12 Since it was not possible to change the growth temperature during the preparation of an SLS, the growth of alloys with x = 0.5 resulted in somewhat lower-quality SLSs. Molecular beam epitaxy (MBE) has been used to prepare Ino.2Gao.8As/ GaAs SLS structures. General features of the MBE technique for GaAs/ AlGaAs growth have been reviewed elsewhere and will not be presented here.g In the growth of In,Ga,-,As alloys, both In and Ga stick at the surface and await the arrival of As atoms (from As, molecules) to complete their nearest-neighbor bonding. The ratio of the In and Ga fluxes determines the composition x, while the sum of the fluxes determines the growth rate. Some downward adjustment in growth temperature from the optimal value for GaAs growth must be made to allow for the weaker In-As bond and a stronger tendency to lose As at elevated temperature. This usually presents no particular problem until thin, multilayer structures are required, as in GaAs/Ino.,Gao.,As SLSs, making it impractical to cycle the substrate temperature to prowde the optimum growth temperature for each material. The optimum growth temperatures for GaAs (near 580" C) and Ino.,Gao,,As (below 500" C) differ greatly. The unavoidable compromise is usually resolved by favoring the lower-bandgap material
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(here, Ino.2Ga,,As), since camer populations will shift toward these regions and, for most electronic and optical functions, high material quality is most important there. The growth conditions settled upon for most of the G a A ~ / 1 n ~ . ~ G aSLS ~ , ~structures As reported here are a growth temperature of 500" C, and a growth rate of 0.7 pm/h. This growth temperature is somewhat above the optimum for this alloy, and requires a corresponding increase in V :I11 flux ratio to more than 5 : 1 to stabilize the surface. The alternate layers are grown by computer-controlled opening and closing of the In shutter only, so that growth rate and V/III flux ratio are not strictly constant. These multilayers were grown on InGaAs buffer/GaAs substrate structures. The presence of interfacial strain appears to cause no problems in the growth of G a A ~ / 1 n ~ . ~ G a ~SLS . ~ Amaterial. s In situ RHEED (reflection high-energy electron diffraction) measurements show well-defined, wellstreaked patterns throughout the growth sequence, even immediately after the opening and closing of shutters to cycle between GaAs and Ino.2Gao.8AS growth. However, when greater strain is induced by attempting to grow structures with greater compositional difference, as in GaAs/ Ino,,Gao,As SLSs, the RHEED patterns become spotty, consistent with nonplanar growth, which we strongly suspect is due to island formation within the epitaxial layers. Island growth under certain growth conditions has also been observed in SiGe SLS growth by MBE.' 3. STRUCTURAL CHARACTERIZATION
Elastic strain accommodation and crystal quality can be evaluated using structural characterization tools. The techniques of transmission electron microscopy (TEM), X-ray diffraction, and ion channeling have been used to characterize the structure of the SLSs prepared by MOCVD and MBE. The results of TEM for a typical GaAs,P,-,/Gap SLS sample are illustrated in Fig. 3. This micrograph was taken at right angles to the growth direction so that the layer thickness could be determined directly from the micrograph (see Ref. 1 1 for sample preparation). The micrograph illustrates the quality and uniformity of the SLS. Similar micrographs are obtained at other positions for the same sample. The micrograph presented in Fig. 4 is a { I 11> lattice image of a GaAs,P,-,/Gap SLS. The { 1 1 1> lattice fringes can be seen to pass without interruption through the successive light and dark Gap and GaAs,P, -, layers. The continuity of the { 111> lattice fringes is evidence for the coherent growth of the successive layers. TEM has also been used to examine the proposal of Matthews and Blakeslee4 that dislocation propagation from underlying graded, buffer, and substrate layers can be blocked by certain SLS structures." Recent results confirm that dislocations from the graded or buffer layers are driven
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FIG.3. Transmission electron micrograph (TEM) of a GaP/GaAs,,,P,,, SLS grown on GaAs0,,,P,,, buffer layer taken at right angles to the growth direction. The layers are 230 A thick.
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to the edge of the sample by the SLS. The dislocation densities were observed to decrease from 2 X lo9 cm-* to less than lo5 ~ m - ~ . ' ~ The SLS samples are routinely examined by x-ray diffraction to determine layer thickness, composition, and quality. As illustrated in Fig. 1,the layer strains induced by the lattice mismatch between the superlattice layers result in a tegragonal distortion of the individual layers. This distortion, together with the periodicity of the superlattice, results in the observed satellite peaks around the (400) substrate reflection. Analysis of the X-ray diffraction pattern for an SLS gives directly the average composition of the SLS and the repeat distance of the SLS layers."J4 If the ratios of the individual layer thicknesses are known, then the tetragonal lattice constants can be calculated using the elastic constants of the binary and the alloy involved in the SLS and the known average composition. If the SLS is grown on top of the buffer layer, an additional peak is present besides the superlattice and substrate peaks. Further peaks arise from the presence of a graded layer. A detailed curve-fitting analysis of double-crystal X-ray rocking spectra gives the SLS repeat distance and the profiles of perpendicular and parallel strains and compositional variation^.'^ SLS results from this last techniaue yield layer-strain values that are consistent with strain accommodation of the lattice mismatch.14 Structural characterization techniques have also been employed to examine the structural stability of SLSs. Many SLS structures have been grown and appear to be stable for at least several years under ambient conditions. However, it is of further importance to investigate the longterm stability of SLS structures under the influence of various external stresses such as high current density, high temperature, etc., which can occur in certain kinds of devices. Studies in this area have begun only recently, and conclusive results have not been obtained yet. Meaningful studies of material stability generally involve the life-testing of a large number of device structures, and great care must be taken to ascertain whether any observed failures are due to instability of the SLS material itself or to other causes such as those often encountered in devices fabricated from bulk material. Some recent work which reflects favorably on the stability of SLSs was carried out by Myers and co-workers.'6 This work involved the doping of GaAs,P,-,/GaP SLSs by implantation of Be+ ions followed by controlled-atmosphere annealing. Two SLS structures were studied, each having fifteen periods ofequal-thickness(- 175 A) alternating layers of GaAsP and Gap. The compositions of the alloy layers in two samples were x = 0.15 and 0.20, corresponding to mismatches of 0.56% and 0.74%, respectively. The Be+ implants were done at 75 keV to a dose of 1 X IOl5 cm-2. Annealing was done at 825°C for 10 minutes in an overpressure
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FIG.4. High resolutionelectron micrograph of the ( 1 1 1) lattice fringes of the same SLS as in Fig. 3. The ( 1 1 1 ) lattice fringes are the diagonal lines passing through the GaF'/GaAsP interface.
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+
of AsH3 PH3 in a H2 carrier. Hall-effect measurements showed that the originally n-type SLSs were converted to p-type by the implantation and annealing, and that acceptor activation and mobilities were compared to values expected for type-converted GaP-based alloys. Furthermore, depthresolved structural characterization by ion channeling demonstrated that there was no signficiant intermixing between adjacent layers in the structure, and that there was no loss of strain in the SLS. These results show that implantation technology can be applied to SLSs and reflect favorably on the stability of SLSs under the quite severe stresses produced by particle bombardment and thermal cycling. 111. Electronic Properties 4. GENERAL FEATURES
Much of the motivation for studying high-quality SLS structures results from the wide range of electronic properties that can be obtained from this new class of materials. This wide range is a consequence of the flexibility in the choice of mismatched layer materials, the influence of quantum size effects, and the effects of layer strains. As a result, SLS electronic properties can be tailored through the appropriate choice of SLS structure^.'^ One example of this tailorability is the capability of independently varying the bandgap (EJ and lattice constant (a'? in SLSs grown from mismatched ternary alloys.I8 Note that this capability does not exist for the bulk materials from which the SLS is grown. Theoretical and experimental values of E, and ull are compared in Fig. 5 for GaP/GaAsP and G a s / GaAsP SLSs. The experimental values (circles) represent photoluminescence measurements of E, for structureswith SLS layer thicknesses 2 60 A. Lattice-constant values were measured using X-ray diffraction. Other details of this figure are discussed later in this section. Although only one set of SLS structures is examined in the figure, theoretical results indicate that all SLS E , values between those shown and the bulk alloy values can be obtained. These specific SLS results are representative of a general feature of ternary SLSs. The large tetragonal elastic strains in the SLS layers can produce marked effects on the electronic properties. The hydrostatic component of the strain causes shifts in the bulk energy levels of the layers, and the uniaxial component causes splittings of certain degenerate levels (e.g., the m, = kj, k4 valence-band maxima). These bulk energy shifts in turn alter the band structure of the SLS through their effects on the quantum-well structure. The resulting energy levels of the SLS are therefore determined by: the particular bulk quantum-well structure associated with the flexible choice
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LATTICE CONSTANT (&
5.46
5.50
1
5.54
I
I
5.58
I
I
5.62
1
I
1
I
2.8 T = 295K
2.6
2.4
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1.4
I 0
I
I 0.2
1
I 0.4
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AVERAGE COMPOSITION
I 1.o
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FIG.5. Summary of measured transition energies for both GaP/GaAs,P,-, (type 11) and GaAs/GaAs,P,-, (type I) as a function of average composition X ~ x / 2(type 11) or X= ( 1 x)/2 (type I). These SLS structures have layer thicknesses 5 60 A. The lattice constant parallel to the interfaces is also labeled on the top horizontal axis. The corresponding curves for the bulk alloy transitions are also included for comparison.
+
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STRAINED-LAYER SUPERLATTICES
469
of layer materials, the additional strain modifications of the well structure, and quantum size effects5*’’The energy versus wave-vector dispersion of the SLS bands can be modified from bulk values in the layer planes as well as along the growth direction. For example, the uniaxial strain component significantly alters the bulkI9 and SLS valence-band effective masseszo*zl near the valence-band maxima. These effectsprovide novel ways in which to tailor the SLS optical and transport properties. Another interesting feature of multilayer structures is the existence of nonzero optical matrix elements which are zero in bulk materials. In particular, certain indirect-gap optical transitions in bulk materials (valence band to [OOl] [OOi] conduction-band minima) become direct transitions in the (001) superlattice due to zone-folding effects. The optical matrix elements associated with these zone-folded transitions are no longer zero due to the symmetry breaking provided by the layered nature of the structure. Enhanced optical absorption can result from this effect.
