2

Intersubband Transitions in Quantum Wells Physics and Device Applications I1

SEMICONDUCTORS AND SEMIMETALS Volume 66

Semiconductors and Semimetals A Treatise

Edited by R. K. Willardson

Eicke R. Weber

CONSULTING PHYSICISTDEPARTMENT OF MATERIALS SCIENCE SPOKANE, WASHINGTONAND MINERALENG~VEERING UNIVERSITY OF CALIFORNIA AT BERKELEY

Intersu bband Transitions in Quanturn Wells Physics and Device Applications I1

SEMICONDUCTORS AND SEMIMETALS Volume 66 Volume Editors

H. C. LIU INSTITUTE FOR MICROSTRUCTURAL SCIENCES NATIONAL RESEARCH COUNCIL OTTOWA, ONTARIO, CANADA

FEDERICO CAPASSO BELL LABORATORIES, LUCENT TECHNOLOGIES MURRAY HILL, NEW JERSEY

A CADEMIC PRESS San Diego San Francisco London Sydney Tokyo

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Contents PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LISTOF CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Chapter 1 Quantum Cascade Lasers . . . . . . . . . . . . . .

1

xi

Jerome Faist. Federico Capasso. Carlo Sirtori. Deborah L . Sivco and Alfred Y. Cho THEORETICAL FRAMEWORK . . . . . . . . . . . . . . . . . . . . . . . ENERGY BANDDIAGRAM. . . . . . . . . . . . . . . . . . . . . . . . MATERIAL ASPECTS. . . . . . . . . . . . . . . . . . . . . . . . . . . OPTICAL CONSTANTS . . . . . . . . . . . . . . . . . . . . . . . . . . STABILITY REQUIREMENTS:WHYTHEINJECTORMUSTBEDOPED . . . . . . . QC LASER WITH DIAGONAL TRANSITION AT i= 4.3 pm . . . . . . . . . . . 1. Characterization of the Active Region: Photocurrent and Absorption Spectra 2. Infiuence of the Doping Projilr . . . . . . . . . . . . . . . . . . . . . 3 . Anticrossing qfthe States in the Active Region . . . . . . . . . . . . . 4 . Band Structure at Threshold . . . . . . . . . . . . . . . . . . . . . . 5. Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Device Processing . . . . . . . . . . . . . . . . . . . . . . . . . . I . Laser Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . VIII. QC LASERS WITH VERTICALTRANSITION AND BRAGG CONFINEMENT . . . . . 1. Quantum Design . . . . . . . . . . . . . . . . . . . . . . . . . . 2. RateEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. OPTIMIZATION OF THE VERTICAL TRANSITION LASERAND CONTINUOUS WAVE OPERATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Spectral Characteristics . . . . . . . . . . . . . . . . . . . . . . . x. VERTICAL TRANSITION QC LASER WITH FUNNEL INJECTOR A N D ROOM-TEMPERATURE OPERATION . . . . . . . . . . . . . . . . . . . . . 1. Pulsed Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . Threshold Current Density . . . . . . . . . . . . . . . . . . . . . . 3 . High-Temperature, High-Power Continuous Wave Operation . . . . . . . 4. Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . XI . LONG-WAVELENGTH ( A 2 8 pm) QUANTUM CASCADE LASERS. . . . . . . . 1. Plasmon-Enhanced Waveguide . . . . . . . . . . . . . . . . . . . . . I1. 111. IV . V. VI . VII.

V

. . .

.

.

6 10 11 12 13 14 14 16 18 20 24 21 21 29 30 35 40 41

.

.

44 46 48 50 51

52 53

vi

CONTENTS 2. Quantum Design of a QC Laser with Diagonal Transition at I = 8.4 pm . . 3. Long- Wavelength Quantum Cascade Laser Based on a Vertical Trunsition . 4. Room-Temperature Long- Wavelength (2 = 11 pm) QC Laser . . . . . . . 5 . Semiconductor Lasers Based on Surface Plasmon Waveguides . . . . . . . 6. Distributed Feedback Quantum Cuscade Lasers . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

Chapter 2 Nonlinear Optics in Coupled-Quantum-Well Quasi-Molecules . . . . . . . . . . . . . . . . . . . . . . .

56 56 61

65 68 81

85

Federico Cupusso. Curlo Sirtori. D . L. Sivco. and A . Y. Cho I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. NONLINEAR OPTICALSUSCEPTIBILITIES IN THE DENSITY MATRIXFORMALISM . . I11. NONLINEAR OPTICALPROPERTIES OF COUPLED QUANTUM WELLS . . . . . . . IV . INTERSUBBAND ABSORPTION AND THE STARK EFFECTI N COUPLEDQUANTUM WELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GENERATION IN COUPLED QUANTUM WELLSAND V. SECOND-HARMONIC RESONANT STARK TUNINGOF x‘2’(20) . . . . . . . . . . . . . . . . . . . VI . FAR-INFRARED GENERATION BY RESONANT FREQUENCY MIXING . . . . . . . . VII. THIRD-HARMONIC GENERATION AND TRIPLYRESONANTNONLINEAR IN COUPLED QUANTUM WELLS . . . . . . . . . . . . . . . SUSCEPTIBILITY VIII . MULTIPHOTON ELECTRON EMISSION FROM QUANTUM WELLS . . . . . . . . . Ix. RESONANT THIRD-HARMONIC GENERATION VIA A CONTINUUM RESONANCE. . . X . CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 3 Photon-Assisted Tunneling in Semiconductor Quantum Structures . . . . . . . . . . . . . . . . . . . . .

85 87 90 93 102 109 112

116 121 122 123

127

Karl Unterrainer I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . TI. THEORYON PHOTON-ASSISTED TUNNELING . . . . . . . . . . . . . . . . . ].General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Perturbative Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Nonperturbative Limit ( Tien-Gordon Model) . . . . . . . . . . . . . . TRANSPORT IN EXTERNAL AC FIELDS . . . . . . . . . . . . . . I11. COHERENT 1. Transport in Minibands . . . . . . . . . . . . . . . . . . . . . . . 2 . The Wannier-Stark Ladder . . . . . . . . . . . . . . . . . . . . . . 3. Classical Description of Minibund Transport in External A C Fields . . . . . 4. Quantum Mechanical Description ojsuperluttices in External A C Fields . . . 5 . Analogy to the A C Josephson Effect . . . . . . . . . . . . . . . . . . METHODS. . . . . . . . . . . . . . . . . . . . . . . . IV . EXPERIMENTAL V . EXPERIMENTS ON PHOTON-ASSISTED TRANSPORT IN A RESONANT TUNNELING DIODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TUNNELING IN WEAKLY COUPLED SUPERLATTICES . . . . . . VI . PHOTON-ASSISTED 1. Photon-Assisted Tunneling between Ground and Excited States . . . . . . . 2. Photon-Assisted Tunneling between Ground States (Dynamic Localization. Absolute Negative Conductance. Stimulated Multiphoton . Emission) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . TEMRTZ TRANSPORT IN SUPERLATTICE MINIBANDS . . . . . . . . . . . .

127 129 129 130 130 134 134 138 140 144 146 147 149 155 155

163 172

CoNTENTs VIII . PHOTON-ASSISTED TUNNELING A N D TBRAHERTZ AMPLIFICATION . . . . . . . . Ix. SUMMARY AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii 182 183 184

Chapter 4 Optically Excited Bloch Oscillations-Fundamentals and Application Perspectives . . . . . . . . . . . . . . . . . . 187 P . Huring Bolivar. T. Dekorsy. and H . Kurz I. INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. HISTORICAL BACKGROUND - BLOCHOSCILLATIONS IN THE SEMICLASSICAL MODEL . . . . . . . . . . . . . . . . . . . . . . . . . 111. WANNIER-STARK DESCRIPTION OF BLOCHOSCILLATIONS . . . . . . . . . . . I v . TIME-RESOLVED INVESTIGATION OF BLOCHOSCILLATIONS . . . . . . . . . . . v. BLOCHOSCILLATIONS AS MODELSYSTEM FOR COHERENT CARRIER DYNAMICS M SEMICONDUCTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . APPLICATIONOF BLOCHOSCILLATIONS AS A COHERENT SOURCE OF TUNABLE TERAHERTZ RADIATION. . . . . . . . . . . . . . . . . . . . . . . . . VII. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS OF VOLUMES IN THISSERIES. . . . . . . . . . . . . . . . . .

187 188 192 197 203

210 214 215 219 225

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Preface Research on intersubband transitions in quantum wells has led to several practical devices, such as the QWIP (quantum well infrared photodetector) and the QCL (quantum cascade laser). These are two of the success stories in using quantum wells for practical device applications. Research activities in this area have been very intense over the past ten years, resulting in many new devices that are presently being developed for the market. We therefore feel that the time is right to collect a comprehensive review of the various topics related to intersubband transitions in quantum wells. We hope that this volume will provide a good reference for researchers in this and related fields and for those individuals- graduate students, scientists, and engineers- who are interested in learning about this subject. The eight chapters in Volumes 62 and 66 of the Academic Press Semiconductors and Semimetals serial cover the following topics: Chapters 1 and 2 in Volume 62 discuss the basic physics and related phenomena of intersubband transitions. Chapters 3 and 4 in Volume 62 present the physics and applications of QWIP. Chapter 1 in Volume 66 reviews the development of QCL. Chapter 2 in Volume 66 studies nonlinear optical processes. Chapters 3 and 4 in Volume 66 introduce two related topics: photonassisted tunneling and optically excited Bloch oscillation. We thank all the contributors who have devoted their valuable time and energy in putting together a timely volume. We also thank Dr. Zvi Ruder of Academic Press for providing assistance and keeping us on schedule.

H. C. LIU FEDERICO CAPASSO

ix

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List of Contributors

Numbers in parentheses indicate the pages on which the authors’ contribution begins.

P. HARINGBOLIVAR(187), lnstitut fur Halbleitertechnik 11, RMTH Aachen, Germany FEDERICO CAPASSO (1, 85), Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey

ALFREDY. CHO(1, 8 9 , Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey T. DEKORSKY (187), lnstitut .fur Halbleitertechnik 11, RWTlrl Aachen, Germany JEROME FAIST(l), Institute of Physics, University of NeuchZltel, Neuchitef, Switzerland H. KURZ(187), Institut ,fur Halhleitertechnik 11, R WTH Aachen, Germany CARLOSIRTORI (1, 85), Thornson-CSF, Laboratoire Centrul de Recherches, Orsay, France DEBORAH L. SIVCO(1, 85), Bell Laboratories, Lucent Technologies, Murray Hill,New Jersey KARL UNTERRAINER (127), lnstitut fur Festkorperelektronik, Technische Universitat W e n , Vienna, Austria

xi

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SEMICONDUCTORS AND SEMIMETALS. VOL. 66

CHAPTER 1

Quantum Cascade Lasers Jerome Faist INSTITUTE OF PHYSICS UNIVERSITY OF N e u c H h L NEUCHATEL, SWITZERLAND

Federico Capasso BELLLABORATORIES, LUCENT TECHNOLOOES MURRAYHILL.NEWJERSEY

Carlo Sirtori THOMSON-CSF LABORATOIRE CENTRAL DE RECHERCHES ORSAY, FUNCE

Deborah L. Siuco and AEfred Y. Cho BELLLABORATORIES, LUCENT TECHNOLOGIES MURRAY HILL, NEWJERSEY

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. THEORETICAL FRAMEWORK . . . . . . . . . . . . . . . . . . . . . . . 111. ENERGY BANDDIAGRAM . . . . . . . . . . . . . . . . . . . . . . . . IV. MATERIAL ASPECTS . . . . . . . . . . . . . . . . . . . . . . . . . . V. OPTICAL CONSTANTS . . . . . . . . . . . . . . . . . . . . . . . . . VI. STABILITY REQUIREMENTS: WHYTHE INJECTOR MUSTBE DOPED . . . . . . . VII. QC LASERWITH DLAGONAL TRANSITION AT I = 4.3 pm . . . . . . . . . . . 1. Characterization of the Active Region: Photocurrent and Absorption Spectra . 2. Influence of the Doping Profile . . . . . . . . . . . . . . . . . . . . 3. Anticrossing of the States in the Active Region . . . . . . . , . . . . . 4. Band Structure at Threshold . . . . . . . . . . . . . . . . . . . . . 5. Waveguide . , . . . . . , . . . . . . . . . . . . . . . , . . . . 6. Device Processing . . . . . . . . . . . . . . . . . . . . . . . . . 7 . Laser Characteristics . . . . . . . . . . . . . . . . . . . . . . . . VIII. QC LASERSWITH VERTICALTRANSITION AND BRAGG CONFINEMENT , . . . . .

2 6 10 11

12 13 14 14 16 18 20 24 21 21 29

1 Copyrigbt C 2000 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-752175-5 ISSN 0080-8784100$30 00

2

JEROMEFAISTET AL. 1. Quantum Design . . . . . . . . . . . . . . . . . . . . . . . . . . 2. RateEquations . . . . . . . . . . . . . . . . . . . . . . . . . . IX. OPTIMIZATION OF THE VERTICAL TRANSITION LASERAM) CONTINUOUS WAVE OPERATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Spectral Characteristics . . . . . . . . . . . . . . . . . . . . . . . x. VERTICAL TRANSITION QC LASERWITH FUNNEL INJECTOR AND ROOM-TEMPERATURE OPERATION. . . . . . . . . . . . . . . . . . . . 1. Pulsed Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Threshold Current Density . . . . . . . . . . . . . . . . . . . . . . 3. High-Temperature, High-Power Continuous Wave Operation . . . . . . 4. Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . XI. LONG-WAVELENGTH ( A 3 8 pm) QUANTUM CASCADE LASERS . . . . . . . 1. Plasmon-Enhanced Waveguide . . . . . . . . . . . . . . . . . . . . 2. Quantum Design of a QC Laser with Diagonal Transition at /1 = 8.4 pm . 3. Long- Wavelength Quantum Cascade Laser Based on a Vertical Transition . 4. Room-Temperature Long- Wavelength ( A = 11pm)QC Laser . . . . . . 5. Semiconductor Lasers Based on Surface Plasmon Waveguides . . . . . . 6. Distributed Feedback Quantum Cascade Lasers . . . . . . . . . . . .

R E F E R E N C E S .. . . . . . . . . . . . . . . . . . . . . . . . . . .

30 35 40 41

44 46 48

. .

. . . . .

50 51 52 53 56 56 61 65 68 81

I. Introduction

Most solid-state and gas lasers rely on narrow optical transitions connecting discrete energy levels between which population inversion is achieved by optical or electrical pumping (Yariv, 1988). In contrast, semiconductor diode lasers, including quantum well lasers, rely on transitions between energy bands in which conduction electrons and valence band holes, injected into the active layer through a forward-biased p-n junction, radiatively recombine across the bandgap. The bandgap essentially determines the emission wavelength. In addition, because the electron and hole populations are broadly distributed in the conduction and valence bands according to Fermi's statistics, the resulting gain spectrum is quite broad and is of the order of the thermal energy kT. The unipolar intersubband laser or quantum cascade laser that we discuss here differs in many fundamental ways from diode lasers. It relies on only one type of carrier (in our case electrons), making electronic transitions between conduction band states (subbands) arizing from size quantization in a semiconductor heterostructure. These transitions are denoted as intersubband transitions. As shown in Fig. 1, their initial and final states have the same curvature and, therefore, if one neglects nonparabolicity, the joint density of state is very sharp and typical of atomic transitions. In contrast to interband transitions, the gain linewidth now depends only indirectly on temperature through collision processes. The gain in an interband transition

3

1 QUANTUM CASCADE LASERS

l

4

.

4

gIa n ,i,

Ene~y

FIG. 1. Comparison between an intersubband transition (a) and an interband transition in a quantum well (b).

is limited by the joint density of states and saturates when the electron and hole quasi-Fermi levels are well within the conduction and valence band, respectively. In contrast, the intersubband gain has no such limitation, the gain being limited only by the amount of current one is able to drive in the structure to sustain the population in the upper state. Another fundamental aspect of our intersubband laser is the multistage cascaded geometry of the structure, where electrons are recycled from period to period, contributing each time to the gain and photon emission. Thus each electron injected above threshold generates N , laser photons, where N , is the number of stages, leading to a differential efficiency and therefore an optical power proportional to N,. The cascaded geometry has some significant advantages over the usual arrangement where the individual gain regions (quantum wells) are electrically pumped in parallel as in conventional diode lasers. In the latter case, a uniform gain across the active region is limited by the ratio of the effective transit time between wells, including capture, of the slower carrires (usually holes) and the recombination time; that is, the reciprocal of the total recombination rate due to stimulated, spontaneous, and nonradiative emission in the gain region. For this reason, the number of quantum wells in a multiquantum well laser is usually limited

4

JEROMEFAISTET AL.

to 5 or 10. In the cascaded scheme, the number of stages N , is only limited by the ratio between the effective width of the optical mode and the length of an individual stage. Depending on the wavelength, our structures have in general a number of stages N , between 16 and 35. Since the gain is proportional to N , , a large value of N , enables us to decrease the inversion density in each stage, strongly reducing electron-electron scattering processes and therefore the broadening of the distribution associated with hot carrier effects. There is also a strong effort to develop interband midinfrared quantum cascade lasers based on type I1 heterostructures, which resulted in the demonstration of the first device operating up to 200 K at 3.8 pm wavelength (Yang et a/., 1997). The idea of a unipolar laser based on transitions between states belonging to the same band (conduction or valence) is quite old, and can be traced to the original proposal of B. Lax (1960) of a laser based on an inversion between magnetic Landau levels in a solid (cyclotron laser). The latter was first demonstrated experimentally in the far infared using lightly p-doped germanium. Interested readers can consult a comprehensive review by Andronov (1987). The seminal work of Kazarinov and Suris (1971) represents the first proposal to use intersubband transitions in quantum wells, electrically pumped by tunneling, for light amplification. In their scheme, electrons tunnel from the ground state of a quantum well to the excited state of the neighboring well, emitting a photon in the process (photon-assisted tunneling). Following nonradiative relaxation to the ground-state electrons are injected into the next stage and so forth sequentially for many stages. Population inversion in this structure is made possible by the relatively long ( 2 10 ps) nonradiative relaxation time associated with the “diagonal” transition between adjacent wells, compared with the short intrawell relaxation. During the 1970s, intense experimental work proceeded on the twodimensional electron gas formed at the Si-SiO, interface in silicon metal-oxide semiconductor field effect transistors (MOSFET), focusing mainly on the electrical properties (Ando et al., 1982).The first intersubband emission spectra in an electron gas, heated by a parallel current, was reported in the Si-SiO, system (Gornik and Tsui, 1976). The advent of molecular beam epitaxy (MBE) (Cho, 1994) in the late 1960s for the first time enabled the fabrication of heterostructures based on 111-V compounds (and now extended to almost all semiconductors) with very sharp interfaces (the size of the interface fluctuations is about a monolayer) and excellent compositional control, and created a great interest in the study of multiquantum well structures. This led to the first observation of intersubband absorption in a GaAs-AlGaAs multiquantum well (West and Eglash, 1985). Early attempts were made to implement experimentally the proposal of Kazarinov and Suris (1972) in a GaAs-AlGaAs superlattice. They led to the observation of intersubband luminescence pumped by resonant tunneling by M. Helm and coworkers (Helm et al., 1988).

1 QUANTUM CASCADE LASERS

5

Kazarinov’s paper spurred an intense theoretical activity in the 1980s and early 1990s, with a large number of proposals for intersubband lasers. In one variant (Capasso et al., 1986), electrons are injected by resonant tunneling into the second excited state of a quantum well, followed by a laser transition to the lower excited state and nonradiative relaxation (via phonon emission) from the latter to the ground state. Electrons are then reinjected into the next stage and so on, sequentially through many quantum wells. The use of a stack of double quantum wells in a sequential resonant tunneling structure was proposed by Liu (1988). Later, many proposals considered structures with resonant tunneling injection from the ground state of a quantum well into the first excited state of an adjacent quantum well (Kastalsky et al., 1988; Borenstein and Katz, 1989; Loehr et al., 1991; Choe et al., 1991, Belenov et al., 1988; Hu and Feng, 1991; Mii et al., 1990; Henderson et al., 1993) and laser action between the latter and the ground state, repeated sequentially through many wells. For population inversion, this scheme requires that the tunneling escape time from the ground state (i.e., the reciprocal of the resonant tunneling rate) be smaller than the intersubband scattering time between the states of the laser transition. This is difficult to implement since it necessitates very short tunneling escape times to compete against the optical phonon scattering time of about a picosecond. Optically pumped structures were also proposed (Sun and Khurgin, 1991; Berger, 1994). To establish population inversion, other proposals relied on the impossibility of emission of optical phonons for subbands spaced by less than an optical phonon. This requires operation of the device in the far infrared, which is a difficult task because waveguides in this wavelength range are very lossy and difficult to manufacture. Apart from an interest stemming from basic physics, these attempts to develop lasers based on intersubband transitions in the mid-infrared and far infrared were motivated by the lack of convenient semiconductor optical sources in this wavelength region. The mid-infrared is a very important spectral region for spectroscopy and gas-sensing since most molecules have their fundamental vibration modes in the 3- to 12-pm wavelength region. The existing technology, based on interband lead-salt lasers (Tacke, 1995), has the major drawback of necessitating cryocoolers, since, in practical systems, these devices operate at temperatures between 50 and 130K. Alternative diode technologies, so far limited to the 3- to 5-pm wavelength band, now include antimony-based quantum well lasers with type I (Choi et al., 1996; Lane et al., 1997) or type I1 (Chow et al., 1995; Meyer et al., 1995) transitions, and, as mentioned, quantum cascade lasers based on interband transitions (Yang et ul., 1997). So far, however, only quantum cascade (QC) lasers based on intersubband transitions have demonstrated good room temperature performance virtually across the whole mid-infared region.

6

JEROMEFAISTET AL.

QC lasers have been made possible by the convergence of two techniques: molecular beam epitaxy and band-structure engineering (Capasso, 1987; Capasso and Cho, 1994). With the latter technique, energy diagrams with nearly arbitrary shapes can be designed using building blocks such as compositionally graded alloys, quantum wells, and superlattices. In this way, entirely new materials and semiconductor devices can be designed and their properties tailored for specific applications. MBE, with its ability to grow atomically abrupt heterojunctions and precisely tailored composition and doping profiles, is the best epitaxial growth technique to fabricate QC lasers.

11. Theoretical Framework

We computed the electronic states of our multiquantum well heterostructures in the envelope function approximation (Bastard, 1990), including the nonparabolicity through an energy-dependant effective mass (Nelson et al., 1987): (1) where ( E - V )is the energy of the electron E measured from the conduction band edge V(z) of the material. In this approach, E,, the material's bandgap, is related to the nonparabolicity coefficient y (Nelson et al., 1987) through

(2)

y - l = 2m*E,/h2

We found that this model predicts the correct resonant energies with a typical accuracy of a few millielectron volts even for scattering states located above the barrier height (Capasso et al., 1992). However, since we solve a one-dimensional Schrodinger equation, which includes the energy dependant effective mass [Eq. (l)], the wave functions that we compute with this approach are not orthogonal, which makes the definition of the dipole matrix element between the state i and j

(3)

zij = (CPiIZlPj)

somewhat problematic. The solution to this problem is to go back to the two-band model and compute the matrix element including the valence band part. The dipole matrix element now reads (Sirtori et al., 1994) 2..= "

ii 2(Ej - Ei)

(CPilpz

1 1 m*(Ei, z) -k m*(Ej, z)

where the momentum operator p z is defined, as usual, p ,

(4) =

-ih(a/az). To

1 QUANTUM CASCADE LASERS

account for the underlying valence band component, the wave functions and ‘pj must be normalized according to

’

+

E - V(Z) E - V ( Z )+ E,(z)

7 ‘pi

(5)

In contrast to the square well case, for which this more accurate approach leads to modification of the value of the matrix elements in the order of only about 5-lo%, we found that the use of Eqs. (4)and ( 5 ) instead of Eq. (3) in our laser structures leads to much more significant differences, up to 40%. The reason is that we compute our dipole matrix elements for transitions in which the nonparabolicity effects are enhanced, such as having at least one high-energy state lying close to the top of the barrier, and those being diagonal in real space. The effect of an applied electric field on the structure is taken into account by digitizing the potential in 0.5- to 1-nm steps and enclosing the whole structure in a large “box.” The latter procedure alleviates the normalization problems associated with pure scattering states. When needed, self-consistency with Poisson’s equation is introduced assuming (i) that all the impurities are ionized and (ii) periodic boundary conditions on each stage of the structure. The main challenge in the design of a laser based on intersubband transitions is to obtain a population inversion. Unlike most other laser systems, population inversion between subbands does not come from an intrinsic physical property of the material system but from a careful design of the scattering times. The dominant scattering mechanism between subbands separated by more than an optical phonon energy (i.e., > 30 meV for InGaAs) is the emission of optical phonons. This process is always allowed and is very efficient, leading to lifetimes of the order of a picosecond. For this reason, it is essential to include the computation of the optical phonon-emission rate in the general design of the laser. The optical phonon-scattering rate between two subbands is computed for electrons emitting bulk phonons (Price, 1981; Ferreira and Bastard, 1989). Since our excited states densities are very low, the electron is always assumed to be at k,, = 0. In this case, the optical phonon scattering rate reads:

where 2m*(Eis - ha,,)

(7)

8

JEROME FAISTET

AL.

is the momentum exchanged in the transition. Absorption of optical phonons at finite temperatures is taken into account by evaluating Eq. (6) using the relevant momentum for optical phonon absorption

Although this procedure neglects the inherent complexity of the phonon spectrum of an InGaAs-AlInAs superlattice, we found that it gave good agreement with our measurement of the intersubband scattering rate in a 10-nm square well (Faist et al., 1993). In general, this lifetime has a minimum (-0.25 ps) for a subband spacing equal to the optical phonon energy and increases monotonically with subband spacing up to a few picosecond for subbands spaced by 300 meV, when phonons with large momentum must be emitted. On the other hand, the radiative rate ( T , ~ ~ ~ ) -for ' spontaneous photon emission in a single polarization mode is given by

where YE is the refractive index, c is the light velocity, E, is the vacuum permitivitty, and q is the electronic charge. The calculated radiative efficiency &d = T ~ ~ Jforz spontaneous , ~ ~ ~ emission of a square well is plotted as a function of the transition energy in Fig. 2. The strong increase above the optical phonon energy is due both to the increase of zopt with transition energy, as described in the previous paragraph and to the E,, dependence of z , contained ~ ~ ~ in Eq. (9). For transition energies below the optical phonon energy and at low temperatures, optical phonon scattering is forbidden and the lifetime, limited by emission of acoustic phonons, is much longer, of the order of hundreds of picoseconds (Faist et al., 1994a). Relatively high values of the radiative efficiency can be obtained (Helm et al., 1988). However, since the acoustic phonon energy is of the order of a millivolt, the Bose-Einstein factor increases very rapidly with temperature. As shown in Fig. 2, the radiative efficiency in the far infrared drops by approximatively two order of magnitude between 0 and 100 K while remaining practically constant in the mid-infrared. This makes the design of a laser structure in the far infrared very challenging since self-heating effects are expected to be strong in a laser close to threshold. The peak material gain between subbands i and j assuming a Lorenzian

1

.9 c m .D

2

QUANTUM

9

CASCADE LASERS

I0"

1o

10"

1 o4

Io

1o - ~

-~

Io8 1o

-~

-~

1 o-'

10

100

10.'

Transition Energy (mev) FIG.2. Calculated radiative efficiency of a quantum well for energies above and below the optical phonon energy. The shaded area comprises the wavelengths for which light cannot propagate (Reststrahlenband) and those at which multiphonon absorption is important.

line reads

where q is the electron charge, 1 is the emission wavelength; n is the mode refractive index; 2yij is the full width at half maximum (FWHM) of the transition in energy units determined from the luminescence spectrum (spontaneous emission); nj and n, are the sheet electron densities in subband j and i; and L, is a normalization length, which in this work is chosen as the length of a period of the active region. The peak modal gain G, is defined as G, = G,T, where r is the confinement factor of the waveguide. The latter is simply the product of the number of periods N , and the overlap factor of the mode with a single period, rp.The overall factor to the left of the population difference in Eq. (10) is called the gain cross section. We denote it g, to distinguish it from the gain coefficient g, used in our papers on QC lasers and defined as G J J , the ratio of the peak material gain to the current density J . Note that n, - n,, in steady state, is proportional to J . It is interesting to note that in contrast to the spontaneous emission, the material gain has no strong wavelength dependence and therefore the variation of threshold current as a function of design wavelength is expected to be weak and mainly determined by free-carrier absorption.

10

JEROMEFAIST ET AL.

111. Energy Band Diagram

Our lasers consist of a cascaded repetition of typically 16 to 35 of an identical stage. As shown in Fig. 3, each stage consists of an active region followed by a relaxation-injection region. The active region is the region where population inversion and gain takes place. As shown in Fig. 3, the electronic spectrum of the active region consists usually of a ladder of three states engineered so as to maintain a population inversion between the third and second states. Assuming a simplified model with no nonparabolicity and a 100% injection efficiency in the n = 3 state, the population inversion condition is simply z32 > z2 where ~3;’is the nonradiative scattering rate from level 3 to level 2 and z;’ is the total scattering rate out of level n = 2. Note, however, that since the nonradiative channel 3-1 is usually not negligible, especially in coupled-well structures, this condition is actually less stringent than the condition t 3 > z2 between the total lifetimes of states n = 3 and n = 2. The lifetime t2 in most QC laser structures is determined by the scattering rate 1/(z2J to a lower subband ( n = 1). The ladder of electronic states with a suitable energy spectrum and scattering properties can be obtained in a variety of multiquantum well structures. We distinguish these structures by the character of the 3-2 transition. The transition is said to be vertical when the two wave functions of states n = 3 and n = 2 have a strong overlap, and diagonal when this overlap is reduced.

............

..."............. 4

Active region

L

A

v v

b

Relaxation + injection

FIG.3. General philosophy of the design: each stage of the multistage active region consists of a gain (or active) region followed by a relaxation-injection region.

1 QUANTUM CASCADE LASERS

11

It is usually beneficial to leave the active region undoped since, as we have shown experimentally, the presence of dopants in the active region significantly broadens and red shifts the lasing transition by introducing a tail of impurity states (Faist et at., 1994b). The relaxation-injection region is the section of the period where the electrons cool down and are reinjected into the next period. This region must be doped to prevent the strong spacecharge buildup that would arise if the electron population was injected from the contact. In the most recent structures, doping was restricted to the center of the injector to separate electrons in the ground state of the latter from the parent donors, as in modulation-doped heterostructures. This reduces scattering thus enhancing the injection efficiency.

IV. Material Aspects In principle, a heterostructure that fulfills the preceding requirement and is based on the design philosophy displayed in Fig. 3 can be realized using almost any semiconductor material. All of our experiments were performed on In,~,,Ga,,,,As-A1,,,,In,,,,As heterojunction material lattice-matched to InP and grown by MBE. This material combination has the following advantages over the more conventional GaAs-AlGaAs heterostructures. First, the electron masses in both well and barriers are lighter, enabling simultaneously larger oscillator strengths and smaller optical phonon scattering rates. The large conduction band discontinuity AE, = 0.52 eV enables the design of lasers in a large wavelength range from 1 = 4.3 to 17 pm. In addition, the InP substrate has very good waveguiding properties: having a lower refractive index than both AlInAs and GaInAs, it can be used as a cladding material. It is a binary material and thus provides a good thermal transport, reducing the thermal impedance of the laser, as opposed to an alloy. QC lasers based on GaAs-AlGaAs require the growth of a thick lower cladding based on AlGaAs with a large A1 mole fraction. However, the GaAs-based material has advantages of its own, namely, a lattice match obtained regardless of the A1 mole fraction, enabling more flexibility in the design and somewhat relaxed growth requirements. Intersubband electroluminescence (Li et al., 1998; Strasser et al., 1997) and laser action (Sirtori et al., 1998a) were demonstrated in quantum cascade GaAs-AlGaAs heterostructures. This will soon enable a comparison of both technologies. The Si-SiGe heterostructure material also has advantages, including the absence of a restrahlen band, which should enable the design and operation of QC lasers in the 20 to 50-pm wavelength range, and compatibility with silicon technology.

12

JEROMEFAISTET

AL.

V. Optical Constants The multistage gain region is inserted in an optical waveguide designed for the laser operating wavelength. Since the optical constants are fixed by the material used (in our case, InGaAs and AIInAs), the physical dimensions of the waveguide scale with the wavelength. This makes the waveguide design in the 4- to 10-pm wavelength range a challenge since one needs a waveguide with low loss and a high confinement factor (to minimize the threshold current) while maintaining the total thickness of the layers grown by epitaxy to a minimum. For a reliable waveguide design, a good knowledge of the optical constants of InGaAs and AlInAs is needed. The values obtained by linear interpolation between the binary compounds = 3.071. To obtain (Lynch and Hunter, 1991) are nlnCaAs= 3.324 and nInAIAs more accurate values, three periods of an optical quarter-wave Bragg reflector (LinCaAs = 320 nm and LinAIAs = 346 nm) were grown by MBE, lattice-matched to a semi-insulating InP substrate. The optical transmission, measured with a Fourier transform infrared (FTIR) spectrometer is shown in Fig. 4.A best fit of this data, also displayed in Fig. 4, leads to the value n l n ~ a= ~ 3.458 s and n l n ~ l A s= 3.198. As shown later in the text, these values gave very good agreement between the measured and calculated spacing of

0.6 c

.-0 cn .v, 0.4 E cn c

E 0.2 0

FIG.4. Solid line: mid-infrared transmission spectrum of a three-period quarter-wave stack with 320-nm-thick InGaAs layers and 346-nm-thick InAlAs layers. Dashed line: computed value of the transmission assuming the refractive index displayed in the figure.

1 QUANTUM CASCADE LASERS

13

the longitudinal modes of QC lasers. The good agreement of the fit up to 1000cm-' shows that no significant dispersion of the indices occurs up to 10 pm.