5. SPECIFIC MATERIAL SYSTEMS In this section we will discuss the SLS electronic properties of specific material systems. We will first consider SLSs fabricated in the GaAsP alloy system. These SLSs have a number of interesting features. The X I-r15 and r,-rl5 energy gaps are 1.907 and 1.425 eV for GaAs and 2.26 and 3. I eV for GaP. The fundamental energy gap in GaAs,P,-, changes from indirect to direct for x > 0.54. Recent experiments indicate the GaAs rI5 valence band lies 0.6 eV higher than the corresponding band in GaPF2This gives rise to type-I superlattices (i.e., electrons and holes localized in the GaAs layers) for GaAs/GaAs,P,-,. These superlatticesare similar to those in the AlGaAs/GaAs system. Alternately, GaP/GaAs,P, -, superlattices with x < 0.54 have holes localized in the GaP layers and electrons localized the GaAsP layers. This spatial separation of carrier types is similar to that occumng in the InAs/GaSb superlattices, which are classified as type-I1 superlattices. The most general form of these superlattices can be represented as GaAs,, P ,- /GaAs,P, -x2. Two limiting forms have been invesand GaAs/GaAsZl -,. tigated experimentally: GaP/GaAs,P To understand the energy levels of these SLSs it is instructive to consider a heterojunction with coherently strained layers. A type-11, (100) Gap/ GaAsP heterojunction will be considered here, but the results of the present discussion are easily extended to type-I heterojunctions. The lattice constant of GaAs is 3.6% larger than that of Gap. Thus the arsenic-rich layer is subjected to biaxial compression while the GaP layer is under biaxial tension. These strains can be resolved into hydrostatic and uniaxial strain components. The hydrostatic component &h shifts the “center of gravity” of a given bulk band, while the uniaxial component E , splits
,
470
G.
c. OSBOURN et al.
degenerate bulk bands. Since the signs of E, and &h are opposite for the two layers, so are the shifts and splittings. In Fig. 6 we have schematically represented these strain-induced modifications of the rl and XI bulk conduction bands and the r, and degenerate r, bulk valence bands. Note that the six equivalent XIbulk-band minima are split into two equivalent band minima for k along [OOl] and [OOT] (perpendicular to the interfaces) and four equivalent minima for k along [IOO], [TOO], [OlO], and [OTO] bands. The former two minima are zone folded to the r point of the SLS Brillouin zone (for structures with an even number of 111-V monolayers per unit cell5),while the latter four minima are folded to other indirectband minima in the SLS zone.
-GaP
GaAs P,,,
i\
'I
001
I':
I
I,
I I
I
I
I
I
I UNSTRAINED
' I
I
I I
BIAXIAL TENSION
I I
I I I
I
I
I
--___LI
-x
0
11/2,*1/2>
r7
I I BlAXlAL
I I
UNSTRAINED
CoMPRESS'oN
I
1
/A
100,010 J
---,x
I I I I
8.
STRAINED-LAYER SUPERLATTICES
47 1
Three sets of valence-band quantum wells and hole states are indicated in Fig. 7 for the GaAsP/GaP SLSs. The upper two sets [labeled P(3) and rv(+)] derive from the highest-lying zone-center Ts valence-band maxima of the bulk, which have spin quantum numbers m, = A $ and -++, respectively. These states are degeneratein the bulk, but split by strain in the SLS. The states labeled P(so) derive from the bulk r, split-off bands. It is important to note that the hole effectivemasses transverse to the superlattice direction are substantially altered from the bulk value^.'^*^' This results in large changes in the transverse mobilities and densities of hole states. A highly schematic representation of the important GaAsP/GaP quantum well transitions resulting from the strained heterostructures is also shown in Fig. 7. Associated with each periodic potential of a given symmetry there exists a well-defined ground-state energy level as well as higher-lying levels. For the particular heterojunction illustrated in Fig. 6 the lowest electron level and highest hole level occur in alternate layers. This spatial separation due to the “staggering” of the energy gap gives rise to an effective bandgap that is lower than the bandgap of either bulk layer material. Because of this particular arrangement of quantum wells, a simple picture can be used to interpret the results of optical In this picture the transitions that
rc
Conduction band states
XC
rv(112)
band states
-- -- -- -Ga(As,P) Gap Ga(As,P) Gap
FIG.7. Schematic of quantum well structure and energy levels for a GaAsP/GaP SLS.
472
G.
c. OSBOURN et al.
define the main features in optical luminescence and absorption occur between the lowest energy states in each well. These transitions determine the onset energies above which new sets of quantum well transitions can occur. The details of other higher-lying transitions within the same wells will be smeared out by the spatial separation of the wells and by alloy broadening. The six lowest transitions are indicated in Fig. 7 and are labeled E g l ,E,, E , A,, Eol,Eoz,and E , A, in Table I. Other transitions involving the indirect [TOO], [loo], [OlO], and [OTO] minima are also listed for completeness. These are expected to be weak transitions and are not experimentally observed. The lowest three transitions in Fig. 7 are expected to be enhanced through the zone-folding/symmetry-reduction effect; however, spatial separation of the electron and hole wells will also significantly reduce these oscillator strengths near the bandgap energy. At higher energies, transitions involving states which are more delocalized (and therefore have larger matrix-element overlaps) become accessible. In particular, detailed tightbinding studies indicate that SLS valence-band states associated with the split-off wells can have significant amplitude in the barrier layers due to valence-band mixing.24It is expected that these transitions will exhibit a net enhancement of the absorption coefficient due to zone folding. In contrast, the GaAs/GaAs,P,-, SLS systems with x < 0.3 result in electron and hole quantum wells both in the GaAs layers. These structures can exhibit a series of quantum well transitions which are not affected by spatial separation and which are not significantly broadened by the alloy fluctuations in the alloy barrier layers. The energies of the SLS states have been calculated for both the type-I and type-I1 cases using an effective-mass (Kronig-Penney) model, with barrier heights at each of the high-symmetry points determined by the band offset^.'^-^^
+
+
TABLE I
SUMMARY OF OPTICAL TRANSITIONS Conduction-bandstates Valence-band states
rw
rvw rv(w) r?
X , folded [001];=
X,[0011" -%I
Esz Eg A,
+
X,[010],[ 1001b Weak, not observed Weak, not Observed Weak, not observed
rl'
E,, Eo2 E, A,
+
[olol,r l W b weak, not ObServed Weak, not observed Weak, not observed
Normally indirect transitions, but direct in SLS due to zone folding. Indirect transitions. Direct transitions.
8.
473
STRAINED-LAYER SUPERLATTICES
We have measured the transition energies in many different MOCVD Ga(As,P) SLSs using photoluminescence (PL), photocurrent, absorption, and excitation spectroscopies. The details of these experiments have been discussed p r e v i ~ u s l y . ~ For ~ ~ ~the ~ -type-I1 ~ * SLSs with x = 0.2, PL spectra always exhibit a dominant band-edge peak due to the transitions Eg, and EB2,which are sometimes resolved. In addition, much weaker Eor EO2 transitions (not resolved) are always observed at higher energy. The PL spectrum for a type-I1 SLS with x = 0.4 is shown in Fig. 8. The band-edge emission again shows Eg,and E , transitions in x = 0.2 samples but shifted to lower energy due to the higher As concentration in the alloy layers. At higher energies, Fig. 8 reveals a weak emission peak at an energy correspondingto the Eg A. transition. At even higher energies there is no evidence of the E,, EO2transitions. This is in contrast to spectra for x = 0.2 samples. Apparently, the emission strength shifts from the Eol EO2transitions to the Eo A. transition with increasing x. This effect is not fully understood and will be discussed shortly. The optical properties of GaAsP/GaP SLSs have also been investigated using photocurrent spectroscopy.26This technique provides a convenient means of obtaining information about optical absorption of thin epitaxial structureswithout the necessity of removing their substrates. In our photocurrent studies we used samples similar to those described above, which had undoped SLSs of 0.5 - 1.5 pm total thickness, lattice-matchingbuffer layers (also undoped) of 1-3 pm thickness, and n-type Gap substrates. Ohmic contacts were alloyed into the substrates, and thin, semitransparent Au dots were evaporated directly onto the SLSs. The SLSs were illuminated, through the Au films, by monochromatic light. In the photocurrent
+
+
+
+
+
-
1.8
2.0
2.2
2.4
PHOTON ENERGY (eV)
FIG.8. Photoluminescence spectrum of SLS with x = 0.4. The peaks near 2.0 and 2.2 eV are due to the SLS,while the structure near 2.1 eV is associated with the underlying buffer layer.
474
G.
c. OSBOURN et al.
technique, free electrons and holes generated by optical absorption are separated by the built-in field produced by the metal-semiconductor Schottky barrier and thereby produce current in an external circuit. Only camers generated in or near the depletion region of the Schottky bamer are collected, so that substrate contributions to the photocurrent are discriminated against. In some of the spectra, features due to the buffer layer were observed and corrected for, as was discussed in a previous publication.26 Room-temperaturephotocurrent data from a GaAs,P, -,/Gap SLS with 320 A thick layers and fifteen superlattice periods are shown in Fig. 9. The top part of this figure is a logarithmic plot of photocurrent versus photon energy, whereas the bottom part shows the derivative of the photocurrent. The photocurrent spectrum has distinct onset features of 2.05, 2.25, and 2.53 eV. These energies agree quite well with the theoretically predicted transition energies of 2.09 eV for the transition Egl, 2.24 eV for the transition Eg A,,, and 2.53 eV for the transition E,, . The theoretically predicted energies are indicated by arrows on the figure. We note that the closely spaced Eg, and EB2transitions are not separately resolved in the room-temperature photocurrent spectra. A striking feature of Fig. 9 is that the photocurrent at 2.2 eV (after saturation of the band-edge transition) is more than a factor of ten weaker than the photocurrent at 2.5 eV (after saturation of the transition from the split-off valence band, but before any transitions to the higher-lying conduction-band states). This feature of the data supports the idea discussed above that delocalization of the quantum
+
I
1.9
2.1
2.3
2.5
2.7
PHOTON ENERGY (ev)
FIG.9. Photocurrent spectrum (top) and its derivative (bottom) for a typical GaAsP/GaP SLS.
8.