VI. Stability Requirements: Why the Injector Must Be Doped In t heir original paper Kazarinov and Suris (1971) proposed to use an undoped periodic superlattice as the gain medium. In this scheme, a strong applied electric field is supposed to position the ground state 1' of well i above the second state 2'" of the following well i 1 (i.e., E'; > E;+'). A photon of energy hv = E: - E? is then supposed to be emitted by photonassisted tunneling. Population inversion is obtained automatically since the upper state of the lasing transition is also the ground state of each well. However, this band alignment is inherently unstable because the current is at a minimum between the two maxima occurring for the band alignments 1' - 2i+l and 1' - 3"'. Indeed it is well known from experiments and theory that superlattices break into different electric field domains within which the energy levels are locked in resonance. In a laser structure, a homogenous and stable electric field distribution is an absolute necessity. Therefore, the structure (i) must be doped so that the integrated negative charge is always exactly compensated by the fixed positive donors even in the situation of strong injection to prevent space-charge formation and (ii) the operating point must be a stable point of the current-voltage (I-V) characteristics to prevent the breakdown of the active region into different field domains. Both conditions are fulfilled in our design. Below the threshold bias, the structure has a high electrical resistance because the band diagram has an overall sawtooth shape. The bias for which the band diagram goes from sawtooth to staircase corresponds approximately to the threshold. Close to threshold, the current is controlled by resonant tunneling between the ground state of the graded gap injector and the excited state of the adjacent active region. The laser threshold occurs before the peak current (full resonance) is reached, thus preventing the formation of domains since the device is always operated in the stable region of the I-V characteristic. As already mentioned, overall charge neutrality must be maintained under operating conditions. Therefore, for the injector, which acts as an electron reservoir, not to be depleted at threshold, its donor sheet density N , must be much larger than the excited state density at threshold nth. On the other hand, free-carrier losses must also be minimized and we choose as a rule of thumb N , r r ~2-5 x nthM 1-4 x 10" cm-2.

'

+

14

JEROMEFAISTET AL.

VII. QC Laser with Diagonal Transition at il = 4.3 pm

1. CHARACTERIZATION OF THE ACTIVEREGION:PHOTOCURRENT AND ABSORPTION SPECTRA We start the discussion of our different QC laser designs with the original structure designed for operation at 4.3-pm wavelength (Faist et al., 1994~). This design has some unique features. First, it is the one that enables the shortest wavelength to be reached for a given band discontinuity. Second, it makes use of a diagonal transition with an upper state anticrossed with another level higher in energy, a scheme that has many interes6ng features. In addition, it is the only structure for which a complete characterization was carried out, which includes, beside electroluminescence, photocurrent and absorption spectroscopy. This detailed characterization was carried out on samples with an active region similar to that of the laser samples but grown with a fewer number of period ( 5 or 10) on a semi-insulating InP substrate (Faist et al., 1994d). A semi-insulating substrate with its high transparency enables an accurate measurement of luminescence, absorption and photocurrent through a polished 45" wedge. Such a coupling is in principle not as efficient as a waveguide coupled through a cleaved facet or a two-dimensional grating because the light is not completely polarized normal to the layer but has the advantage of having a well-controlled coupling efficiency. The band diagram of one period of the structure with and without bias applied is displayed in Figs. 5a and 5b, respectively. At zero applied bias, charge transfer between the doped injector region (short period superlattice at the right-hand side of the potential profile) was taken into account in our calculations, which solved Poisson's and Schrodinger's equation self-consistently. In the same figure, the photocurrent spectrum of the sample, grown by MBE using InGaAs-AlInAs lattice-matched on an InP substrate, is displayed for various applied biases. The three observed photocurrent peaks, corresponding to the 1-3, 1-4, and 1-5 transitions in the active region are easily identified and correspond closely to the calculated transitions displayed in the band diagram. In addition, the strong Stark shift of the 1-3 transition is expected since level 3 and level 1 belong to two distinct wells and have not yet anticrossed with levels 2 and 4. This anticrossing is discussed later in the text. Absorption from the active region was also measured in a large mesa (800 x 800pm) by a differential absorption technique with the light incident and exiting through a 45" polished wedge. In our structure, at an electric field -50 kV/cm (applied bias U 1 V) the injected current is negligible (< 100 pA) but the active region is already depleted. As a resuIt, the differential transmission AT/T= [T(U = 1 V) -

-

1 QUANTUM CASCADE LASERS

r

0.8

i

I

-

200 (bl

15

300 400 Photon Energy (rneV)

FIG. 5. Quantum cascade emitter structure: (a) calculated conduction band diagram of a portion of the AlInAs (barriers)-GaInAs (wells) structure under positive bias conditions and an electric field of lo5V/cm. The dashed lines are the effective conduction band edges of the digitally graded 18.1-nm-thck electron injector. The latter comprises six 3-nm-thick AIInAsGaInAs periods of varying AlInAs duty factor to grade the average alloy composition. Electrons are tunnel-injected through a 7.4-nm-thick barrier into the n = 3 subband of the active region. The latter comprises 1.1- and 3.7-nm quantum wells separated by a 3.5-nm-thick barrier and a 3-nm well sandwiched between 3-nm-thick barriers. The third 3.0-nm-thick well is sandwiched between 3-nm barriers. The 1.1-nm-thick well and its barriers are n-type doped to 1017c ~ I - This ~ . gives rise to a small band bending. (b) Measured photocurrent spectra at various bias voltages below the onset of strong electron injection and band diagram at 0 bias. The positions of the energy levels with respect to the bottom of the flat well are E , = 175meV, E , = 230meV, E , = 410meV, E , = 525 meV, and E , = 620meV. The peaks in the spectra correspond to the optical transitions indicated by the arrows in the inset.

16

JEROME FAISTET

2.0

1

AL.

- - - Photocurrent - Absorption

1.0

0.5 0

L

200

250

300

350

Photon Energy (mew FIG. 6. Absorption (full line) and photocurrent (dashed line) spectrum from the active region at zero bias from the structure of Fig. 5.

T(U = O)]/T(U = 0) is a measure of the absorption from the active region at U = 0 V. Here A T / T measured at TL= 10 K for our samples is shown in Fig. 6. The same 1-3 and 1-4 transitions are again easily identified. 2. INFLUENCE OF

THE

DOPING PROFILE

A study was also carried out to identify the best doping profile (Faist et al., 1994b). To this end, two otherwise identical samples were grown with a similar sheet density per period but a different doping profile. The reference sample, similar in active and injector region design, to that of Fig. 5, was doped with Si to n = 1 x 1017cm-3 across the graded injector, the smallest (0.8-nm) well of the active region and its adjacent barriers. In the second sample, however, the dopants are set back from the active region and only the injector is doped to n = 1.5 x l O I 7 emp3. A comparison of the electroluminescence spectra performed at an identical injection current was carried out at low temperature with a FTIR spectrometer by a lock-in and step-scan technique. The luminescence quantum efficiencies, which are directly proportional to the radiative efficiencies, are measured to be identical in the two structures. This is a strong indication that, for transitions energies above the optical phonon threshold, the impurity potential has a negligible influence on the intersubband lifetime, as expected from theoretical calculations (Ferreira and Bastard, 1989). In contrast, the spectra taken at an identical injected current ( I = 50 mA, corresponding to an injected current density of J = 1.1 kA/cm2), and compared in Fig. 7, are very different. The spectrum of the sample with setback shows a dramatic narrowing of the luminescence

17

1 QUANTUM CASCADE LASERS

150

200 250 300 350 Photon Energy (mev)

400

FIG. 7. Electroluminescence spectrum for the reference sample (left peak) and the sample with setback; that is, with an undoped active region and injection barrier (right peak). The structures and corresponding energy diagrams are similar in design to those of Fig. 5. The diagonal transition is between 0.8- and 3.5-nm-thick wells separated by a 3.5-nm barrier. The third 2.8-nm-thick well is sandwiched between 3-nm barriers. Electrons are injected into state 3 through a 4.5-nm barrier. The drive current is I = 50mA. The spectra are fitted with a Gaussian and Lorentzian line shape (dashed curves) for the reference sample and sample with doping setback, respectively.

line down to 21 meV from its value of 50 meV for the reference sample. The sample with doping setback shows a very good agreement between the calculated (E23 = 293 meV) and measured (294meV) values of the transition energy. On the contrary, the luminescence peak of the sample with a doped barrier has a transition energy of E,, = 245 meV, 30.5 meV lower than the calculated value ( = 275.5 meV). Moreover, the electroluminescent spectra compared in Fig. 7 exhibit a very different line shape: while the spectrum of the sample with doping setback is nearly Lorentzian, the spectrum of the reference sample is well fitted by a Gaussian line shape. These observations are in qualitative agreement with a band tail picture of impurity disorder. Because of the doping level used ( - 1017cm-3), the impurity states merge to create a tail on the low-energy side of the two-dimensional density of states. We therefore interpret the discrepancy between the calculated and measured E , , transition energy as well as the broadening of the electroluminescence in the sample doped in the barrier as a red shift induced by the dopant impurities. We stress here that for both samples, the optical transitions we observed in both absorption and electroluminescence follow the polarization selection rules for intersubband transitions. They are not transitions between discrete impurity states.

18

JEROMEFAISTET AL.

Since the peak gain [Eq. (lo)] is proportional to the inverse of the luminescence linewidth, an undoped active region is needed to reduce the threshold current density and to achieve laser action.

3. ANTICROSSING OF THE STATES IN

THE

ACTIVEREGION

As shown in Fig. 8, the line shape and the peak position of the luminescence of the sample with doping setback, for relatively low injected currents is strongly dependent on the injected current and therefore on the applied bias. As the current is increased, the peak blue shifts and narrows significantly. Measurement of the peak position and linewidth as a function

1.oo

0.75

b

0.50

g

a

0.25

-220

n

240

260

280

300

320

Photon Energy (meV) I (mA) 10

s

20

50

100

I

I

I

200

300

30

p v

A

e

E v

295

I

B 25

c

.-0 4-

LL

v)

c

E

I-

20

1.80

1.85

1.90

290

FIG.8. (a) Luminescence spectra of the sample with doping setback for different drive current: 7.5 mA (dashed line), 30mA (dotted line), and 150 mA (full line). (b) Peak position and FWHM of the luminescence spectrum as a function of applied bias.

19

1 QUANTUM CASCADE LASERS

of applied field gives insight into the narrowing mechanism. The electroluminescence spectrum shows a strong narrowing of the line from an FWHM of 32 meV at I = 7.5 mA to 20.5 meV at I = 150 mA as the peak position shifts from 290 to 297 meV. We interpret these observations in the following way. The reader is referred to the diagram of Fig. 5, which is qualitatively identical and of similar dimensions. Because they are very thin, the wells on both sides of the center well support only one bound state. The center well, being thicker has an excited state very close to the barrier edge, as shown experimentally in Figs. 5 and 6, in our photocurrent and absorption measurements. The position of the energy level differences E , - E , = E,, and E , - E l = E l , as a function of the applied field is displayed in Fig. 9. The ground states of the 2.8- and 3.5-nm quantum wells anticross as they are brought into resonance at a field of about 60 kV/cm. Since the barrier between these two wells is only 3nm, both the minimum splitting energy is relatively large (E'$ = 20meV) and the two states remain very strongly coupled over a broad range of fields. The Stark shift of the E,, transition is clearly sublinear in the applied electric field. This sublinearity originates from the repulsion-

40

2 (u

"9 :-.

w (u

c

A .-

0

5 200 A= a

100

30

mt

0

1-3

1 u 60 90

30

I 0

120

Electric Field (kV/cm) FIG. 9. Computed transition energies El, and El, of the 1-3 and 2-3 transitions and of the products of the square of the matrix elements of the latter with E,, and El,, which are proportional to the oscillator strengths, as a function of applied electric field for the sample with doping setback. These calculations are also valid for the L = 4.3 pm quantum cascade laser of Fig. 10.

20

JEROMEFASTET AL.

between level 3 in the 0.8-nm well and the second excited state of the center 3.5-nm well, close to the top of the barrier. Level 3 is clearly being “pushed” into the 3.5-nm well. This sublinear dependence of the 2-3 transition is clearly observed experimentally in the electroluminescence data (Fig. 8, bottom). Due to this shift of the center of charge of the n = 3 wave function, the energy of this state is now less sensitive to the thickness fluctuations in the 0.8-nm well, yielding a narrowing of the intersubband luminescence and therefore a higher peak material gain. Figure 9 also shows the field dependence of the oscillator strength for the two transitions. The matrix element of the 2-3 transition rapidly increases for fields >50kV/cm as a result of the anticrossing between the n = 3 and n = 4 states, as the latter is pushed into the 3.5-nm well, thus enhancing the overlap between these states. This increase of the oscillator strength is at the expense of that of the 3-1 transition, which rapidly decreases, as required by the well-known sum rule. Laser designs based on a diagonal transition with anticrossing between the upper level and an excited state close by in energy are attractive because they still lead to a longer lifetime for the upper state, compared to a vertical transition, which yields a good population inversion, while at the same time providing a narrower linewidth than a purely diagonal transition. This can be easily seen in a tight-binding picture. One easily shows that when states n = 3 and n = 4 are completely anticrossed (i.e., when the splitting between the bonding and antibonding states is minimum), the upper state lifetime is about twice the value of a single well for the same optical transition energy. This is because at the anticrossing point the wave functions of the n = 3 state is spread nearly equally between the two wells, decreasing the squared matrix element for emission by optical phonons by a factor of 2. As we shall see, this design has also the advantage of a better injection efficiency even at high temperatures.

4. BAND STRUCTURE AT THRESHOLD

The overall band structure of two periods of the active region with an applied bias corresponding to the approximate threshold field ( z 95 kvjcm) is displayed in Fig. 10. A transmission electron micrograph of the structure is shown in Fig. 11. The active region design is identical to that of the samples with undoped active regions used in the luminescence experiments discussed in the two preceding section. Lasing occurs between states n = 3 and n = 2. Parallel to the layers, these states have plane-wave-like energy dispersion. The corresponding energy subbands are nearly parallel (Fig. lob) because of the small nonparabolicities for wave number k lI corresponding

21

1 QUANTUM CASCADE LASERS

U

FIG. 10. (a) Conduction band energy diagram of a portion of the 25-period section of the quantum cascade laser based on a diagonal transition and operating at I = 4.3 pm, corresponding to the energy difference between states 3 and 2. The dashed lines are the effective conduction band edges of the digitally graded electron-injecting regions. Electrons are injected through an AlInAs barrier into the n = 3 energy level of the active region. The wavy arrow indicates the laser transition. (b) Schematic representation of the n = 1, 2, and 3 states parallel to the layer; kll is the corresponding wave vector. The bottoms of these subbands correspond to the energy levels n = 1, 2, and 3 indicated in (a). The wavy arrows indicate that all radiative transitions originating from the electron population (shown as shaded) in the n = 3 state have essentially the same wavelength. The straight arrows represent the intersubband optical-phonon-scattering processes; note the fast (subpicosecond) relaxation processes with zero momentum transfer between the n = 1 and n = 2 subbands separated by one optical phonon.

22

JEROMEFAISTET AL.

GalnAs

AllnAs

wells

(nm) L

-0.8 ___

-3.5 ____. ----2.8

Active

3.0 region 30 1

Injection region

FIG. 11. Transmission electron micrograph of a portion of the cleaved cross section of the quantum cascade laser based on a diagonal transition and operating at 2 = 4.3 pm (see Fig. 10). Three periods of the 25-stage structure are shown. The superlattice period of the digitally graded region is 3 nm and the duty cycle of the AlInAs barrier layers varies from 40 to 77% top to bottom, creating a graded gap pseudoquaternary alloy.

to a small Fermi energy E , < 1-3 meV of the electrons injected in the n = 3 state. As a result, electrons making radiative transitions to a lower subband (for example, from n = 3 to n = 2) will all emit photons of essentially the same energy hv = E , - E,. It is worth noticing that, at low temperatures, a narrow emission line is also expected even when the nonparabolicity is not negligible (i.e., for large values of Ef)because the electron-electron interaction condenses the intersubband absorption in a collective mode with a narrow spectrum. This effect was clearly demonstrated in absorption experiments in heavily doped InAs wells (Gauer et al., 1995) and subsequently theoretically explained (Nikonov, 1997). In a single electron picture, the joint density of states of these transitions is therefore similar to a delta function in the absence of broadening. If a population inversion is the created between these excited states, the gain spectrum will be correspondingly narrow, nearly symmetric, and much less sensitive to thermal broadening of the electron distribution, unlike the gain spectrum associated with interband transitions in diode lasers. Recent experiments have shown convincingly that, in the mid-infrared, the broaden-

1 QUANTUM CASCADE LASERS

23

ing of intersubband transitions is dominated by interface roughness (Campman, 1996) and therefore the broadening increases strongly for narrower wells. The coupled wells are engineered so as to provide a reduced spatial overlap between the initial and final states, n = 3 and n = 2 of the laser transition. This reduces tunneling out of the n = 3 level into the broad quasi-continuum of states, thus enabling a large enough electron population buildup in this state to achieve laser action. The resulting intersubband nonradiative scattering rate of the initial state ( r 3 ) - ' , equal to sum of the phonon relaxation rate (T~~)-’= 0.25 ps-' computed using Eqs. (6)-(8) = 0.17 ps-l, is relatively small. and the escape rate to the continuum (TJ’ The resulting lifetime of the upper state is z 3 = 2.5ps, calculated at the threshold electric field ( F = 95-100kV/cm) at which laser action was observed in these structures. The reduced spatial overlap of states 3 and 2 also enhances the scattering time from state 3 to state 2. The calculated value at the bias field of Fig. 10 is 2 3 2 = 4.3 ps; this ensures population inversion between the two states ( ~ 3 2> zZ1) since the lower of the two ~. inelastic relaxation by empties with a relaxation time T~~ 0 . 6 ~ Strong means of optical phonons with near zero momentum transfer occurs between the strongly overlapped and closely spaced n = 2 and n = 1 subbands since their separation E , , is chosen by design to be equal to an optical phonon (34 meV). The calculations of Fig. 9, in fact, show that their splitting becomes equal to the optical phonon energy at an electric field ( F r 90 kV/cm) close to the value for which the energy diagram acquires a staircase shape and that these two states remain strongly coupled (i.e., anticrossed) with a separation slightly greater than 34 meV over a relatively broad range of fields (90-120 kV/cm). This feature ensures population inversion in a range of operating fields that includes the threshold for laser action, making the design of the laser more robust. For a given active region and injector layer sequence, the threshold current density Jthdepends on the waveguide design (mode confinement factor and waveguide losses) and on the mirror losses, which are proportional to the inverse of thc cavity lcngth. Note also that in this range of fields, the oscillator strength is high, to enhance the gain for a fixed population inversion (Eq. 10) and weakly dependent on the electric field, along with E 2 , . Finally, the tunneling escape time out of the n = 1 state is extremely short (about 1ps), further facilitating the population inversion. Electron injection in the n = 3 state is achieved via a digitally graded injector region followed by a 4.5-nm-thick AlInAs tunnel barrier. The applied electric field flattens the average conduction band edge in the injector (the dashed horizontal lines in bold, Fig. lo), converting the overall sawtooth-like conduction band diagram of the structure at zero bias into an

-

24

JEROMEFAISTET AL.

energy staircase. This sawtooth-to-staircase transition, which allows a rapid increase of the injected current once the correct band alignment has been achieved, was first introduced in the design of staircase avalanche photodiodes and solid-state photomultipliers (Capasso et al., 1983). In our first report (Faist et al., 1994c) we neglected size quantization in this 18.3-nmthick graded region. A more careful subsequent analysis of these effects showed however that the bound states in the injector region do not play a significant role in the operation of this laser (Faist et al., 1998a). In further designs with more complex injector designs we will however take into account the quantization of the ground state of the injector. We conclude this section with the observation that laser designs based on a diagonal transition and anticrossing of the upper excited state are attractive because they provide a relatively long lifetime for the latter, which yields a good population inversion, but at the same time they provide a narrower linewidth than a purely diagonal transition. As we shall see, they are particularly useful at longer wavelengths since at smaller transition energies the electron-optical phonon intersubband scattering times become shorter due to the smaller momentum transfer.

5. WAVEGUIDE

The complete layer sequence of the laser, including the waveguide, is shown in Fig. 12; the corresponding refractive index profile and calculated mode profile is shown in Fig. 13. The waveguide must fulfill the requirements of low optical losses while maintaining at the same time a minimum thickness of grown material for a TM propagating mode, as required by the optical selection rules for intersubband transitions. The latter requirement is important to optimize the thermal transport across the device and to minimize the growth time and the number of defects. The waveguide comprises, on both sides of the 25-periods active region-injector region, two 300-nm-thick GaIn As guiding layers, which enhance the optical confinement by increasing the average refractive index difference between the core and cladding regions of the waveguide. The bottom cladding consists of a 500-nm-thick AlInAs layer grown on top of the InP substrate. The top cladding consists of a 2500-nm-thick AlInAs layer followed by a 670-nmthick GaInAs cladding region. The purpose of this GaInAs layer is to decouple the high loss (a = 140cm-') metai contact-semiconductor interface plasmon mode from the laser mode by enhancing the difference of the effective refractive indices of the two modes. In this case, we raised the refractive index of the plasmon mode. It is actually usually more efficient to reduce the latter, by heavily doping the top layer, depressing its refractive

25

1 QUANTUM CASCADE LASERS

GaInAs Sn doped

n = 2.0x 1020 cm-3

20.0nm c

0

GaInAs

1.O X 101*

AlGaInAs Graded

1.ox 1018

AlInAs

5.0 x l O l 7

1500.0

AlInAs

1.5 x 1017

100o.o

670.0

30.0

stl s2

0-

a -00 3 .= 05

:x p

Digitally graded

Active region

undoped

21.1

GaInAs

1.0 X 10"

300.0

AlGaIn As Digitally graded

i . 5 x 1017

AlInAs

1.5 X 1 Oi7

I

33.2

Q)

500.0

rrn,

3 c

0%

a-0 Doped n+ InP substrate

7.0 x 10l8

’rn

9"

FIG. 12. Schematic cross section of the complete laser. The overall structure has a total of about 500 layers.

index. The latter approach was used to develop our so-called plasmonenhanced waveguides for longer wavelengths, and is discussed in more detail ~ ) of further in the text. By using low-doped ( n = 1-5 x 1017~ m - instead heavily doped InP substrate (where n = 5-9 x 10l8cmP3) we were able, in later designs, to remove the bottom 500-nm-thick AlJnAs, shortening the growth and improving the heat dissipation. In all our waveguide designs, the transitions between the barrier and well materials are graded (either with

26

JEROME FAISTET AL.

1.o

7

-

0.8 -

I

I

AllnAs

I

1

MQW

I

I

I

I

-

InP substrate 1

4.0

3.5 $ U K

Q)

3.0 .1 4-l

0

a, 0.4

m

U

t

0

=

2.5

0.2 0

2

2

2.0 6

Distance (pm) FIG. 13. Calculated refractive index and mode profile for the waveguide of the I QC laser structure of Fig. 12.

= 4.3 pm

analog or digital graded regions) over a distance of about 30nm. This grading has the purpose of reducing the series resistance by forming a smooth band profile, preventing the formation of a barrier between the cladding layers. Another important feature of the waveguide design is to minimize the optical losses due to free carriers. This is obtained by reducing to its minimum value the doping level around the waveguide region while maintaining low resistivity. It is also important to prevent intersubband absorption at the laser wavelength, particularly in the injector regions where most of the electron density of each period resides. The optical loss is computed by solving the wave equation for a planar waveguide with a complex propagation constant, modeling each layer with its complex refractive index. The imaginary (loss) part of the refractive index was obtained through a Drude model (Jensen, 1985), where the dielectric constant E is

with a plasma frequency

1 QUANTUM CASCADE LASERS

27

where n, is the electron concentration, E , is the (high-frequency) dielectric constant, and m* is the electron’s effective mass. We used a scattering time z = 0.2-0.5 ps, depending on the doping level and wavelength (Jensen, 1985). This value of scattering time gives a good agreement with the measurement of free-carrier absorption in bulk GaAs or InP, as reported in the literature (Jensen, 1985) or measured by us. Note, however, that we systematically underestimate our waveguide losses. This discrepancy remains basically unexplained.

6. DEVICE PROCESSING The samples were processed into 10- to 1Cpm-wide mesa waveguides by wet etching through the active region down to about l p m into the substrate. An insulating layer was then grown by chemical vapor deposition to provide an insulation between the contact pads and the doped InP substrate. For minimum losses, we chose SiO, for short wavelengths (A < 5 pm) and at long wavelengths (A = 11pm) and Si3N4 for all the intermediate ones. Windows are defined through the insulating layer by plasma etching, exposing the top of the mesa. For this process, Si3N, is preferred over SiO, whenever possible because it is easier to remove by plasma etching in a CF, gas. Ti-Au nonalloyed ohmic contacts were provided to the top layer and the substrate. A scanning electron microscope picture of the cleaved facet of a processed device is displayed in Fig. 14. The devices are then cleaved in 0.5- to 3-mm-long bars, soldered to a copper holder, wire-bonded, and mounted in the cold head of a temperature controlled He flow cryostat.

7. LASERCHARACTERISTICS A set of electroluminescence spectra of a 500-pm-long and 1Cpm-wide laser is displayed in Fig. 15 for various injected currents. The drive current consisted of 80-ns-long electrical pulses with a 80-kHz repetition rate. The spectrum below a 600-mA drive current is broad, indicative of spontaneous emission. Above a drive current of 850-mA, corresponding to a threshold current of 15 kA/cm2, the signal increases abruptly by orders of magnitude, accompanied by a dramatic line narrowing. This is direct manifestation of laser action. A plot of the optical power versus drive current for various temperatures is displayed in Fig. 16 for a longer device (1 = 1.2mm). The optical power is measured by focusing the light with a fll.5 optics on a fast, calibrated, room-temperature HgCdTe detector. The threshold current

28

JEROME FAISTET AL.

FIG. 14. Scanning electron micrograph of the cleaved facet of a processed device.

density was lowered to f,, = 5.4 kA/cm2 due to the lower mirror losses of this longer device. The current-voltage characteristics at 10 and 100 K are displayed in the inset. As expected, no significant current flows for bias below 8 V , when the band diagram still has an overall sawtooth shape. Above this voltage, a current of several hundred milliamps flows across the device. These early devices already showed a fundamental property of intersubband QC lasers: a weak temperature dependence of the threshold current density. As shown in the inset of Fig. 16, the threshold current has the typical J exp( TIT,) temperature dependence. The value of To = 112 K is much larger than the value typical for interband lasers (To 20-50 K). The weak temperature dependence of QC laser threshold can be ascribed to the following: (a) the material gain is insensitive to the thermal broadening of the electron distribution in the excited state since the two subbands of the laser transition are nearly parallel; (b) Auger intersubband recombination rates are negligible compared to the optical phonon scattering rates; (c) the variation of the excited-state lifetime with temperature is small, being controlled by the Bose-Einstein factor for optical phonons; and (d) the measured luminescence linewidth is weakly temperature dependent.

-

-

1 QUANTUM CASCADE LASERS

29

FIG. 15. Emission spectrum of the laser at various drive currents. The strong line narrowing and large increase of the optical power above I = 850mA demonstrate laser action. The spontaneous emission and the laser radiatron are polarized normal to the layers. The I = 4.26 prn emission wavelength is in excellent agreement with the calculated value.

VIII. QC Lasers with Vertical Transition and Bragg Confinement

In summary, population inversion in structures based on a diagonal transition is obtained through the combination of two design features. First, the laser transition proceeds by photon-assisted tunneling; that is, it is diagonal in real space between states with reduced spatial overlap. This increases the lifetime of the upper state and also decreases the escape rate (zest)-' of electrons into the continuum. Second, a third state, located approximately one phonon energy below the lower state of the lasing transition, is added. The resonant nature of the optical phonon emission between these two states reduces the lifetime of the lower one to about 0.6 ps. However, being less sensitive to interface roughness and impurity fluctuations, a laser structure based on a vertical transition (i.e., with the initial and final states centered in the same well) would exhibit a narrower gain spectrum and thus a lower threshold, provided that the resonant phonon-emission scheme is sufficient to obtain a population inversion and that electrons in the upper state can be prevented from escaping into the continuum.

30

JEROMEFASTET AL.

0

0.5

1.o

1.5

2.0

Current (A) FIG. 16. Measured peak optical power P from a single facet of the QC laser versus drive current at different heat-sink temperatures. The temperature dependence of the threshold current is shown in one of the insets. The solid line is an exponential fit, A exp(T/T,). In the other inset, the drive current is shown as a function of applied bias at 10K (solid line) and 80 K (dashed line).

1. QUANTUM DESIGN The main challenge in designing a vertical transition QC laser is to suppress tunneling out of the n = 3 state. Sirtori et al. (1992) showed that electronic quarter-wave stacks can be designed to confine an electronic state in the classical continuum. To enhance the confinement of the upper state in a structure based on vertical transitions, a straightforward idea would be to substitute the digitally graded region with a quarter-wave stack. However, this design would suppress the escape from the state ( n = l) due to the formation of localized states in the quarter-wave stack above the n = 1 level (Sirtori et al., 1992), preventing population inversion. Instead, we chose to keep the effective conduction band edge of the digitally graded superlattice flat under the applied field, as was done in the previous devices, while now requiring that each well and barrier pair accommodates a half electron de

1 QUANTUM CASCADE LASERS

31

Broglie wavelength, thus satisfying the Bragg reflection condition (Faist et al., 1995a). The barrier length 1, and well length 1, will, however, depart individually from a quarter wavelength. In mathematical terms, we require that the effective conduction band potential V ( x j ) of the injector at the position x j of the j t h period, be approximated by

where AE, is the conduction band discontinuity between the barrier and well material (= 0.52 eV). This creates a quasi electric field, which exactly cancels the applied field at threshold Fth:

We also have for each layer pair l,,j and lb,j the Bragg reflection condition:

where k,,j and kb,j are the wave numbers in the well and barrier materials. This condition ensures the constructive interference of the electronic waves reflected by all the periods. For our given upper state energy, this set of equations is solved iteratively for each consecutive layer pair 1, and 1, of the graded superlattice. This procedure yields successive values of 1, = 2.1, 2.1, 1.6, 1.7, 1.3, and 1.0nm and I , = 2.1, 1.9, 2.0, 2.3, and 2.7 nm, right-to-left in Fig. 17. In a solid-state picture, a region is created that has, under bias, an electronic spectrum similar to the one of a regular superlattice, with a miniband facing the lower states of the active region for efficient carrier escape from the ground state of the lasing transition and a minigap facing the upper state for efficient carrier confinement (Fig. 17a). This confinement is clearly apparent in Fig. 17b, where the calculated transmission of the superlattice is plotted versus electron energy at the field Fth= 85 kV/cm corresponding to the laser threshold. The transmission is very small at the energy E , corresponding to the upper state n = 3, while remaining sufficiently large (> 10- ') at the energy E l (Fig. 17b) to ensure a short escape time (<0.6 ps) from the n = 1 state into the superlattice. As shown in Fig. 18, the full width at half maximum of the luminescence peak of the structure based on a vertical transition is narrower (13meV) than the one based on a diagonal transition (20meV). This

32

JEROMEFAISTET AL.

I

h

4

85 kVhm

I

Active

Region

(b)

1

Eo

jj

'g

0.1

C

"t

g 0.01 0.001 0.0001

0

0.2

0.4 0.6 Energy (eV)

0.8

FIG.17. (a) Schematic conduction band diagram of a portion of the active region of the AlInAs-GaInAs QC laser based on a vertical transition at an applied electric field of 85 kV/cm. The dashed lines are the effective conduction band edges of the 20.8-nm-thick superlattice electron injector. As shown, this region is designed so that a minigap blocks electron escape from level 3. The two 4.5- and 3.6-nm quantum wells of the active region are separated by a 2.8-nm barrier. Tunneling from the injector is through a 6.5-nm barrier and electrons escape out of the n = 1 state through a 3.0-nmbarrier. (b) Calculated transmission of the superlattice as a function of energy. The position of the relevant energy states E,, E,, and El are also indicated. The energy level differences are E , - E , = 271 meV and E , - El = 30rneV.

33

1 QUANTUM CASCADE LASERS

1.oo Diagonal :(FWHM= 21meV)

I

0.75

(FWHM = 13meV)

'

I

'

I

0.50 0.25

I

'

n U

150

200

250

300

350

400

Photon Energy(meV) FIG. 18. Comparison of the intersubband electroluminescence spectrum from the laser based on a diagonal transition and operating at 4.3 nm and the structure based on a vertical transition operating at 4.6 nm. The temperature is T = 10 K. Being less sensitive to interface roughness, the spectrum of the structure based on a vertical transition is much narrower.

narrower peak is the result of the vertical transition being less sensitive to interface roughness than an diagonal one. We also verified the effectiveness of the Bragg reflector in preventing the electron escape by comparing the structure described here with one that had the injector of the 4.3-pm laser based on a diagonal transition, which was not optimized for Bragg reflection. In Fig. 19, we compare the intersubband electroluminescence efficiency of both structures. The strong difference in luminescence efficiency (about a factor of 3) is related to the decrease of the upper state lifetime in the non-Bragg device due to tunneling into the continuum. This lead to the conclusion that a QC laser based on a vertical transition without a Bragg reflector would have a prohibitively large threshold by the reduction of the upper state lifetime from the 1.2ps limited by optical phonon emission to about 0.3 ps limited by tunneling into the continuum. The lasers were processed as described for the 4.3-pm devices. Figure 20 displays the peak optical power versus drive current obtained in pulsed mode. At a temperature of T = 10 K (solid line), and above a drive current of It,, = 1.0 A, the signal increases abruptly from microwatts levels to about lOmW at I = 1.5A.This is a direct manifestation of laser action corresponding to a threshold density of Jth= 2.4 kA/cm2. The measured threshold bias of 8.5V corresponding to an electrical field in the active region F,, 85 kV/cm is in good agreement with the value calculated for resonant

-

34

JEROME FAIST ET AL.

12

T

5 cn s 5

1

1

1

1

[

1

,

1

1

,

1

1

1

1

[

, , [ [ I [ -

-

- 12 -

-

7

-

8-

-

With Bragg Reflector

-

F C

FIG.19. Intersubband electroluminescence from two non-lasing structures grown without optical waveguide and having the same active region based on a vertical transition. One structure (upper line) has an injector designed with a Bragg reflector (see Fig. 17), the other one (lower line) has an injector not optimized for Bragg reflection (see Fig. 10). Note the strong difference in luminescence efficiency related to the decrease of the upper state lifetime in the non-Bragg device due to tunneling into the continuum.

tunneling injection. In fact, a detailed study of the I-V curves of a shorter, nonlasing device shows that for biases around 9 V, a negative differential region appears in the latter, accompanied by quenching of light emission corresponding to the 3-2 transition. This gives a strong indication that injection occurs in our devices through resonant tunneling. In the lasing structure, negative resistance is not observed because the electron density in the upper state remains essentially locked at the threshold value. The same peak optical power versus drive current measurements repeated at T = 50 K (long-dashed lines) and T = 62 K (short-dashed lines) show a relatively strong increase in the laser threshold with increasing temperature, along with a decrease of the differential quantum efficiency. We attribute this large temperature sensitivity of the threshold and low slope efficiencies to backfilling of the carriers from the graded gap region into the state n = 2 by thermal activation. This effect can be best understood by considering the rate equations of the laser.

1 QUANTUM CASCADE LASERS

35

Bias Current (A) FIG.20. Peak optical output power from a single facet versus injection current at a heat sink temperature of 10 K (solid line), 80 K (long-dashed line), and 100 K (short-dashed line) of the vertical transition QC laser of Fig. 17. The device is 3.1 mm long. Inset: high-resolution spectra of the sample above threshold. The longitudinal mode spacing (0.494cm-I) is in good agreement with the calculated value (0.5 cm- ').

2. RATEEQUATIONS a.

Threshold Current

We now investigate the transport across one period of the structure in a rate-equation approach. This model is valid for almost any type of active region by simply inserting the correct numerical values for the parameters. For simplicity, we neglected the state n = 1 of the active region and restricted the analysis to a three-level system that includes the ground state of the injector with sheet density ng and the two states of the lasing transition with densities n2 and n3. The relevant scattering mechanisms, characterized by rates, are indicated schematically in Fig. 21. From level 3, electrons scatter nonradiatively to levels 2 and 1 and to the injector with a T&!, where T~~ and z31 are optical-phonontotal rate zT1 = t;i : ;z emission processes calculated as discussed in a previous section and T~~~ is the electron escape by tunneling into the continuum. The scattering from level 3 to level 1 is included in the calculation of z3 but, for simplicity, level 1 is merged with the injector in the rate equations. Similarly, the total

+

+

36

JEROME FAST

ET AL.

U FIG.21. Schematic diagram of a quantum cascade laser with the various tunneling and relaxation processes indicated by arrows; E,, is the quasi-Fermi level in the injector.

scattering rate out of level 2 7;’ = 7;; + 7;; is the sum of the rates zz;' into level 1 by resonant optical phonon emission and z i t into the graded injector by direct tunneling. As previously discussed, in our designs the injector is never depleted; that is, n3,n2 << n, so that the fluctuations of n4 can be neglected. We assume that the graded injector is characterized by a single temperature T, which includes both the lattice and electronic contribution. This will induce a thermal population in level 2 nihermE n, exp( - A/ kT), where A is the energy difference between level 2 and the quasi-Fermi energy in the adjacent relaxation region. The rate equations relating the sheet densities n2 and n3, the photon density S (per unit length per period) and their time derivatives read:

n;

n3

=-

+ Sgc(n3- n2) - n2

-

nge- A / k T z2

=32

[y,(n3 - n2) - a] S

+ P-

where g, is the gain cross section, c is the velocity of light in vacuum, and n

37

1 QUANTUMCASCADE LASERS

is the mode refractive index. The total optical losses a = aW + a, are the sum of the waveguide aW and mirror a, losses. We assumed that kT is much larger than the Fermi energy in level 2 (i.e., no final state bottleneck) so that the spontaneous emission can be described as ~ n 3 / T s p o , , where zSponis the spontaneous emission time [Eq. (6)] and /? 0.001 the fraction of the spontaneous light emitted in the lasing mode. Note that most coefficents in Eq. (14) are temperature-dependant, including the gain cross section y,. The temperature dependance of the opticalphonon-emission-scattering rates, including absorption, stimulated, and spontaneous emission of optical phonons, can be approximated by

-

t-'(T)

= sC'(0)

c

1

+ exp(hw,,/kT)

-1

where hwlo is the optical phonon energy. As we show in the experimental section, the peak gain can be fitted by a quadratic dependence in temperature: y(T) = g(O)/(l

+ aT + bT2)

(18)

where a and b are fitting coefficients measured experimentally. Both Eqs. (15) and (16) introduce temperature dependencies that are not very strong, and typically represent a factor of 2 between liquid helium and room temperature. The threshold condition is obtained from Eq. (16) by setting the derivatives and S to zero. Neglecting the spontaneous emission, the threshold current J,, is

The temperature dependence of the threshold current density can be obtained by inserting Eqs. (17) and (18) in Eq. (19). Note that as long as ntherm << a/g,, the temperature dependence of the threshold current will be weak (about a factor of 4 between helium and room temperature). Because of its exponential dependence, the most efficient way of obtaining a low ntherm is by maximizing the activation energy A. Note also that a large g ' , which is a desirable feature for low threshold, will require even lower value of ntherm for high-temperature operation. Because of this, a laser based on vertical transitions, which exhibit larger gain coefficients g at the expense of shorter scattering times than lasers based on transitions diagonal in real space, are more sensitive to backfilling. '

38

JEROME FMST ET

AL

In a similar fashion, the slope efficiency per facet dP/dI is dP dl

1 dS 1 hv CI, N,hva, - = - N, 2 dJ 2 q CI,+CI,

-= -

~

where hv is the laser photon energy. b. Injection E$ciency

In the rate equations just described, we assumed implicitly that the injection efficiency is unity into state 3 (i.e., each electron leaving the injector is going into state n = 3). This assumption may be wrong in the presence, for example, of strong elastic or inelastic scattering, which may scatter the electrons directly into state n = 2, or if the injector is not well aligned energetically to state n = 3. We then must assume that only a fraction of the electron qin is injected into the n = 3 state, with the remaining fraction (1 - sin)being injected into the n = 2 subband. A nonunity injection efficiency has a strong influence on the threshold current density (written here at T = 0):

and on the slope efficiency

dP di

-

N, 2 a,

CI,

+ ct,

Equations (19) and (20) have little predicting power since the injection efficiency is not known a priori. However, they are relevant by stressing the importance of a close-to-unity injection efficiency. As an example, for typical values of the parameters, the threshold increases by 50% for qin = 0.8 and goes to infinity for rin<0.45. Similarly, the injection efficiency has a very strong effect on the differential efficiency. c.

High-Frequency Response

Although we do not have any experimental data on the high-frequency response of a QC laser, some theoretical insight can be gained very easily

1

QUANTUM CASCADE

LASERS

39

from the rate Eq. (16). The small-signal frequency response around an operating point defined by a current density J, and photon flux So is derived from Eq. (16) (Yariv, 1988). Neglecting the second-level population, and using the definition of the stimulated lifetime z,li; = gS, and photon lifetime T& = (+)a, the modulus square of the frequency response is

where the normalized frequency

has been used. As expected, the classical result for a semiconductor laser (Yariv, 1988) is recovered by setting T~~ = co in Eq. (23). Note first that the high-frequency rollover point (w'= 1) does not depend on z j 2 and has the same value than for a regular semiconductor laser. However, as soon as Zphot 2 2 3 2 , the prefactor of the d 2 is positive and there is no resonance o’= 1 regardless of the value of T , (i.e., ~ regardless ~ ~ of the ouput power). In the devices considered here, Tphot/TsZ z 2-10 and so the frequency response is strongly damped. This damping shows very clearly in Fig. 22, where we compared h(w) for different values of Tstim comparing the cases Tphot z 7 3 2 (QC laser) and Zphot << T~~ (conventional semiconductor laser). In contrast to the conventional semiconductor laser, the QC laser has a damped frequency response for all output powers. This behavior can be understood from the equations in the following way. In the conventional semiconductor laser, only the photon population is damped by the cavity losses, expressed by the constant ac/n prefactor in Eq. (16c). The carrier population is only damped through its coupling to the photon population. This is why the resonant behavior of the frequency response appears at low powers (i.e., when T , >~Tphot), ~ that ~ is, when the frequency response, controlled by the longest time constant, is undamped. On the contrary, in the QC laser case where Tphot Z 7 3 2 , both electron and photon population are independently damped regardless of the output power. Since the photon lifetime and T~~ are typically a few picoseconds and the stimulated emission time is much less than 1 ps, above laser threshold, one concludes that it should be possible to modulate QC lasers at frequencies of several hundreds of gigahertz and possibly up to -1 THz (assuming negligible parasitics). This modulation bandwidth is more than one order of magnitude greater than that of the fastest diode lasers. Similar conclusions have been reached by Mustafa et al. (1999) through a rate-equation analysis.

40

JEROMEFAIST ET AL.

-

1

2 0.1

r

0.01

....... ---

%ti 'stim= 'phot

.""

.

0.01

= '32 1

0.1

10

w FIG. 22. Small-signal frequency response of the QC laser; w' is the modulation angular frequency of the laser normalized by the geometric mean of the stimulated emission rate &,,,-I and the reciprocal of the photon lifetime T ~ T~~ ~is the ~ scattering ~ ; time from state 3 to state 2. In the limit of very long T ~ the ~ typical , resonant frequency response of a conventional semiconductor laser is recovered.

IX. Optimization of the Vertical Transition Laser and Continuous Wave Operation Considerations based on the rate equations, and more precisely, on the temperature-dependent threshold current density of Eq. (19) lead us to believe that the early lasers based on a vertical transitions suffered from the adverse effect of backfilling. The combination of a relatively large doping n, corresponding to an injector sheet density n, = 4.5 x lo1' cm-2 and a low value of the separation between level n = 2 and the Fermi energy in the injector (A z 35 meV) lead to a large population of thermally activated carrier in the latter state, and, therefore, to the relatively poor performance as a function of temperature displayed in Fig. 20. Further design changes (lengthening the injector by adding more well-barrier periods to the superlattice) lead to an increase of this value to A E 100meV (Faist et al., 1995b). The general philosophy of the design including a vertical transition and a superlattice injector tailored for Bragg reflection at the energy of state n = 3, however, remained the same. We plot the temperature dependence of the threshold current density for devices with various values of A in Fig. 23. The improvement in performance with A is quite striking, and is attributed to the progressive decrease in backfilling. The values of A that we report in

1 I

n N

P v

rn Q) L c I-

I

-

5-

-0

0

I

0 --

JZ

QUANTUM CASCADE LASERS

/ I

~

'

~=30me~ A A

'

'

'

~

'

'

=1OOme"

T,=100K

1

1

1

1

1

1

1

1

1

"

-

2p 1

:

41

1

1

l

1

FIG. 23. Measured threshold current density in a vertical transition QC laser (Fig. 17), as a function of temperature, for various values of A, the energy separation between the ground state of the injector and the n = 1 level of the preceding active region.

Fig. 23 for the first two samples are lower than the one we reported in our original publication (Faist er al., 1995a). The reason is that we originally quoted the value computed from the band diagram at threshold. We felt later that it was more accurate to estimate directly A from the measured bias U at threshold, assuming that the Fermi energy of the injector is lined up with the state n = 3 at this bias. Then A is simply obtained by the relation A = ( U - hv/q)/N,, neglecting the small series resistance. Samples with large values of A enabled us to operate our device in continuous wave above liquid nitrogen temperature with tens of milliwatts of optical power, as shown in Fig. 24.

1. S PECTRAL CHARACTERISTICS

The continuous wave optical spectra of these lasers were measured with a high-resolution (0.09 cm- ') Nicolet Fourier transform infrared spectrometer in rapid-scan mode. In the vast majority of the temperature and current ranges, the spectrum was found to be monomode, with sidelobes below 20 dB, limited by our spectrometer's resolution. Setting the injected current at a fixed value I = 450 mA, the typical tuning curve of a 2.25-mmlong device as a function of holder temperature from a laser is displayed in Fig. 25. The total continuous tuning range is 1.8cm-', or 4nm. Three

42

JEROMEFAIST ET AL.

20 15

zE

v

10 $

B

5

0

0.2

0

0.4 Current (A)

0.6

a

n-

FIG. 24. Continuous optical output power from a single facet versus injection current for various heat sink temperatures, as indicated. The laser stripe is 9 pm wide and 2.25mm long.

-

2147

2149 2151 2153 Wavenumber (cm-')

-

FIG. 25. Tuning curve of a vertical transition QC laser versus temperature for a fixed injection current I = 450 mA. The spectrum is monomode in the whole temperature range. Inset: three representative spectra at T = 12, 25, and 35 K. The sidelobes on either side of the main peak are artifacts of the Fourier transform used to compute the spectrum.

1 QUANTUM CASCADELASERS

43

typical spectra are also reported in the inset. The device is also tunable with current, in which case a total continuous tuning range of 1.2cm-' at T = 13 K is obtained. The gain below threshold can be evaluated through a measurement of the Fabry-Perot oscillations in the luminescence spectra using the HakkiPaoli technique (Hakki and Paoli, 1975). This technique measures the peak net modal gain minus waveguide losses G, - mW or net gain. The results for a 1.3-mm-long device at T = 15 K are reported in Fig. 26. Using the

a9. E

.-c mm

P

C .-

mCJ)

Ia Current density ( W d ) FIG. 26 (Upper) Peak net modal gain G , measurements using the Fabry-Ptrot fringes of the luminescence spectra for increasing current. The temperature is 15 K and the device length 1.3mm. (Lower) G , as a function of current density.

44

JEROME FAIST ET

AL.

Kramers-Kronig transformation of this absorption data, one also obtains the change of the real part of the refractive index. From this data, we deduced a linewidth enhancement factor (Henry, 1982) c( = An'/An" < 0.1 at the lasing wavelength. This low value was predicted for our laser, which exhibits a delta-like joint density of states characteristic of gas lasers. It is a very important feature of such a laser since it implies the ability of a modulation with low chirp and a narrow linewidth. The plot of the peak gain versus injected current density J (Fig. 26) shows a linear dependence with a slope of 9 cm/kA. The extrapolation at J = 0 yields a waveguide loss of 12cm-'. As mentioned, we do not have a good explanation for the difference between this value and the calculated value (5 cm- ').

X. Vertical Transition QC Laser with Funnel Injector and Room-Temperature Operation As shown in Fig. 23, significant improvements were obtained in the vertical transition design by increasing the value of A. However, for A > 100meV, no further improvement was obtained. Further increases in the value of A had the following drawbacks: (a) by lengthening the injector, it reduced the number of periods that could fit under the optical mode; this translated into a reduction of the gain per period; and (b) the bias at threshold is increased, which then increased heat dissipation and the electron heating. The choice of a large A was dictated by the relatively large doping of the injector, the latter being required by a relatively thick injection barrier. At this point, a radical change in design was decided, as shown in Fig. 27 (Faist et al., 1996). The injection barrier thickness was strongly decreased to about 5 nm. The strong coupling between the ground state injector wave function and the upper state of the lasing transition enabled a reduction of the integrated sheet density introduced by doping to about 1-2 x 10'l cm-' while keeping the maximum injection current density to J 10-15 kA/cm-' as required for high-temperature (300 K) operation. The value of A was kept to about 100 meV and, due to the small doping, the backfilling was kept to a minimum. In addition, compared with previous lasers based on a vertical transition this new design has an additional 0.9-nm-thick CaInAs quantum well coupled to the active region by a 1.5-nm barrier, which selectively enhances the amplitude of the wave function of level 3 in the 5.0-nm injection barrier (Fig. 27). This maximized the injection efficiency by increasing the overlap between the n = 3 wave function and the ground state wave function g of the injector while reducing that of the latter with the

-

45

1 QUANTUM CASCADE LASERS

Injection barrier

(b)

- w2OPm-3 I

graded

30

,=7yin18

n=blol* AllnAs

~

0

1

1201) 7

3x1017

graded InGaAs

700 601)

n = 2 ~ 1 0 ' ~ 30 n = 1 ~ 1 0 ' ~ 300

InP

,,,,,,,,,Mn!L--zQ. graded InGaAs

n = 2xIOl7 n=1xlnl7

30 400

Active region (25x) n = IxIO" graded n=2x1017

1200 400 25

InGaAs

Active region (25x) InGaAs n= lxIOl7 graded n=2x10L7 1nP

1200 300 25

n = m 1 8 c m - 3 130Onm ~ 0 1 7 1500

InP

n =2 ~ 0 ' ~

n = 1x10'~

FIG. 27. (a) Schematic conduction band diagram of one period of the 1 = 5.2 pm QC laser optimized for room-temperature operation at an electric field of 7.6 x 104V/cm. The layer sequence of one period of the A1,,,,Ino,,,As-Gao,,,In~,~3As structure, in nanometers, left to right and starting from the injection barrier is (5.0-0.9), (1.5-4.7), (2.2-4.0), (3.0-2.3), (2.3-2.2), (2.2-2.0), (2.0-2.0), (2.3- 1.9), and (2.8- 1.9). The structures are left undoped, with the exception of layer numbers 14 to 16 for sample D-2122 and 11 to 13 for sample D-2160, which are n-type doped with Si to N , = 2 x 10'7cm-3. (b) Layer structure of the waveguide of the reference sample D-2160. (c) Layer structure of the waveguide of the sample D-2122. The dashed line indicates the interface where the growth was interrupted. Active region here indicates the total stack of 25 periods.

46

JEROME FAIST ET

AL.

n = 2 and n = 1 states by elastic or inelastic scattering. However, in contrast to QC structures based on a diagonal transition, the presence of this additional 0.9-nm well did not reduce the oscillator strength of the lasing transition ( z 3 2= 1.6 nm). As in previous designs, the chirped superlattice of the relaxation-injection region acts as a Bragg reflector to suppress the escape of electrons from the n = 3 excited state into the continuum while allowing their extraction from the lower state (n = 1) into the miniband of the relaxation region. However, the width of this “miniband decreases toward the 5.0-nm barrier to minimize injection into the active region of electrons thermally excited to higher states of the miniband (Fig. 27). This has the effect of funneling electrons into the ground state g of the relaxation-injection region. Ternary materials such as AlInAs have a thermal resistance 15 to 20 times larger than binaries such as InP. We compared devices with two different optical waveguide designs. The reference sample (D-2160), grown in one step, uses a top AlInAs cladding similar to the one of previous QC lasers (see Fig. 27b). In the second device (D-2122) and as shown in Fig. 27c, the first MBE growth is stopped after the growth of the core of the waveguide is completed. The sample is then arsenic-capped and transferred in a second MBE chamber fitted with a phosphorus solid source. There, the arsenic is thermally desorbed under ultra-high-voltage (UHV) conditions and the growth of a InP cladding is immediately started. The resulting interface is sharp and free of defects as shown by the transmission electron microscopy cross section of Fig. 28.

1. PULSED OPERATION Figure 29 shows the optical power versus drive current from a single facet of the InP-clad sample D-2122 obtained using f/O.8 optics and a calibrated, room-temperature HgCdTe detector. Maximum peak output powers of about 200 mW at 300 K and 100 mW at 320 K are obtained. The slope efficiency, dP/dI = 106mW/A at 300K and 78mW/A at 320K, is only slightly less than one half of its low-temperature value (dP/dI = 250 mW/A). In comparison, at T = 300 K, sample D-2160 (AlInAs cladding) exhibits about one-tenth of the power levels and slope efficiencies. We attribute this difference, which appears only above T 200 K, to the enhanced heat removal during the pulse by the top InP cladding as compared to the AlInAs cladding. The devices were also tested at room temperature in pulsed mode with a relatively large (3.3%) duty cycle. The pulse length was 5011s and the repetition rate was 670 kHz. The light was collected by a nonimaging energy

-

1 QUANTUM CASCADE LASERS

47

FIG. 28. Transmission electron micrograph of the center of the waveguide of sample D-2122. An arrow indicates the interface between the growth of the InGaAs- A1InAs stack and the subsequent InP regrowth. The regrowth process is free from visible dislocations or defects.

48

JEROMEFAISTET AL. 300 h

P

E

v L

g 200 0

a

0 0

2

4

6

Current (A) FIG.29. Collected pulsed optical power from a single facet versus injection current for heat sink temperatures of T = 300 K and T = 320 K and for the InP-clad sample D-2122 (stripe dimensions 14pm x 2.9mm). The collection efficency is estimated to be 70%. Inset: Same characteristics, but for sample D-2160 (9 pm x 3 mm).

concentrator and its average intensity measured by a broadband laser power meter. As shown in Fig. 30, the high peak power observed in sample D-2122 translates into an average power in the 2- to 10-mW range.

2. THRESHOLD CURRENTDENSITY In Fig. 31, the threshold current density Jthis plotted as a function of temperature for both samples D-2160 and D-2122. At T = 300K, the threshold current densities are very similar for both samples and are about 8-10 kA/cm2. Between 160 and 320 K, the data can be described by the usual exponential behavior J exp(T/T,) with an average To = 114K. These performances show a very significant improvement over ther previous results on QC lasers with a vertical transition and a thick injector barrier shown in Fig. 23. On the same graph, we also reported the computed threshold current density Jth [Eq. (21) taking qin = 11. In this expression, CI = a, + a, with CI, = -ln(R)/Lcav = 4.5 cm-' for L,,, = 3 mm, CI, = 10 cm-' extrapolated from an analysis of the subthreshold luminescence, 1 = 5.2 ,urn, L, = 45 nm, rp= 0.022, N , = 25, and n = 3.22. We assumed that the gain has a broadening 2y,, given by the experimental FWHM of the luminescence. We

- -

49

1 QUANTUM CASCADE LASERS

F E 12

v

ti

280K

-

/ /

Pulse length 5011s Duty cycle = 3.3%

-

m 0

.c

/ 3

0a, "4 :rn

2

9

L-,

FIG. 30. Average power as a function of current for sample D-2122 (14 pm x 3 mm) for various temperatures.

obtain a relatively good agreement at low temperatures especially for device D-2160. However, our model predicts a larger To of 153K. Hot electron effects that translate into a higher population of state n = 2 are the most probable explanation for this discrepancy.

10 D-2122

- 5

1

c

2-- 2

To = 114K

.

' D-2160

a Fh

...a

..Ar

7

1

0.5

0

100

200

300

400

Temperature (K) FIG. 31. Threshold current density in pulsed operation for samples D-2122 and D-2160 as a function of temperature. The solid line is the theoretical prediction. The dotted lines indicate the range over which the To parameter is derived.

50

JEROMEFAISTET AL.

3. HIGH-TEMPERATURE, HIGH-POWER CONTINUOUS WAVEOPERATION Continuous wave (cw) operation of these devices is shown in Fig. 32. The devices operate cw up to 140 K for sample D2122 and 121 K for sample D-2160 with a few milliwatts of optical power. Similar devices coming from a different growth run (D2161) displayed extremely large optical powers (Faist et al., 1998b). Figure 33 shows the optical power versus driving current from a single facet of one such lasers. In this measurement, the light is collected by a parabolic aluminum cone acting as a nonimaging energy concentrator with near unity collection efficiency, and the intensity is measured with a broadband laser power meter. A maximum power of more than 700 mW is achieved at 22 K for a drive current of about 2.1 A, corresponding to a current density J 6.2 kA/cm2. The slope efficiency per facet dP/dZ in the single mode portion of the L-I is 582 mW/A, with a lasing threshold of approximately of 1.3 kA/cm2. This large value of slope efficiency is a direct consequence of the cascade geometry of the structure which allows more than one photon emitted per electron above threshold. The value of the slope efficiency agrees very well with the value (624mW/A) predicted by Eq. (20), using the measured value of the waveguide loss

-

30

20

10

0 0

0.2

0.4

0.6

0.8

1

Current (A)

FIG.32. Continuous optical output power from a single facet versus injection current for various heat sink temperatures: (a) sample D-2122 (7 pm x 2.9 mm) and (b) sample D-2160 (9 pm x 3 mm).

51

1 QUANTUM CASCADE LASERS 0

2

J (kA/cm2)

4

6

I4

$0

a

Current (A) FIG.33. Continuous wave light-current (L-I) curve from a single facet of a laser (12 pm x 3 mm) from sample D-2161, of design identical to D-2160, at a temperature of 22 K. In the inset, the corresponding “wall plug” efficiency of the device is reported.

13.8 cm-’ (Faist et al., 1998b). The “wall plug” efficiency (ratio of optical power to electrical power) is plotted in the inset of Fig. 33 as a function of injected current: a maximum of about 8.5% is obtained at -1.7A. A clear feature in the L-I characteristic emerges around 1.2A (400mW); the spectral analysis of the emission shows that it is related to the laser switching from single-mode to multimode operation. c1 =

PROPERTIES 4. SPECTRAL

The spectra properties of the same laser, measured using a high-resolution (0.1 cm-’) FTIR in rapid-scan mode are plotted in Fig. 34. In the region of single mode operation, which extends to more than 300 mW at low temperatures (10 K), the sidelobe suppression is greater than 20 dB, prob-

JEROME FAISTET

52

0

10

20

30

40

AL.

50

60

70

80

90

Temperature (K) FIG. 34. Spectral characteristic of emission for a laser as a function of substrate temperature and emitted power.

ably limited by the resolution of the spectrometer. Note, however, that the boundary between single-mode and multimode operation is not very sharp and exhibits some hysteresis. The measurement of the spectral linewidth of the laser operating in the single-mode regime ( T 20 K, I = 500 mA) with a plane-parallel Fabry-Ptrot interferometer gives a value of 70 Mhz. This value is much larger than the Schawlow-Townes theoretical limit (-kilohertz). This measurement is still limited by the resolution of our spectrometer.

-

XI. Long-Wavelength (A >, 8 pm) Quantum Cascade Lasers After our demonstration of the first quantum cascade laser operating at

1 = 4.3 pm, the first question that arose was whether it would be possible, by a modification of the design, to operate the QC laser in the second atmospheric window (8-13 pm). The answer to this question was very

1 QUANTUM CASCADE LASERS

53

important because it would determine whether the main claim of the QC laser was true: the possibility to design lasers with very different operating wavelengths from of the same semiconductor material. From the beginning, it was quite clear that the longer wavelength would actually bring some advantages to the quantum design. The states would be deeper in the band, with a lighter effective mass and thus a larger oscillator strength and wider wells meant a lower sensitivity to interface roughness. However, the waveguide losses would increase significantly [the free-carrier loss term grows approximately with R2, see Eqs. (11)-(12)] and the lifetime tg limited by optical phonon emission would decrease, along with t32;because the momentum exchanged during the scattering would be decreased. This would increase the threshold current density fthor equivalently reduce the population inversion (J/q)z3(1 - t2/tg2)for a given J . The problem of the optical losses was solved by refining the original waveguide design and using ~ ) substrates. The problem of the reduced low-doped ( n 1017~ m - InP lifetime and population inversion can be alleviated using a diagonal transition with upper state anticrossing (Sirtori et al., 1995), as discussed in a previous section, which gives upper state lifetimes of a few picoseconds. Designing a structure with vertical transition at comparable wavelengths (1 8 pm) appears at first sight to be very difficult since in this case z3 < 1ps. This problem was solved by using a suitably optimized coupledquantum-well active region, in which the nonradiative relaxation by emission of optical phonons, is split into two paths (3-2 and 3-1) of comparable relaxation times of which only one (3-2) fills the lower state of the lasing transition. In addition, 721 can be made shorter than in a structure with diagonal transition by increasing the spatial overlap of states 2 and 1. For this reason, the ratio t 3 2 / ~ 2 1is still very favorable (- 10) even for a laser based on a vertical transition at long wavelengths. Indeed, a laser based on a vertical transition was demonstrated with a lasing wavelength as long as 11 pm (Sirtori et al., 1996a).

-

-

WAVEGUIDE 1. PLASMON-ENHANCED As already mentioned, the problem was to design a waveguide that would provide at the same time low waveguide losses and keep the thickness of the grown MBE material to a reasonable value. In particular, it was realized that growing on a low-doped ( n 1017cmW3)InP substrate instead of the doped substrates available at that time, which had a doping level in the high 10l8cm-3 would bring an enormous advantage. A highly doped substrate would necessitate the growth of a lower AlInAs cladding about 2 pm thick.

-

54

JEROMEFAETET AL.

This layer would increase the growth time, add to the thermal resistance, and dramatically increase the number of defects below the active regmn. However, since these low doped substrates were commercially available only by mid-1995, the first samples were grown on samples that had been grown in-house during a previous study. As for the longer wavelengths, it was imperative to decouple the guided mode from the interface plasmon propagating at the interface between the top metal contact and the semiconductor, which is also TM polarized. However, to limit the thickness of the top cladding layer, it was found that it was much more efficient to decrease the refractive index of the top layer, instead of increasing it as we did in the first QC laser. At these long wavelengths, a low-refractive-index layer can easily be obtained using the dispersion provided by a heavily doped semiconductor layer. The schematic of a waveguide designed using these design rules is shown in Fig. 35 (Sirtori et al., 1995). As shown in this figure, the growth finishes with a 600-nm-thick GaInAs layer Si-doped to n = 7 x lo1*cm-3. The calculated plasma frequency for such a doping level corresponds to a wavelength A = 9.4 pm. The anomalous dielectric dispersion near the plasma frequency (Jensen, 1985) strongly depresses the refractive index of the layer at the laser wavelength of A = 8.4pm from n = 3.5 refractive index of an undoped layer, to n = 1.1, as shown in Fig. 35b, where the calculated optical mode is displayed along with the refractive index profile. This low refractive index has a twofold advantage: it completely decouples the guided mode from the interface plasmon and increases the confinement factor of the mode by “pushing” the mode away from the metal contact. The calculated waveguide loss for such a waveguide is only a, = 7.8 cm-’. Waveguide loss measurements performed with the HakkiPauli technique in similar waveguide at such wavelengths show, however, much higher losses (a, = 30 cm- ’) (Sirtori et al., 1996b). The origin of this discrepancy is still not known. Part of this discrepancy could arise from material and substrate quality since more recent results tend to show somewhat lower losses (a, = 24 cm- ’) (Gmachl et al., 1998a). This waveguide design was kept in all subsequent QC lasers. The only significant modification was to increase the thickness of the confining InGaAs layer adjacent to the active region with a concomitant decrease of the AlInAs top cladding layer. Even if this modification does not lead to an improvement of the calculated waveguide loss, it is believed to improve the device performance since, in general, InGaAs has a much better material quality (lower defect concentration) than AlInAs and therefore can be doped much more lightly. This modification was especially important for the realization of the QC laser based on GaAs material because AlGaAs cannot be doped below 2-5 x loi7 cmP3 without encountering carrier freezeout problems (Sirtori et al., 1998a).

-

55

1 QUANTUM CASCADE LASERS n++ n

GalnAs

n = 7 x 1018 ~ r n - 600 ~ nrn

AlGalnAs Graded

5 x 1017

n

AllnAs

5 x 1017

20

n

AllnAs

3x1Ol7

1200

n

AllnAs

1.2 x 1017

1200

n

AllnAs

1 x 1018

n

AlGalnAs Graded

2 x 1017

n

GalnAs

6 x 1016

n

Injector

undoped

- (a)

30 al

.’Do)

.r

> m

2.p

10 40

500

1 . 5 ~ 1 0 ~ ~ 19.6

Active region

3

m'D a-0

T 2

7

0 0

27.3

n

GalnAs

6 x 1016

700

n

AlGalnAs Digitally graded

1.2 x 1017

25

i

h

cn

c

X

3 *

c -

1

FIG. 35. (a) Schematic cross section of the quantum cascade laser operating at I = 8.4pm. (b) Refractive index profile of the waveguide structure. The calculated profile of the fundamental mode is also shown.

56

JEROMEFAIST ET AL.

2. QUANTUM DESIGNOF A QC LASERWITH DIAGONAL TRANSITION AT II = 8.4 pm To a large extent, the first active region design (Sirtori et a!., 1995) followed the same design philosophy of the first QC laser at II = 4.3 pm. As shown in Fig. 36b, the structure is based on an periodic repetition of a doped injector followed by a three-well active region. At the threshold current, the structure is designed in such a way that the upper state of the lasing transition (state n = 7 in Fig. 36b) is anticrossed with the next upper state ( n = 8 in Fig. 36b). In addition, we noted that a structure based on an upper state, which is anticrossed, results in a narrower luminescence linewidth. This is confirmed by the data of Fig. 36a, which show a FWHM of 10 meV. To ensure population inversion the wave functions and energy levels are designed so that the electron intersubband scattering time from the higher ( n = 7) to the lower ( n = 5) state of the laser transition (z7, = 3.4 ps), which is dominated by optical phonon emission, is longer than the scattering time between the n = 5 and n = 4 energy levels. The latter is estimated to be d 1 ps due to the near resonance of the energy difference E,, with the typical optical phonon energies ( 30 meV) in the AlInAs-GaInAs alloys. The fact that the n = 7 and n = 5 states are not centered in the same well (“diagonal” transition) enhances the relaxation time z7, and the nonradiative lifetime (z7 = 2.7 ps) of the upper state, which increases the population inversion at given injection current density. The calculated matrix element for the 7-5 radiative transition is 1.7 nm. As shown in Fig. 37, these first devices exhibited already a low threshold current density of 2.1 kA/cm2 at T = 10 K and operation up to T = 130 K in pulsed mode. This low value of threshold current at low temperature allowed continuous wave operation in these devices (Sirtori et al., 1998b). However, the poor substrate quality prevented the manufacture of a sufficient number of devices. Good cw operation was however realized with qc lasers based on a vertical transition (Sirtori et al., 1996b).