475
STRAINED-LAYER SUPERLATTICES
well states derived from the split-off valence band should lead to an enhanced transition strength for the E, A. transition compared with the Egl and Egztransitions. The relative oscillator strengths for the various FaP/GaAs,P --x SLS transitions in samples with x = 0.2 and 0.4 have also been examined by absorption spectroscopy. Shown in Fig. 1Oa is the measured absorption coefficient (solid line labeled SLS 434) for an undoped GaP/GaAso,.,Po,, SLS with 180 A thick layers. Also shown are curves for the absorption coefficient of a direct-gap material (curve labeled 1) and an indirect-gap material (curve labeled 2). The latter two curves are actually absorption curves for GaAs and Gap, respectively, which have been shifted in energy to align the onset of absorption with the SLS band edge. It is apparent that near the band edge at 1.95 eV, the SLS absorption is not enhanced with respect to the indirect-gap material. However, near the E, A. transition at 2.05 eV, there is a significant increase in the oscillator strengthcomparable to that for the direct-gap material. This increase in oscillator strength suggests that this SLS transition has become direct due to the zone folding of [OOl], [OOi] conduction-band states to the Brillouin zone center. The observed weak oscillator strength for the E, transitions is due to the spatial separation of electron and hole states that give rise to this transition. To increase the spatial overlap of electrons and holes in these states, SLS samples were grown with lower As content in the alloy layers. This has the effect of lowering the quantum-well barriers which confine the electrons and holes. The absorption spectrum for such a sample is shown in Fig. lob. The measured absorption coefficient near the band edge (hv = 2.15 eV) for this sample with x = 0.2 is enhanced about 10 times over that of the SLSs with x = 0.4.These data indicate that the strength of the band-edge absorption can be adjusted by changing the As composition in the alloy layer. Although the band-edge absorption in Fig. 10b has been increased, the E, A. transition near 2.35 eV is about 6 times weaker than the corresponding transition near 2.1 eV in Fig. 1Oa. These data indicate that the E, A. transition strength is increased with increasing As composition from x = 0.2 to 0.4. This is exactly the same result noted earlier in the discussion of the PL data. The presence of this interesting effect in both absorption and emission data underscores the need for a more detailed band-structure treatment of optical matrix elements as a function of energy and structure in these SLS systems. It should be noted that calculational techniques have recently been developedzgfor lattice-matched systems which will allow these types of studies in the GaP/GaAsP system. In Fig. 11 we show the compositional dependence of the onset features observed in the photocurrent experiments. This figure contains information similar to the left-hand part of Fig. 5, with the addition of data on the
+
+
+ +
476
G.
c. OSBOURN et al.
5
10
T=295 K
1.8
2.0
2.2
2.4
2.6
2.8
3.0
PHOTON ENERGY (eV)
FIG.10. (a) Absorption spectrum for SLS 434 along with those for direct-gap GaAs (curve 1 ) and indirect-gap GaP (curve 2). The latter two curves have been shifted in energy to align the energy gaps at 1.92 eV equal to that for the SLS. (b) Absorption spectra of SLS 449 with x = 0.2.
transitions from the split-off valence-band states. Again we have used bands to indicate the range of energies predicted for a range of layer thicknesses from 60 A to the infinite-thickness limit. A typical pair of experimental error bars is shown on one point. The uncertainty in composition is due mainly to the fact that there is some variation in composition across the wafer, and different pieces of the wafer were used for the X-ray and photocurrent studies. The vertical error bars come from the uncer-
8.
1.8
477
STRAINED-LAYER SUPERLATTICES
2.0
2.2
2.4
2.6
2.8
3.0
PHOTON ENERGY (eV) FIG. 10 (Continued
tainty in precisely determining on-set energies for the relatively broad features observed. In general, the observed features agree with the experimental predictions within experimental uncertainty. Up to this point, all of the optical data have been for type41 SLS samples. Results have also been obtained for type-I GaAs/GaAs,P,-, SLSs. These samples are electronically similar to the well-known GAS/ Al,Ga,-,As superlattices where the electrons and hole states are direct in both real and momentum space. Aside from the obvious chemical difference, the primary difference is the presence of the large biaxial strain in the
478
G . c. OSBOURN
et al.
2.8
2.6
2.4
Y
w 2.2
2.0
1.8
0.1
0.2
0.3
0.4
0.5
X
FIG. 1 1 . Energy of onset features observed in photocurrent spectra of GaAs,P,-,/Gap
SLSs as a function of x compared to theoretical predictions.
GaAs/GaAs,P,-, SLSs. We have studied these SLSs with PL and excitation spectroscopies.22A low-temperature PL spectrum from such a sample having x = 0.56 and equal layer thicknesses of 210 is shown in Fig. 12. The spectrum shows a very strong band-edge emission peak. The peak energy is shifted approximately 40 meV above the GaAs band edge. This is a direct consequence of hydrostatic compression arising from the large lattice mismatch (- 2%) between alternating layers. The peak full width at half-maximum is considerably narrower (- 6 meV) in the strained binary GaAs layers of this sample than that in the type-I1 sample of Fig. 8, which is likely broadened by ternary alloy disorder. At higher temperatures, additional emission peaks are present in the PL spectrum. These peaks are due to parity-allowed transitions between excited electron and hole states localized in quantum wells. At room temper-
A
8.
479
STRAINED-LAYER SUPERLATTICES
T=42K
L(11) 2 . 2
1.50
155
1.60
PHOTON P E R O Y (eV)
FIG.12. Low-temperature photoluminescence spectra of SLS 679 at two different levels of laser irradiance. The large peak is associated with excitonic recombination and is shifted -40 meV from similar emission in bulk GaAs (indicated by the arrow).The schematic at the left shows the effect of biaxial compressive strain on the conduction (CB) and valence (VB) bands.
ature several of these transitions can be observed. The observed transition energies are in agreement with Kronig - Penney calculations. Figure 13 shows an excitation spectrum of the same SLS. To obtain this spectrum, the PL was detected at a fixed wavelength while the excitation laser wavelength was scanned. The spectrum shows a number of transitions, including strong parity-allowed transitions, as well as some weaker transitions which were not observed in the PL spectra. These weaker peaks may be associated with parity-forbidden transitions. The two peaks labeled 1 and I’ involve the stress-split It,t)and It,+)hole states, respectively. The energy separation between these two peaks is primarily due to the large (- 40 meV) valence-band splitting. Their relative intensity is altered from that expected from lattice-matched superlattices due to the uniaxial strain modification of the valence-band densities of states. These effects are in qualitative agreement with the stress-dependent effective-mass theory for degenerate valence bands. l9 To summarize the results of the optical transitions for both type-I1 GaP/GaAs,P --x and type-I GaAs/GaAs,P --x SLS, we have plotted the optical transition energies as a function of average As composition Tin Fig. 5 , where F= (xldl x,d,)/(dl dz).In the indirect-gap range TS 0.5, the
+
+
480
G.
c. OSBOURN et al.
ENERGY (absome urits,ev)
-i
E
1.55
1.60
I
I
1.65
'
1.70 SLS 67914
80-
40-
-
I
80
3 20
20 0
' 0
I
I
I
50
100
150
0
ENERGY blative units, mev)
FIG. 13. Excitation spectrum for SLS 679 with d = 230 A (upper trace) with detection at 1.545 eV (for the n = 1 peak) and 1.554 eV (for higher-energy peaks). The peaks correspond to transitions between quantum well states for electrons and holes. The peak labels are defined by the quantum well states. The upper horizontal scale applies t? the upper trace only. The lower trace is the excitation spectrum for SLS 680 with d = 202 A. The two t r a m have been aligned at the n = I transitions to show the effect of layer thickness on the transition energies. The lower scale is in energy units relative to the n = 1 transition energy.
bandgap for the SLS is lower than that for a GaAs,P,-, alloy of the same composition due to the staggering of quantum wells for electrons and holes. In the direct range x 2 0.5, the SLS bandgap is higher than that for GaAs, due to the compressive hydrostatic strain component present in the GaAs layers. Various transport studies have been carried out in the GaAsP systems. To investigate the electrical characteristics of these SLSs electrical transport measurements (Hall effect and conductivity) have been made on various samples. The measurements were made by the standard Van der Pauw technique on structures similar to those discussed above with regard to the optical studies. Special care must be taken in these measurements to ensure that the properties of the SLS alone are being measured without any contribution from the underlying substrate or buffer layer. In general the SLS can be isolated electrically from the rest of the structure by having the substrate plus buffer be of the opposite camer type from the SLS. Some typical electrical results are shown in Table 11. The first five samples listed are GaAs,P,-,/Gap SLSs with x between 0.16 and 0.3.
8.
481
STRAINED-LAYER SUPERLATTICES
TABLE 11 TRANSPORT DATAOBTAINED FROM HALL MEASUREMENTS ON SELECTED SLS SAMPLES OF THE TYPEGaAs,P,-,/GaPOR GaAs,-,,P,/GaAs
Sample number
Layer thickness Composition
(A)
x=0.16 x=0.16 x = 0.24 x = 0.28
160 I60
Numberof SLSperiods
Doping profile"
T (K)
n orp (cm-')
20 20 20 20 20 40
UD UD UD UD UD MD
300 300 300 300
p = 1 . 8 1017 ~ p = 1 . 4 1017 ~ n=2.1 x 1017 n = 4 . 9 X 1017 n = 1.1 x 1017 p = 1.1 x 1017 p = 7.6 X 10l6 p = 1.8 x 1017 p = 1.0 x 1017 n = 1.1 x 1017 n = 1.1 x 1017 n = 1.2 X 1OI6 n = 1.2 X 10I6
Y (cmz/V s) ~~~
553 602 600 60 1 607
170
737
y = 0.33
230 I20 210
737
y = 0.22
180
40
MD
300
826
y = 0.2
190
35
MD
77 300 77
865
y = 0.2
I70
35
MD
300
x = 0.30
300 300
77
77
50 55 157 183 182 220
805 209 I240
5750 25000 6200 26000
UD, uniform doping; MD, modulation doping.