-

-

QUANTUM CASCADE LASERBASEDON 3. LONG-WAVELENGTH VERTICAL TRANSITION

A

The improvement in performance obtained with the lasers based on a vertical transition at shorter wavelengths stimulated the development of similar structures at longer wavelength. The optimized design is shown in Fig. 38 (Sirtori et at., 1997a). Compared to an earlier vertical transition version (Sirtori et al., 1996b), it has a significant improvement: the strong

1

-

QUANTUM

57

CASCADE LASERS

(4 1.0

.-

I

c

2

Y

5 0.5

2 -

.-8 c

0

50

100 150 200 Photon Energy (rnev)

250

n r

1'

FIG. 36. (a) Electroluminescence spectra of a quantum cascade structure with diagonal transition at L = 8.4 pm and five active-plus-injector regions with no cladding layers, and processed into a mesa, to avoid gain effects. The drive current is 150mA corresponding to a bias electric field across the active region of 6 x 104V/cm. The position of the (7-5) peak agrees with the calculated separation between the n = 7 and n = 5 energy levels ( E 7 5 = 145.4meV). The shift between the two peaks in the spectrum at 200 K is in excellent agreement with the calculated energy difference between the n = 8 and n = 7 anticrossing states ( E 8 , = 41.7meV). (b) Conduction band diagram of a portion of the Ga,~,,In,,,,AsAlo,481no,5zAsquantum cascade laser with diagonal transition at under positive bias corresponding to an electric field of 6.0 x 104V/cm. The design is identical to that of the electroluminescence sample active regions and injectors. The solid curves are the moduli in the active region and in the injector. The baselines of the squared of the wave functions /$Iz indicate the positions of the energy levels. The states of the laser transition are labeled 7 and 5. Electrons in the ground state 1' of one of the injectors tunnel into state 7 of the adjacent active region through the 4.5-nm-thick AlInAs barrier. The 25 active regions each comprise three GaInAs quantum wells of thicknesses 3.5, 7.5, and 5.8 nm, from right to left, separated by 2-nm-thick AlInAs barriers. Electrons tunneling into the next injector through a 2-nm-thick AlInAs barrier encounter in sequence 4.8-, 4.2-, 4.0-, and 4.3-nm GaInAs quantum wells, separated by 0.5-, 0.8-, and 1-nm AlInAs barriers, respectively. The calculated energy separations are El, = 36.8 meV, E , , = 52.6 meV, E,, = 25.5 meV, E45 = 29.9 meV, E , , = 21.8 meV, E,, = 123.6meV, and E , , = 41.7 meV.

58

JEROME FAIST ET AL.

Current (A) FIG. 37. Measured peak optical power as a function of injected drive current for various temperatures for the QC laser of Fig. 36b operating at 1, = 8.4pm.

reduction of thickness of the barrier coupling the two quantum wells of the active region from 1.5 to 1.0 nm. This modification leads to a much stronger coupling of the lower n = 1 and n = 2 states of the active region, maximizing their spatial overlap (Fig. 38) and leading to a further reduction in zzl (=0.2ps) compared with a diagonal transition structure ( T ~ 1 ~ps) or vertical one with thicker barrier (zzl = 0.4 ps, as in the work of Sirtori et al., 1996b). It also enhances T~~ to a value approximately double ( 1 . 4 ~ s )the lifetime z3( = 0.8 ps) by splitting the nonradiative channel from the upper level of the lasing transition n = 3 into paths of comparable strength to the n = 2 and n = 1 states without changing significantly the n = 3 state lifetime. This design strategy maximizes the population inversion An for the vertical transition design although An is still smaller than that achieved in a diagonal transition structure (by a factor s 2 compared with the QC laser of Fig. 36) for the same current density. This factor, however, can be more than compensated by the enhancement of the matrix element of the vertical transition ( z =~ 2.6 ~ nm in Fig. 38 compared to z75 = 1.7 nm in Fig. 36). The net effect is that the threshold current densities [Jth= (aw + ol,)/gT), for the same total loss (waveguide and mirrors) to confinement factor (r) ratio, should be comparable in the designs of Figs. 36 and 38. This is indeed confirmed by experiments, which give -2kA/cm2 for the threshold at

-

1 QUANTUM CASCADE LASERS

59

F = 68kVlcm

lniector Active Region

h

r

..

..

-.

FIG. 38. Schematic conduction band diagram of a portion of the Ga,,,,In,,,,As1 = 8.4pm quantum cascade laser based on a vertical transition under an A10,481no,52As applied positive bias corresponding to an electric field of 6.2 x 104V/cm. The solid curves represent the moduli squared of the relevant wave functions in the active region. In the injector, the dashed lines indicate the positions of the energy levels. The latter are expected to be broadened by impurity scattering and interface roughness. The states of the laser transition are labeled 3 and 2. The calculated energy level differences are E,, = 145 meV and E,, = 41 meV. Each of the 25 periods comprises the following layer sequence (AIInAs-GaInAs), in nanometers, left to right and starting from the injection barrier: (4.5-KO), (1.0-5.7), (2.4-4.4), (1.4-3.6), (1.2-3.6), (1.2-3.4), and (1.0-3.4). The structure is left undoped with the exception of the 4 layers in the center of the injector C(1.2-3.6), (1.2-3.4)] region, which are n-type doped with Si to obtain N , = 2 x lo', C I K ~ The . waveguide design is based on the plasmon enhanced confinement previously discussed. The lower cladding layer is the InP substrate.

cryogenic temperatures in the two structures (Sirtori et al., 1995; Sirtori et al., 1997a). It is worth noting that in coupled-well active regions of the type shown in Fig. 38, the state n = 2 has a significant oscillator strength only to the states n = 1 and n = 3. On the other hand, we know that in an isolated well, the oscillator strength of the 1-2 transition fij" is almost equal to the maximum oscillator strength available from the n = 1 state A,, = Xifli. Keeping in mind that f12 = -fil, we conclude immediately than compared to the single well case, the oscillator strength of the 3-2 transition is increased by an amount approximately equal to the oscillator strength

60

JEROMEFAISTET AL.

between states 2-1 (i.e., f 2 3 2 ff’j” + f i 2 ) . The reduction of the coupling barrier thickness in a vertical transition structure will increase this oscillator strength, decreasing the threshold current density. One could describe the superlattice active region QC laser (Scamarcio et al., 1997) as an extreme case where one adds not one but many levels below the lower state of the laser transition, increasing even further the oscillator strength of the laser transition. Another modification in the design shown in Fig. 38 to previous vertical designs is the reduction of the injection barrier thickness. This increase in the coupling of the n = 3 state to the injector follows the same philosophy than already described in the paragraph dedicated to the description of the room temperature design at 2 = 5 pm. The advantages brought by the so-called “strongly coupled injector” are discussed in more detail in (Sirtori et al., 1998a). This analysis, confirmed by experiments, shows that in an injector design optimized for high peak power in the L-I curve, the maximum possible resonant tunneling current density is J,,, E qn,/2z3 (i.e., the current is limited by the injector doping and the lifetime of the upper state). In this design, the tunnel splitting between the ground state of the injector and the upper state of the laser transition is large enough that they remain in resonance for a large range of drive currents. At the same time, the splitting cannot be too large, otherwise the laser transition would lose oscillator strength to the transition between the ground state of the injector and the lower state of the laser transition. The best compromise is a splitting of the order of half the luminescence width of the laser transition (i.e., E 5 meV for the design of Fig. 38). From the preceding discussion it should be clear that the vertical transition design has some key advantages over the diagonal one, for the same wavelength, losses, and cavity design. First the slope efficiency is higher 2 Second because [see Eq. (20)] of the significantly smaller ratio ~ 2 1 / ~ 3ratio. they have a higher power, for the same doping sheet density in the injector, since they can be driven to higher currents without the injector going out of resonance with the active region, since the lifetime of the upper state of the laser transition is considerably shorter. If the structure is designed correctly, vertical transition structures have a threshold current density comparable to that of diagonal transition designs. These long-wavelength lasers based on a vertical transitions with an optimized design have demonstrated a very high level of performance. Shown in Fig. 39 are the optical power versus injected current of such devices at T = 300 K. The lasing threshold is below 3 A, corresponding to a threshold current density of 8 kA/cm2, the same order of magnitude than obtained in the shorter wavelength lasers. The devices operated still at T = 320 K.

1 QUANTUM CASCADE LASERS

61

Current (A) FIG. 39. Measured peak optical power from a single facet versus drive current at 300 K for two vertical transition QC lasers with different lengths: 1.8mm (a) and 2.5mm (b). Insets: high-resolution spectra at 3.0 A (top) and 3.6 A (bottom). The active region design is shown in Fig. 38.

Lasers operating at longer wavelengths (A = 11 pm) were made following basically the same design as the one shown in Fig. 38 (Sirtori et al., 1996a). The larger losses at this longer wavelength, however, did allow operation only up to a maximum temperature of 220K in pulsed operation. Figure 40 shows the operation of such a laser in continuous wave operation at T = 20 K. Optical output power up to 10 mW were obtained. These results demonstrated that a laser based on a vertical transition could be operated with good performances down to wavelengths longer than 11 pm. 4. ROOM-TEMPERATURE LONG-WAVELENGTH (A = 11 pm) QC LASER

A few design changes were then made to increase the maximum operating temperature of QC lasers with wavelengths between 11 and 12 pm (Faist et

62

JEROME FASTET AL.

6 A

2

F E

4 -

v

4.5 &

-

nB

Q)

3 : 0

>

2r

0’ 0

10.8

10.5

11.1

11.4

11.7

Wavelength (w) ‘

I

I

0.2

0.4

I

0.6 Current (A)

I

0.8

1

FIG. 40. Continuous optical power versus injected current for a device with vertical transition design operating at 1 = 11pm wavelength. The design used is very similar to the 1 = 8.4 pm device shown in Fig. 38.

al., 1998~).As shown in Fig. 41, we chose to use an active region consisting of three coupled quantum wells. For a long-wavelength laser, this choice has two main advantages: it allows a longer lifetime of the upper level of the lasing transition (n = 3) by reducing its overlap with the lowest states n = 1 + z;i)-’ = 1.3 ps and n = 2. The resulting lifetime of level 3 is z3 = ;7;( (with T~~ = 2 ps). Even if this is accompanied by a concomitant decrease in optical matrix element z 3 2 = 2.4nm (instead for z 3 2= 3.1 nm for lasers based on vertical transitions; Sirtori et al., 1996a) threshold current densities and slope efficiencies benefit from a more favorable population inversion between states 3 and 2 since the ratio of lifetimes is T 3 2 / T 2 = 5, instead of z32/z2 = 3.4 for the vertical transition design (note that t2 E zZ1 0 . 4 ~ ~ ) . In addition, this geometry will improve the injection efficiency by reducing the overlap between the ground state wave function of the injectionrelaxation region and the states 1 and 2 of the active region while maintaining a good injection efficiency into state 3, similar to the 5.2-pm room-temperature laser with a funnel injector. As in all our previous designs, the injection-relaxation region is designed with a minigap facing level 3 to prevent escape from the latter into the injector. Moreover, the width of the lower miniband decreases toward the injection barrier to “funnel” electrons into the ground state g of the relaxation-injection region.

63

1 QUANTUM CASCADE LASERS

injection barrier

U -‘ (b)

D-2286

In,&a,,As -GaamAs

InP

5x10’’

1100

7 ~ 1 0 ’ ~1100

InP

8x10”

2000

6x10’’

800

6x10‘’

35 stage active In,&a,,,As

0-2297

10

IxlOM

Ino~szAlo,,As 1.5~10” In, &anA,As

(C)

1Nlo

800 1676

6x10’’

700

In, &a,,

A7

As

stage active +35iniector @n In,&aOA7As

1675 6x10’’

700

InP substrate

InP substrate

FIG. 41. (a) Schematic conduction band diagram of one stage (active plus injector region) of the room temperature 1 = 11 pm QC laser structure under an applied electric field of 4.6 x 104V/cm. The energy separation between states 3 and 2 corresponds to I = 11 pm and that between level 2 and 1 is slightly larger than one optical phonon. The layer sequence of one period of structure, in nanometers, left to right and starting from the injection barrier where Ino,52Alo,48As is 4.2-3.4-0.8-6.8-0.9-6.2-2.8-4.21.0-4.0-1.0~3.9-1.0-3.9-1.1-3.9, layers are in bold, In,,,,Ga,,,,As layers are in roman, and underlined numbers correspond to doped layer with Si to N , = 2.5 x 10” ~ m - (b) ~ ,and (c) Layer sequences of the waveguide. The layer thicknesses are in nanometers. The dashed line indicates the interface where the growth was interrupted.

To maximize the gain, the active region consisted of 35 stages. As shown in Fig. 41, two different optical waveguides were grown. They consists of the InP substrate on one side and of AlInAs (sample D-2286) or InP (sample D-2297) grown by solid-source MBE. As discussed earlier, the latter has a lower thermal resistance than the random alloy, enabling larger powers. Figure 42 shows the optical power versus drive current from a single facet of the InP-clad sample D-2297 obtained using f/O.8 optics and a calibrated,

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JEROMEFAISTET AL.

2

0

6

4

8

10

Current (A)

10

5 7

2

1

t 0

* '-

+ 0

1

D2286 L = 1.5 mm D2286 L=0.75mm D2297 L=2.25mm

100 200 Temperature (K)

300

FIG.42. (a) Collected pulsed optical power from a single facet versus injection current for heat sink temperatures of T = 275, 300, and 320K and for the InP-clad sample D-2197 (28 pm x 2.25 mm). The collection efficiency is estimated to be 50%. (b) Threshold current density in pulsed operation for samples D-2286 (ternary cladding) and D-2297 (binary cladding) as a function of temperature. The solid line is the theoretical prediction for sample D-2297. Cavity lengths are also indicated.

room-temperature HgCdTe detector. Maximum peak output powers of about 55 mW at 300 K and 35 mW at 320 K are obtained. In comparison, at T = 300 K, sample D-2286 (AlInAs cladding) exhibits very similar thresholds but about one-tenth of the power levels and slope efficiencies. We attribute this difference to the enhanced heat removal during the pulse by the top InP cladding as compared to the AlInAs cladding. In Fig. 42b, the

65

1 QUANTUM CASCADE LASERS

threshold current density Jthis plotted as a function of temperature for both samples D-2297 and D-2286. At T = 300 K, the threshold current densities are very similar for both samples and are about 8-10kA/cm2. Between -160 and 320K, the data can be described by the usual exponential behavior J exp(T/To) with an average To = 172 K. This value is similar to that reported ( T = 180 K) by an other group working on intersubband QC lasers (Slivken et al., 1997) at a somewhat lower temperature range T = 200-280 K and is much larger than that obtained for the competing technologies (lead-salts or 111-V). The gain was obtained from an Hakki-Paoli analysis of the subthreshold luminescence spectrum, yielding a value of the waveguide loss of IX, = 74 cm-'. This number allows a computation of the threshold current density Jth [see Eq. (19)], plotted along the experimental points in Fig. 42. The good agreement between calculated and measured high-temperature behavior shows that in these devices, our simple model, which does not include hot carrier effect (Gelmont et al., 1996), nevertheless predicts the correct thresholds. Physically, it means that the electron density is low enough for the carrier-carrier scattering to be ineffective in heating the electron population of the injector. It is maybe no surprise that we observe such a good agreement in a device where the voltage drop, and therefore the energy loss per period is the smallest due to the relatively small photon energy.

-

5.

SEMICONDUCTOR LASERS BASEDON SURFACE PLASMON

WAVEGUIDES

So far, for all the waveguides used for the QC lasers, we tried to avoid a coupling of the guided mode with the interface plasmon propagating at the semiconductor-metal interface. A radically different approach consists, on the contrary, of using this interface plasmon as the main guiding mechanism (Sirtori et al., 1998~).The main advantage of this approach is the complete suppression of the cladding layers. The losses are controlled by the correct choice of the metal. For electromagnetic surface waves at a metal-dielectric interface, the modal losses are strongly dependent on the dielectric constants of the materials. The attenuation coefficient a can be written as

where n and k are the real and imaginary parts of the complex refractive index of the metal, nd is the refractive index of the dielectric, and A is the

66

JEROME FAST ET AL.

wavelength in vacuum. In the previous equation, k is assumed to be much larger than n and nd. From this expression, it is apparent that the losses at the interface are inversely proportional to /2. and can be minimized by choosing metals having a refractive index with a strong imaginary component ( k >> n, i.e., the dielectric constant is almost real and strongly negative). Around the 10-pm wavelength, palladium is the most suitable material for our application since it has very large k and can be easily deposited. Its complex refractive index at the 11.3-pm wavelength, corresponding to the emission frequency of one of our laser structure, is ii = 3.85 + i49.2. Introducing in Eq. (25) the n and k values for Pd and the value of the refractive index of the active region (nd = 3.38) we obtain CI = 14cm-'; using the complex refractive index of gold we would obtain CI 30cm-'. In fact, our waveguide structure is more complicated than a pure surface waveguide and the simple Eq. (25) offers only qualitatively results, which are nevertheless important for the choice of the metal to use at the interface. A significant difference is caused by the InP substrate, which has a lower refractive index than the laser active region. This strongly reduces the penetration of the mode into the semiconductor material and enhances the overlap factor with the active region. To obtain a more accurate prediction of the shape and the losses of the confined TM interface mode a numerical solution based on the transfer matrix method has been used. The results of the mode calculations are presented in Fig. 43. It is important to note that in these waveguides, in spite of the strong increase of the optical losses, there is an enhancement of the overlap factor (r E 70%) between the guided mode and the active material over a regular dielectric waveguide (r z 40%) with the same thickness of waveguide core. The laser structure indicated in Fig. 43a corresponds to that of sample D-2295 and consists of a 700-nm GaInAs buffer layer grown on a low-doped InP substrate, followed by a 25-period QC active material. The active material is identical to that presented in the preceding paragraphs (samples D-2286 and D-2297). The top contact is a 10-nm-thick GaInAs, highly doped layer (1 x lo2' crn-')). The lasers are fabricated as 20-pm-wide, deep-etched ridge waveguides. Si,N, is used for electrical insulation at the sidewalls of the ridges. For sample D-2295 (Fig. 43b), at the center of the top contact, into a 10-pmwide window Pd is deposited directly onto the semiconductor. The Pd does not cover the whole top surface but leaves two narrow stripes (-2 pm) on the sides, which are filled with Ti-Au (300&3000A) during the final deposition for the nonalloyed electrical top contact. This procedure is necessary, since Pd did not prove to be a suitable contact material. It is important to note that the etching of the stripe is not a necessary step for laser fabrication, we chose to do it only to avoid spreading of the current in

-

-

67

1 QUANTUM CASCADE LASERS

(4 n = 3.31 a = 57 ern-’ r = 70%

InP Substrate

I

-2

2

0

4

6

Distance ( prn)

5

Substrate

*

w=14-20vm FIG. 43. (a) Mode profile of the waveguide in the direction perpendicular to the layers for laser D-2295. Given are the calculated values of the effective refractive index (n), total internal waveguide loss (aW),and the overlap factor (r).The complex refractive index of palladium at 1, 11.2 pm is ii = 3.85 + i49.2. (b) Schematic of the device structure. Note the palladium layer encapsulated between the gold and the top surface of the laser.

the lateral direction. In our waveguides, the optical confinement occurs where the metal is deposited, consequently the whole laser fabrication could be reduced, in principle, to a simple metal stripe deposition on an unprocessed waver. Figure 44 reports the light output (at different temperatures) and voltage versus current characteristics of a laser processed from sample D-2295. The threshold current density is Jth 11 kA/cm2 at 10 K and lasing action has been observed up to 110 K. The maximum output power is 1 mW. The threshold current density is about 3.3 times higher than the lasers with the

-

-

JEROMEFAISTET

68

AL.

Wavelength (vm)

Current (A) FIG.44. Applied bias and measured peak output power from a single facet as a function of injected current for a 20-pm-wide and 0.8-mm-long laser at three different heat sink temperatures. Inset: High-resolution laser spectrum of the device at 30 K. The maximum peak output power is estimated to be several milliwatts.

same active materials and dielectric waveguides. From this comparison and the ratio between the overlap factors, we can derive a rough estimation of 100 to 150cm-'. the waveguide losses a,

-

6. DISTRIBUTED FEEDBACK QUANTUM CASCADE LASERS In the preceding subsection, we described quantum cascade lasers operating in pulsed mode with high optical powers at and above room temperature. Under pulsed operation, these lasers exhibited a relatively broadband (10-20 cm- I), multimode operation, as expected for devices based on Fabry-Ptrot cavities. On the other hand, applications such as remote chemical sensing and pollution monitoring require a tunable source with a narrow linewidth. For most sensor applications, the linewidth must be narrower than the pressure-broadened linewidth of gases at room temperature, which is about one wave number, and the source must be tunable over a few wave numbers. By incorporating a grating in a QC laser structure, we have demonstrated a tunable distributed feedback (DFB) laser

69

1 QUANTUM CASCADE LASERS

operating at and above room temperature with a linewidth consistent with . these applications. a.

Loss-Coupled QC-DFB Lasers

The active region of our QC lasers were optimized for high-temperature operation and included a three-well active region with vertical transition, a funnel injector, and low doping. Two samples were designed, D-2189 and D-2195 for operation at 5.4- and 8-pm wavelengths, respectively (Faist et al., 1997). Only the results from the 5.4-pm laser are discussed in this chapter; results on the A = 8 pm laser can be found in the original publication (Faist et al., 1997). The active region of sample D-2189 comprises 29 periods and included a MBE-grown InP top cladding for optimal heat dissipation. The layer sequence is shown in Fig. 45. The energy band diagram of D-2189 is shown in Fig. 27. In DFB lasers, the coupling constant K quantifies the amount of coupling between the forward and backward waves traveling in the cavity. In the coupled-mode theory of DFB lasers (Kogelnik and Shank, 1972), K is written as

where n, is the amplitude of the periodic modulation of the real part of the effective index (neff) of the mode, induced by the grating of periodicity A. The corresponding modulation of the absorption coefficient has an amplitude cxl. The wavelength AB is determined by the Bragg reflection condition 1, = 2neffA(first-order grating). For optimum performance in slope eficiency and threshold current, the product K L ~ ~where ,, L,,, is the cavity length, must be kept close to unity. For good room-temperature perform2-3 mm so the coupling constant should ance in our QC structures, L,,, for an be designed with a value / K / 5 cm- I , corresponding to y1, index-coupled or a, 2.5 cm- for a loss-coupled structure. To this end, both devices incorporated a plasmon-enhanced waveguide designed for an optimal coupling with a corrugated grating on the surface of the device (Fig. 43). The amplitude of the confined mode at the grating surface is about 0.5-1% of its maximum value in the center of the guide. The first-order grating of periodicity A = &/(2neff)( = 850 nm) for the A = 5.4 pm and 1250 nm for the A = 8 pm devices) was exposed by contact photolithography and subsequently etched to a depth of -250nm by wet chemical etching as the first processing step. In the grating grooves, the thickness of

- '

-

70

JEROMEFASTET AL.

the heavily doped plasmon-confining layer is reduced and the guided mode interacts more strongly with the metal contact, therefore locally increasing the loss. We expect therefore the coupling constant to be complex (i.e., to exhibit both a real and imaginary parts). Our estimate for both samples is n, 5 x and a, N 0.5-2 cm- '. Complex-coupled DFB lasers exhibit, in general, better single-mode yield because the loss component lifts the degeneracy between the two modes on each side of the stopband (Kogelnik and Shank, 1972). In addition, since the loss component originates from the metal and is therefore off-resonant, we do not expect any saturation behavior with increasing optical intensity. A scanning electron micrograph of the cleaved facet of a processed device is shown in Fig. 45a. Control samples without grating exhibited performances comparable to the results shown in Fig. 29 with 300 K threshold current densities of 7-9 kA/cm2 and optical powers above 100 mW for the sample 5.4 pm. Therefore the reduction of the cladding D-2189 operating at thickness did not decrease the performances in an appreciable manner. Figure 46 shows the L-I characteristic two 3-mm-long DFB laser from sample D-2189. Laser A exhibits optical powers of about 12mW with a slope efficiency dP/dI = 10 mW/A at T = 300 K. At the same temperature, laser B exhibits much larger powers up to 60mW, with a slope efficiency dP/dI = 44 mW/A. The laser spectrum is single mode over a temperature range from 80 to 315K for laser A. As shown on Fig. 46b, in this temperature interval, the device tunes continuously from A = 5.31 to 5.38 pm. Laser B exhibits single-mode operation from 324 to about 260 K, where a second mode appears. From our data, the presence of the DFB grating seems to reduce the slope efficiency by a factor of 3-10 as compared to Fabry-PQot devices, but leaves the threshold current density essentially unchanged. We attribute this reduction in slope efficiency to an understimate of K, implying that our devices operate in the regime of S L > 1. Assuming a simple rate-equation model with a constant injected current density, the (total) differential quantum efficiency for one period of the active region is N

-

where G,, is the total threshold modal gain, a, is the waveguide loss, and I(z) is the (average) intensity along the DFB cavity of length L, normalized such that (1/JL) [L'& f ( z ) dz = 1. Equation (27) predicts a strong reduction of slope efficiency compared to the Fabry-Pkrot case when kL >> 1.

1 QUANTUM CASCADE LASERS

71

(4

0-2189 n = 7x101scm” 9OOnm

FIG,45. (a) Scanning electron micrograph of a cleaved facet end of loss coupled DFB-QC laser. (b) Layer structure of the waveguide of the sample D-2189. The dashed line indicates the interface where the growth was interrupted. The layer sequence of sample D-2189 is described in Fig. 27.

At T = 300 K, our lasers dissipate too much power ( - 80 kW/cm2) to be operated in coutinuous wave. This large amount of thermal power heats the device during the electrical pulse, causing its emission wavelength to drift. For this reason, the spectra of these devices was found to be very narrow, limited by our spectrometer’s resolution (0.125 cm-l) for very short pulses (5-1011s) just above threshold and increased with pulse length up to 1 cm-’ for a 100-ns-long pulse. To quantify this effect and check that the broadening was caused only by the thermal drift, we performed time-resolved spectra of our lasers. To this end, laser B of sample D-2189 is excited with a 100-ns-long electrical pulse with a current I = 3.14A. The peak optical

72

JEROME FAIST ET AL.

FIG. 46. (Upper) Collected pulsed optical power from a single facet versus injection current for heat sink temperatures for two primarily DFB lasers from sample D-2189. (Lower) Tuning range of the same two lasers.

-

power at this current is P 50mW. The laser emission is detected with a room-temperature MCT detector with a subnanosecond time constant. The signal is sampled by a boxcar integrator, the output of which is fed back into the FTIR spectrometer. The resulting spectra taken at a 10-ns interval with a 3-ns-long gate are shown in Fig. 47. From these spectra, it is clear that the laser keeps a narrow emission line, which drifts with time a t a rate of 0.03 cm-'/ns. From this and the value of the tuning rate shown in Fig. 47, we extrapolate a temperature increase of 20K during a 100-ns-long pulse.

73

1 QUANTUM CASCADE LASERS

Wavelength(pm)

5.370

1862

5.365

1864

5.360

1866

5.355

1868

Wavenumber (cm I) FIG. 47. Time-resolved spectra of laser B from sample D-2189 at T = 300 K. The length of the electrical pulse is 100 ns. The spectra are taken 10 ns apart with a 3-ns-long gate. The peak optical power is 50 mW.

The rate of wavelength shift is proportional to the power dissipated in the active region and therefore will decrease with operating point and threshold current density.

b. Index-Coupled DFB Lasers Loss-coupled DFB lasers with an etched grating on the surface have some very advantageous features: because they are loss-coupled, they are much more likely to be monomode regardless of the position of the cleaved facet. Most importantly, their manufacture is very simple and requires only one growth step at the beginning. The grating periodicity can be adjusted u posteriori by trial and error on the grown layer. Although this grating strongly influences the QC through its plasmon-enhanced waveguide, it is located away (>2 ym) from the active region and the region of maximum intensity of the laser mode. Strong coupling is attainable only with a

74

JEROMEFAIST ET AL.

reduced upper-cladding-layer thickness. This increases the waveguide loss through the absorbing top metal layer and in turn decreases the performance of the DFB laser. A still higher level of performance can be reached using a grating etched directly inside the active region because it allows strong coupling with negligible additional loss. The scanning electron microscope (SEM) cross section of the structure is shown in Fig. 48 (Gmachl et al., 1997). In the first MBE step, the QC active regions section is grown embedded between two InGaAs layers. The upper InGaAs layer serves as the host region for the first-order grating. The latter is transferred by contact lithography and wet chemical etching. In the second MBE step, InP is epitaxially grown directly on top of the grating. The grating strength is controlled by the grating depth and duty cycle during grating fabrication (etching) and the reflow of material in the regrowth process. It has an approximately trapezoidal shape with a duty cycle of 30-50%. The tuning curve of two such devices with an active region designed for the 8.5-mm wavelength is displayed in Fig. 49 (Gamchl et al., 1998b). Continuously tunable single-mode operation is achieved from 8.38 to 8.61 pm (AAem = 230 nm) in the same wafer using two different grating periods: A, = 1.35pm (8.38 to 8.49 pm) and A, = 1.375 pm (8.47 to 8.61 pm). The tuning is achieved by changing the heat-sink temperature

FIG.48. Scanning electron micrograph of a cross section of an index-coupled grating QC-DFB laser designed for a wavelength near 8 pm. The grating (period A = 1.350pm) is clearly visible just above the active region.

75

1 QUANTUM CASCADE LASERS

F Y

0

8.5

8.6

WAVELENGTH

8.7

m]

TEMPERATURE [Kj FIG. 49. Single-mode tuning characteristics of two lasers with two different grating periods A and operating at I. 8.5 pm. The squares and circles are data obtained under pulsed operation, the triangles under cw operation. The laser with A = 1350 nm displays lasing on Fabry-Perot modes below a temperature of 80 K. This is due to the detuning of the peak gain with respect to the Bragg resonance. This allows Fabry-Perot modes, which are lasing at the peak gain, to reach laser threshold under high pumping conditions. The data shown for A = 1350nm are from a shallow etched device (see inset of Fig. 50). Inset: Single-mode spectrum of a laser operating in pulsed mode at room temperature. A side-mode suppression ratio of better than 30 dB is obtained.

-

between 20 and 320 K mainly through the temperature dependence of the material refractive indices (thermal red shift). Above ~ 2 0 K0 (average) linear tuning coefficient of +0.58 nm/K is obtained. The thermal tuning coefficient of the peak gain has been measured from the subthreshold luminescence spectra as z 1.2 nm/K. With regard to this relative detuning of the peak gain wavelength and the Bragg resonance, the latter has been designed (by choice of the grating period) such that their overlap improves in the high-temperature operating range. We attribute the fact that the device can be operated with such a wide temperature range while maintaining a dynamical single-mode operation with a side-mode suppression ratio larger than 30dB to the stronger coupling provided by the index grating. The index modulation is estimated as the difference between the modal refractive indices of the “undisturbed” waveguides at the location of the grating grooves (InP) and plateaus (InGaAs). The deviation of the grating

76

JEROME FAIST ET AL.

shape from a sinusoidal shape as well as a duty cycle other than 50% reduces the modulation amplitude of the grating. A correction factor of 0.8 (estimated for a trapezoidal grating with 50% duty cycle) finally results in An = 1.79 x From this value, and using Eq. (26), we obtain a coupling coefficient of K~~ % 33 cm- A clear Bragg stopband with width ALBragg= A n . A % 24 nm follows from the strong index-coupling of the QC-DFB laser. The two Bragg resonances on either side of the stopband-located on the slope of the narrow gain spectrum-experience a strong discrimination with respect to each other due to the large value of AABraggleading to single-mode operation. Tunable single-mode lasing has also been achieved in cw operation. The lasing resonance has been tuned from 8.47 to 8.54 pm by changing the heat sink temperature from 20 to 120 K. Finally we discuss the L-I characteristics of the QC-DFB devices at various heat sink temperatures. Devices cleaved to a length of 1.5 mm are compared. In pulsed operation (pulse duration: 50 ns; frequency: 5 kHz) the deep etched ridge waveguide devices (see the inset in Fig. 50 for a schematic of the device cross section) display a low threshold current density of 2.2 kA cm-2 (at 20 K) and 8.8 kA cm-2 (at 300 K). A collected peak output power of 13 mW (slope efficiency 23 mW/A) is obtained for a 11-ym-wide ridge at 300 K and 23 mW (30 mW/A) for a 17-ym-wide ridge. Figure 50 shows the continuous wave (cw) L-1 curves of the same devices operated up to 125K with z l 0 m W of cw output power at 120K. The maximum power measured at 80 K is 50 mW. From the data taken from deep etched QC-DFB devices, it: appears that wider stripes result in an higher peak output power without degrading the single-mode yield. Therefore shallow etched ridge waveguide lasers have been fabricated. Owing to the subsequent current spreading higher threshold currents are obtained. Assuming the same threshold current densities as for deep etched devices, an effective lasing stripe width of ~ 6 pm 0 can be assumed. Figure 51 shows the pulsed L-I characteristics of a 1.5-mm-long device with shallow etched ridge around room temperature, resulting in 60-mW (single-mode) peak output power at 300 K and 44 mW at 320 K. It should be noted that the index-coupled 8-pm-wavelength QC-DFB lasers described in this section had a three-well vertical transition design with a funnel injector, which further improves the performance of the vertical transition design discussed in a previous section for devices of the same wavelength. In fact, further optimization of this structure led to the recordings of cw optical powers (200mW at 80K) and pulsed room temperature powers (325 mW) (Gmachl et al., 1998a) for a 30-stage device. / T (O.l), ~ ~a short lifetime 't3 = 1.65 ps, a This structure has a small T ~ ~ ratio large matrix element of the laser transition z 3 2 = 2nm, and a further

'.