These structures were uniformly doped with Zn for the p-type samples and with Se for the n-type samples. Doping levels were in the low 10'' cm-3 range. The observed hole mobilities of 50 and 55 cm2/V s are somewhat lower than would be expected based on mobilities in bulk GaAsP alloys. The reason for this is not known, but a similar discrepancy has been observed for uniformly doped AlGaAs/GaAs superlattices. On the other hand, the mobilities of the three n-type samples (samples 600, 60 1, and 607) are comparable to bulk values. Results for four GaAs,P,-,/GaAs SLSs are shown in the bottom part of Table 11. Modulation doping, with the donors confined to the central 25% of the wide-gap GaAsP alloy layers, was utilized for these samples. The mobilities of the two p-type samples, both at room temperature and at liquid nitrogen temperature, are comparable to literature values for bulk GaAs. The use of modulation doping should make it possible to study high-mobility hole transport in GaAs at low temperature without the limitations caused by carrier freeze-out and impurity banding that are encountered in bulk material. The mobilities observed for the modulationdoped n-type samples are comparable to those observed in similar AlGaAs/GaAs superlattices.m The anisotropic nature of minority carrier diffusion in these SLSs has been quantitatively examined using optical technique^.^' Diffusion lengths
G.C. OSBOURN et
482
al.
perpendicular and parallel to the interfaces (L, and L!, respectively) have been measured in (001) GaAs,,P,, /Gap SLSs. Recent band-offset measurements2’ indicate that carrier motion perpendicular to the interfaces should be inhibited for both electrons and holes. The measurement of L , allows a determination of whether tunneling or thermionic emission is important for the perpendicular direction. In contrast, the parallel transport should provide a test of the crystalline quality of the strained bulk materials which comprise the layers. The crystals examined in these experiments were composed of alternating, equally thick layers of Gap and GaAs,P -,, where the composition x is nominally 0.2. We find that the perpendicular diffusion length (- 0.1 pm) is more than an order of magnitude smaller than the parallel diffusion length (- I .5 pm). The measured values are summarized in Table 111. The parallel diffusion length is comparable to that measured in the bulk materials that compose the SLS. The small value for the perpendicular diffusion length is consistent with the existence of large potential barriers in both the conduction and valence bands that inhibit transport normal to the SLS interfaces.
TABLE III SUMMARY OF PARALLEL AND PERPENDICULAR MINORITY CARRIER DIFFUSION LENGTHS FOR GaP/GaAs,,P,, SLS ~
~
~~~~
~~~~
Parallel Diffision Lengths Sample 449 553 554 598 606 642
X= x/2
Layer thickness (A)
0.096
320 150 150 250 360 60
0.084
0.08 1 -0.1 0.079 0.096
Doping (cm-3)
Lll(pm)
Undoped p , mid-10I7
n, mid-10” n, low-10‘8 n, low-10‘8 Undoped
1.6 1.5 1.5 1.4 1.5 1.6
Perpendicular Diffision Lengths Sample 553
561 564
Junction type
p s ~(1s x 10’7 on n-buffer (1 X 1Ol8) n-SLS(1 X 101*)onpSLS (1 x 10”) pSLS (1 X 10”) on n-SLS ( 5 x 1017)
x
Layer thickness (A)
LL(pm)
0.084
150
0.12
0.119
100
0.06
0.070
260
0.08
8.
STRAINED-LAYER SUPERLATTICES
483
There are several conclusions to be made from these observations. First, the lateral transport properties are not degraded in these highly strained materials. This is of particular interest for SLS p-type materials where the layer strain splits the heavy- and light-hole bands and alters the lateral masses. The lowest band can be either light or heavy depending on the sign of the layer shear strain. Second, the perpendicular transport, as expected, is largely inhibited by the potential barriers. Estimated values of the perpendicular masses are much too large to account for the observed values of LL and reveal that thermionic emission is likely responsible for the observed values. Thus, it would be interesting to study LI versus temperature. The short values for L , indicate that surface recombination in SLS structures is largely eliminated. This is an attractive feature for several device applications, such as emitters and detectors. The electronic properties of SLSs grown in the GaAs/InGaAs system have also been studied experimentally and theoretically. These samples were grown by MBE, as discussed in a previous section. The emphasis in this work has been on the transport properties of these structures. Photocurrent spectroscopy has been used to investigate the band-edge optical absorption of a number of In,Ga, -,As/GaAs SLSs with x = 0.2. A typical spectrum3*(obtained at room temperature) is shown in Fig. 14. In this spectrum we note a sharp onset of photocurrent at 1.26 eV, which we interpret as the fundamental bandgap of the SLS. For the particular struo ture used to obtain the spectrum.of Fig. 14 a theoretically calculated gap of 1.26 eV has been obtained, in good agreement with the measured result. Most of the samples studies showed some structure above the bandgap, such as can be seen in Fig. 14 above 1.3 eV. This structure has not been investigated in detail yet, but it is anticipated that features due to higherlying subbands of the SLS as well as absorption features in the buffer and substrate may be observed. The large electron mobility of InGaAs alloys suggeststhat InGaAs/GaAs SLSs have potential applications in high-speed devices. To obtain the best possible electrical transport characteristicsit is desirable to use modulation doping to separate the carriers from their parent donors or acceptors. Modulation doping has been achieved in In,Ga -,As/GaAs SLSs using Si donors for n-type material33and Be acceptors for p-type.** A schematic representation of two doping schemes used to study n-type SLSs with x = 0.2 is given in Fig. 15. As indicated, samples with both uniform and modulation doping were grown. The modulation-doped structures had 120 A layer thicknesses and 45 A spacer layers between the Si donors and the undoped quantum wells. Figure 16 is a plot of room temperature Hall mobility versus average carrier concentration for the two series of samp l e ~ A. ~significant ~ enhancement in mobility of the modulation-doped
,
1.2
I
I
1.3
1.4
1.5
PHOTON ENERGY (ev)
FIG. 14. Typical room-temperaturephotocurrent spectrum of an In,,2Ga,,~,As/GaAs SLS showing a sharp band-edge feature at 1.26 eV.
Uniform doping (UD)
vB
Modulation doping
(MD)
--
GaAs InGaAs GaAs InGaAs FIG. 15. Illustration of uniform- and modulation-doped structures used to study electrical transport in InGaAs/GaAs SLSs.
8.
485
STRAINED-LAYER SUPERLATTICES
I no. Gao.gAs/ GaAs SLS
0
\
.
\ \
GaAs
I
10'6
I
loq7
'
I
lOl8
n (cm-7 FIG.16. Mobility versus carrier density for uniform doping (UD) and modulationdoping (MD) of InGaAs/GaAs SLSs compared to bulk GaAs.
samples compared to the uniformly doped ones is observed. Room-temperature mobilities are limited by phonon scattering; so much larger mobility enhancements are observed at low temperature. Figure 17 shows a comparison of low-temperature Hall mobility for two samples, one uniformly doped and one modulation doped. The samples have comparable carrier densities. The modulation-doped sample has a peak low-temperature mobility of over 30,000 cm2/V s. This value is comparable to peak mobilities observed in AlGaAs/GaAs superlattices with comparable spacer-layer thicknesses.30 The low-temperature electrical transport results discussed above show that InGaAs/GaAs SLSs can have electrical quality comparable to that obtained in similar structures made from lattice-matched materials. This demonstrates that the many highly strained interfaces (>100) in the SLS structures do not significantly degrade electrical characteristics, and that SLSs should therefore be useful for high-speed device applications.
486
G.C. OSBOURN
et al.
Ino,,Gao.8As/GaAs
Si-doped
SLS
n %l~lO''crn-~
Y
a 10
0
-
0
"
"
~
"
"
" " 100
"
200
3 0
T (K) FIG. 17. Temperature-dependent mobility for UD and MD samples doped at the same level.
As was mentioned in a previous section, interesting valence-band effects are predicted to occur in SLSs due to the strain-induced degeneracy removal. In Fig. 18 we present tight-binding calculations of cyclotron 2D hole masses of the uppermost pair ofvalence bands in model monolayer SLS structures as a function of hole energy [&& = E(k = 0) - E(k)]and the splitting of the SLS valence bands at the zone center.% The model I11- V layer materials were chosen to have band offsets which result in hole confinement in the biaxially compressed layer, and the tight-binding material parameters were selected to give hole masses similar to those in the GaAs/Ino.2Gao.8As system." The strain value in the compressed well layer was treated as a variable parameter to illustrate the dependence of the hole mass on the valence-band splitting. Figure 18 illustrates several interesting features. First, the uppermost valence bands exhibit quite small in-plane masses at small E,, values. These low-mass bands can be preferentially populated by holes at low temperatures, low electric fields, and low hole concentrations. The magnitude of this mass near &,ole = 0 approximately tends toward a large stress limit. Second, the two distinct mass values for each strain value of the SLS indicate a spin splitting of these bands. This
8.
487
STRAINED-LAYER SUPERLATTICES .4
1
I
I
10
20
30
.3
0
40
EHOLE (meW FIG. 18. Calculated cyclotron hole masses versus hole energy for the m, = ?$ valence bands of several model SLS structures with hole confinement in biaxially compressed layers. The hole energy is defined to be the positive difference between the valence-band energy of interest and the valence-band maxima. The different pairs of curves correspond to different strain magnitudes in the holecontaining layers, and the values of the m, = +-$, zonecenter valence-band splitting are given for each pair.
+&
splitting is a consequence of the absence of inversion symmetry in the 111-V zincblende materials, and the magnitude of this splitting is influenced by the uniaxial component of the layer strains. Finally, the mass values significantly increase with Eholcabove a value around one-fourth of the valence-band maxima splitting. This increase is a consequence of band mixing of the split valence bands, and it limits the energy range over which small-mass 2D holes can be observed. Experimental evidence for this small-hole-mass effect has recently been obtained” from high-field magnetotransport experiments on Be-doped In,,Gao,As/GaAs SLSs. For these measurements structures with different ratios of layer thicknesses were employed to vary the strain, and therefore the valence-band splitting, in the InGaAs quantum wells. The experimental results are summarized in Table IV. The first two columns of this table give the layer-thickness ratio and the corresponding compressive biaxial strain in the alloy layer for each sample. The third and fourth columns give the carrier densities and mobilities at 4 K, as determined from low-field Hall measurements. The last three columns give the results of the highfield magnetotransport measurements, with F being the Shubnikov- de Haas frequency, P the two-dimensional carrier density, and m*/mo the
488
G.C. OSBOURN
et al.
TABLE IV STRUCTURAL AND TRANSPORT PROPERTIESOF TYPE In,,Gao,,As/GaAs STRAINED-LAYER SUPERLATTICES Layer thickness ratio d(GaAs)/d(InGaAs)
Parallel strain in InGaAs
P
P
(%)
( loi7cm-*)
(cm2/Vs)
F (kG)
0.5 1 .o 2.0
0.48
I .4 1.7 2.3 3.3 2.7
6,000 12,000 9,700 7,900
66 55 61 94
14,100
75
3.6
3.0 4.0
0.73 0.97 1.09 1.16
P (10" cm-2)
m*/m,
3.2 2.7 2.9
0.14
4.5
0.13 0.13
0.14 0.15
effective-mass ratio. The effective mass is found to be small and independent of strain, suggesting that the large-splitting limit of the effective mass has been reached. The effective mass is somewhat larger than the expected value near Ebole= 0, and reasons for this discrepancy are being investigated. Theoretical work has been camed out on a number of other proposed SLS systems which have not yet been grown. For example, theoretical studies of the bandgaps of InAsSb SLSs have recently been carried and the results indicate that certain of these structurescan have wavelength cutoff values (A, = 1.24/EJ greater than 12 pm at 77 K. This is larger than that of any bulk I11 - V alloys at 77 K, including InAsSb alloys, which exhibit A 5 9 pm at 77 K. The extension of certain InAsSb SLS & values beyond the maximum 1,values for the bulk InAsSb alloys is made possible through the effects of the layer strains in the SLSs. As a result, these materials should be I11 - V candidates for long-wavelength detector applications. The proposed large 1, SLS structures are chosen to contain (100) oriented layers of InAs,-,Sb, with x = 0.6 1, since this alloy composition corresponds to the longest bulk A,. The larger-gap alloys InAs,-,Sb,r 0.61 are chosen because the resulting SLS structures contain expansive hydrostatic and compressive (100) uniaxial strain components in the InAso,39Sbo,,layers. The expansive hydrostatic strain component causes a reduction in the bulk E , of the InAso.39Sbo.6, layers through its influence on the conduction-band minimum, and the compressive uniaxial strain component contributes through the splitting of the degenerate valence-band maxima. The m, = *HA$) valence band is split toward (away from) the conduction band in the InAso.,,Sbo,, layer. The magnitude of the total reduction depends on x and on the ratio of layer thicknesses in the SLS.
8. STRAINED-LAYER SUPERLATTICES
489
Figure 19 presents calculated values of the bulk A, value (i.e., without the quantum size effect of the SLS) at 77 K of the strained InAso.39Sbo.61 layers in an InASo.39Sbo.61/InAs,-,Sb, SLS with three ratios of layer thicknesses as a function of x. The corresponding lattice mismatch between the layers is also given at the top of the figure. These values were calculated using the techniques described by 0 s b 0 u r n . ~ ~ The bulk A, variation in Fig. 19 is entirely due to the increase in the layer strain. It can be seen from the figure that the bulk A, values of the strained InAso~,9Sbo,,layer at 77 K can be extended beyond 12 pm for all of the thickness ratios considered. It should be noted that the strain values required for these results are well within the range of strains which have been experimentallyemployed in the GaAsP and InGaAs SLS material systems. The Ac value for the SLS itself will also be influenced by the quantum size effect and by the band offsets of the InAsSb alloy system. The quantum size effect in these structures will increase the SLS bandgap, and so this effect competes with the bandgap reduction due to strain. However, the effect can be minimized by employing relatively thick InAso,39Sbo.61 layers in the SLS. Also, the effect can be counteracted by employing greater strain values. In order to compute specific superlatticeA, values for various L A T T I C E MISMATCH 0.5
0.0
(%I 1.5
1.0
12.0
11.0 A,
(pm) 10.0 9.0
8.0
I
0.6
I
I
I
0.7
I
0.8 X
FIG. 19. Calculated values of the bulk A, value (without quantum effects) at 77 K of the strained InAs,,,Sb,,, layers in an InAs,,,Sb,,, /InAs,-,Sb, SLS with three ratios of layer thicknesses (2 : I, 1 :1, 1 :2) as a function of x. The corresponding lattice mismatch between the two types of SLS layers is given at the top of the figure.
490
G.C. OSBOURN
et al.
InAso.39Sbo,61 /InAs,_,Sb, SLS structures, it is also necessary to know the conduction-band offsets of the layers. These values are not known for InAs/InSb, and different schemes for estimating them (electron affinities, values based on photoemi~sion~~) differ by 0.3 eV. However, calculations of SLS Ac values have been camed out using a variety of band-offset values. These results35(not shown) indicate that SLS Ac values reaching 12 pm at 77 K are possible regardless of the precise band-offset values.
-
IV. Applications of Strained-Layer Superlattices
6. GENERAL FEATURES The general motivation for using SLSs in device structures is the capability of tailoring the SLS material properties. Appropriately designed SLS structures could have one or more optical and transport properties which are advantageous for optimized device performance. A number of examples of SLS features which could be useful have already been discussed in the previous sections. The device implications of some of these are discussed below. The capability of independently varying the bandgap and lattice constant in ternary SLSs makes possible (for the first time) the growth of high-quality, thick-layered heterostructures from lattice-mismatchedmaterials. In this case, one or more SLSs are used as the thick “layers” in the heterostructure. The thin individual layers within the SLS prevent misfit generation within the SLS itself. If the thick layers (SLSs and/or alloys) are lattice matched as a whole to each other, then dislocations are also not generated at the interfaces between them. Thus appropriately designed thick layered structures of this type can also be free of misfit dislocations. Of course, a number of device concepts are based on the use of heterostructures. The simplest superlattice heterojunction which can be formed as a superlattice/alloy interface as shown in Fig. 20a. Noting that electrical and optical properties of an SLS are tailorable functions of layer thickness and composition, superlattice/superlatticeheterojunctions may also be formed, as shown in Fig. 20b. For SLS/SLS heterojunctions care must be taken that the thickness or compositional changes do not alter the bulk lattice constant. An interesting method for forming a superlattice/alloy heterojunction involves impurity or implantation-inducedintermixing of a portion of the layers of a superlattice, as shown in Fig. 20c. Zinc diffiusion3’ and Si i m p l a n t a t i ~ nunder , ~ ~ appropriate energy, fluence, and annealing conditions, have been shown to disorder AlAs/GaAs superlattices. Implantation in SLS materials39*“has also been shown to offer the possibility of impurity introduction without disordering, as discussed in a previous section. As the location of the implantation or diffusion can be selectively determined by
8.
STRAINED-LAYER SUPERLATTICES
491
a)Superlattice/Alloy Heterojunction
Buffer
I
b)Superla ttice/Superlattice Heterojunct. by Thickness by Composition
c)Diffusion and/or Implantation Induced Disordering
Ez
Implantation also offers tbe possibility o f isolated impurity introduction m'thout disordering
FIG.20. (a) Schematic of a superlattice/alloy heterojunction. (b)Schematic of two superlattice/superlattice heterojunctions. (c) Schematic of a disorder-inducedalloy/superlattice heterojunction
masking and photolithography, a three-dimensional heterojunction can be formed. Because of the highly anisotropic nature of transport in superlattice materials, strong differences are expected in the injection and collection characteristics of parallel and perpendicular junctions. Many of the implications of these differences are yet to be determined. Another useful feature is the possibility of exploiting modulation-doping techniques in lattice-mismatched material systems with larger mobilities than those available in the matched GaAs/AlGaAs system. Field-effect transistors (FETs) fabricated from such materials should show improved characteristics. A prototype SLS FET is discussed below. Zone-folding enhancements of the optical absorption coefficient of SLSs grown from indirect-gap materials may make the use of these materials more attractive for certain optoelectronic or photovoltaic applications. An interesting material system which might benefit particularly from this effect is the SiGe SLS system.' The combination of Si processing technology and SiGe SLSs with tailorable absorption properties (if compatible), could lead to important extensions of the optoelectronic capabilities associated with Si-based technology. Much further work will be required to determine the feasibility of this approach.
492
G.C. OSBOURN
et al.
The intentional use of strain in the SLS materials to modify their properties has a number of potential applications. One example is the predicted extension of the InAsSb SLS 1, to 12 gm at 77 IS.These 111-V materials are expected to exhibit several advantages over the I1 - VI alloy HgCdTe for long-wavelength intrinsic detector array applications. The most important advantage is the use of 111-V materials with greater bond strengths than those of HgCdTe. This should make the SLS materials more stable and more amenable to device processing steps. Other potential advantages have been discussed elsewhere.35 Another example is the strain-induced preferential population of small m* valence bands in selecte d SLSs, which was discussed in a previous section. This effect might be used to improve the characteristicsof p-channel I11- V FETs (at reduced temperature) and make them more compatible with n-channel FETs in the same material system. Improved complementary logic circuits in 111- V materials might ultimately result from this. Finally, it should again be noted that certain SLSs can block the propagation of threading dislocation^.^^^^*'^ This metallurgical effect allows highquality epilayers to be grown on an SLS/graded layer/substrate structure in which the substrate has a different lattice constant from the epilayer. In this case, the first few periods of the SLS are used to inhibit the misfit dislocations in the graded layer from propagating into the epilayer during growth, so that the epilayer and the uppermost part of the SLS can be of device quality. This approach has the potential disadvantagethat the graded-layer defects may still propagate into the active epilayer during operation in certain high-power device structures (e.g., lasers). For SLS applications of this type, it is necessary to consider SLS structures which are closely matched to some available substrate. Figure 21a-c shows SLS structural types which are not bulk mismatched with respect to the substrate and which do not utilize a damaged buffer. The first of these structures, shown in Fig. 2 la, is as SLS in which the lattice constants of the constituent layers, weighted by the layer thicknesses, are centered about the lattice constant of the substrate. An example of this is an In,Ga,-,As/GaAs,P,-, SLS (Bedair et aL41;Ludowise et a/.,42]grown directly lattice matched to GaAs. A second bulk-matched SLS grown without a buffer is the unequal layer thickness structure shown in Fig. 2 1b. In an SLS of this type the layer with the lattice constant furthest away from that of the substrate is made thin with respect to the other layer. This forces most of the strain into the thinner layer and allows the bulk lattice constant of the SLS to more closely approach that of the substrate. An example of this structure is a GaAs/Ino.13Go,8,As(layer thicknesses 190 and 1 10 A, respectively) SLS grown on G ~ A sA. special ~ ~ case of the unequal layer thickness SLS is the single, strained quantum well or strained multi-quantum well structure
8.