-

1

QUANTUM CASCADE

77

LASERS

T=20K

0

0.5

1 .o

-

1.5

FIG. 50. Light output-to-current characteristics at various heat sink temperatures of a deep etched ridge waveguide QC-DFB driven in continuous wave (A = 1.375 pm; length: 1.5 mm; width: 1 1 pm). The light is collected from one facet with near unity collection efficiency. The kink at high power levels in the temperature range <80K indicates the switching from a single-mode to a Fabry-Ptrot (FP) type emission spectrum. The detuning at low temperatures was chosen on purpose to optimize the devices for room temperature operation. Inset: Schematic cross section of a QC-DFB device with deeply etched ridges. The dashed circle indicates the region of highest modal intensity.

optimized injector (which uses nonequilibrium transport across its length to inject electrons directly from the n = 1 state of an active region into the n = 3 state of the active region of the adjacent period), to enhance the slope efficiency and the maximum optical power. Gmachl et al. (1998a) also shows that by using a vertical transition design with a three-well active region and an optimized injector the performance of the QC laser does not suffer in scaling the design from 1 = 5 pm to 2 = 8 pm. d. Applications The realization of a QC-DFB laser operating at room temperature had a very important consequence for the possible use of the QC technology for practical applications. Because the QC-DFB laser exhibits dynamic singlemode operation even at room temperature, it enables gas spectroscopy with

78

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JEROMEFAISTET 120 L

F E Y

’

"

"

90

II

60

3 Q

'

AL.

+0

30 n u

0

5

10

15

CURRENT [A]

-

FIG. 51. Light output-to-current characteristic of a shallow etched QC-DFB driven in pulsed mode at various heat sink temperatures around room temperature (A 1.35 pm;length 1.5 mm). The light is collected from one facet with an estimated collection efficiency of 50%. Inset: Schematic cross section of the device.

a laser source with a linewidth of a few megahertz. As we will show, an average optical output power of about a milliwatt is sufficient for most detection schemes. We implemented a sensitive detection measurement using such a device (Namjou et al., 1998). The QC-DFB laser, designed to emit near 8pm is mounted in a temperature-controlled, evacuated chamber. The laser is driven by a 2.6-A amplitude, 1-MHz train of 11-ns-wide current pulses (- 1% duty factor) resulting in 10-pW collected average laser output power. The thermoelectric cooler temperature is set to 1°C. This tunes the laser resonance close to the N,O absorption line of interest. We fine-tune the laser by superimposing on the laser driving waveform a 4-Hz current ramp, which alone would be insufficient in amplitude to bring the laser above threshold but which causes a temperature modulation on the laser sufficient to sweep the lasing frequency by about one wave number. Finally, we combine the ramp with a 1.8-kHz sinusoidal dither waveform, which adds a second, higher frequency temperature modulation, resulting in a concomitant wavelength modulation. The modulated laser beam is collimated and then directed through a 10-cm-long gas cell equipped with wedged ZnSe windows. The transmitted light is focused onto a mercury cadmium telluride photoconductive detector, cooled to 77 K. The detector is coupled to a matched pre-amp whose output

1 QUANTUM CASCADE LASERS

79

is detected by a lock-in amplifier. The wavelength modulation dither serves as the lock-in reference and the lock-in output is recorded using a computer equipped with a 14-bit digital signal processor. The computer records individual ramp scans and can be set to average these scans. The data reported here were mostly recorded using 16-scan averaging with a lock-in bandwidth of 250 Hz, resulting in a net detection bandwidth of 16 Hz. Since the dither frequency is much lower than the underlying pulse repetition frequency, this arrangement results in wavelength modulation absorption spectroscopy. The gas cell could be filled with commercially premixed volumes of diluted gas, then further diluted by admitting high-purity dry nitrogen. For most of the measurements reported here, we used samples of 10, 1, and 0.1% N,O in N, diluted in steps of five as many as three times and then analyzed with the laser at pressures from 1 to 20Torr. By setting the temperature of the thermoelectrically cooled laser head we obtained sets of scans, using the 4-Hz current ramp, whose center frequency could be estimated using the wavelength calibration of the QC-DFB laser obtained independently using an FTIR spectrometer. Figure 52 shows several such scans overlaid to provide a continuous temperature scan from

FIG. 52. Wavelength modulation signal vs laser frequency obtained by overlaying a series of scans through 1 Torr of a 10% N,O-90% N, sample and 10 Torr of a 10% CH,90% N, sample. The dotted vertical lines show the locations and relative line strengths of the corresponding lines from the HITRAN database. The scans were pieced together using an FTIR calibration of the laser frequency as a function of temperature.

80

JEROMEFAISTET AL.

-15 to 5"C, corresponding to a 2.5-cm-' laser frequency scan starting at 1280cm- '. The spectral data show the characteristic derivative line shape of wavelength modulation spectroscopy obtained from a 10% N,O-90% N, premixed sample with a total pressure of 1 Torr. The frequency calibration was checked by comparing the data with the known frequencies and line strengths of the N,O obtained from the HITRAN database. These are shown at the bottom of the figure. The FTIR calibration and HITRAN frequency values agree to better than 1% over the entire scan. This experiment demonstrated sensitive detection of dilute samples of N,O using a quantum cascade distributed feedback laser operated near 1°C. The noise equivalent absorbance of 5 x 10-5/flz is limited by two nonfundamental factors and could be readily improved by modification of the stability of the drive electronics. In another experiment (Sharpe et al., 1998), high-resolution spectra of NH, and NO were obtained with index-coupled grating QC-DFBs of 8 pm at 5.2-pm wavelengths operating in cw at 80 K. Direct absorption spectroscopy was performed by ramping the laser current with 10-kHz sawtooth waveforms and averaging many sweeps. From these data a laser linewidth, limited by technical noise, of tens of megahertz over a few milliseconds was obtained. The instantaneous linewidth was estimated to be a few megahertz. The latter estimate was obtained without a superimposed ramp by driving the laser in cw at 80 K. This important result shows that Q C lasers are much more immune to wavelength drifts and jitters, compared to lead-salt lasers and therefore better suited for high-resolution spectroscopy. Paldus et al. (1999) reported the first use of QC lasers in photoacoustic spectroscopy using a resonant acoustic cavity. Traces of ammonia diluted in nitrogen were detected with a sensitivity of 10 parts per billion. Our opinion is that the fundamental linewidth of QC lasers should be Schawlow-Townes limited as, for example, in gas lasers, because the linewidth enhancement factor or CI parameter for QC lasers is expected to be near zero if the DFB Bragg wavelength is positioned at the peak of the gain spectrum. In QC lasers, the gain spectrum has basically the same shape of the absorption spectrum unlike interband diode lasers. Thus at the peak of the gain spectrum, the fluctuations of the refractive index, caused by electron density fluctuations, are zero because of the Kramers-Kronig relationships, leading to a = 0.

-

ACKNOWLEDGMENTS Collaborations with C. Gmachl, A. L. Hutchinson, J. Baillargeon, S. N. G. Chu, and E Whittaker are gratefully acknowledged.

1 QUANTUM CASCADE LASERS

81

REFERENCES Ando, T., A. B. Fowler, and F. Stern. (1982). Rev. Modern Phys. 54, 437. Andronov, A. A. (1987). Sou. Phys. Semicond. 21, 701. Bastard, G. (1990). Waue Mechanics Applied to Semiconductor ffeterosfructures(Les Editions de Physique, Paris). Belenov, E. M., P. G. Eliseev, A. N. Oraevskii, V. I. Romanenko, A. G. Sobolev, and A. V. Uskov. (1988). Sou. J . Quantum Electron. 18, 995. Berger, V. (1994). Semicond. Sci. Technol. 9, 1493. Borenstein, S. I., and J. Katz. (1989). Appl. Phys. Lett. 55, 654. Campman, K. L., H. Schmidt, A. Imamoglu, and A. C. Gossard. (1996). Appl. Phys. Lett. 69, 2554. Capasso, F. (1987). Science 235, 172. Capasso, F., and A. Y. Cho. (1994). Surface Sci. 299-300, 878. Capasso, F., W. T. Tsang and G. F. Williams. (1983). IEEE Trans. Electron. Deuices 30, 381. Capasso, F., K. Mohammed, and A. Y. Cho. (1986). IEEE J . Quantum Electron. QE-22, 1853. Capasso, F., C. Sirtori, J. Faist, D. L. Sivco, S. N. G. Chu, and A. Y. Cho. (1992). Nature 358, 565. Cho, A. Y. (ed.). (1994). Molecular Beum Epitaxy (AIP Press, Woodbury, NY). Choe, J.-W., A. G. U. Perera, M. H. Francombe, and D .D. Coon. (1991). A p p l . Phys. L e f t . 59, 54. Choi, H. K., G. W. Turner, M. J. Manfra, and M. K. Connors. (1996). Appl. Phys. Lett. 71,2936. Chow, D. H., R. H. Miles, T. C. Hasenberg, A. R. Kost, Y. H. Zhang, H. L. Dunlap, and L. West. (1995). Appl. Phys 67, 3700. Faist, J., F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, S. N. G. Chu, and A. Y. Cho. (1993). Appl. Phys. Lett. 63, 1354. Faist, J., C. Sirtori, F. Capasso, L. Pfeiffer, and K. W. West. (1994a). Appl. Phys. Lett. 64, 872. Faist, J., F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (1994b). Appl. Phys. Lett. 65, 94. Faist, J., F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho. (1994~).Science 264, 553. Faist, J., F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, S. N. G. Chu, and A. Y. Cho. (1994d). Appl. Phys. Lett. 64, 1144. Faist, J., F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (1995a). Appl. Phys. Lett. 66, 538. Faist, J., F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (1995b). Appl. Phys. Lett. 67, 3057. Faist, J., F. Capasso, C. Sirtori, D.L. Sivco, J. N. Baillargeon, A. L. Hutchinson, S. N. G. Chu, and A. Y. Cho. (1996). Appl. Phys. Lett. 68, 3680. Faist, J., C. Gmachl, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon A. L. Hutchinson, and A. Y. Cho. (1997). Appi. Phys. Lett. 70, 2670. Faist, J., F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, S. N. G Chu, and A. Y. Cho. (1998a). Appl. Phys. Lett. 72, 680. Faist, J., A. Tredicucci, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho. (1998b). IEEE J . Quantum Electron. 34, 336. Faist, J., C. Sirtori, F. Capasso, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, A. Y. Cho. (1998~).IEEE Photon. Technol. Lett. 10, 1100. Felix, C. L., J. R. Meyer, I. Vurgafdman, C. H. Lin, S. G. Murray, D. Zhang, and S. S. Pei. (1997). IEEE Photon Technol. Lett. 9, 734.

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Fcrreira, R., and G. Bastard. (1989). Phys. Rev. B 40, 1074. Gauer, C., A. Wixforth, J. P. Kotthaus, M. Kubisa, W. Zawadski, B. Brar, and H. Kroemcr. (1995). Phys. Rev. Lett. 74, 2772. Gelmont, B., V. Gorfinkel, and S. Luryi. (1996). Appl. Phys. Lett. 68, 2171. Gmachl, C., J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho. (1997). IEEE Photon. Technol. Lett. 9, 1090. Gmachl, C., A. Tredicucci, F. Capasso, A. L. Hutchinson, D. L. Sivco, J. N. Baillargeon, and A. Y. Cho. (1998a). Appl. Phys. Lett. 72, 3130. Gmachl, C., F. Capasso, J. Faist, D. L. Sivco, J. N. Baillargeon A. L. Hutchinson, and A. Y. Cho. (1998b). Appl. Phys. Lett. 72, 1430. Gornik, E., and D. C. Tsui. (1976). Phys. Rev. Lett. 37, 1425. Hakki, B. W., and T. L. Paoli. (1975). J . Appl. Phys. 46, 1299. Helm, M., P. England, E. Colas, F. DcRosa, and S. J . Allen, Jr. (1988). Phys. Rev. Lett. 63, 74. Henderson, G. N., L. C. West, T. K. Gaylord, C. W. Roberts, E. N. Glytsis, and M. T. Ansom. (1993). Appr. ~ h y sLett . 62, 1432. Henry, C. H. (1982). IEEE J. Quantum Electron. QE-18, 259. Hu, Q., and S. Feng. (1991). Appl Phys. Lett. 59, 2923. Jenscn, B. (1985). In Handbook of Optical Constants, ed. E. D. Palik (Academic, Orlando), Chap. 9. Kastalsky, A,, V. J. Goldman, and J. H. Abelcs. (1988). Appl. Phys. Lett. 59, 2636. Kazarinov, R. F., and R. A. Suris. (1971). Fiz. Tekh. Poluprov 5, 797; transl. in Sou. Phys. Semicond. 5, 707 (1971). Kazarinov, R. F., and R. A. Suris. (1972). Fiz. Tekh. Poluprov. 6, 148; transl. in Sou. Phys. Semicond. 6, 120 (1972). Kogelnik, H., and C. V. Shank. (1972). J . Appl. Phys. 43, 2327. Lane, B., D. Wu, A. Rybaltowsky, H. Yi, G. Diaz, and M. Razcghi. (1997). Appl. Phys. Lett. 70, 443. Lax, B. (1960). In Proceedings o f t h e International Symposium on Quantum Electronics, ed. C. H. Townes (Columbia Univ. Press, New York), p. 428. Li, Y. B., J. W. Cockburn, M. S. Skolnick, J. P. Duck, M. J. Birkctt, I. A. Larkin, R. Grey, G. Hill, and M. Hopkinson. (1998). Appl. Phys. Lett. 72, 2141. Liu, H. C. (1988). J . Appl. Phys. 63, 2856. Lochr, J. P., J. Singh, R. K. Mains, G. I. Haddad. (1991). Appl. Phys. Lett. 59, 2070. Lynch, D. W., and W. R. Hunter. (1991). In Handbook of Optical Constants of Solids II, ed. Edward D. Palik (Academic Press Inc., San Diego, CA). Meyer, J. R., C. A. Hoffman, F. J. Bartoli, and L. R. Ram-Mohan. (1995). Appl. Phys. Lett. 67, 757. Mii, Y. J., K. L. Wang, R. P. G. Karunasiri, and P. F. Yuh. (1990). Appl. Phys. Lett. 56, 1046. Mustafa, N., L. Pcsqucra, C. Y. L. Cheung, and K. A. Shore. (1999). IEEE Photon. Technol. Lett. 11, 527. Namjou, K., S . Cai, E. A. Whittaker, J. Faist, C. Gmachl, F. Capasso, D. L. Sivco, and A. Y. Cho. (1998). Opt. Lett. 23, 219. Nelson, D. F., R. C. Miller, and D. A. Klcinmann. (1987). Phys. Rev. B 35, 7770. Nikonov, D., A. Imamoglu, L. V. Butov, and H. Schmidt. (1997). Phys. Rev. Lett. 79, 4633. Paldus, B. A., T. G. Spence, R. N. Zare, J. Oomcns, F. J. M. Harren, D. H. Parker, C. G. Gmachl, F. Capasso, D. L. Sivco, J. N. Baillargeon, A. L. Hutchinson, and A. Y. Cho. (1999). Opt. Lett. 24, 178. Price, P. J. (1981). Ann. Phys. 133,217. Scamarcio, G., F. Capasso, C. Sirtori, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (1997). Science 276, 773.

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Sharpe, S. W., J. F. Kelly, J. S. Hartman, C . Gmachl, F. Capasso, D. L. Sivco, J. N. Baillargeon, and A. Y. Cho. (1998). Opt. Lett. 23, 1397. Sirtori, C., F. Capasso, J. Faist, D. L. Sivco, S. N. G. Chu, and A. Y. Cho. (1992). Appl. Phys. Lett. 61, 898. Sirtori, C., F. Capasso, J. Faist, and S. Scandolo. (1994). Phys. Rev. B 50, 8663. Sirtori, C., J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (1995). Appl. Phys. Lett. 66, 3242. Sirtori, C., J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (1996a). Appl. Phys. Lett. 69, 2810. Sirtori, C., J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (1996b). Appl. Phys. Lett 68, 1745. Sirtori, C., J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (1997a). I E E E Photon. Technol. Lett. 9, 294. Sirtori, C., J. Faist, F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (199713). IEEE J . Quantum Electron. 33, 89. Sirtori, C., P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, and U. Oesterle. (1998a). Appl. Phys. Lett. 73, 3486. Sirtori, C., F. Capasso, J. Faist, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho. (1998b). I E E E J . Quantum Electron. QE-34, 1722. Sirtori, C., C. Gmachl, F. Capasso, J. Faist, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (1998~).Opt. Lett. 23, 1366. Slivken, G., C. Jelen, A. Rybaltowski, J. Diaz, and M. Razeghi. (1997). Appl. Phys. Lett. 71 2593. Strasser, G., P. Kruck, M. Helm, J. N. Heyman, L. Hvozdara, and E. Gornik. (1997). Appl. Phys. Lett. 71, 2892. Sun, G., and J. B. Khurgin. (1991). In lntersubband Transitions in Quantum Wells, eds. E. Rosencher, B. Vinter, and B. Levine. NATO AS1 Series, Series B: Physics, Vol. 288, p. 219. Tacke, M. (1995). Infrared Phys. Technol. 36, 557. West, L. C., and S. J. Eglash. (1985). Appl. Phys. Lett. 46, 1156. Yang, R., B. H. Yang, D. Zhang, C.-H. Lin, S. J. Murry, H. Wu, and S. S. Pei. (1997). Appl. Phys. Lett. 71,2409. Yariv, A. (1988). Quantum Electronics, 3rd ed. (Wiley, New York).

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SEMICONI>IJC’lORS AND SEMIMETALS, VOL 66

CHAPTER 2

Nonlinear Optics in Coupled-Quantum-Well Quasi-M olecules Carlo Sirtori LABORATORE CENTRAL DE RECHERCHES THOMSON-CSF FRANCE ORSAY.

Federico Capasso, D. L. Sivco, and A. Y Cho BELLLABORATORIES LUCENTTECHNOLWIES MURRAY HILL.NEWJERSEY

I.

INTRODUCTION.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

11. NONLINEAR OPTICAL SUSCEPTIBILITIES IN THE DENSITY MATRIXFORMALISM . . 111. NONLINEAROPTICAL PROPERTIES OF COUPLED QUANTUM WELLS . . . . . . Iv. lNTERSUBBAND ABSORPTIONAND THE STARK EFFECTIN COUPLED QUANTUM WELLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. SECOND-HARMONIC GENERATION I N COUPLED QUANTUM WELLSAND RESONANTSTARKTUNING OF ~‘~’(2to) . . . . . . . . . . . . . . . . . . VI. FAR-INFRARED GENERATION BY RESONANT FREQUENCY MIXING . . . . . . . VII. THIRD-HARMONIC GENERATION A N D TRIPLY RESONANTNONLINEAR SUSCEPTIBILITY IN COUPLED QUANTUM WELLS . . . . . . . . . . . . . . VIII. MULTIPHOTON ELECTRONEMISSION FROM QUANTUM WELLS . . . . . . . . Ix. RESONANTTHIRD-HARMONIC GENERATION VIA A C O N T W M RESONANCE . . . X. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 87 90

93 102 109 112

116 121

122 123

I. Introduction The tailoring of wave functions and energy levels using bandgap engineering and molecular beam epitaxy (MBE) has played an important role in the design of quantum semiconductor structures (Capasso, 1991; Capasso and Datta, 1990). During recent years infrared intersubband transitions in quantum wells have been the subject of considerable study both from a 85 Copyright (c) 2000 by Academic Press All rights of reproduction in any form resened. ISBN 0-12-752175-5 ISSN ~ R O - X ~ R s3o.00 ~ K I ~

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ClWO SIRTORI ET AL.

physical and technological point of view in light of their large dipole matrix elements (1-3 nm) and oscillator strengths ( f - me/m* E 15 - 20) (West and Eglash, 1985; Rosencher et al., 1992). For example, high-performance infrared detectors at 1z 10 pm, based on bound-to-continuum transitions, have been developed (Levine et al., 1988; Levine, 1992; Bethea et al., 1991). Modulators have also been investigated due to their potential for high-speed operation and high on-off ratio (Karunasiri et al., 1990; Mii et al., 1990). Large second-order nonlinearities based on intersubband resonant enhancement were first predicted by Gurnick and De Temple (1983). More recently, several groups have demonstrated that quantum wells made asymmetric by application of an electric field (Fejer et al., 1989) and asymmetric step quantum well structures (Rosencher and Bois, 1991; Boucaud et al., 1990, 1991; Yo0 et al., 1991; Rosencher et al., 1989, 1990) exhibit large second-order nonlinear susceptibilities for pump wavelengths 1 E 10pm. In this chapter, we report an in depth study of these properties in the AlInAs-GaInAs and GaAs- AlGaAs heterostructure material systems grown by MBE. We show that by judicious control of the tunnel coupling between wells and of the thickness of the latter one can design the wave functions and the energy levels in such a way that these new structures behave as quasi-molecules with strongly field tunable nonlinear optical properties. Coupled quantum wells present unique opportunities for engineering new semiconductors with large optical nonlinearities associated with intersubband transitions in the infrared. These represent an excellent model system to investigate optical nonlinearities. Results on various aspects of this ongoing research have been reported elsewhere (Capasso and Sirtori, 1992; Capasso et al., 1992a, 1992b; Sirtori et al., 1991, 1992a, 1992b, 1992c, 1994a). In this chapter, we give a detailed theoretical and experimental analysis of various nonlinear optical phenomena -secondand third-harmonic generation (SHG and THG), difference frequency mixing (DFM), multiphoton electron escape -associated with these structures and of their tunability by an applied electric field. Other results presented here include a theoretical and experimental study of the large linear Stark shifts typical of coupled quantum wells; low-temperature measurements of their nonlinear susceptibilities, which have yielded secondand third-order susceptibilities significantly larger than at 300 K due to line-narrowing effects; and the observation of the enhancement of x(3) by continuum resonances. Structures with giant nonlinear susceptibilities ~ " ) ( 2 0 )and ~ ' ~ ' ( 3 0 ) (compared to the bulk constituents of the quantum wells) have been designed and demonstrated (Boucaud et al., 1990, 1991; Capasso and Sirtori,

2 NONLINEAR OPTICSIN COUPLED-QUANTUM-WELL QUASI-MOLECULES 87

1992; Capasso et al., 1992a, 1992b; Fejer et al., 1989; Gurnick and De Temple, 1983; Rosencher et al., 1989, 1990, 1992; Sirtori et al., 1991, 1992a, 1992b, 1992c, 1994a; Yo0 et al., 1991). They exhibit large linear Stark shifts of the intersubband transitions that have been used to efficiently tune the nonlinear susceptibilities. In structures designed for secondharmonic generation, the nonlinear susceptibility 1~(')(20)1exhibits a peak as a function of the electric field corresponding to the energy levels being made equally spaced via the Stark effect. Difference-frequency mixing experiments in a GaAs- AlGaAs modulation doped coupled quantum wells has shown extremely large values of the second order susceptibilities. In a three-coupled-well structure triply resonant third harmonic generation has been observed. This process is associated with four equally spaced bound states. The corresponding I x ( ~ ) ( ~ o(m/V)2 )~ at 300K and 4x (m/v)2 at 30 K) is the highest measured third-order nonlinear susceptibility in any material. The equivalent of multiphoton ionization of a molecule has also been investigated in this structure. Electrons are photoexcited to a continuum resonance above the barrier via a three-photon transition enhanced by intermediate energy levels. The effect of this resonance on ~ ( ~ ' ( 3 0 . 1as ) the electric field is varied is also investigated.

11. Nonlinear Optical Susceptibilities in the Density Matrix Formalism

In this section, we present a compact formalism for the derivation of high-order nonlinearities in coupled quantum wells deduced from Boyd (1992), Rosencher and Bois (1991), and Shen (1984). We perform our calculation for the nonlinear polarization only up to the second order. The calculations of the third-order susceptibility proceeds along lines that are analogous to those followed by the present derivation but with an increased number of indices in the formulas, which make the calculation tedious and somewhat cumbersome. Let us consider the Hamiltonian H = H , + V (t ) describing the system composed by the coupled quantum wells (H,) plus the perturbative term representing the energy of interaction of the coupled quantum wells with the extremely applied radiation field (V(t)).Here V ( t ) is the dipole interaction term -qzE(t), where q is the electron charge, z is the position operator in the direction parallel to the growth, and E(t) is the electromagnetic field. At thermal equilibrium, the density matrix p(O) is diagonal, and the diagonal elements pii are the thermal population of the level Ei, given by the Fermi level in the quantum well. When the system is excited by an electromagnetic field at frequencies cop and oq along the

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CARLO SIRTORI ET AL.

direction of growth z such as

+

E(t) = Empeiwpt EmpeiWpf + C.C.

(1)

we can use the Heisenberg equation of motion, with the phenomenological inclusion of damping, to describe the temporal behavior of the density matrix: Pnm

=

~

[IH,p l n m - Y m n ( p n m - pi?) ih

1 =

CHO - qzE(t), ~

I n m - ynm(P

- P(O))nm

(2) For simplicity, we assume in our derivation only two different values of relaxation rates: y1 = l/Tl for n = m,is the diagonal (or inelastic) relaxation rate and y z = l/Tz is the off-diagonal (or elastic) relaxation rate. In most cases, Eq. (2) cannot be solved analytically. We therefore seek a solution in the form of a perturbation expansion in terms of a power series of p: p,, =

1 pi$ = pnm+ pi2 + pi2 + ... (0)

(3)

j

By substituting Eq. (3) in Eq. (2) we obtain infinite set of coupled first-order differential equations for the density matrix elements as

The electronic polarization of the quantum well can be expressed also as a power series as Eq. (3). We are interested here only in the first two terms, and therefore we expand the polarization up to the second order, obtaining

+

+

P ( t ) = E O X ~ ~ E m P e i m~p t~ ~ ~ ~ + + , ~ E , ~ E... ~ ~ e ~( 5 () ~ p + ~ q

where x(l) and x(’) are the linear and second-order coefficient of the susceptibility expansion. For simplicity, we have neglected the term linear in EOq on the right-hand side of Eq. (5) because we have assumed EOq << Emp. To calculate the second-order polarization, we first must know p(2), which can be computed by integrating Eq. (4) twice:

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 89

where

By substituting the expression of ph:) in Eq. (6), then calculating the commutator and finally the integral, we obtain an expression for = pnm(wp (2) w,) p;?(2wp). The first term on the right-hand side corresponds to the sum frequency (or difference depending on the sign of up,a,),the second is related to second-harmonic susceptibility. We carry on the derivation of the second-order susceptibility in the more general case of sum frequency, by recalling that the second harmonic can always be recovered by imposing w p = w q :

+

+

The electronic polarization of the second-order oscillation at w p given by

+

P ( 2 ) ( o p w q )= N ( p ( w ,

+ w,))

=

+

qNTr[p(2)z(wp wq)]

+ w, is (8)

where N denotes the volume density of electron in the quantum well. Now using Eq. ( 5 ) for the definition of x?:+~, and comparing with the result of Eq. (8) we find the expression for the second-order susceptibility as

This sum has sharp maxima when the pump frequency cop and w, are resonant with the energy level separations and electrons are only in the ground state m (i.e., pip) = pi:) = 0). In these special situations, all the terms of the sum, except the resonant one, can be neglected. The expression of x( ) is strongly simplified, as shown in formulas (lo), ( l l ) , and (12) of the next section. The rest of the chapter provides experimental evidence of huge nonlinearities associated with the resonances mentioned here.

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CARLO SIRTORI ET AL

111. Nonlinear Optical Properties of Coupled Quantum Wells This section describes the design of coupled-well structures with extremely and ~ ( ~ ’ ( 3 ~ these 0 ) ; coefficients are the nonlarge x(’)(2w), x(’)(w, - 02), linear susceptibilities for SHG, DFM, and THG, respectively. Our strategy to optimize 1x()( and 1x(3 1 is, of course, applicable to other nonlinear susceptibilities as well. Nonlinear resonant susceptibilities of order n, x ), are, in general, the sum of several terms each containing in the numerator the product of (n 1) dipole matrix elements and in the denominator products of linear combinations involving the photon energies participating in the nonlinear interaction, the energy differences between the excited states and the initial state of the system (usually the ground state), and the linewidths of the transitions (Shen, 1984). To optimize (x( )(and 1x(3)1in our quantum wells we therefore need to maximize the product of the dipole matrix elements of the transitions and minimize the energy denominators using resonant effects. For SHG and THG the latter is achieved by structures that either by design or by application of an electric field have equally spaced energy levels (El through E , for SHG and El through E , for THG; the subscript 1 denotes the ground state). In the case of DFM, the resonances are obtained by having E , - El and E, - E , equal to the photon energies of the two (CO,) lasers beams impinging on the sample. In this structure, we have made use of modulation doping in high-quality AIGaAs-GaAs heterostructures (Pfeiffer et al., 1991) to minimize the intersubband transition linewidths. For the resonant case, ~ ( ~ ) ( 2 o ) , x(’)(o, - w,), and ~ ( ~ ’ ( 3 are w ) well approximated by Eqs. (lo), (ll), and (12) (see later) where N is the electron density in the wells, c0 is the permittivity of the vacuum, q is the electron charge, AEij = E j - Ei is the separation between subbands j and i of the conduction band quantum well, and Tijand ( z i j ) are the half width at half maximum and the matrix element of the i -,j intersubband transition, respectively. Equations (lo), (1 l), and (12) (see later) imply a Lorentzian broadening (i.e., homogeneous) of the transitions. Note that the matrix elements ( z i j ) are typically, by design, in the 10- to 20-A range, leading to high oscillator strengths (Sirtori et al., 1994b). These values are about three orders of magnitude larger than the ones found in molecules (a few picometers). This fact and the multiple resonance effects are responsible for the extremely large nonlinearities in the infrared. The preceding matrix elements are somewhat larger than or comparable to (depending on the layer thicknesses, the bandgap of the bulk quantum well material, etc.) those of interband transitions in the bulk constituents of the quantum wells, such as GaAs and Gao,,,In,,,,As (Burt, 1993). However, in the latter case, one cannot exhibit resonance effects since the relevant single photon, two-photon (2ho) and three-photon transitions

+

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 91

(3ho) at ho = 120 meV are strongly detuned from the energy gaps between the various conduction band edges and the valence band edge (Burt, 1993). Figures la, lb, and l c represent the conduction band diagrams of one period of the structures designed for SHG, DFM, and THG at the pump photon energies of the CO, laser (116-134meV). The states are calculated by solving Schrodinger's equation using the envelope function approximation. For the A1,,,,In,,,,As-Ga,~,,Ino,~~As parameters we used AEc = 0.52 eV for the barrier height; rn*(GaInAs) = 0.043 m,, m*(AlInAs) = 0.07m0 for the effective masses; and y = 1.13 x 10-"m2 for the nonparabolicity coefficients. Nonparabolicities were taken into account using the method of Sirtori et al. (1994b). In the structures designed for SHG and THG, only the thicker well is assumed to be doped (uniformly) so that space charge effects on the band diagrams of Fig. 1 can be neglected. In the case of DFM, the effects of space charge, introduced by the modulation doping,

(4 E3 = 334 E2 = 228

El = 92 meV

fb) E4 = 242 EQ= 197 Ep = 75 El = 6 1 meV

E4 = 506 E3 = 386 E2 = 270 El = 151 meV

FIG. 1. Conduction band energy diagrams of a single period of the coupled quantum well nonlinear optical structures. Shown are the positions of the calculated energy subbands and the corresponding modulus squared of the wave functions. (a) The GalnAs wells have thicknesses of 6.4 and 2.8 nm and are separated by a 1.6-nm AlInAs barrier. (b) The GaAs wells barrier. (c) The GaInAs wells are 6.1 and 7.0 nm thick, separated by a 2-nm A1,,,Ga,,,,As have thicknesses of 4.2, 2.0, and 1.8 nm, respectively, and are separated by 1.6-nm AlInAs barriers.

92

CARL0 SIRTORI ET AL.

are apparent from the band diagram of Fig. lb. The asymmetry of the two-well potentials is essential for SHG and DFM since in a symmetric well (z,,) = 0 due to the same parity (even) of the ground and second excited states so that x(,) = 0. This point can be easily understood from Eqs. (10) and (11). In Fig. la, E, - E , and E, - El are made intentionally different so that an electric field can be used to achieve equal level spacing for maximum (~'~'(2w)I. For the SHG structure (z12) = 15.4A, (z,,) = 22.3 A, and (z,,) = 12.1 A.While (z,,) and (z,,) are sensitive to the thickness of the tunneling barrier due to the evanescent wave coupling between the two wells, ( z , , ) is not, since states 1 and 3 reside essentially in the same well. To enhance the nonlinear susceptibility, the structure for DFM is designed so that the pump frequency w1 and the generated frequency w, = w , - w, are in resonance with the energy separations E, - E l and E , - El, respectively. At the same time, the wave functions are tailored to maximize the product of the relevant matrix elements: ( 2 1 3 ) = 8 A, (z,,) = 40 A, and (z2,) = 22 A.To minimize the linewidths of the relevant transitions we have made use of modulation doping. In this process, electrons are introduced in the well by doping only the barrier regions. The attendant spatial separation between ionized donors and electrons strongly reduces scattering and thus the linewidths of the intersubband transitions, particularly at low temperature. It is important to stress that Eq. (11) is valid only in the limit of a negligible electron population on the n = 2 state. Therefore the structure for DFM is optimized only at cryogenic temperature and for low pump powers. To design a structure with a triply resonant ~'~'(301) a harmonic oscillator potential cannot be used since (z14) = 0, so that x’,’ =0, due to the linearity of electron oscillations. For compositionally graded parabolic or semiparabolic wells of finite depth the deviation from the ideal harmonic oscillator potential caused by the finite well depth and band nonparabolicities give rise to (z14) # 0. Nevertheless, (zI4) is always much less than (zl,), (z,,), or (z,,). For the AlInAs-GaInAs structure of Fig. l c instead, all the relevant matrix elements are large: (z,,) = 8.6& (z,,) = 13A, (z,,) = 22.7 A, and (z,,) = 22.6 A. The calculated energy levels are E l = 15lmeV, E2 = 270meV, E , = 386meV, E, = 506meV.

~ ' ~ ' ( 3 w=) 4, N E~

(z12)(z23)(z34)(z41)

(hw - A E , ,

-

iy,,)(2hw - A E 1 3 - iy1,)(3hw - AE14 - iy14)

(12)

2 NONLINEAR OPTICSrn COUPLED-QUANTUM-WELL QUASI-MOLECULES 93

AlInAs-GaInAs rather than AlGaAs-GaAs may have few advantages for what concerns the physics of intersubband transitions: (1) the effective mass in the barrier material (AlInAs) is significantly smaller than in the high A1 concentration compositions of AlGaAs required to confine four equally spaced bound states (see Fig. lc) so that the coupling barrier does not need to be problematically thin ( < lo&; and (2) the electron effective mass in GaInAs (35% smaller than in GaAs) has the advantage of larger dipole matrix elements for the same intersubband transition energies. However, the smallest linewidths measured by absorption spectroscopy have been found in the GaAs- AlGaAs heterostructures.* Our AlInAs-GaInAs structures, grown by MBE lattice matched to a semi-insulating (100) InP substrate, consist of 40 coupled-well periods (as in Figs. l a and lc), separated from each other by 150-A undoped AlInAs barriers. Only the thickest wells are doped n-type with silicon in both structures (3 x l O " ~ m - ~and 1 x l O l ' ~ m - ~for the two-well and threewell structure, respectively). Undoped 100-8, GaInAs spacer layers separate the multiquantum well structure from n+ 4000-A-thick GaInAs contact layers. The layer thicknesses were verified by transmission electron microscopy. For the DFM (Fig. lb), we used GaAs-A10~3,Gao,,,As grown by MBE on a semi-insulating GaAs substrate. The coupled quantum wells of Fig. l b are repeated 50 times and are separated by a 950-81 barrier. To supply the electron charge in the wells a delta-doped Si layer (1 x 101Zcm-2)is inserted in the separating barrier to ensure a symmetric charge transfer. The resulting electron sheet density in the coupled wells is 4 x 101'cm-2.

IV. Intersubband Absorption and the Stark Effect in Coupled Quantum Wells

For the T H G and SHG the sample transmission was measured at room temperature, in the case of DFM, the absorption spectra were taken at cryogenic temperature. Data are collected using an infrared Fourier transform interferometer. Only the component of the polarization of the incident radiation normal to the layers contributes to intersubband absorption in the absence of strain (West and Eglash, 1985; Rosencher et al., 1992). Thus, to increase the net absorption we fabricated a multipass (six) waveguide by cleaving a bar and polishing both cleaved ends at 45" angle (Mii et al., 1990). One of these edges was then illuminated at normal incidence. The measured *Recent FTIR measurements by J. Faist (1994) on a modulation doped coupled well AlGa-GaAs structure grown by L. Pfeiffer have demonstrated linewidths (FWHM) z 2 meV at 5 K and 4 meV at 300 K, for intersubband transition energies , . 130 meV.

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CARL0 SIRTORI ET AL.

mid-infrared absorbance (= -log transmission) spectra of the SHG, DFM, and THG structures are shown in Figs. 2, 3, and 4, respectively. The absorbance for polarization in the plane of the layers -log IT;, is subtracted from the absorbance for polarization in the plane of incidence -log TL.The latter polarization corresponds to half the radiation intensity in the sample being polarized normal to the layers. This procedure removes most of the free-carrier absorption in the heavily doped buffer layers and in the quantum wells.* It is easy to show that the absorption coefficient of the quantum wells is related to the absorbance byt

where Lintis the total interaction length in the quantum wells defined as

*More precisely the plotted spectra are obtained by taking the difference of L o g l O ( ~ , / T L with ) and without samples. ?This definition of a is more physically meaningful than the one often used by many authors, i.e., apsriad = u w [ L w / L , + L,] where L, is the thickness of the barrier layer.

0.410

i'p

WAVE NUMBER (cm-I ) 15pO 20pO

2 7

0.340

0.060 100

I

200

300

ENERGY (mev)

FIG.2. Measured absorbance of the two-coupled-well structure for second-harmonic generation at room temperature.

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 95

2.01

I

I

I

I

PHOTON ENERGY (mev) FIG. 3. Measured absorbance of the two-coupled-well structure for difference frequency mixing at 5 K (solid line) and 60 K (dashed line) temperatures.

i0

PHOTON ENERGY (mev)

FIG. 4. Measured absorbance of the three-coupled-well structure for third harmonic generation at room temperature.

96

CARLO SIRTORI ET AL.

where L , is the total quantum well thickness plus that of the thin coupling barriers, N is the number of periods, np is the number of passes, and 6 is the angle of incidence with respect to the normal to the plane of the layers (45" in our case). The area under the absorption peak corresponding to the i - j intersubband transition multiplied by In( 10) is the integrated absorption strength I,. In electron volts, the latter is given by I,

27?qp, Nn, =

&,nl

sin' 0 (zij)

where p, is the sheet electron density of the ith state, n = 3.34 the refractive index, and ;i is the peak wavelength of the intersubband transition (West and Eglash, 1985; Sirtori et al., 1991). Thus from the integrated absorption strength and the knowledge of the other quantities in Eq. (15), one can deduce the matrix element (zij) of the transition. The peaks at 137.2 and 238.2 meV in Fig. 2 are due to the 1 + 2 and 1-+ 3 transitions, respectively; their position is in excellent agreement with our calculations for the two-well structure (see Fig. la). For the peak related to the 1 -,2 transition (137.2 meV) one obtains, from the data of Fig. 2, I , = 9.5 meV. Using Eq. (15) and the nominal value of p,(=2 x 10"cm-') one finds (z12) = 15.51$ in good agreement with the calculated value. From the 1 -,3 peak one finds from Fig. 2, I , = 5 meV. From the ratio of the integrated absorption strength ( I , ) for the two transitions one obtains (z12)/(z13) = 1.38. Thus (z13) = 11.2& in good agreement with the theoretical value. Note that the 1-+3 transition is forbidden in a symmetric quantum well; by properly engineering the asymmetry we have been able to create a large dipole matrix element. Figure 3 shows the absorption spectrum of the structure for the DFM at two temperatures ( 5 and 60K). The position of the 1 -, 3 and 2 -+ 3 transitions are in agreement with the calculated energy level differences. The increased height of the 2 -,3 peak at high temperature is a consequence of the increased population of the n = 2 state. At T = 5 K (i.e., when the thermal population on the n = 2 state is negligible), we find the measured dipole matrix elements from the n = 1 state to the n = 3 and n = 4 states in very good agreement with those calculated. (See footnote on p. 375.) In the absorbance spectrum of the three-coupled-well structure (Fig. 4) the positions of the peaks at 102, 232, and 344 meV are related to 1 3 2, 1-, 3, and 1 -,4 intersubband transitions, respectively, and are in excellent agreement with the calculated values.* From the areas under the absorbance *Transmission electron microscopy measurements give the following values for the layer thicknesses (from left to right in Fig. lc): 45 8, (GaInAs), 15 8, (AIInAs), 22 8, (GaInAs), 13 A (AlInAs), and 20 8, (GaInAs).

2 NONLINEAR OPTICSIN COUPLED-QUANTUM-WELL QUASI-MOLECULES 97

peaks, fitted with a Lorentzian, and the nominal electron sheet density in the well ( r 5 x 10"cm-2) one finds values of the matrix elements ((z12) = 16 (zlJ = 6.4 A,(z14) = 8.8 A), in good agreement with the calculations. The "shoulder" at 130 meV in Fig. 4 is due to the 2 -+ 3 transition. At 300 K, the second level is populated by the thermal tail of the Fermi distribution. In addition, the matrix element of the 2 + 3 transition is large ( ( z ~ ~=) 23 A). At 30 K the "shoulder" disappears. At this temperature the peaks shift higher in energy (by a few percent) and become significantly narrower (by a factor E 1.6). The coupled-well structures of Fig. 1 exhibit large linear Stark effects. Figure 5 shows the calculated intersubband separation energies as a function of electric field for the SHG structure. The intersubband energy separation AE13is found to be weakly dependent on the electric field, while the other two transitions (AE12and AEZ3)are strongly affected. Approximating their shift with the potential drop between the centers of the two wells gives good agreement with the actual behavior. This behavior is physically understood by noting that, approximately speaking, the first and third states are confined by the thick well, while the second state is confined by the thin well (see Fig. la). It is well known that within a first-order approximation the energy of a confined state of a quantum well with respect to the well center is independent of the electric field as long as the potential drop across the well is small compared to the confined state energy; that is, the linear Stark effect is zero and only a small quadratic effect exists (Harwit

A,

180

r

I

I

1

I

1

1

1

>.

g - 1601.2 5L

1

I

'

I

I

'

' ' ' '

I

E3 - E l

I

'

-I

zo $"

Eh ? L*

I

240

- 220

n z 140-

53

'

120-

- 200 100-

80

"

I

' 1

" " I

'

"

"

'

I

'

' I " '

' 180

FIG. 5. Calculated intersubband separation energies as a function of the electric field for the two-coupled-well structure (Fig. la). Positive bias polarity corresponds to the thin well being lowered in energy with respect to the thick well.

98

CARLOSIRTOFU ET AL.

and Harris, 1987). Thus as the electric field is increased, the first and third energy levels track the center of the thick well, while the second energy level tracks the center of the thin well. The net effect is that the energy of the 1 -, 3 transition is weakly dependent on the electric field while the energy of the 1 --* 2 and 2 -,3 transitions are shifted by an amount equal to the potential drop between the centers of the wells. The latter behavior is similar to that of the so-called “local-to-global state” transitions in step quantum wells (Karunasiri et al., 1990; Mii et al., 1990; Yuh et al., 1989). Our calculations also show that the dipole matrix elements vary weakly (by a few percent) with the electric field in the range used in our experiments. The absorption spectrum of the structure for the DFM as a function of an applied bias, has been studied in details and shows interesting interference effects between the subbands. Its discussion is reported in Chapter 2 of this book. Figure 6 shows the energy level separation as a function of electric field in the three-coupled-well sample. The 1 -,2 and the 3 -,4 transitions exhibit a large linear Stark effect because the first and fourth states are mainly localized in the largest well, while the second and third states are localized in the thinner wells, similar to the situation of the two well structure. The

=

I

0 . .-

L

-30

-10

0

10

30

50

70

ELECTRIC FIELD (kV/cm)

FIG.6. Calculated energy level separations as a function of the electric field for the three-coupled-well structure. Positive bias polarity corresponds to the thickest well being lowered in energy with respect to the other wells.

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES99

energy of the 2 + 3 transition (and likewise the 1 + 4 transition) is instead weakly dependent on the electric field since the initial and final state are concentrated in the same wells. Note that in the calculations of Figs. 5 and 6, we neglected the space charge effect arising from the separation between the centroids of the donor and electron charge in the doped wells. This is a good approximation, since these wells are thin and the doping density relatively small. On the structures for SHG and THG the Stark shifts were measured using a stabilized CO, laser as the source. The samples were processed into 350-pm-diameter mesas and ohmic contacts were provided to the two n+ cladding layers. They were then cleaved in narrow strips and the cleaved edges were polished at 45" to provide a two-pass waveguide for the pump beam similar to the six-pass structures used for the absorption measurements. In this geometry, after entering the sample at normal incidence (on one of the polished edges) and traversing the multiquantum well region, the laser beam is reflected off the top surface of the mesa and passes a second time through the multilayers (Fig. 7). The linear polarization of the laser was oriented in such a way as to maximize the component of the electric field of the incident wave ( F j ) normal to the layers (i-e., Fi/$). The incident laser power was 100 mW and the beam was chopped at 10 kHz. The transmitted signal was detected using a calibrated pyroelectric detector followed by a lock-in amplifier. The absorption coefficient was obtained from these measurements as a function of the applied electric field, at various laser wavelengths, in the temperature range 10-300 K. Some representative data are shown in Figs. 8a and 9a at 30 K for the SHG and THG structures, respectively. The Stark shifts are virtually independent of temperature since they depend primarily on the wave functions and energy levels, which are weakly dependent on temperature in these structures. In Figs. 8a and 9a the peak of the absorption coefficient at a given incident photon energy ho corresponds to

FIG. 7. Cross section of multipass sample geometry used in the experiments. Indicated are also the pump beam trajectory and its polarization.

100

CARLO SIRTORI ET AL.

01

- 60

I

- 30

I

I

0

30

I

60

90

ELECTRIC FIELD (kV/cm)

(4

(b) FIG. 8. (a) Measured absorption coefficient as a function of the applied electric field at various pump photon energies for the two-coupled-well structure. The dashed curves are Lorentzian fits to the data. (b) Position of the measured absorption peak versus electric field. Note the large linear Stark effect. The solid line is the calculated E,-E, transition energy taken from Fig. 5 .

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 101

-

6000 I T=30K

r

I

-E I-

z

W 4000

0 LL LL

w

s Z

Q 2000

ka

z:

m

a

I

01

-30

- 60

I

I

I

30 ELECTRIC FIELD (kVlcm)

0

3

60

(4

140

?E v

w’ I

120

w" I

0

I

I

I

I

40 60 ELECTRIC FIELD (kV/crn)

20

I

80

(b) FIG.9. (a) Measured absorption coefficient as a function of the applied electric field at various pump photon energies for the three-coupled-well structure. The dashed curves are Lorentzian fits to the data. (b) Position of the measured absorption peak versus electric field. The solid line is the calculated E,-El transition energy obtained from Fig. 6 .

102

CARLOSIRTORI ET AL.

an electric field F such that E , - El = hw. Thus a plot of hw versus F gives directly the Stark shift (Fig. 8b). We find 4.7 meV per 10 kV/cm in very good agreement with the calculated Stark shift of the 1 + 2 transition for the SHG structure (solid line in Fig. 8b). Similar plots were obtained for the three-coupled-well structure (Fig. 9b) using the measured absorption coefficient as a function of the electric field (Fig. 9a). The slope of the data (Fig. 9b) gives the Stark shift of the 1 + 2 intersubband transition; the solid line is the calculated intersubband separation ( E , - El). The agreement with the data, giving a Stark shift 5 meV per 10 kV/cm, is excellent. Our absorption measurements show that the linewidths of the transitions are reduced in going from room temperature to low temperature. At low temperature in our SHG and THG structures, the dominant broadening mechanisms are interface roughness scattering associated with monolayer fluctuations in the well width (Sakaki et al., 1987) and impurity scattering. At these temperatures ( 30 K) the contribution to the broadening of intersubband optical phonon scattering is 1.5meV, as shown by recent measurements of the electron scattering rate between subbands separated by 120 meV in GaInAs wells (Faist et al., 1993). Acoustic phonon broadening is completely negligible. It is worth noting that interface roughness scattering (Sakaki et al., 1987) gives a homogeneous broadening contribution to the linewidth, as d o the other two mechanisms. This is because the de Broglie wavelength of electrons involved in the intersubband transition is significantly greater than the length scale of the interface roughness (ea few tens of angstroms in samples without growth interruption; Sakaki et al., 1987). At low temperatures the smallest in plane electron wavelength is determined by the Fermi wave vector, Amin = 271/KF,where K , = ( 2 7 ~ p , ) ~ ’ ~ . Thus for the two and three coupled well structures Amin is respectively 700 and 400 A. In this respect it is interesting to note that our absorption curves are routinely best fitted by Lorentzian lineshapes, supporting the notion of a homogeneously broadened intersubband transition. N

N

V. Second-Harmonic Generation in Coupled Quantum Wells and Resonant Stark Tuning of f2)(2w) For these experiments the same sample geometry of the Stark effect measurements was used (Fig. 7). After passing through a telescope to reduce its divergence, the pump beam (CO, laser) traverses two high-extinctionratio polarizers. By appropriately varying the angles between the axes of the latter, one can vary the power and the linear polarization of the light incident on the sample. To accurately measure the laser wavelength, a beam-

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 103

splitter deflects a portion of the light into a monochromator. A lens focuses the light normal onto one of the 45" edges of the sample. The secondharmonic radiation is collected by a lens, followed by a sapphire window to cut the pump beam, an analyzer and appropriate filters to select the secondharmonic radiation. The signal is detected with a calibrated, nitrogen-cooled InSb detector. We also adjusted the polarization of the pump beam so as to maximize its component normal to the layers, as shown in Fig. 7. The measured second-harmonic power accurately follows the expected squarelaw dependence on the pump power, shown in Fig. 10 for two bias conditions. For this measurement the laser photon energy hw was adjusted to 122meV. At this energy, 2hw is very close to the measured separation between the third and first energy level ( E 3 - El = 238.2meV, Fig. 2). Figure l l a represents a graph of the second-harmonic power as a function of the pump photon energy hw at zero bias; the position of the maximum corresponds to an energy such that 2ho matches E , - E , = 238.2meV. A second peak (single-photon resonance) is not observed within the detection limits of the apparatus since the energy E , - El (137.3 meV) is greater than the maximum photon energy of the CO, laser. Figure l l b is a similar plot of P(2w) from a different part of the wafer. Note that this time the single

PUMP POWER (mW) FIG. 10. Second-harmonic power as a function of pump power at room temperature for different bias conditions for the two-coupled-well structure. The pump wavelength is I 1 10.15 pn. The solid lines represent least square fits with quadratic law. The bias polarity is the same as in Fig. 8b.

104

CARLO SIRTORI ET

500 I

-B

I

I

I

I

200 100

$

0

-

a

a

400

4

I

300 0

!200

a a .I

0 500

5

I

300

2

2

I

I

AL.

-

. a

100

0 110

120 130 PHOTON ENERGY (mev)

140

FIG. 11. Second-harmonic power as a function of pump photon energy for two different sections of the sample, at zero applied bias. In (a) and (b) the peak at E 118 meV corresponds to the two-photon resonance condition 2hw = E , - E , . In (b) the peak at z 132meV corresponds to the resonance conditions hw = E , - El. This resonance is accessible due to in-plane variations of the layer thickness.

photon resonance peak at ho = 132meV is observed since the E , - E , transition is now within the range of the CO, laser. Thickness variations of one monolayer or so are not uncommon across a 2-h-diameter MBE wafer. Such a variation will affect much more the energy of the 1 + 2 transition than the 1 -,3 transition since the second state is spatially centered on the thin well, while the third state is centered on the thick well. Note, in fact, that the position of the second-harmonic peak associated with the 1 + 3 resonance hardly changes in the two samples. The distance traveled by the pump and second-harmonic beams in each pass through the superlattice is much smaller than the coherence length 1,( E 120 pm). In addition, the phase mismatch acquired by the fundamental

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 105

and second-harmonic waves in the top InGaAs cladding layer between the two passes through the superlattice, is negligible since the distance travelled ( E 1.1 pm) is << 1,. Also the relative phase shift upon reflection from the top surface is negligible because of the near equality of the refractive indexes at w and 2w. The equation for P(2w) can then be well approximated by (Singh, 1986)* P(2w) = p y & ; ' 2

w2

[l

-

where 6' is the angle of incidence with respect to the normal to the plane of the layers; L is the thickness of the active region of the superlattice (i.e., the wells and thin coupling barrier) times the number of passes (two); S is the area of the laser spot on the sample (diameter = 150 pm); n is the refractive index [n = n(w) z n(2w)I; R(2w) is the reflectivity at 2 0 ; T ( w ) and 5720) are the transmission of the CO, laser and of the second-harmonic power, respectively, in traversing twice the coupled well region, respectively; and P(w) is the pump power. Note that Eq. (16) for the second-harmonic power is an approximation of the physical situation. In general, P(2w) arises from the coherent superposition of the polarization of the well and that of the bulk (dominated by the InP substrate) (Fejer et al., 1989; Yo0 et al., 1991). This gives rise to additional contributions to P(2w) z 20pW for a pump power of 70 mW; that is, a negligible fraction of the measured P(20) (Sirtori et al., 1991). Note also that the detected fluorescence at 2w (i.e., the intersubband spontaneous emission occurring when 2 h o = AE13) can be shown to be completely negligible also in the most favorable conditions (AEI2 = AE23), that is, for a bias = 4 V . This is because the radiative efficiency of the 3 + 1 transition is very small (dlo-') since the spontaneous emission lifetime is many orders of magnitude longer than the intersubband relaxation time and the emission is nondirectional. To verify the selection rules for SHG we rotated the polarization of the pump beam. *A rigorous derivation of the second-harmonic power in the case of phase matching shows that the spatial dependence of P(2w) is given by

4 [ a- exp

-(al

+

[ (q)

a 2 ) L] exp

- exp

r+)y

where CI = u1 - ci2/2;aI and a2 are the absorption constants at the pump and second-harmonic frequency, respectively; and L is the interaction length. For interaction lengths L such that a L / 2 << 1 (which is well satisfied in our structure) the preceding expression reduces to a 2 / 2 ) L ] ,that is, LZT(to)[T(2w)]''2, which appears in Eq. (16). L?exp[-ul

+

106

CARLO SIRTORI ET AL.

The signal is zero when the pump beam has no component of the electric field normal to the layers (4 = 0). As the angle 4 is increased, the second-harmonic power exhibits the sin44 dependence expected for intersubband transitions (Sirtori et al., 1991). We also verified that the secondharmonic signal is linearly polarized in the plane defined by the propagation direction and the normal to the layers (Sirtori et al., 1991). From the analysis of the data of Fig. 10 and Eq. (16) we find lx(2)(20)l= 3 x lO-'m/V at zero bias and I~(~)(20)1 = 7.5 x 10-*m/V at 4.0V, corresponding to a field F % 3.8 x 104V/cm. The second-harmonic power and the susceptibility show a pronounced peak for positive polarity at fields in the range 3.5-4.5 x lo4 V/cm (Figs. 12 and 13). by) an electric field can be understood in The enhancement of ~ ( ~ ' ( 2 w terms of the Stark shifts (Fig. 5). For the photon energies shown, hw < AE,,, so positive bias must be applied to achieve the resonance condition ho = AEI2 and a peak in Ix(2)(20)1(see Vodjdani et al. 1991). The higher ho,the lower the field required to achieve this resonant condition and thus the peak of will shift to lower bias; this trend is clearly manifested in the data of Figs 12 and 13. The maximum value of Ix"'(2w)l will occur when AE,, = AE13= ho.Our calculations predict that this should occur at ho x 120 meV corresponding to an electric field 3.2 x lo4 V/cm (Fig. 5), in good agreement with the experimental values (122.2 meV and 3.7 x 104V/cm). From the curves at 122.2 and 120.4meV (Fig. 13), we

0 2.5 APPLIED BIAS (V)

5.0

FIG. 12. Second-harmonic power as a function of applied bias at various pump photon energies at room temperature. The pump power is 60mW.

2 NONLINEAR OPTICSIN COUPLED-QUANTUM-WELL QUASI-MOLECULES 107 I

I

-3

0 3 APPLIED BIAS (V)

I

‘

I

K W

0

U

0

6

FIG. 13. Measured second-order susceptibility 1~‘~’(2w)l as a function of bias at different pump photon energies at room temperature. These data are obtained from those of Fig. 1 1 using Eq. (6).

obtain a shift of 4 meV for an electric-field increase of 10 kV/cm, in good agreement with the calculations (Fig. 5). Using for r the experimental value of 15meV obtained from the Fourier transform infrared (FTIR) measurements (Fig. 2) and the calculated values of the transition matrix elements ((zI2) = 16.1A, ( z ~ ~= )20.4A, ( z 3 ] )= 12.4A at F = 3.7 x 104V/cm), one then finds from Eq. (10) the maximum value I~(~’(2w)l = 1x m/V, in good agreement with the experimental data (0.75 x 10-7m/V). The experimental error in the measured If2)(2w)l is estimated to be *20%. Note that typical values for bulk GaAs, InAs, and InP are 5 x 10- l om/V at A % 10pm pump (Shen, 1984). Figures 14 and 15 show the second harmonic power as a function of pump power at a pump photon energy ho = 122.7meV and as a function of bias at various pump photon energies, at cryogenic temperature. In Fig. 16, I ~ ( ~ ) ( 2 0is) 1plotted as a function of bias using the data of Fig. 15 and Eq. (16). Note that the peak of J ~ ‘ ~ ’ ( 2 wis) labout twice the room temperature value. This enhancement is due to the reduction of the linewidths of the intersubband transitions caused by the reduced scattering at low temperatures. From Eq. (10) one then expects an increase of the maximum of 1f2’(2w)l (obtained when the levels are equally spaced and tio = AE,, = AEI3) by a factor -2, in agreement with the observed experimental enhancement.

- -

108

CARL0 SIRTORI ET AL.

I

I

'

3

" ' I

F C v

a

5

1

g 2

z

0.1

0

P

a

4 9 0 Y

(I)

0.01

0.001 10

20

50 100 PUMP POWER (mW)

200

FIG. 14. Second-harmonic power as a function of pump power at cryogenic temperature for different bias conditions for the two-coupled-well structure. The pump wavelength is 1 r 10.25pm. The solid lines represent least-square fits with a quadratic law. The bias polarity is the same as in Fig. 7.

6

-2 CT

W

3

g4 0

6

z

$ 2

n

I

0

8 (0

0 -2.5

0

2.5

5.0

APPLIED BIAS (V)

FIG. 15. Second-harmonic power as a function of applied bias at various pump photon energies at cryogenic temperature. The pump power is = 150mW.

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 109

z m

F

a w

% 3 (II

1.6~10-~1

I

1.2x

0.8 x

-3

0

3

6

APPLIED BIAS (V)

FIG. 16. Measured second-order susceptibility I ~ ' ~ ' ( 2 0 )as 1 a function of bias at different pump photon energies at cryogenic temperature. These data are obtained from those of Fig. 14 using Eq. (6).

VI. Far-Infrared Generation by Resonant Frequency Mixing For the difference frequency measurements the sample was processed into a trapezoidal two-bounce waveguide, wedged at 45" of base widths 0.5 mm on the substrate sides and 1.5 mm on the epilayer side and 0.5 mm thickness. After entering the sample at 45" incidence the laser beams traverse the epilayer, bounce off the grown surface of the wafer, and emerge from the opposite sides of the waveguide, collinear with the incidence direction, after having traversed the layers a second time. The wafer is clamped between two polished copper surfaces. The sample is mounted in a continuous flow Dewar equipped with ZnSe entrance window and polypropylene exit windows. For the experiment, we used two highly stabilized CO, lasers and care was used to ensure an excellent overlap of the two beams. The spot size radius on the sample is 110 pm. The optical path between the exit window and the detector is contained in a plexiglass cylinder, flooded with dry nitrogen to minimize spurious atmospheric absorption effects. The difference frequency signal is collected and focused onto a silicon, helium cooled bolometer by two gold plated 2-in.-diameter off-axis parabola mirrors with 3-in. length. the beam are chopped at a rate of 500 Hz, and the signal from the detector is detected with a lock-in amplifier. The dependence of the detected signal at A = 60 pm as a function of the product of the power of the two pump beams is shown in Fig. 17. For this experiment the

110

CARLO SIRTORI ET AL.

POWER PRODUCT (rnW)2 FIG. 17. Measured power of the far-infrared DFM ( w 3 )signal as a function of the product of the power of the pump beams at frequencies w1 and w 2 at 7 K temperature. The solid line represents a least-squares fit to the data. The inset shows the angular dependence of infrared signal as a function of polarization angle.

wavelength of the two lasers were kept fixed at the photon energies that maximize the detected signal; the polarization of the two beams is in the plane of incidence to maximize coupling to the intersubband transitions. The linearity of the far infrared power as a function of the power product (Fig. 17) demonstrates that the signal originates from difference-frequency mixing. To check the polarization selection rule we varied the beam polarization from 4 = 0 (normal to plane of incidence) to 4 = 90 (in the plane of incidence) and beyond. The results are shown in the inset of Fig. 17. The detected signal is proportional to s i n 4 ~This . dependence is easily understood by noting that the far-infrared (far-IR) power is generated by the components normal to the layers E,(w,) and E,(w,) of the incidents fields. As the polarization angle 4 is changed E,(w,) and E,(w,) vary proportionally to sin 4 and thus the far-IR power, which is proportional to E:(w,) * E:(w,), scales as sin4+. Figure 18 shows the far-IR as a function of the photon energy difference between the two pump beams. The difference frequency signal peaks slightly above (20.5 meV) the renormalized transition energy AEl ,. The latter is larger than the calculated value (14 meV), because of the depolarization shft (Sirtori et al., 1994a). To analyze the data, we solve the wave equation for

2 NONLINEAR OPTICSIN COUPLED-QUANTUM-WELL QUASI-MOLECULES 111

.

l % I

.

I 0

3

0.4

0

0.2

PHOTON ENERGY DIFFERENCE (meV)

Yi"l z u

[r

w LU n L O15

17

19

21

23

PHOTON ENERGY DIFFERENCE (mev)

FIG. 18. Far-IR power as a function of photon energy difference between the pump beams at 7 K temperature. The power of each pump beam is 100mW. From these data the - oz)lis obtained (inset). difference-frequency dependence of the susceptibility Ix(')(w,

the difference-frequencyfield E(w3)(Shen, 1984). The final expression for the power P(oJ can be well approximated by

where P ( w l ) and P ( w 2 )are the powers of the pump beams in the material assumed to be linearly polarized in the plane of incidence; L is the interaction length (= active material layer thickness/cos 8, where 8 = 57.6" is the angle between the propagation direction in the material and the normal to the layer); n,,, nwz,and nwo are the refractive indices of the three waves; rl and r2 are the radii of the pump beams; a,, and aa2 are the absorption coefficients of the pump beams, and aa3 is that of the far-iR wave; c is the velocity of light in vacuum; ymax is half the maximum angle subtended by the detector; the Bessel function term describes the effect of = ka sin y, where l/a = diffraction from a circular aperture where l/r, + l/r2; and k is the wave number of the far-IR radiation in the material.

112

CARLOSIRTORI ET

AL.

In our sample geometry, the interaction length is much smaller than the coherence length (1, x 30 pm). From the measured far-IR power (Fig. 18) and absorption coefficient, one can obtain an estimate for I X ( ~ ) ( W ~ - oz)lusing Eq. (17). The resulting plot as a function of difference frequency is shown in the inset of Fig. 18. The position of the peak X ( ~ ) ( W-, o,)is in excellent agreement with the 1 + 2 transition energy obtained from the absorption data, as expected from Eq. (11). Our experiments have accurately verified the expected dependence on the powers and on the polarization of the pump beams. In addition the - 0 2 )peaks ( at the energy AE,,, renornonlinear susceptibility I X ( ~ ) ( O , malized by the depolarization shift, in excellent agreement with the theoretical expectations. However, the experimental peak value of is significantly smaller (by about one order of magnitude) than the predictions based on Eq. (11). This might be due to a collapse of the time-dependent perturbative expansion of the nonlinear coefficients (Almogy and Y ariv, 1995) and surely to a number of experimental reasons associated with wellknown uncertainties in the far-IR difference-frequency measurements which have been amply discussed in the literature. There are experimental inherent difficulties in achieving high accuracy in these measurements due to such effects as absorption, diffraction, and walk-off due to double refraction and collection efficiency.

VII. Third-Harmonic Generation and Triply Resonant Nonlinear Susceptibility in Coupled Quantum Wells The experimental arrangement and sample geometry used to observe THG in the structure of Fig. l c are identical to those used for the SHG experiments, except that in the present case, the C O , beam was more tightly focused on one of the 45" edges of the sample to increase the third-harmonic signal. The third-harmonic power is expected to increase as the third power of the pump beam (Shen, 1984). In addition, only the component normal to the layers of the electric field of the incident pump wave contributes to intersubband THG. Thus if we denote with 4 = 90 the polarization direction of the incident pump beam (Fig. 7), rotating it will reduce the third-harmonic power according to sin6#. The measured third-harmonic power (Figs. 19 and 20) verifies the preceding dependence pump power and polarization angle. Spontaneous emission of the frequency 3w corresponding to transitions between the third excited state and the ground state is

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 113 1001

F

Q

v

50

80-

hw=118.3meV T=300K

-

a 60-

0 z 0 z n a

-

-

-

40-

-

I

n 20-

-

u:I I-

om

,;

7

FIG. 19. Third-harmonic power as a function of pump power P(w) in the three-coupledwell structure; the solid Line represents a Et with a cubic.

POLARIZATION ANGLE,

0 (deg)

FIG. 20. Third-harmonic power as a function of pump polarization. The data follow the expected sin6d dependence (solid line).

114

CARLO SIRTORI ET AL.

completely negligible due to the small radiative efficiency of these transitions (40-5). To derive the expression for the third-harmonic power generated in our structure, we observe that the distance traveled by the pump and the thirdharmonic beam in each of the two passes through the superlattice is much smaller than the coherence length (1, = 28 pm at il = 10.6 pm). In addition, the phase mismatch acquired by the fundamental and third-harmonic waves in the top InGaAs cladding layer, between the two passes through the superlattice, is negligible since the distance traveled in this region ( z 1.1pm) is << 1, and the relative phase shift on reflection from the top surface is negligible due to the near equality of the refractive indexes at 3w and w. The expression for P(30) can then be simply derived from Maxwell’s equations including the absorption losses at w and 30, for the case of phase matching and no pump depletion (Sirtori et al., 1992a)

where Pfw) is the pump power (i.e., the incident power minus the reflected power); S is the area of the laser spot (diameter = 42pm); n, and n30 are the refractive indices at w and 30; M , and aJo are the absorption coefficients for the pump and third-harmonic signal, respectively (obtained from the FTIR data), M = (3/2)a, - (1/2)a3,; 8 is the angle of incidence with respect to the normal to the plane of the layers (45” in our case); L is the interaction length, which for our double-pass structure reduces to 2 x (total quantum well thickness plus that of the thin tunneling barriers between the wells)/ cos8; and p,, is the vacuum permittivity. In deriving Eq. (18) we have considered the case of a pump linearly polarized so that the component of the electric field normal to the layers is maximized. Since this component is proportional to sine, the contribution of the pump power to THG comes in as [P(w) sinZ8I3in Eq. (12). The additional sin’ 6 factor in Eq. (17) is a result of the transformation from the crystal coordinate system to the laboratory frame used to describe the field propagation. By best-fitting the data of Fig. 19 and using Eq. (17), one obtains 1~(~)(30)( = 0.6 x 10- l4 (m/V)2. The dependence of THG on pump wavelength was also investigated. The data are shown in Fig. 21a. The power increases rapidly as the photon energy approaches 115 meV, corresponding to the resonance condition 3hw = E , - El. The range of photon energies is limited by the CO, laser.