STRAINED-LAYER SUPERLATTICES
493
a)Cen tered SLS Example: InGaAs/GaAsP on GaAs L
I
b )Un eq u a1 Layer Thickness SLS Example: GaAs/lnGaAs on GaAs
c)Strained Quan turn Well Example: InGaAs QW Confined by G a s
FIG.2 1. (a) Schematic of an SLS grown on a lattice-matched substrate. (b) Schematic of an unequal-layer-thicknessSLS grown on substrate which is the same material as the thick SLS layers. (c) Schematic of a strained quantum well sandwiched between thick layers which are matched to the substrate.
shown in Fig. 2 Ic. If a single strained layer or a small number of strained layers are sandwiched between thick layers matched to the substrate, all of the strain is confined to the thin layers, and the bulk lattice constant of the structure is hardly disturbed. In structures of this sort the critical layer thickness of the strained quantum well would be half that calculated for an equal layer thickness SLS because of the localization of the strain. An example of this is the strained, multi-quantum well, injection laser described by Laidig et al." 7. SLS DEVICE EXAMPLES
As a demonstration of the utility of SLS materials, three devices, including a phot0detector,4~an injection and an FETa have been constructed. The active layers of these devices were formed from MBE-grown, Ino.2Gao.8As/GaAs SLS (equal layer thicknesses, repeat distance = 240 A, interlayer, bulk lattice mismatch of approximately 1.4%) material with an effective bandgap of approximately 1.21 eV. For optical and electrical confinement, many of the devices utilize SLS/alloy heterojunctions formed between the Ino.,Gao.8As/GaAsSLS and Ino.lA~o.3Gao.6As, a bulk material with a bandgap of approximately 1.65 eV. All of these SLS structures were grown lattice matched on either Ino,,Gao.9As or Ino.,Alo.,Gao.6AS buffer layers which, in turn, were grown mismatched (by approximately 0.7%) on GaAs substrates. In this configuration, the metal-
494
G.C. OSBOURN et
al.
lurgical aspects of SLSs are critical to efficient device operation and a dominant consideration in discussions of stable, long-term life. A section view of a double-heterostructure (DH), Ino,Gao.8As/GaAs, SLS photodetector is shown in Fig. 22. The external quantum efficiency (uncorrected for surface reflection) as a function of wavelength, at zero volts reverse bias, of the structure operated as a photodetector is shown in Fig. 23. The device is observed to have a peak external quantum efficiency of approximately 50% at 770 nm and an optical absorption edge at 1050 nm. The oscillatory nature of the quantum efficiency curve is the result of a stepped absorption coefficient as a function of wavelength, which is characteristic of superlattice material^.^' The poor quantum efficiency close to the band edge is caused, in part, by a short perpendicular diffusion length (estimated to be approximately 0.1 pm for this SLS material,@which effectively limits the active absorption volume of the detector to the volume of the junction depletion region itself. This can be readily overcome by employing lower doping levels, which serves to extend the depletion layers, and by optimization of the metallurgical p-n junction depth within the SLS. A double-heterostructure, stripe-geometry, injection laser formed from a 12-period, Ino,2Gao.8As/GaAs SLS cladded above and below by the widebandgap quaternary, Alo.31no.lGao.6As, is shown in Fig. 24. The diodes had cleaved cavities approximately 300 pm long, and the lateral extent of the lasing medium was determined by spreading resistance in the anode material. The metallurgical p-n junction was contained within the SLS, implying that the superlattice was utilized as a “bulklike” material with tunable, anisotropic properties, and not simply as a quantum well region. This work
FIG.22. Section view of a double-heterostructweIn,,Ga,,As/GaAs
SLS photodetector.
8.
STRAINED-LAYER SUPERLATTICES
495
Wa velength(nm) FIG.23. Measured external quantum efficiency as a function of wavelength for the unbiased SLS photodetector in Fig. 22.
follows previous photopumped laser studies by Ludowise et in InGaAs/Gas, GaAsP/GaAs, and InGaAslGaAsP SLS material, injection laser studies by Laidig et al.50 in InGaAs/GaAs strained multi-quantum well material, and photopumped studies by Temkin and Tsang5' in GaSb/ AlGaSb strained multi-quantum well material. Emission spectra, both below and above threshold, for the SLS laser are shown in Fig. 25. Operated pulsed at room temperature, the devices had a
Cleaved Facet
FIG.24. Schematic of a double-heterostructure, stripgeometry,injection laser. The active region is a 12-period In,,Ga,,As/GaAs SLS, and the cladding layers are A1o,,Ino,,Gao.,As alloys.
496
G.C. OSBOURN et Room Temperature Pulse Width = loons Pulse Rate = 4.OkHz 1.026nm or
al.
c w Operation at 76K
975.5nm
,
6.0 x i '
miw=~. 7A 1.271eV Mode S p s c . = 4 . d
I = lOOmA 974
915
I t h = 95mA
0.4 r
:.
4.0
2.0
2
0.0 97€
1
40
FIG.25. Emission spectra for the SLS laser shown in Fig. 24.
peak emission wavelength near 1026 nm with a threshold current of 460 mA. Operated cw at 77 K, the devices had peak emission near 976 nm with a threshold current of 65 mA. This corresponds to a 77 K threshold current density in the range of 1000-3000 A/cmZ. Immersed directly in LNz, the diodes lased continuously for approximately l+ hours before the optical output power at a fixed drive current had dropped to one-half of its initial value. This prototype laser structure is a severe test of the quality of SLS material, in that the active lasing medium was grown directly on a damaged Alo.31no.,Gao~,As buffer layer. This forces the device to utilize SLS periods close to the SLS/buffer interface as optically active material and also places the lasing medium in close proximity to a large pool of damage which may propagate during device operation. Improved structures would separate the lasing and confining media from the damaged buffer, perhaps by use of an SLS placed in the structure for purely metallurgical purposes. A plan view, section view, and equivalent circuit of an InGaAsIGaAs, modulation-doped SLS FET are shown in Fig. 26. The structures utilized 17 periods of equal thickness, alternating GaAs and Ino,zGao,,Asto form the channel region of the device, followed by two periods of heavily Si-doped material to aid contacting of the source and drain metallizations.
8.
STRAINED-LAYER SUPERLATTICES
497
ej Section a t A-A' A1 Schottky Drain Gate
AuGe/Ni/Au Source 2 Period n+Contact 17 Period n Channel lp+ Ino,,G%,#s Buffer
p GaAs Substrate A uBe/A u
Equivalent Circuit Drain Scbottky
Source
FIG.26. Plan view, section view, and equivalent circuit of an In,,,Ga,,As/GaAs tiondoped SLS FET.
modula-
A 1 pm buffer layer of p+-In,,Gao.,As was grown between the SLS channel and the p+-GaAs substrate. The SLS channel material was modulation doped by shuttering the Si flux on during growth of the central 40 A of the wide-bandgap (GaAs) layers. The spike concentration is estimated to be approximately 1 X 1018/cm3.This procedure leaves spacer layers, approximately 55 A thick, of undoped GaAs between the doped regions and the conducting, narrower-bandgap (In,,Ga,,As) wells. A close inspection of Fig. 26 will show that the A1 gate metal contacts both the n-type SLS channel and the p-type In,,Ga,,As buffer. The A1- to -buffer contact forms a parasitic Schottky diode which effectively clamps the upper and lower gates together electrically for any appreciable negative bias on the upper gate. As described, the resultant transistor is structurally a double-gate hybrid device composed of both a metalsemiconductor FET and a junction FET.
G.C. OSBOURN et
498
al.
Representative InGaAs/GaAs FETs with gate lengths ranging from approximately 2.5 to 10 pm and Van der Pauw (VDP) patterns formed from ungated material, fabricated as described above, were electrically evaluated both at room temperature and at 77 K. Room-temperature and 77 K, dc common-source output characteristics of a transistor with a 2.5 pm gate length and with the upper and lower gates externally shorted are shown in Fig. 27, and drain current and transconductance versus gate-source voltage are shown in Fig. 28. At room temperature the device is observed to have a characteristic drain saturation current, IDS, of 64 mA ( V , = 4 V, V& = 0 V), a double-gate pinch-off voltage, V p , of 3.1 V ( VDs = 4 V, I D = 5% of IDS),and a maximum, normalized, double-gate extrinsic transconductance, g,,, of 84 mS/mm ( V , = 4 V). The drain current is also observed to vary approximately linearly with gate bias. Separate roomtemperature measurements yielded a value of 14 0 for both the total parasitic source resistance and the total drain resistance, and a value of 200 pA for the reverse leakage current of the paralleled Schottky gate and junction gate at -2 V. Separate measurements on the VDP patterns formed from ungated channel material yielded room-temperatureelectron Hall mobilities of 6700 cm2/V s for this sample. Excluding the effects of resistive parasitics, a normalized, room-temperature, double-gate, intrinsic transconductance, g,, may be calculated to be 120 mS/mm ( V , = 4 V) for this transistor. At 77 K, IDS was observed to increase to 98 mA, V p remained essentially constant at 3.2 V, and g, increased to 140 mS/mm. Separate measurements showed the parasitic source and drain resistance decreased to 7 and 6 0, respectively, the gate leakage current at - 2 V decreased to 40 PA,
b)Liquid Nitrogen
a)Room Temperature 1OO.r
\
100.1
I D s s ( v D s = 4 v , V G S =ov) v p ( v D s = 4 v , Ig=5%
Of
1
64mA
I D s s ) -3.1V
98mA
-3.2V
8.
STRAINED-LAYER SUPERLATTICES
V I ,=4V, ~ Unilluminated
Maximum Normalized g (VDS'4v)
RT
499
77K
84mS/mm I44mS/mm
FIG.28. (a) Room-temperature and (b) 77 K, drain current and transconductance versus gate-source voltage for the FET in Fig. 26.
and the measured Hall electron mobility in ungated material increased to 28,000 cm2/V s. The calculated g,, likewise increased to 190 mS/mm. These results demonstrate that electronic and optoelectronic devices can now be fabricated from SLS materials with tunable optical and transport properties. The results also make a strong statement about the high crystalline quality that can be obtained from SLS material with significant interlayer mismatch (- 1.4%) and many interfaces. V. Summary
The capability of growing high-quality superlattice structures from lattice-mismatched materials is now well established. A number of materials studies have indicated that the electronic properties of these new semiconductors can be tailored through the flexible choice of superlattice materials and structure. These materials are of both scientific and technological interest, and a number of prototype device structures have been fabricated utilizing them. The majority of the many potential SLS systems and many interesting electronic properties have yet to be explored. The strained-layer superlattice field should provide exciting materials and device research opportunities for years to come. NOTEADDEDIN PROOF.The field of SLS research has expanded very rapidly. There now exist hundreds of publications which report growth and characterization of SLSs in many materials systems including groups 111-V, IV, and 11-VI semiconductors. Much of this newer work is referenced here. Additional references can be found in brief reviews of 111-V SLS work which have appeared in the l i t e r a t ~ r e . ~The ~ - ~more ~ recent work includes studies of and structural analysis by x-ray diffraction," TEM,59.6' Auger p r ~ f i l i n g , ~ ~ . ~ ~ and ion channeling.64 Other studies have investigated structural stability against thermal cycling," effects of ion implantation With65 and WithoutM disordering, and dislocation
500
G.
c. OSBOURN et al.
hltering.67-69Studies of energy band structure have investigated effective masses and valenceband Energy levels and crystalline quality have been investigated by optical transitions observed in absorption and As well, electroabsorptionS2and nonlinear optical effects83have been reported. Waveguiding in InGaAs SLS has been ~ t u d i e d . * ~ . ~ - ~ ~ Transport properties of p S L S materials8788as well as n-SSQWa9and pSSQW90 have been reported for InGaAs SLSs. Anisotropic hole diffision,9’ recombination lifetimes:’ and Gunn ~ ~ deep ~~’ oscillations9*have also been reported for this material. Ion implantation d ~ p i n g and level^^.^^ have been investigated. In device technology areas, both n - t ~ p e %and , ~ ~p - t y ~ e ~ ~ channel InGaAs SSQW FETs have been fabricated. Long-lifetime SLS L E D s ~and ~ stimulated emissionIm in SSQWs have been reported. Avalanche photodectors have been fabricated and Finally, these 111-V SLS materials have been applied to such varied optical interference devices,106and buffer layers for filtering uses as photoelectrodes.lw~loS dislocations before the growth of devices.107-*w
REFERENCES 1. See,for example, A. G. Milnes and D. L. Feucht, “Heterojunctions and Metal-Semicon-
ductor Junctions.” Academic Press,New York, 1972.