2 NONLINEAR OPTICSIN COUPLED-QUANTUM-WELL QUASI-MOLECULES 115

n

0

(u

>

0

2 E

0.

0.6

0 0 0

0.4

110

0

120

130

PHOTON ENERGY

140

(mev)

FIG. 21. (a) Third-harmonic power and (b) third-order susceptibility I X ( ~ ’ ( ~ O ) ~ as a function of pump photon energy at room temperature. The pump power is 300 mW.

Note that peaks corresponding to hw = AE,, and 2hw = AE31 cannot be resolved since the energy differences between AE,,, AE23, and AE34 are less than the broadening, as shown by Fig. 4. From the data of Fig. 21a and substituting in Eq. (17) the experimental values of all the quantities involved, one obtains I~(~’(3w)l as a function of ho (Fig. 21b). Using the calculated ( z i j ) in Eq. (12) and the half widths at half maximum of the FTIR = 16, and rI4 = 18 meV) as an estimate absorbance peaks (rl,= 20, r13 of the broadenings Tij in Eq. (12), one obtains I ~ ( ~ ) ( 3 or) 1x 1.3 x (m/V)2 at hw = 115 meV, which compares favorably to the experi(m/V)’ (Fig. 21b). mental value: 0.9 x The preceding experiment was repeated at 30 K. The measured thirdharmonic power as a function of pump photon energy is shown in Fig. 22. Because of the slight shift of the intersubband transitions to higher energies the peak of the third-harmonic signal is now observable. It corresponds to a 1~(~)(3w =) 4 l x (m/V)’. The enhancement is approximately a factor of 4 compared to the room temperature and is due to the narrowing of the

116

CARL0 SIRTORJ ET AL.

0 0

F n g 300 -

0

0

2

0

g

4 0 B 9

n I

T=30K

0

0

200

-

0

0

0

100 -

0

0

I-

0

0

I

1

.

1

I

I

FIG.22. Third-harmonic power as a function of pump photon energy at cryogenic temperature. The pump power is 300 mW.

intersubband transitions, since the peak of ( ~ ( ~ ' ( 3 wscales ) l as l / r 3 . To our knowledge this is the largest third-order nonlinear susceptibility reported in any material. It is five to six orders of magnitude greater than I ~ ( ~ ' ( 3 w ) l associated with bound electrons in InAs and GaAs (Shen, 1984), at comparable wavelengths. The value reported by Sa'ar et al. (1992) for the thirdI% 10.6pm in a GaAs-AlGaAs order intersubband DC Kerr effect at , multiwell structure is 1.7 x (m/v)2. Walrad et al. (1991) have reported intersubband nondegenerate four-wave mixing in AlGaAs-GaAs quantum wells and Grave et al. (1992) phase conjugation associated with intersubband nondegenerate four-wave mixing, demonstrating very large ~ ( ~ ' ' s .

VIII. Multiphoton Electron Emission from Quantum Wells In the three-well structure previously discussed (two periods are shown in Fig. 23a) electrons can be promoted to a resonant state above the barriers via a three-photon transition, giving rise to a photocurrent (Fig. 23b), in a process conceptually similar to multiphoton ionization of a molecule. Depending on the bias polarity this effect can be enhanced by the presence of intermediate energy levels (Fig. 23b).

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 117

FIG. 23. (a) Energy band diagram at zero bias of two periods of the three-coupled-well structure. Shown are the positions of the calculated energy subbands and the corresponding modules squared of the wave functions. (b) Energy band diagram with positive bias polarity (3V). Note the Stark shift of the energy levels and the three photon transition (3hv with hv 130meV) into a resonant state localized in the barriers.

The samples were processed into 350-pm-diameter mesas and ohmic contacts were provided to the two ’1 GaInAs cladding layers. They were then cleaved in narrow strips and the cleaved edges were polished at 45" to provide a two-pass waveguide for the pump beam (Fig. 7). The samples were cooled down to cryogenic temperatures in a Helitran flow Dewar to minimize competing effects of electron thermionic emission out of the wells. Since our measurements are performed at relatively high fields (lo4V/cm), the Stark effect plays an important role.

118

CARLO SIRTORI ET AL.

To observe the photocurrent generated by the three-photon bound-tocontinuum transition in positive-bias polarity, the incident photon energy must be tuned so that 3hw 2 AE, - El, where AE, is the conduction-band discontinuity, implying hw 2 125 meV. The three-photon transition rate can be strongly increased by the intermediate energy levels E , and E,. To exploit this resonant enhancement, we have made use of the Stark effect. At zero bias AE12< AE23; BE,, strongly increases with increasing positive bias (Fig. 9(b), while our calculations show that AEZ3is weakly dependent on the electric field (Fig. 6). At 3 V and ho = 130 meV the detunings from the intermediate energy levels are large enough to minimize the population of the second and third levels by absorption (ho- AE,, = 13 meV; 2ho - AE,, = 15 meV), but sufficiently small to produce a resonant enhancement of the three photon transition 1 + 5. Note that the final state of the latter is a resonant state above the barriers (Fig. 23b). This state is centered in the barrier layers and resembles a Fabry-Perot cavity resonance in optics, due to the sizeable reflection coefficients between the barrier and the three-coupled-well regions. Note, however, that there is still a substantial dipole matrix element between this state and the third confined state of the well ((z,~) = 3.8 A at 3 V). States localized in the barrier layers have previously been theoretically investigated by Jaros and Wong (1984) in AlGaAs-GaAs multiquantum wells. The photocurrent generated by the three-photon transitions is expected to increase with the third power of the laser beam (Shen, 1984). In addition, only the component normal to the layers of the electric field of the incident wave contributes to the photocurrent. Thus if # = 90 represents the polarization of the incident laser shown in Fig. 7, rotating the polarization of the laser will reduce the photocurrent according to sin6# (Fig. 20). The photocurrent measured under the optimum conditions of photon energy and bias described earlier, demonstrates very clearly this angular dependence (Fig. 24a). The same dependence is found with opposite-bias polarity (Fig. 24b) but the signal significantly decreases. This is a result of the asymmetry of the structure; in this polarity the Stark effect increases the energy detunings from levels E, and E, (ho- AEl, = 43meV; 2ho - AE13= 46 meV, at - 3 V bias), thus decreasing the resonant enhancement of the 1 + 5 three-photon transition. Figure 25 shows the photocurrent spectral response at 3 V bias. The data show that the photocurrent (for photon energies 124meV, such that the final state is above the barrier) first rises and reaches a maximum around ho x 130meV. As previously discussed, at this energy the three-photon transition is strongly enhanced by the intermediate states and is resonant with the state localized in the barrier. At higher energies increased detuning from resonance is responsible for the photocurrent decrease. In reverse

2 NONLINEAR OPTICSIN COUPLED-QUANTUM-WELL QUASI-MOLECULES 119

15

I

I

I

I

I

I

BIAS = +3V

I

I

/a\

10

0

45

90

135

180

POLARIZATION ANGLE, 4 (deg) FIG. 24. Photocurrent as a function of laser polarization for opposite bias polarities at an incident power of 100mW. The data follow closely the sin64 dependence (solid curve) expected for three photon intersubband transitions.

polarity (- 3 V), we found that the photocurrent increases for Rw 2 119 meV. The rise of the photocurrent is gradual and no peak is observed since the multiphoton transition is only slightly enhanced by intermediate states (Fig. 26). The photocurrent in the multiphoton emission regime can be expressed in terms of the three-photon emission rate W similar to three-photon ionization of an atom (Shen, 1984; Chiu and Lambropoulos, 1984)

where I is the laser intensity and u3 is a generalized three-photon cross

120

CARLOSIRTORI ET AL. I

0 1 0

BIAS = +3V

z %W

40

-

0 O

.

00

0

30-

0

a a

; 3

20-

0

B

10

-

0 0.

. 0 . . . @

01

I

I

120

110

I

I

130

I

140

PHOTON ENERGY (mew

FIG.25. Photocurrent spectral response at 15 K for the three-coupled-well structure of Fig. 20 obtained with the available transitions of a tunable CO, laser for positive bias polarity. The incident power (100mW) is the same at all wavelengths. The peak is a manifestation of the resonant enhancement of the three-photon electron emission out of the well (see bottom of Fig. 23).

section expressed in cm6s2 units (Shen, 1984; Chiu and Lambropoulos, 1984). The expression for o3 contains several terms. Each term has in the numerator three matrix elements connecting the initial and final states via intermediate states and in the denominator two appropriate frequency factors responsible for the resonant enhancement previously discussed (Shen, 1984; Chiu and Lambropoulos, 1984). The photocurrent in our structure is then expressed as, using Eq. (19),

I,,

=

2qy, N , np,(P, sin28) (hw) 3 s2

where vc is the collection efficiency, 2 N , is the number of wells traversed in a double pass, ns the electron sheet density, S is the laser spot area on the sample (diameter E 42 mm); 8 is the angle of incidence with respect to the normal to the plane of the layers (45" in our case), and we have assumed that the laser is linearly polarized so that the component of the electric field normal to the layers is maximized. The collection efficiency y, is assumed to

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 121

a

6-

a

7 5-

v

ia

4-

2

3-

a

a@

a@

IT

8E 2 10

a

mu.#

*@ I

I

be s 1, which has been found to be the case in studies of quantum well infrared detectors (Levine, 1988). We have furthermore neglected the attenuation of the laser since the transmission of the quantum well region in our double-pass geometry is 0.8 at the above photon energy. Taking I,, E 50 nA (i.e., the value measured at hw = 130 meV), one can estimate from Eq. (20), a three-photon cross section C T ~z I x cm6 s2. This value is in reasonable agreement with theoretical estimates and is many orders of magnitude larger than values found in atoms for three-photon ionization processes (Shen, 1984; Chiu and Lambropoulos, 1984). This is not surprising since g3 is proportional to the product of three matrix elements squared, each much larger than atomic values. In addition, the vicinity to two resonances further enhances 0 3 .

IX. Resonant Third-Harmonic Generation via a Continuum Resonance

It is clear from the band diagram of Fig. 23b that at a bias of 3 V and for a pump photon energy h o 130meV one should also observe a resonant enhancement of third-harmonic generation since at this energy, 3hw =

122

CARLO SIRTORI ET AL.

Fn 2 60

w = 129.2 meV

5

2

u

2 40

0

BQ

I

g -

20

I I-

0 -5.0

-2.5

2.5

0

5.0

7 5

BIAS (V)

FIG.27. Third-hamonic power as a function of applied bias at various pump photon energies at 15 K for the three-coupled-well structure.

E , - El,2ho z E , - El, and hw E , - E l . As the pump photon energy is increased above E 120meV the third-harmonic power at and near zero bias strongly decreased, a manifestation of detuning from the resonant situation (see Fig. 22). However, another peak develops at a bias z 3 V as the pump photon energy approaches 130 meV (Fig. 27), in accordance with the preceding prediction of a resonant enhancement of I~(~)(3o)l. From an analysis of the data using Eq. (17), we are able to deduce that ~ ( ~ ’ ( 3= 0) (m/V)’ at ho = 129.2meV and 3 V bias.

X. Conclusions To summar ize, in this chapter we have presented the results of an extensive research program on the IR nonlinear optical properties of coupled quantum well semiconductors. By careful tailoring of layer thicknesses and compositions, the nonlinear susceptibilities can be maximized so that the structures effectively behave as artificial molecules with giant IX(”1 and I x ‘ ~ ) ~ . One of the most interesting aspects of these new systems is the extremely large linear Stark effect. This phenomenon has been used to electric-field-tune f 2 ) and x(3)and is of interest also for the realization of fast IR modulators with high on-off ratio.

2 NONLINEAR OPTICS IN COUPLED-QUANTUM-WELL QUASI-MOLECULES 123

REFERENCES Almogy, G., and A. Yariv. (1995). “Resonantly-enhanced nonlinear optics of intersubband transitions.” J . Nonlin. Opt. Phy. Mater. 4, 401-458. Bethea, C. G., B. F. Levine, V. 0. Shen, R. R. Abbott, and S. J. Hsieh. (1991). “10pm GaAs/AlGaAs multiquantum well scanned array infrared imaging camera.” IEEE Trans. Electron Devices 38, 1118- 1123. Boucaud, P., F. H. Julien, D. D. Yang, J. M. Lourtioz, E. Rosencher, P. Bois, and J. Nagle. (1990). “Detailed analysis of second harmonic generation near 10.6pm in GaAs/AlGaAs asymmetric quantum wells.” Appl. Phys. Lett. 57, 215-218. Boucaud, P., F. H. Julien, D. D. Yang, J. M. Lourtioz, E. Rosencher, and P. Bois. (1991). “Saturation of second harmonic generation in GaAs/AlGaAs asymmetric quantum wells.” Opt. Lett. 16, 199-202. Boyd, R. (1992). Nonlinear Optics. San Diego, CA: Academic. Burt, M. G. (1993). “The evaluation of the matrix element for interband optical transitions in quantum wells using envelope functions.” J . Phys. C 5, 4091-4098. Capasso, F. (1991). “Bandgap interface engineering for advanced electronic and photonic devices.” M R S Bull. 16, 23-29. Capasso, F., and S. Datta. (1990). “Quantum electron devices.” Phys. Today 43, 74-82. Capasso, F., and C. Sirtori. (1992). “Bandgap engineering of coupled-quantum well molecules with large field-tunable nonlinear optical properties.” In Quantum Electronics and Laser Science Conf, OSA Tech. Dig. Series, vol. 13, pp. 108-109. Washington, DC: Opt. SOC. Amer. Capasso, F., C. Sirtori, A. Y. Cho, and D. L. Sivco. (1992a). “A new class of coupled quantum well semiconductors with large electric-field tunable nonlinear susceptibilities in the infrared.” In Int. Con$ Quantum Electronics Tech. Dig. Series, vol. 9, pp. 60-61. Capasso, F., C. Sirtori, D. Sivco, and A. Y. Cho. (1992b). “Nonlinear optics of intersubband transitions in AlInAs/GaInAs coupled quantum well: Second harmonic generation and resonant Stark tuning of f2’(2w).” In Intersubband Transitions in Quantum Wells, E. Rosencher, B. Vinter, and B. F. Levine, Eds., NATO AS1 Series, Series B, Physics, vol. 288, pp. 141-149, New York Plenum. Chiu, S. L., and P. Lambropoulos, Eds. (1984). Multiphoton ionization of Atoms. Orlando, FL: Academic. Faist, J., F. Capasso, C. Sirtori, D. L. Sivco, A. L. Hutchinson, S. N. G. Chu, and A. Y.Cho. (1993). “Measurement of the intersubband scattering rate in semiconductor quantum wells by excited state differential absorption spectroscopy.” Appl. Phys. Lett. 63, 1354-1356. Faist, J., C. Sirtori, F. Capasso, L. Pfeiffer, and K. West. (1994). “Phonon limited intersubband lifetimes and linewidths in two-dimensional electron gas.” Appl. Phys. Lett. 64, 872-874. Fejer, M. M., S. J. B. Yoo, R. L. Byer, A. Harwit, and J. S. Harris. (1989). “Observation of extremely large quadratic susceptibility at 9.6-10.8 pm in electric field biased AlGaAs/ GaAs quantum wells.” Phys. Reo. Lett. 62, 1041-1044. Grave, I., M. Segev, and A. Yariv. (1992). “Observation of phase conjugation at 10.6pm via intersubband third order nonlinearities in a GaAs/AIGaAs multiquantum well structure.” Appl. Phys. Lett. 60, 2717-2719. Gurnick, M. K., and T. A. De Temple. (1983). “Synthetic nonlinear semiconductors.” IEEE J . Quantum Electron. QE19, 791-796. Harwit, A,, and J. S. Harris. (1987). “Observation of Stark shifts in quantum well intersubband transitions.” Appl. Phys. Lett. 50, 685-687. Jaros, M., and K. B. Wong. (1984). “New electron states in GaAs/AI,Ga,_,As superlattice.”

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J . Phys. C17, L765-L769. Karunasiri, R. P. G., Y. J. Mii, and K. L. Wang. (1990). “Tunable infrared modulator and switch using Stark shift in step quantum wells.” I E E E Electron Device Lett. 11, 227-231. Levine, B. F. (1992). “Recent progress in quantum well infrared photodetectors.” In Intersubband Transitions in Quantum Wells, E. Rosencher, B. Vinter, and B. F. Levine, Eds. Nato AS1 Series, Series B, Physics, vol. 288, pp. 43-55. New York: Plenum. Levine, B. F., C . G. Bethea, G. Hasnain, J. Walker, and R. J. Malik. (1988). “High-detectivity D* = 1 x 10’’ c m m GaAs/AlGaAs multiquantum well I = 3.3 pm infrared detector.” Appl. Phys. Lett. 53, 296-298. Mii, Y. J., R. P. G. Karunasiri, K. L. Wang, M. Chen, and P. F. Yuh. (1990). “Large Stark shifts of the local-to-global-state intersubband transitions in step quantum wells.” Appl. Phys. Lett. 56, 1986-1988. Pfeiffer, L., E. F. Schubert, K. W. West, and C. W. Magee. (1991). “Si dopand migration and the AlGaAs/GaAs inverted interface.” Appl. Phys. Lett. 58, 2258. Rosencher, E., and P. Bois. (1991). “Model system for optical nonlinearities asymmetric quantum wells.” Phys. Rev. B 44,11315-11327. Rosencher, E., P. Bois, J. Nagle, E. Costard, and S. Delaitre. (1989). “Observation of nonlinear optical rectification at 10.6 pm in compositional asymmetrical AlGaAs/GaAs multiquantum wells.” Appl. Phys. Lett. 55, 1597-1599. Rosencher, E., P. Bois, B. Vinter, J. Nagle, and D. Kaplan. (1990). “Giant nonlinear optical rectification at 8-12 pm in asymmetric coupled quantum wells.” Appl. Phys. Lett. 56, 1822- 1824. Rosencher, E., B. Vinter, and B. F. Levine, Eds. (1992). Intersubband Transitions in Quantum Wells, NATO AS1 Series, Series B, Physics, vol. 288. New York Plenum. Sa’ar, A., N. Kuze, J. Feng, L. Grave, and A. Yariv. (1992). “Third order intersubband Kerr effect in GaAs/AlGaAs quantum wells.” In Intersubband Transitions in Quantum Wells, E. Rosencher, B. Vinter, and B. F. Levine, Eds. NATO AS1 Series, Series B, Physics, vol. 288, pp. 197-207. New York Plenum. Sakaki, H., T. Noda, K. Hirakawa, M. Tanaka, and T. Matsusue. (1987). “Interface roughness scattering in GaAs/AlAs quantum wells.” Appl. Phys. Lett. 51, 1934- 1936. Shen, Y. R. (1984). The Principles of Nonlinear Optics. New York: Wiley. Singh, S. (1986). In CRC Handbook Series of Laser Science and Technology: Optical Materials, Nonlinear Optical Properties, Radiation Damage, vol. 111, M. Weber, Ed. Boca Raton, FL: CRC. Sirtori, C., F. Capasso, D. L. Sivco, S. N. G. Chu, and A. Y. Cho. (1991). “Observation of large second order susceptibility via intersubband transitions at I 10 pm in asymmetric coupled AlInAs/GaInAs quantum wells.” Appl. Phys. Lett. 59, 2302-2304. Sirtori, C., F. Capasso, D. L. Sivco, and A. Y. Cho. (1992a). “Giant triply resonant t h r d order nonlinear susceptibility x(”(3w) in coupled quantum wells.” Phys. Rev. Lett. 68, 1010-1012. Sirtori, C., F. Capasso, D. L. Sivco, and A. Y. Cho. (1992b). “Resonant multiphoton electron emission from a quantum well.” Appl. Phys. Lett. 60, 2678-2680. Sirtori, C., F. Capasso, D. L. Sivco, A. L. Hutchinson, and A. Y. Cho. (1992~).“Resonant Stark tuning of second-order susceptibility in coupled quantum wells.” Appl. Phys. Lett. 60, 151-153. Sirtori, C., F. Capasso, J. Faist, L. Pfeiffer, and K. W. West. (1994a). “Far-infrared generation by doubly resonant difference frequency mixing in a coupled quantum well two-dimensional electron gas system.” Appl. Phys. Lett. 65, 445-447. Sirtori, C., F. Capasso, J. Faist, and S. Scandolo. (1994b). “Nonparabolicity and a sum rule associated with bound-to-bound and bound-to-continuum intersubband transitions in quantum wells.” Phys. Rev. B 50, 8663-8674.

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2 NONLINEAR OPTICSIN COUPLED-QUANTUM-WELL QUASI-MOLECULES 125 Vodjdani, N., B. Vinter, V. Berger, E. Bockenhoff, and E. Costard. (1991). “Tunneling assisted modulation of the intersubband absorption in double quantum wells.” Appl. Phys. Lett. 59,555-557. Walrad, D., S. Y. Auyang, P. A. Wolff, and M. Sugimoto. (1991). “Observation of third order optical nonlinearity due to intersubband transitions in AlGaAsiGaAs superlattices.”Appl. Phys. Lett. 59, 2932-2934. West, L. C., and S. J. Eglash. (1985). “First observation of an extremely large dipole infrared transition within the conduction band of a GaAs quantum well.” Appl. Phys. Lett. 46, 1156- 1157. Yoo, S. J. B., M. M. Fejer, R. J. Byer, and J. S. Harris Jr. (1991). “Second order susceptibility in asymmetric quantum wells and its control by proton bombardment.” Appl. Phys. Lett. 58, 1124-1726. Yuh, P. F., and K. L. Wang. (1989). “Large Stark effects for transitions from local states to global states in quantum well structures.” f E E E J . Quaiituni Electron. 25, 1671-1676.

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SEMICONDIJC'TOKS AND SEMIMETALS. VOL. 66

CHAPTER 3

Photon-Assisted Tunneling in Semiconductor Quantum Structures Karl Unterrainer INSTITUT FUR FESTK~RPERELEKTRONIK TECHNISCHE UNIVERSITAT WEN VIENNA, AUSTRIA

I. INTRODUCTION . . . .

.

.

. . . . . . . . . .

. .

. . . .

.

. . .

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.

. . .

127 I29 129 130 130 134 134 138 140 144 146 147

,

149

11. THEORY ON PHOTON-ASSISTED TUNNELING . . . . . . . . . . . . . . . .

111.

IV. V. VI.

VII. VIII.

Ix.

1.General.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Perturbative Limit . . . . . . . . . . . . . . . . . . . . . . . . 3. Nonperturbative Limit (Tien-Gordon Model) . . . . . . . . . . . . COHERENT TRANSPORT IN EXTERNAL AC FIELDS , . . . . . . . . . . . I. Transport in Minibands . . . . . . . . . . . . . . . . . . . . . . 2. The Wannier-Stark Ladder . . . . . . . . . . . . . . . . . . . . 3. Classical Description of Miniband Transport in External A C Fields . . . 4. Quantum Mechanical Description of Superlattices in External A C Fields . 5. Analogy to the A C Josephson Efect . . . . . . . . . . . . . . . . EXPERIMENTAL METHODS . . . . . . . . . . . . . . . . . . . . . . EXPERIMENTS ON PHOTON-ASSISTED TUNNELING IN RESONANT TUNNELING D I O D E S . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . PHOTON-ASSISTED TUNNELING IN WEAKLY COUPLED SUPERLATTICES . . . . I. Photon-Assisted Tunneling between Ground and Excited States . . . . . 2. Photon-Assisted Tunneling between Ground States (Dynamic Localization. Absolute Negative Conductance, Stimulated Multiphoton Emission) . . . . . . . . . . . . . . . . . . . . . . . . . . . . TERAHERTZ TRANSPORT IN SUPERLATTICE MINIBANDS . . . . . . . . . . PHOTON-ASSISTED TUNNELING AND TERAHERTZ AMPLIFICATION . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . .

k i v l M A R Y AND OUTLOOK . REFERENCES..

. . . . .

155 155

163 172 182 183 184

I. Introduction

Semiconductor quantum structures are at the center of both fundamental and device-oriented research. These structures are very attractive, since the possibility of bandgap engineering enables tailoring their optical and elec127 Copyright Q 2000 by Academic Press All rights of reproduction in any form reserved ISBN 0-12-752175-5 ISSN 0080-8784100 $30 00

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trical properties. The existence of quantized energy levels in such structures is the basis for resonant tunneling and for the enrichment of the optical absorption spectrum by intersubband transitions. Resonant tunneling has been studied in great detail since the very early stage of molecular beam epitaxy in 1974 (Esaki et al., 1974). The same is true for measurements of the optical properties. Surprisingly, it took nearly a decade before the first intersubband absorption was reported by West and Eglash (1985). A lot of theoretical work was devoted to intersubband transitions and many device proposals were based on them. Among these proposals are intersubband infrared detectors, which have been realized and have reached a quite high level of performance (see Chapter 3). Nonlinear optical elements were also suggested based on intersubband resonances due to a large degree of freedom for the design of the symmetrical properties (Rosencher et af., 1996). In 1972 a very important proposal was made by Kazarinov and Suris (1972), who predicted that electrically biased quantum wells should emit radiation due to photon-assisted tunneling. The first experimental proof was the observation of spontaneous terahertz (THz) radiation from a multiple quantum well under sequential resonant tunneling (Helm et a!., 1989). In 1994 the quantum cascade laser was realized based on photon-assisted tunneling between quantized states (see Chapter 5). Photon-assisted tunneling connects transport measurements with optical experiments to explore the properties of semiconductor quantum structures for the development of future THz electronic devices. Esaki and Tsu (1970) proposed high-frequency oscillators by tailoring the nonlinear electronic transport properties of semiconductors by fabricating superlattice structures. Electrons accelerated by a moderate, constant electric field E should exhibit negative differential velocity as they are accelerated up to the zone boundary of a miniband and Bragg-reflected. This repetitive motion of acceleration and Bragg reflection is called Bloch oscillation, characterized by the Bloch frequency, wB= (eEd/h), where d is the superlattice period. Bloch oscillation is a well-defined normal mode of the system at THz frequencies, where oB z > 1 (Krieger and Jafrate, 1986; Bouchard and Luban, 1993). However, it took more than two decades before the first observation of Bloch oscillations was made in optical experiments (see Chapter 8). The observation of continuous Bloch emission from electrically injected carriers in a superlattice has escaped observation so far. Photon-assisted tunneling experiments can answer the question of whether Bloch oscillations couple to THz radiation. In addition, several exciting effects have been predicted for miniband superlattices in an external THz field. Strong THz radiation should suppress the miniband conductivity due to dynamic localization. Connected to this is the occurrence of absolute negative conductivity and self-induced transparency (Ignatov and

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129

Romanov, 1976a, 1976b; Bass, 1977). The experimental investigation of these effects is very interesting from a physical point of view and for technological applications in THz electronics.

11. Theory on Photon-Assisted Tunneling

1. GENERAL We want to study the vertical transport through semiconductor heterostructures under the influence of externaI radiation. We restrict ourselves to radiation with a photon energy much less than the bandgaps of the materials involved. Thus, no bandgap transitions and no generation of electron-hole pairs must be considered. We start with a simple double-well structure under bias (see Fig. 1). The current in this structure is given by (Weisbuch and Vinter, 1991)

where z is the growth direction, v, is the electron velocity in the z direction, T is the transmission probability, and f;: and ff are the distribution functions of the initial and final states. This is the well-known tunnel current. For a resonant tunneling structure, resonances in the current occur when the quantized levels in both wells align (right side of Fig. 1). The voltage regions above the resonances are characterized by a sharp drop of the conductivity, leading to negative differential conductivity.

7

FIG. 1. Conduction band diagram of a double quantum well. Left side, no applied bias; right side, applied bias voltage V so that levels E i and E , are aligned.

130

KARL UNTERRAWR

2. FERTURBATIVE LIMIT In the presence of an external field, an additional current can flow due to optical transitions form one well to an adjacent well (see Fig. 2). This current is related to the optical transition rate K,,which can be calculated in first-order perturbation theory according to Fermi's golden rule

The interaction Hamiltonian H is given by - e r ' E , in the dipole approximation, and the matrix element for intersubband transitions reduces to (flHli) =

-

eE2

s

q~,(z)zcpr(z)dz

=

-eE:zi,

(3)

where the q ( z ) are the wave functions of the quantized states. For intersubband transitions, an electric field component parallel to the growth direction is necessary. The same is true for the observation of a photon-assisted tunneling current parallel to growth direction. A photon-assisted current through the structure is expected when the photon energy is equal to the difference between the energy levels ho = E , - Ei and for a nonvanishing matrix element z i f .

3. NONPERTURBATIVE LIMIT(TEN-GORDON MODEL) When the external electric field E , is high (i.e., when the voltage drops across one well), edE, (d is the thickness of the well) becomes comparable

Z

FIG. 2. Photon-assisted tunneling between Eiand E,. The photon energy hv is equal to the energy difference of the levels.

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131

to the applied DC voltage V , edE, z e x perturbation theory is no longer valid and a more rigorous solution for photon-assisted transport is needed. The model of Tien and Gordon (1963) was originally formulated to describe photon-assisted tunneling in superconducting junctions. The model assumes two electron systems separated by a barrier. An external electromagnetic field at frequency o incident on the junction can be represented as a timedependent voltage V, applied across the junction (see Fig. 3). In addition the tunnel junction is biased with a constant voltage V,, so that V ( t )=

v, + V,cos(ot)

(4)

The Hamiltonian of the modulated system is given by H

=

H,

+ e(V, + V,cos(wt))

(5)

where H , is the Hamiltonian in absence of any external fields with energy levels Ei and eigenfunctions Yi given by Y (x,t ) =

(x)exp [ - ($5) Eit ]

(6)

The time-dependent voltage is assumed to adiabatically modulate the potential energy of each energy level on one side of the barrier. The other side is supposed to be grounded and thus not affected. The time dependence of the wave function for every electron state is then modified in the following way:

Yi(x, t)

= Y,(x) exp

[ j: -

[E,

+ eV(t')]dt'

1

(7)

FIG. 3. Double quantum well with applied AC electric field. The left well is assumed to be grounded.

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KARLUNTERRAINER

where 'Pi are the wave functions and Ei are the eigenvalues of the electrons in the unperturbed system. This equation can be expressed in terms of Bessel functions:

The influence of the radiation can thus be described in terms of a probability amplitude J,(eV,/ho) for every energy level to be displaced in energy by n k o . The situation is shown in Fig. 4. All electrons are modulated with the same phase. Thus, the virtual states created by the external radiation are equivalent to bias voltages (V, nhw/e) applied across the junction with a probability 51( e V , / h o ) . The resulting tunnel current can therefore be described by the expression

+

where I,( V ) is the I-V characteristic without external radiation. The tunnel current is now a sum of the original (n = 0, zero-photon) part and contributions from photon-assisted tunneling ( n = 1, one-photon tunneling; n = 2, two-photon tunneling): The relative strength is given by the Bessel functions. In Fig. 5 the resulting tunneling current is shown for intermediate field strength. The unirradiated I-V curve and the one-photon and twophoton contributions are shown as thin solid lines. The resulting current is shown as a thick solid line. The photon-assisted contributions are replicas

FIG.4. The external radiation leads to the formation of photon sidebands (dashed lines) separated by + n . hv from the original level. In the situation shown, the electron can tunnel from level E , in the right well to the left well through the n = - 1 photon sideband.

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133

I

V FIG. 5. Schematic tunnel current in an external AC electric field. The central peak is due to the alignment of the quantized levels. The peaks on both sides are due to photon-assisted tunneling and are separated in voltage by &- hv/e from the main peak. The unirradiated I-V and the individual contributions that form the photon sidebands are shown by thin lines. The thick solid line shows the resulting tunnel current. The original value of the tunnel current of the main peak is suppressed in the presence of an external field.

of the zero-photon current-voltage characteristic shifted by f nhw/e to higher or lower voltages. For small amplitudes V,(eV, << hw), we can expand the Bessel functions using

Considering only the lowest (quadratic) terms in V,,, we obtain the following expression for the tunnel current

This expression shows that the tunnel current under external radiation is modified by adding photon-assisted channels. The third term is assisted tunneling by absorbing one photon, and the fourth term is an additional current due to stimulated emission of one photon. The magnitude of the additional current is determined by the amplitude of the AC field strength el/,/(hw). The second term corresponds to a reduction of the original DC tunnel current due to a decrease of the probability amplitude of the ground state.

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We can rewrite Eq. (11) to a form of a second difference:

In this form, the second difference reduces to the second derivative of the I-V curve, and for small photon energies compared to the voltage scale of the nonlinearities the standard classical result for rectification is obtained:

Tucker (1985) showed that even for large V,, a classical expression for the tunnel current can be obtained for small photon energies. Equation (9) reduces to the classical AC rectification:

This expression for the tunnel current is valid only for photon energies hw/e smaller than the voltage scale of the tunnel resonances. In the following, we discuss these general results for the case of our double-quantum-well system. We assume two energy levels and two different bias situations. In Fig. 6a, the bias is small and the difference between the first level in the first well and the second level in the second well is still large compared to the photon energy. The energy difference between the ground states is equal to the photon energy. For this situation, current flow is possible only through the n = - 1 sideband which corresponds to the emission of one photon. In Fig. 6b, the bias is higher and the difference between the ground state of the first well and the excited state of the second well becomes comparable with the photon energy. An additional current flow is possible through the n = 1 sideband, which corresponds to the absorption of a photon.

III. Coherent Transport in External AC Fields 1. TRANSPORT IN MINIBANDS

To describe transport in superlattices formed by the periodic growth of quantum wells and barriers we must distinguish two cases. When the barriers are thick so that the tunneling time through a single period is longer

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

135

FIG.6 . Photon-assisted tunneling in a double well with two energy levels. (a) For this bias situation, photon-assisted tunneling is possible only through the n = - 1 photon sideband. The electron can only tunnel from the E , to the E’, level by stimulated emission of one photon. (b) For this bias situation, an electron can tunnel from the left to the right only through the n = 1 photon sideband. An electron in E , can tunnel to the excited level E2 by absorbing one photon from the external field.

than the typical scattering time, the transport can be described by sequential tunneling processes through single periods. If the barriers are thin enough, the wells are strongly coupled and the wave functions are extended throughout several periods. The coupled states form minibands with an energy dispersion relation given by the miniband bandstructure (see Fig. 7). The transport is dominated by coherent tunneling processes. The tunneling time

h

n

N

P a c

a

FIG. 7. (a) Conduction band of a superlattice. The strong coupling lifts the degeneracy of the energy levels. (b) A typical band structure of the first and second minibands.