2. F. C. Frank and J. H. van der Merwe, Proc. R. Soc. London, Ser A 198,2 16 (1949). 3. J. H. van der Merwe, CRC Crit. Rev.Solid State Mater. Sci. 7,209 (1978). 4. J. W. Matthews and A. E. Blakeslee, J. Cryst. Growth 27, I18 (1974); 29,273 (1975);32, 265 ( 1976). 5. G. C. Osbourn,J. Appl. Phys. 53, 1586 (1982). 6. I. J. Fritz, S. T. Picraux, L. R. Dawson, T. J. Drummond, W. D. Iaidig, and N. G. Anderson, Appl. Phys. Lett. 46,967 (1985). 7. J. C. Bean, L. C. Feldman, A. T. Fiory, S. Nakahara, and I. K. Robinson, J. Vac. Sci. Techno].A [2] 2,436 (1984). 8. P. D. Dapkus, Annu. Rev.Mater. Sci 12,243 (1982). 9. M. B. Panish, Science 208,916 (1980). 10. P. D. Dapkus, J. Cryst. Growth 68,345 (1984). 1 1 . R. M. Biefeld, G. C. Osbourn, P. L. Gourley, and I. J. Fritz, J. Electron. Mater. 12,903 (1983). 12. R. M. Biefeld, J. Cryst. Growth 56,382 (1982). 13. R. M. Biefeld and C. R. Hills, unpublished results. 14. A. Segmeller and A. E. Blakeslee,J. Appl. CYystaZlogr. 6, 19 (1973). 15. V. S. Speriosu, M.-A. Nicolet, S.T. Picraux, and R. M. Biefeld, Appl. Phys. Lett. 45,223 (1984). 16. D. R. Myers, R. M. Biefeld, I. J. Fritz, S. T. Picraux, and T. E. Zipperian, Appl. Phys. Lett. 44, 1052 (1984). 17. G. C. Osbourn,J. Vac. Sci. Technol. B [2] 1,379 (1983). 18. R. M. Biefeld, P. L. Gourley, I. J. Fritz, and G. C. Osbourn, Appl. Phys. Lett. 43, 759 (1983); P. L. Gourley and R. M. Biefeld, J. Vuc. Sci. Technol. B [2] 1,383 (1983). 19. G. L. Bir and G. E. Pikus, “Symmetry and Strain-Induced Effects in Semiconductors” (P. Shelnitz, trans., D. Lourish, ed.),Chapter 5. Keter Publ. House, Jerusalem, 1974 (distributed by Witey, New York). 20. J. E. Schirber, I. J. Fritz, and L. R. Dawson, A&. Phys. Lett 46, 187 (1985). 21. G. C. Osbourn,Superlattices Microstruct. 1,223 (1985). 22. P. L. Gourley and R. M. Biefeld, Appl. Phys. Lett 45, 749 (1984). 23. G. C. Osbourn,J. Vac. Sci. Technol. 21,649 (1982). 24. G. C. Osbourn, unpublished.
8.
STRAINED-LAYER SUPERLATTICES
501
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Index A Absorption nonlinear due to excitons, 29 1 - 293 spectra, semiconductors, 283 AlGaAs, variable-gap, electron velocity measurements, 365 -368 (Al,Ga)As/GaAs ring-oscillator Circuits, 196 (Al,Ga)As/(In,Ga)As MODFET current-voltagecharacteristics, 185 electron mobility, I84 electron velocity, 184 Avalanche photodiode graded-gap, 340-341 low-noise multilayer, 338 339 multiquantum-well, 34 1 - 344
-
B Band-edge discontinuities, doping interface dipoles, 388-392 Bandgap engineering, 3 19-320, see also specific devices chirp superlattice devices, 356-357 pseudoquaternary semiconductors, 357 360 repeated velocity overshot devices, 353-354 sawtooth superlattices, 354-356 staircase potentials, 352 Bloch-like envelope wave function, 25 Bloch oscillation, 93-96 Bohr radius, 18- 19,20 Buried-heterostructurelasers, 431,433
C Channeling diode, 33 1 -338 avalanche photodiode, 336-338 current-voltage characteristics, 333-335 space-charge regions, 332 - 334 ultra-low capacitance, 331 336
-
Charge-injectiontransistor, 328 - 329 Chemical-beam epitaxy, 444-446 Chemical vapor deposition, 444 CHINT, 328- 329 Chirp superlattice devices, 356-357 Complex wave vector, 292 Continuum states, 111-V semiconductor quantized structures, 29- 30 D
Degenerate four-wave mixing, multiple quantum well structures, 301 -302 Doping interface dipole, 388 - 392 Double-heterostructure, 397 - 398 Double-heterostructure laser Auger depth profile, 45 1 Fabry- Perot, threshold current density, 420,422-423,425 gain spectra, 7 1,73 growth, 45 1 lasing spectra at room temperature, 455 light output versus pulsed current amplitude, 45 I -452 performance characteristics, 454 Drude model. 108
E Elastic strain, accommodation, 459-46 1 Electron -electron interaction, 34 Electron velocity, measurements in variable-gap AlGaAs, 365 - 368 Electrostatic potential, 111-V semiconductor quantized structures, 33 - 34 Emitter grading, heterojunction bipolar transistor, 368- 373 Envelope wave functions, 308- 309 approximation, 9 Bloch-like, 25 Exciton binding energy, 52, 54
505
INDEX
dynamics, 66-69 effects, 52 - 62 luminescence, 63 nonlinear absorption and refraction due to, 291 -293 parameters describing, 284 in quantum wells, 18-2 1
F Fabry- Perot DH lasers, threshold current density, 420, 422-423, 425 Fang-Howard variational wave function, 35 FET calculating noise figures, 235 fabrication, 138- 140 heterojunction, see Heterojunction FET high speed, 135 threshold voltage, 195 Fukui fitting factor, TEGFET, 239
G GaAs/(Al,Ga)As MODFET, I83 - 185 GaAs/GaAMs interface, 8 quantum Hall effect, 1 12 GaAs MESFET, see also TEGFET electron mobility, 2 14 market demand, 203-205 G~,.,,I~~53As/InP quantum wells, 443 -455 deposition, 444-445 photoluminescence spectrum, 447-450 Graded-gap base transistor, 36 1 - 365 GRIN- SCH band-energy levels, 76 circuit, 432,434 cw threshold distribution, 429 -430 device characteristics,419 -434 energy-band diagram, 419-421 far-field intensity distributions, 428 -429 hybridcrystal growth, 430,433 light-current characteristics,43 1, 433 light output as function of input voltage, 433-434 optical confinement factor, 42 1 optical gain, 424 superlattice buffer layer, 425, 427
threshold current density, variation with cavity length, 426, 428
H Hartree potential, 43 HEMT, see also MODFET circuit diagram, 270 clock frequency, 250 comparison with other high-speed device approaches, 253-254 device modeling, 261 -264 energy-level schemes, 36 -37 future VLSI prospects, 274-276 integrated circuits, 264-274 logic circuits, 264-267 memory circuits, 268 - 274 material technology, 255-258 principles, 250-252 readJwrite operation waveforms, 27 1 -272 self-alignmentdevice fabrication technology, 258-261 technological advantages, 250-254 threshold voltages, 260-261 Heterojunction bipolar transistor, 36 1, see also specific transistors emitter grading, 368-373 variable-gap AIGaAs, 365 - 368 doping superlattice, 46 -47 modulation-doped, charge transfer, 3 1 - 33 tunable banier heights, 388-392 Heterojunction FET charge control, 144- 146 current -voltage characteristics, 146- 149 operation, 140- 141 two-dimensional electron-gas concentration, 141- 144 velocity versus electric field, 147- 148 Heterostructure, see also 111-V semiconductor quantized structures electrical properties alloy disorder scattering, 84 - 85 Bloch oscillation, 93 96 control of electron mobility by gate-controlled deformation, 89 90 Coulomb interaction matrix element, 82
-
-
507
INDEX
doping, 8 1- 82 electron drift velocity, 90, 92 electron heating, 9 1- 93 hot electron effectsin parallel transport, 90-93 interface-roughness-limited mobility, 85 - 86 intersubband scattering, 87-88 mobility in parallel transport, 78-90 perpendicular transport, 91 -98 phonon scattering mechanisms, 80-81 quantum transport, see Quantum transport resonant tunneling effect, 94-97 scattering mechanisms, 78 - 80 temperature dependence of electron mobility, 83,85 tunneling hot electron transistor, 97 -98 modulation doping, 30-43 charge transfer, 31 -33,41-42 electrostatic potential, 33 - 34 energy-level calculation, 34-37 gate-voltage dependence, 38 -39 sheet electron concentration, 39-40 thermal ionization energy of Si donor, 41 thermodynamic equilibrium, 37 -43 optical properties bandgap discontinuity, 53 exciton dynamics, 66-69 exciton eff'ects, 52-62 inelastic light scattering, 69-70 laser action, 7 1-78 llght propagating dong layers, 52 light propagating perpendicular to layers, 5 1-52 linewidth versus confining energy, 58-60 linewidth versus substrate temperature, 60-61 low-temperature luminescence, 62-65 optical matrix element, 47-49 selection rules, 49- 52 transmission curves for passive waveguides, 62 types, 3 Hilsum-Ridley- Watkins mechanism, 32 1- 322
I 111- V, semiconductor quantized structures, see also Heterostructures absorption coefficients,22 communicating multiplequantum-well structure, 24-29 conduction electron energy levels, 9 - 13 continuum states, 29 - 30 double-well structure, 24-25 excitons and shallow impurities in quantum wells, 18 - 23 hole dispersion curve, 14 hole energy levels, 13- 17 n- i - p - i structures, 43 - 46 quantum wells energy level, 9- 17 SchrGdinger-like equation, 10- 11 semiconductor purity and interfaces, 5 -9 transverse dispersion curves, 16 tunneling structures, 23 -24 two-dimensional density of states, 17- 18 Impurity gettering, 63 Inelastic light scattering, by electronic excitations, 69-70 (In,Ga)As/(Al,Ga)As MODFET drain voltage effect, 19 1 equivalent-circuit parameters, 189, 192 gate voltage effect, 188 190 pseudo-morphic, 181- 183 supply voltage effect, 187- 188
-
In,,lG~,4,As-InP quantum wells, 434-438
J Joint density of states, 283
L Laser, SLS, 494 -496 Lead-chalcogenide lasers, 439 -440 Lead-salt diode lasers, 439 Liquid-phase epitaxy, quantum confinement heterostructure lasers, 450 Logic circuits, HEMT, 264-267 Low-temperature luminescence, 62- 65 Luttinger Hamiltonian, 13
508
INDEX
M
schematic cross section, 168- 170 self-aligned modulationdoped, 175- 176 sheet-electron areal density and electron Memory circuits, HEMT, 268 -274 mobility, 176- 177 Memory devices, real-space transfer strucsmall-signal circuit-element values, tures, 329 - 330 162- 163 MESFET, see also GaAs MESFET small-signal transconductance, 162,164 channel region, 136 S parameters, 160- 161 current-gain cutoff frequency, 163, 165 supply voltage effect, 158 - 159 electron mobility, 249 threshold voltage shifts, 194 gate capacitance, 162, 164 transconductance, 150- 155, 172 small-signal transconductance, 162, 164 yield, 195 transconductance, 242 Modulation doping, 136- 138 MOCVD, 5-8,461-462 Molecular beam epitaxy, 5-6,444,447,450 Mode-locking, multiple quantum well strained-layer superlattices, 462-463 structures, 302 - 303 MODFET surface defect problem, 255 -256 activation energy, 194 Multiple quantum well advanced technology requirements, absorption spectra, 66-67 175-180 applications, 30 1 - 304,3 11 - 3 14 anomolies in current-voltage band-energy levels, 74,76 characteristics, 168- 175 band structure, 28 1 current-gain cutoff frequency, 162- 163, bound-state energies, 285 degenerate four-wave mixing, 301 - 302 165, 167, 169 electric field degradation processes, 179 180 drain current voltage characteristics, parallel to layers, 305 - 306 150-152, 173-174 perpendicular to layers, 306-3 10 drive threshold voltage effect, 157 158 envelope wave function, 308 309 energy-level schemes, 36- 37 excitation spectrum, 57 - 58 equivalent circuits, 160- 170 low-temperature experimental results, feedback capacitance, 165- 166 287-288 gate-bias stress measurements, 185, 187 mode-locking, 302 - 303 gate capacitance, 162, 164 modified groupV/III ratio, 175 band-energy levels, 76 low-noise performance, 167, 170 device characteristics,4 13-4 19 low-power operation, 156 nonlinear absorption and refraction due low-temperature photoluminescence to excitons, 29 1 -293 spectra, 177- 179 observations of nonlinear excitonic maximum drain current, 152- 154 effects, 293 - 300 microwave performance, 160- 170 optical bistability, 304 noise margins, 159 optical modulators, 3 1 1 -3 12 optimization, 149- 155 optical properties induced by static field, output power and power-added efficiency, 304-305 193 origin of excitonic saturation at room output resistance, 165- 169 temperature, 300- 301 pumpprobe experiments, 296 -297 performance in logic Circuits, 155 160 photoconductivity, 171- 172, 193 quasi-2D exciton binding energy theory, 286-287 power-gain cutoff frequencies, 162- 163 problems and projections, 191 , 193- 196 readsorption effect on edge luminescence, 77 pseudo-morphic, 180- 191 reliability, 195 resonant Rayleigh scattering, 68
-
-
-
-
-
509
INDEX
saturation of excitonic absorption, 293-296 self-electroopticeffectdevices, 3 12- 3 14 shifts of light- and heavy-hole excitons, 309-310 single-particle light scattering spectrum, 70 temperature dependence and roomtemperature exciton resonances, 288-291 theory of linear absorption and band structure, 282-286 Multiquantum well laser growth, 450-45 I lasing spectra at room temperature, 455 light output versus pulsed current amplitude, 453-454 performance characteristics,454
N Negative differential resistance fieldeffect transistor, 324- 328 current-voltage characteristics, 327-328 n-i-p-i structures, 43-46 Noise figure, 231 -243 TEGFET, 229-230 Nonstationary and microscopic model, 221, 223-226
0 Optical matrix element, 47-49 Optical modulators, multiple quantum well structures, 3 1 1 - 3 12 Organometallicchemical vapor deposition, 444-445,447
P Parallel transport hot electron effects, 90-93 mobility, see Heterostructures, electrical properties PbTe quantum well, 439-441
Perpendicular transport, 9 1,92-98 Perturbative Hamiltonian, 15,20, 22 Phonon scattering mechanisms, 80- 8 I rates via Frohlich mechanism, 91 Photocapacitive detector, ultra-low capacitance, 33 1-336 Photodiode, see also Avalanche photodiode channeling, avalanche, 336-338 current-voltage characteristics, 335- 336 ultra-low capacitance, 331 -336 Photomultiplier, staircase-solid-state, 344- 349 Pump-and-probe experiments, 66 -67
Q Quantum bubble laser, gain spectra, 402 -405 Quantum confinement heterostructure lasers, 397 402 electron bound energy states, 399 electron density distributions, 404 gain spectra, 402-405 hole bound energy states, 400 In0.53Ga.47As-I~.,~,4,As, 438 -439 I I I + ~ ~ G ~ .InP, ~~A 434-438 Slong-wavelength, 434- 439 parabolic compositional profiles, 400-40 1 photoluminescence spectrum, 446 -448 quality, 446 quantized energy levels, 399 quantum noise and dynamics, 407-409 short-wavelength, 409 -434 MMQW device characteristics, 413-419 MQW device characteristics, 41 1-413 temperature dependence of threshold current, 406-407,419 very-long-wavelength,4 19-442 Quantum Hall effect, 107- I15 Drude model, 108 finite width, 109- 110 fractional, 1 13- 1 15 GaAs/GaAlAs, 1 12 Quantum transport, 98- 1 15 allowed states and density of states, 99-100 Fermi level, 102
-
510
INDEX
Landau levels, 99 - 101 low-field oscillatory conductivity, 105 magnetic field effect on 2D electrons, 99- 104 oscillatory phenomena, 102- 104 quantum Hall effect, 107- 115 Shubnikov-de Haas measurements, 104- 106 Quantum well absorption spectra, 54, 56 base, resonant tunneling transistors, 372, 374-381 electroreflectance,47 -48 energy level, 9- 17 fit of observed transitions, 53, 55 lasers, 71 -78 luminescence, 64-65 parabolic, excitation spectrum, 53 - 54 photoluminescence and circular polarization spectrum, 5 1 Quasi-2D exciton binding energy theory, 286 - 287
R Real-space transfer structures, 320-33 1 analytical model, 323-324 charge-injection transistor, 328 - 329 definition, 320 fast switching and storage of electrons, 322 memory devices, 329 - 330 modulation doping 320- 321 negative differential resistance field-effect transistor, 324- 328 time constants, 322-323 tunneling devices, 330-33 1 Reflection high-energy electron diffraction, 5 Refraction, nonlinear due to excitons, 291-293 Repeated velocity overshoot devices, 353-354 Resonant Rayleigh scattering, multiple quantum well, 68 Resonant tunneling transistor band diagram, 374-377 circuit diagram, 376 - 378 current-voltage characteristics, 379- 380 with quantum-well base, 372,374-381 Ring oscillator, HEMT, 264-267
5
Sawtooth superlattices, electrical polarization effects, 354-356 Schrodinger equation, 10- 11, 3 1 SDHT, 36 - 37, see also MODFET Self-aligned gate fabrication, HEMT, 258 -26 1 Self-electrooptic-effect devices, multiple quantum well structures, 3 12- 3 14 Shallow impurities, quantum wells, 20-23 Shubnikov-de Haas measurements, 104- 106 S parameters MODFET, 160- 161 TEGFET, 230-231,233-234 Stark broadening, 305 Strained-layer superlattice alloy heterojunction, 490-49 1 applications, 490-499 crystal growth, 461 -463 double-heterostructure photodetector, 494-495 double-heterostructure, strip-geometry, injection laser, 494-496 elastic strain accommodation, 459-46 1 electronic properties, 467 -490 absorption spectrum, 475 -477 diffusion lengths, 481 -482 energy levels, 469-47 1 excitation spectrum, 480 general features, 467-469 mobility, 485-486 modulation doping, 483 -485 optical transistors, 472,479-480 photocurrent spectroscopy, 473-474, 478,483-484 photoluminescence spectrum, 473, 478-479 small-hole-mass effect, 486-487 transport data, 480-481 transport properties, 487-488 type-I GaAsIGaAsP, 477 -479 type-I1 GaASP, 469-477 FET, 491 -492 modulation-doped, 496 -499 graded-layer/substratestructure, 492 - 493 structural characterization, 463 -467 Superlattice allowed energy bands, 28-29
51I
INDEX
density of state (DOS), 28 Sawtooth, 354-356 tight-binding model, 25- 27
-
T TEGFET, see also MODFET analytical model, 2I6 221 characteristics, 218-222 electron mobility, 210-21 1, 214-215 electron velocity, 2I 1 -213 energy-level schemes, 36-37 equivalent circuit, 234 Fukui fitting kctor, 239 low-noise, 244 noise figure, 231 243 assoCiated gain, 229-230 frequency dependence, 240-24I semiempiricalapproach, 236-240 theoretical approach, 232-236 pinch-off voltage, 221-222 recessed gate, 208 screening effect, 213-216
-
-
sheet density in ZDEG,216-218 S parameters, 230-23I, 233- 234 structures, 206-209 subbands and charge control determination, 226-229 transport properties, 210 216 Thermodynamic equilibrium, I11-V semiconductor quantized structures,
31-43 Threshold current seealso density, 405-408,415-416,451, GRIN-SCH temperature dependence, 406-407,419,
440-442 Transconductance, MODFET, 150- 155 Transferred electron effect, 321- 322 Tunneling, real-space transfer structures,
330-331 U Ultrathin, well-controlled semiconductor heterostructures, 1 5
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