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KARLUNTERRAINER

between the wells is much shorter then the typical scattering time and the electrons can propagate through several periods without scattering. The band structure of a superlattice in tight-binding approximation for the growth direction parallel to the z direction is given by Ei(k) = Ei

+ Ai/2(1 + ( -

1)' cos(k,d))

h2 + 2m* (k: + k,Z)

(15)

where Ei is the bottom edge of the ith miniband, Ai is the miniband width, and d is the superlattice period. The electron group velocity within one subband is given by

u(k,)

1 dE h dk,

Ad 2h

= - - = - sin(k,d)

The motion of an electron within a miniband for an applied electric field E in the z direction is given by hk,

= - eE

* k,(t)

= k,O - eEt

(17)

Substituting in Eq. (16) results in Ad . v(t) = - sin(k,Od/h - eEt d/h) 2h Thus, the velocity oscillates with a typical frequency oB= eEd/h- the so-called Bloch frequency. This corresponds to a Bragg reflection of the electron into the opposite direction at the end of the first Brillouin zone of the miniband. Integration of Eq. (18) leads to

describing an oscillation in real space with an amplitude dAl(2eEd). The amplitude in real space decreases for increasing applied field. When the voltage drop across one well eEd becomes comparable to the miniband width A, the oscillation in real space becomes confined within a single well. The occurrence of Bloch oscillations in a biased superlattice, of course, influences the DC conductivity. At low bias, when the Bloch frequency is < l), the electrons never not high enough to overcome scattering (oB.z reach the inflection point and the current in the superlattice direction is

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

137

given by the Drude conductivity. When the bias is high enough to make Bloch oscillations possible, the electrons start to oscillate in space and can no longer contribute to the DC current. For a quantitative description of the influence of scattering, a Boltzmann's equation approach is used, which for a nondegenerate Maxwell-Boltzmann distribution, leads to the following relation between miniband current and applied electric field (ho,< A) (Lebwohl et al., 1970)

. ne2z

J=-

E msL 1 + o;z2

I,(A/2kT) Zo(A/2kT)

where z is the scattering time, n is the carrier concentration, mSL is the effective mass at the bottom of the miniband, and I i are the modified Bessel functions. From Eq. (20), we see that for wBz<<1 the current increases linearly with field. Scattering dominates, and the electrons cannot perform coherent Bloch oscillations. The situation changes for w,' z > 1.The current decreases for increasing field because now some of the electrons can perform Bloch oscillations. Electrons in a Bloch oscillation mode do not contribute to the DC current (assuming that the real-space amplitude is smaller than the contact distance). The decrease of the current leads to a region of negative differential conductivity in the superlattice I-V curve above eEd > h/z. The peak in the I-V curve is thus given by o,’z = 1 which corresponds to a value for the electric field of h/(ze). This behavior is plotted in Fig. 8 for a superlattice period of d = 10nm and a scattering time of z=1x s. For electric fields smaller than 200V/cm, the current increase is linear. At 600V/cm, a maximum occurs where the Bloch frequency equals the inverse scattering time followed by a region of negative differential conductivity. The high-frequency properties of a superlattice for small AC fields (EAc << EDc) were calculated by Ktitorov et al. (1972). The high frequency conductivity is given by

with Go

I '(A/ 2kT) msL Io(A/2kT)

ne2z =-

The high-frequency conductivity reduces for wB-+ 0 to the classical Drude

138

KARLUNTERRAINER

1

0

2

E (kV/cm) FIG.8. Current through a superlattice according to Esaki and Tsu (1970). The superlattice period is d = 10 nm. The scattering time is assumed to be 7 = 1 x 10-12s. The maximum of the current is observed for wB’7= 1; above that, negative differential conductivity is observed.

conductivity and for o -,0 to the static differential conductivity cd= dj/dE, wherej is given by Eq. (20) (Shik, 1975). The main result of Eq. (21) is that amplification is predicted for frequencies o for 0 < o < wB,where the real part of the high-frequency conductivity is negative (Grondin et al., 1985). The amplification has a pronounced maximum around wBfor the condition z > 1 and o.z > 1. This means that a superlattice is a high-frequency amplifier in the negative differential conductivity region given by Eq. (20). However, the negative conductivity for low frequencies can cause domain oscillations that prevent the existence of a well-defined Bloch frequency throughout the whole superlattice. Such domain oscillations were observed by Schomburg et al. (1997), demonstrating the potential of a semiconductor superlattice to work as a “classical oscillator.”

2. THEWANNER-STARK LADDER The quantum mechanical description of a superlattice in the presence of an applied electric field shows that the miniband is split into a series of discrete levels (Wannier, 1962). These levels are called the Wannier-Stark ladder and their eigenvalues are given by (Bleuse et al., 1988)

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES 8:

= E'

139

h2 + ueEd + (k: + k,Z) 2m

where E' are the energy levels of an isolated well and v is an integer. The wave functions of the Wannier-Stark states can be written as

where the # Q are ~ ~ the wave functions of the isolated wells and the coefficients c,, are given by Cn,, i

) ' 2(

J,

-u

where A' is the ith miniband width, and J,-" are Bessel functions. The electric field causes a localization of the nth Wannier-Stark wave function to the nth well. When eEd becomes comparable to the miniband width A, the wave functions are isolated from each other. The dynamical behavior of the system can be described by the probability P, of finding the electron at time t in the &Jz - nd) state when its wave function was &,,(z) at time t = 0:

P ( t ) = J,"

Ai sin(o,t/2) [eEd __

1

where wB is again the Bloch frequency. Thus, the Bloch frequency also determines the time dependence in the quantum mechanical Wannier-Stark picture. Finally we can calculate the matrix element for inter-Wannier-Stark transitions

This shows that optical transitions are allowed only for Av = & 1 and that the matrix elements for up and for down transitions are identical (Ferreira and Bastard, 1990; Bastard and Ferreira, 1991). This leads to the trivial consequence that as many photons are being absorbed as photons being emitted and the net inter-Wannier-Stark absorption will be zero. We see in the following that scattering and finite level width change this trivial consequence and enable gain and absorption in a superlattice.

140

KARL UNTERRAINER

DESCRIPTION OF MINIBAND TRANSPORT IN EXTERNAL AC 3. CLASSICAL FIELDS For a superlattice in an AC electric field the current can be calculated by a Boltzmann equation approach where the distribution function has the form (Tsu and Esaki, 1971; Romanov, 1972; Ignatov, 1976a; Pavlovich and Epshtein, 1976) (27) where f , is the equilibrium distribution function, t’ is the time of the most recent scattering event, and Akz is the momentum acquired since the last scattering event:

AkZ =

-

f

l

E ( t ” ) dt“

Substituting Eq. (28) into Eq. (27) with an electric field of the form E ( t ) = E , E , cos(wt) and using a Maxwell distribution function for f, results in the following expression for the current:

+

with w, = eE,d/h, j , = (ho,/edz), and 6, is the linear static conductivity. We can expand this in a Fourier series and obtain the expression for the current of the superlattice in external fields k = m

where the J , are Bessel functions.

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

141

This general result contains several very surprising results. The expression for the stationary current can be written as m=a

j = j , J$(eE,,d/hw)

* It

1

+ WgT2

+j,

1 Jk(eE,d/ho)

m= t

For no external radiation this expression is identical to the superlattice ILV curve predicted by Esaki and Tsu (1970). In Fig. 9, the stationary current through the superlattice is plotted as a function of the AC field strength e E , d / h o and of the applied DC electric field strength eE,d/hw. Without an external AC field, a linear increase of the current with a maximum and a negative differential conductivity region is observed. With increasing amplitude of the AC field two remarkable features are observed. First, the current at the initial maximum together with the low field conductivily decreases.

FIG.9. Stationary current through a superlattice in a DC electric field and in an AC electric field. The fields are scaled by the photon energy of the AC electric field. At low DC electric field localization effects are seen for increasing AC field strength. In the negative differential conductivity region, photon-assisted peaks occur when the Bloch frequency = eEd/ h = n‘hw, that is, when the Bloch frequency equals the drive frequency or multiple integers of it.

142

KARLUNTERRAINER

Second, a second peak occurs in the negative differential conductivity region at a position for the DC electric field where the Bloch frequency w B equals the frequency w of the external field. This additional maximum in the current is caused by the external field and is thus called a photon assisted peak. For an even higher AC electric field a second and third additional peak occur at the DC bias position where the Bloch frequencies w B equals 2 0 and 3 0 , respectively. For increasing amplitude, the induced current at the photon-assisted peaks decreases again following the Bessel function behavior. The low-field conductivity has a minimum for the value of the AC electric field where the two-photon-assisted current has a maximum. Actually, for a certain value of the AC field strength, the current is driven to zero. This effect is the dynamic localization where an external AC field inhibits the transport through a periodic structure. For hgher amplitudes E,, a small region of negative low-field conductance is observed. This is another fascinating effect of the interaction of an external field and a superlattice. The DC current for low bias turns negative where the positive peak of the nonradiated I-V curve is situated. This is a remarkable prediction because for this situation, the electron moves in the opposite direction of the applied field. This absolute negative conductivity in a superlattice was predicted by Ignatov and Romanov (1976b). It is important to note that this result is obtained for a structure with no internal built in field such as in a photodiode. This absolute negative conductivity is of pure quantum mechanical origin. We have seen that the AC field causes photon-assisted peaks in the negative conductivity region of a superlattice. In the following, we see that these photon-assisted peaks are caused by stimulated emission of photons. To do so we must investigate the high-frequency conductivity of the superlattice. In Fig. 10, the stationary current together with high-frequency conductivity is shown for a field strength of eE,d/hw = 1.3. In the negative conductance region, a photon-assisted peak rises at a voltage position shifted by hole relative to the peak of the DC I-V curve. It is obvious from the calculation of the AC conductivity at the frequency w from Eq. (31) that this peak corresponds to the stimulated emission of a photon. As shown in Fig. 10, the high-frequency conductivity becomes negative for exactly those values of the DC bias for which an increase of the DC current-a photon-assisted peak -is observed. In addition, this calculation predicts amplification of radiation of the frequency w in a superlattice biased above the first photon step. This means a superlattice represents gain if the Bloch frequency w B is larger than the drive frequency. Right at the resonance w B = w,the AC conductivity is zero. For wB< w the superlattice absorbs the external radiation. This is a very important result for the development of THz radiation sources. Quantitative

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

0

1 2 3 DC Bias (w$w)

143

4

FIG. 10. Calculation of the superlattice current as a function of the applied bias for eE,d/ hw = 0 (dotted curve) and for eE,d/hw = 1.3 (solid curve). The absorption coefficient (dashed curve) is shown for eE,d/ho = 1.3. THz gain is predicted when the induced DC current shows an increase.

calculations concerning the emission from Bloch oscillations can be found in Grondin et al. (1985) and Korotkov et al. (1994). This behavior can also be understood in the Wannier-Stark level picture, where an increased current is caused by stimulated emission of photons through intra-Wannier-Stark level transitions and a suppression is caused by absorption. In a single superlattice miniband, this is an unambiguous method of interpreting the induced current since no transport channels through higher subbands exist where a positive induced current can also be caused by absorption and subsequent inelastic scattering. Without applied DC bias ( E = 0), the expression for the time dependent current reduces for wz >> 1 to j ( t ) = j o J oeE,d ( F& +)ni s

eE,d smwl) .

(33)

From this expression, we see that the amplitude oscillates with increasing field strength eE,d/hw. At the zeros of the zeroth-order Bessel function, the total current becomes zero. Therefore, the superlattice should be transparent at certain values of the field strength; this transparency was also predicted by Ignatov in the 1970s. In addition, for some values of the field strength, the amplitude of the harmonic currents can be larger than that of the fundamental current. For this, we must rewrite form Eq. (31), which reduces to

144

for k

KARLUNTERRAINER = 21

+ 1, and to A,

=0

for k = 21, which shows that only odd harmonics exist. Equation (34) enables an exact summation, which, for w >> z,leads to eE,d

= iJ.4

eE,d

(7 ( ) 7)

(35)

JO

This shows that odd harmonics can have larger amplitudes than the linear current at certain values of the field strength where the Bessel function of the linear current is in the vicinity of a zero. Therefore the current in a superlattice can be dominated by harmonic currents. This is an important issue for frequency conversion, for example, for third-harmonic generation (Bass and Rubinshtein, 1977; Lam et al., 1990 Wanke et al., 1996).

MECHANICAL DESCRIPTION OF SUPERLATTICES IN EXTERNAL 4. QUANTUM AC FIELDS The quantum mechanical approach of a strongly driven superlattice is based primarily on the temporal periodicity of the driving laser field, rather than the spatial periodicity of the superlattice. The periodicity in time leads to a formulation in terms of quasi-energy eigenvalues, and due to the periodicity in space, the quasi-energies form minibands (Holthaus, 1992). A superlattice interacting with strong far-infrared radiation is described by the following Hamiltonian:

where E, is the amplitude of the AC electric field of the laser, and o is the frequency. The Hamiltonian is periodic in time: H ( x , t ) = H ( x , t T), T = (27qo). Solutions of Eq. (36) are given by Floquet wave functions

+

Y , ( x , t ) = exp( - iEt)u,(x,t )

(37)

+

with quasi-energies E and time-periodic functions u,(x, t ) = u,(x, t T). As for the crystal momentum, there is also a “Brillouin-zone” for the quasienergies with a zone boundary at w.

3 TUNNELING IN SEMICONDUCTOR

QUANTUM STRUCTURES

145

Assuming a dispersion relation E ( k ) = E , - (A/2) cos(kd) and a harmonic electric field E(t) = E , , sin(wt). The velocity u(t) is given by

And the velocity averaged over one laser period is given

where J , is the zeroth-order Bessel function. Thus, if the ratio between the Bloch frequency and the driving frequency is equal to a zero of J , the average electron velocity vanishes. The average electron velocity is a measure for the coupling between individual wells. The bandwidth of the miniband is affected by the laser field in the same way as is the averaged velocity. When the initial bandwidth is given by

A

a quantum mechanical calculation yields an approximate expression for the quasi-energies E, = E ,

- -AJ , ( ~ ) c o s ( ~ ) m o d ( w ) 2

As for the averaged velocity, the width of the quasi-energy miniband vanishes when the argument of the Bessel function is equal to one of the zeros of J,. Grossman et al. (1991) used a numerical analysis to study tunneling in a symmetric double well in an external AC field and also observed dynamic localization for specific amplitudes of the external field. J. Zak (1993) extended this approach to give a more general expression for the miniband width of a superlattice driven by a strong harmonic field E , and biased by a constant electric field E, so that eE,d = nhco, which means that the Wannier-Stark spacing is equal to the driving frequency:

where J , are the Bessel functions. In case of one-photon resonance, the

146

AC-localization

0 ,DC;localizatipn

0

,

I

I

2 4 6 AC field (eEcod/hv)

FIG. 11. Quasi-energy miniband width for DC bias equal to zero ( J , ) and for a bias E for that the condition eEd/R = wB= Rw is fulfilled ( J t ) . For zero bias, the miniband width decreases with increasing AC field strength, which leads to localization. With applied bias, the miniband width increases with increasing AC field accordmg to the n = 1 Bessel function. This corresponds to the formation of a photon-assisted tunneling channel. The DC localization is released by the external AC field.

collapse of the quasi-energy miniband occurs at the zeros of the J , . Figure 11 shows that the quasi-energy miniband width for the one-photon resonance is zero without an external field. This is due to the Wannier-Stark localization of the superlattice in a DC electric field E,. Since no scattering or level broadening is included in the full quantum mechanical model, this localization appears for all nonzero DC electric fields. An additional external AC field lifts the localization and the miniband width reaches a maximum at the maximum of J , . For higher n-photon resonances the miniband opens when the external field E , reaches a maximum of the nth Bessel function J,(eE,d/ho). The miniband collapse for higher n-harmonic transitions appears at the zeros of higher n-Bessel functions. Thus, the fully quantum mechanical solutions lead for these special cases to the exact same results as the quasi-classical calculation. 5. ANALOGYTO THE! AC JOSEPHSON EFFECT

The problem of a superlattice with an applied DC electric field E , and an AC field E, can be treated in analogy to superconducting AC Josephson junctions (de Gennes, 1966; Ignatov et al., 1993) where the current is given by

I = I, sin(@)

and

d, = 2eV/h

(43)

3 TUNNELING I N SEMICONDUCTOR QUANTUM STRUCTURES

147

The velocity of an electron in a superlattice miniband is given by

Ad u(k) = -sin(kd) 2h

and

kd

e h

= -E(t)

d

(44)

The solution of the Josephson junction gives the well-known Shapiro steps at V , = N . h o / 2 e . This simple analogy predicts Shapiro steps in the DC current of a semiconductor superlattice driven by an AC field at resonances E,d = N .ho/e, which indicates that the high-frequency field couples to Bloch oscillations. Josephson-type effects in resonant-tunneling diodes were discussed by Gurvitz (1991).

IV. Experimental Methods For the successful realization of photon-assisted tunneling experiments it turns out that the coupling of the external radiation to the device is crucial (Keay et al., 1995~;Unterrainer 1996a). Wire bonds have been used as antennas (Guimaraes et al., 1993) and whiskers together with corner cube reflectors were employed (Ignatov, 1994). The use of coplanar antenna structures integrated to the quantum structures has proven to be the most efficient coupling scheme. The problem is to focus 100-pm radiation to micron-sized devices and to keep the polarization of the radiation perpendicular to the tunneling barriers. The proper polarization is very important since even a small component of the electric field parallel to the tunneling barriers would induce Drude heating by accelerating the electrons in the wells within their subbands. In the following, the fabrication of the samples into bow-tie-coupled devices is described. The mesas containing the quantum structure with areas of 4-10 pm2 are formed by dry etching through the top n+ region and the quantum structure followed by an ion (H +) implantation-isolation process. Ohmic Au-Ge-Ni contacts are fabricated on the top and the bottom of the mesas. The THz radiation is coupled to the superlattice by a coplanar broadband bow-tie antenna (see Fig. 12). The radiation pattern of the bow-tie antennas is characterized by two lobes that are orthogonal to the metal bows of the bow tie (Compton et al., 1987). The polarization is parallel to the metal bows. The impedance of the antenna is determined by the opening angle. Since the device impedance changes with applied bias and with the external field, however, no impedance matching could be maintained for the photon-assisted tunneling experiments. This antenna technique ensures that the polarization of the AC electric field inside the mesa is parallel to the growth direction independently from

148

KARLUNTERRAINER

FIG.12. Semiconductor quantum structure with a coplanar bow-tie antenna attached to the top and bottom contact. With this device geometry, the polarization is maintained perpendicular to the tunneling barriers. Except for the actual device area, the whole GaAs chip is proton isolated to prevent electrical contact between the bows.

the polarization of the incident field. A high-purity Si hemisphere is mounted on the substrate side of the sample acting as a lens and focusing the radiation onto the area of the antenna. Gold wires are bonded to pads at the bow-tie antennas to allow for conductivity measurements. In the following sections, we discuss experiments on photon-assisted tunneling in semiconductor nanostructures. The first experiments were reported about two decades after the first theoretical consideration. Significant developments in two different fields had to be accomplished prior to the realization of the first experiment. The first development was the fast progress of the materials science for the production of quantum wells and superlattices. The improvement of molecular beam epitaxy enabled the growth of high-quality semiconductor nanostructures with controllable low-doping concentration. The second development was the successful realization of free electron lasers, which provide tunable, high-intensity THz radiation (Ramian, 1992). Photonassisted tunneling was observed earlier in superconductor-isolator-superconductor junctions and lateral semiconductor quantum dots, where microwave sources in the GHz range could be used (Dayem and Martin, 1962;

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

149

Hamilton and Shapiro, 1979; Richards and Shen, 1980; Wyss et ul., 1993; Arnone et al., 1995; Kouwenhoven et al., 1994; Blick et al., 1995). However, the work for these systems did not cover the high field limit, where most of the exciting phenomena discussed in Sections I and I1 should appear. This becomes clear from a simple estimate of what power is needed to reach the first zero of the zero-order Bessel function. Let us assume the photon energy of 100meV for an infrared photon instead of 10meV for a far-infrared or THz photon. Since the argument in the Bessel functions is not the amplitude of the radiation but the ratio of the amplitude and the photon energy, 100 times higher power is required for an infrared experiment to observe the same effect as for THz radiation. The higher power requirements for the infrared are not only demanding for the laser sources but also generate a problem with unwanted heating of the samples. Thus, with the availability of convenient THz free-electron lasers (FEL), the THz frequency range was selected for the experimental observation of the effects predicted in the first section.

V. Experiments on Photon-Assisted Tunneling in Resonant Tunneling Diodes For the observation of photon-assisted tunneling, a strong nonlinearity of the current voltage characteristics is required, which means that the width of a tunneling peak in voltage must be narrow compared to the photon energy hw/e. Since the width of a transmission peak of a tunneling structure is given by the inverse of the tunneling time, the requirement of a narrow peak in the I-V curve means that the tunneling time must be larger than 27c/w. Resonant tunneling diodes have shown very narrow resonant tunneling peaks, whose width is around 10 meV or even below (Reed et al., 1989). This is comparable to photon energies in the THz range. The first observation of photon-assisted tunneling in an triple-barrier resonant tunneling device was performed by Drexler et al. (1995). In the structure for this work, the current is given by the tunneling of electrons between two two-dimensional (2D) electron systems. Tunneling between two 2D systems is allowed only if the 2D subbands are perfectly aligned in energy, assuming no scattering in the barrier or other processes lifting the momentum conversion. Thus, for relatively small barriers where scattering in the barrier can be neglected, the peaks in the I-V curve are expected to be very sharp. Due to the discrete structure of the subbands, no thermal broadening is possible. The only broadening mechanisms are device imperfections (i.e., monolayer fluctuations) and scattering-induced broadening of the subbands.

150

KARLUNTEWNER

The samples used in that study were GaAs-Al,Ga, -,As (x = 0.3) structures grown by molecular beam epitaxy. The structure consisted of two wells that were 180 and 1008L wide, respectively, enclosed by AlGaAs barriers, which were 35-A thick. The inner barrier was chosen to be 60 A to ensure that the tunneling is mostly dominated by 2D-2D tunneling between the wells and that the peaks in the I-V curve are originating from this process. The two wells are supplied by two 3D electron systems on the other side of the outer barriers. The 3D electron system consisted of highly doped loi8 cm-3 GaAs layers, which also served as contact layers. The structure is shown in Fig. 13 with the three lowest energy levels and the corresponding wave functions. The DC I-V curve of a bow-tie processed device is shown at the bottom of Fig. 14 where two resonances can be seen at -86 and at 88mV. For positive bias, the electrons tunnel from the 180-A well into the 100-A well. The energies of the lowest subbands in the two wells are 13 and 33meV, respectively. A resonance should be observed when the ground state of the narrow well is pulled into resonance with the ground state of the wide well. For negative bias, the narrow well becomes the emitter, and we expect a resonance when the first excited state of the wide well is pulled into resonance with the ground state of the narrow well. The energy of the first excited subband of the wide well is 51 meV. For both bias directions, we would expect the resonance for a energy difference between the wells of about 20 meV. However, the measured bias voltage to observe resonance is about 80meV, which means that the voltage does not drop only across the

h

N Y

>

I

I

.

3500

3700 z(A)

3900

FIG. 13. Energy band diagram of the triple-barrier resonant tunneling diode. The three lowest energy levels and their envelope wave functions are shown (after Drexler et al., 1995).

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

151

120

100

3.4THz 2.5THZ

1.STHZ 1 .OTHz

-0.2

-0.1

0

v (V)

0.1

0.2

FIG. 14. The DC current-voltage characteristics of the triple-barrier structure are shown at the bottom. Positive bias means positive voltage at the narrow well. The I-V curves of the irradiated sample for 1.0, 1.5, and 3.4 THz have been vertically displaced (after Drexler et al., 1995).

middle barrier, and we have to assume a lever arm between applied voltage and energy difference between the wells, which is about 0.25 meV/mV. The measured width of the tunneling peaks of the I-V curve are 25 and 35 mV. Considering the lever arm, we can estimate the corresponding linewidth to be 6 and 9 meV. Thus, we expect to see distinct photon-assisted tunneling peaks for photon energies above 6 and 9 meV. In Fig. 14, the results for the irradiated device for different THz frequencies are shown. As expected, at a frequency of 1 THz, which corresponds to a photon energy of 4 meV, photon-assisted side peaks are observed. When the THz frequency is increased, the photon-assisted peaks become clearer, and at the 3.4 THz, they are completely separated from the main peaks. The position of the photon-assisted peaks shifts linearly with frequency, and the position is independent of the power. The difference between the photonassisted peaks and the main tunneling peak scales with 0.25 meV/mV. This is exactly the same value we determined from the DC I-V curve, which proves that these induced resonances are due to photon-assisted tunneling between the quantized states of the two wells in the resonant tunneling diode (RTD) structure. The peak that is observed at the lower voltage side of the direct tunneling peak corresponds to photon-assisted tunneling due

*

152

KARL UNTERRAINER

to absorption of one photon. The photon energy ho is equal to the energy difference between the ground state of the wide well and the ground state of the narrow well. An electron from the wide well can tunnel to the narrow well only by absorbing a photon with energy ho. A completely different situation is true for the peak at the higher voltage side. Here an electron can tunnel only from the wide well to the narrow well under the emission of a photon with a photon energy equal to the energy difference between the two ground states. This is a very important result, since the occurrence of this “emission” peak indicates that there is population inversion between the wells For a more quantitative analysis, the results must be compared to the model for photon-assisted tunneling of Tien and Gordon in Section I. In Eq. (9), the lever arm must be considered in the term nhwle. From this model, the heights of the photon-assisted peaks on both sides of the fundamental peaks should be the same. This is clearly not the case in the experiment; the “emission” peak is always higher than the absorption peak at the low voltage side. The Tien and Gordon model accounts only for a single barrier and does not include real populations. In the experiment, the triple-barrier resonant tunneling barrier has three barriers with two of them having a different barrier width. As a result, the voltage drops not only across the middle barrier but also over the outer barrier. As a result, only part of the voltage is available to change the energy difference between the wells, which leads to the observed lever arm. The part of the voltage that drops over the outer barriers changes the difference between the Fermi energy of the emitter (collector) and the ground level of the wells. As a result, the carrier injection into the emitter (wide) well as well as the extraction from the collector (narrow) well changes, and the electron concentration in the wells changes with applied bias, and the assumption of the Tien-Gordon model that the number of electron is constant becomes invalid. The experimental results show that the absorption peak decreases in relative height compared to the emission peak for increased photon energy. Since the absorption peak moves to lower voltages for increased photon energy, we conclude that the electron concentration in the emitter well decreases with decreasing voltage. An even more pronounced dependence on the bias voltage is observed on the negative bias side, where hardly an absorption peak is found. These discrepancies from the Tien-Gordon model lead to two consequences. First, a more quantitative model is needed for photon-assisted tunneling that must include the influence of the electron concentration in the wells on the strength of the photon-assisted tunneling channels. The second consequence is that photon-assisted tunneling can help to determine the local populations in the wells of a resonant tunneling diode. This is an

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

153

important issue for the development of more accurate models for the calculation of the DC I-V curves of such devices. In Fig. 15, I-V curves of the resonant tunneling diode are shown for different power levels of the external field at a frequency of 1.5THz. The lowest trace in Fig. 15 shows only one photon absorption and emission side peak. Two photon peaks can be observed in the middle curve, which was recorded at about twice the power. The one-photon peaks are also stronger in this trace. The curve at the top of the figure even shows a three-photon peak for the emission channel at an intensity five times higher than that of the lowest curve. The one- and two-photon side peaks are even more pronounced at this intensity. The peaks at the higher voltage side are attributed to stimulated emission of one, two, and three photons. At the lower voltage side, absorption of one and two photons is observed. A side peak due to three-photon absorption is missing. This is again explained by the low electron concentration at the expected low bias position of the three-photon peak. The power inside the resonant tunneling diode is not measured directly and a rough estimate can be made by setting the argument of the Bessel functions of the Tien-Gordon model equal to one where we expect to see

-0.2

-0.1

0

0.1

0.2

v (V) FIG. 15. I-V curves at different field strengths of the external field at a frequency of 1.5 THz. The curves at 2.11, and 5.41, have a vertical offset. The dashed vertical lines show the expected distance of the photon-assisted peaks from the main tunneling peak (short dash) (after Drexler et al., 1995).

154

KARLUNTERRAINER

the onset of the two photon channel. This results in a power density of about 50 kW/cm2. In fact, the Tien-Gordon model does not explain the observed intensity dependence of the relative peak heights. At the highest intensity where the three-photon side peak occurs there should be only a strongly decreased one-photon peak. This is not the case, and the overall intensity dependence cannot be explained by the Bessel function behavior. The difference from the Tien-Gordon model is again explained by the dependence of the strength of the photon sideband on the electron concentration in the emitter well. The carrier concentration in this well depends also on the temperature of the highly n-doped GaAs contact regions. The number of available electrons for tunneling is very sensitive to the position of the Fermi energy in the setback region and to the width of the distribution. The temperature dependence of the DC I-V curve shows a small shift of the main tunneling peak to lower voltages, indicating that the injection into the emitter well becomes more efficient. The tunneling peak for negative bias does not shift temperature, which could be explained by the higher confinement energy of the narrow well. The intensity-dependent I-V curve shows for positive bias also a shift of the tunneling peak to lower voltages with increasing power, which can be explained by heating of the electrons in the highly n-doped contact regions and thus changing the carrier concentration in the wide well. The external radiation can be absorbed in the 3D contact regions by free-carrier absorption and the power density is high enough to induce considerable heating. On the other hand, the temperature does not influence the linewidth of the main tunneling peak and that of the photon-assisted tunneling peak. This is due to the quantized 2D subbands, which for negligible nonparabolicity, allow tunneling only for perfect alignment in energy. This is obvious from the experiment with an external radiation at a frequency of 1THz. The sidebands can still be resolved, although the thermal energy at 100K is 8.6 meV and the photon energy is only 4.1 meV. For the same reason, no thermal broadening is observed in the intensity dependence measurements. The heating of the n-doped contact region can only be inferred from the small shifts of the main tunneling peak. The results for the resonant tunneling diode shows that photon-assisted tunneling can be observed in these structures. Surprisingly, we find that the photon emission channel is very pronounced. The photon absorption channel is weaker since it should occur at smaller voltages where the carrier density in the emitter well is low. The observation of the photon emission channel shows in a unique way that such devices are a suitable gain medium for possible THz lasers based on the concept of the quantum cascade lasers. It is very remarkable that the gain and the inversion are still present at a temperature of 100 K for which the difference of the energy levels in the wide

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

155

and narrow wells is less than 1 kT. This result is different from the conclusion of Helm et al. (1989), who argued from spontaneous emission measurements in an RTD that the populations of the energy levels are thermalized even at 10K. An extension of the Tien-Gordon model to include the population of the wells could help to extract quantitative values of the occupation in the wells from the photon-assisted tunneling experiments. The small value of the photon absorption channel shows that the structure used is not very well suited for detection. To address this issue, the design of the RTD must be changed in such a way that there are enough electrons in the emitter well at low voltages.

VI. Photon-Assisted Tunneling in Weakly Coupled Superlattices

1. PHOTON-ASSISTED TUNNELING BETWEEN GROUND AND EXCITED STATES The first results on THz photon-assisted tunneling in semiconductor quantum structures were reported on weakly coupled superlattices (Guimaraes et al., 1993; Keay et al., 1994). There was not very much previous experimental work since the condition for high-frequency 02 > 1 and intense radiation eE,d/hw > 1 must be fulfilled. The first requirement is that the electrons should be submitted to at least one cycle of the applied radiation before being scattered. The second requirement is that the electric field E , of the radiation applied across one period of the superlattice be comparable to the photon energy ho/e. Typical scattering times in semicons, which means that the frequencies must ductor superlattices are z = be at least in the THz range. The second condition eE,d/ho > 1 requires increasing intensities with increasing frequency and, thus, makes the THz range an optimal choice for observing nonlinear effects. The power levels to achieve nonlinear effects for superlattices with periods of about 10 nm are of the order of E , = 1 kV/cm. This value does not look quite demanding; however, only the advent of high-power free-electron lasers made nonlinear THz spectroscopy practical. The free-electron lasers at the University of California, Santa Barbara (UCSB), routinely provide tunable radiation at the kilowatt power level from 0.12 to 5THz. The pulse length is several microseconds and the repetition rate is a few hertz. In contrast to highpower CO, laser pumped gas lasers the intensity is constant within several percent during the pulse. This enables quasi-continuous wave measurements of the THz-radiation-induced conductivity at a constant power level, a very important requirement to clearly observe the predicted nonlinear effects,

156

KARLUNTERRAINER

which should occur at a very narrow intensity range ( e g , dynamic localization). In the following I will discuss a series of experiments on photonassisted tunneling by the UCSB group. The first report came from Guimaraes et al. (1993) on a GaAsAl,,,Ga,,,As superlattice consisting of 100 periods of 33-nm-wide quantum wells separated by 4-nm barriers. The structure was grown on n f GaAs substrate and the superlattice was doped with Si to n = 2-3 x l O I 5 ~ m - ~ . A 100-nm-thick GaAs contact layer was grown on top of the superlattice . (200 x 200pm) mesas were doped with Si to n = 2 x 1 0 ' * ~ m - ~Square produced by standard wet chemical etching and ohmic contacts were fabricated at the top of the mesa and on the substrate. Gold wires were bonded to the contacts, which served as electrical contacts for the bias voltage as well as to couple the THz radiation into the sample. By this simple technique, it was possible to couple in the radiation with the correctly polarized field parallel to the growth direction since the electric field components normal to the growth direction where shorted by the top contact metalization (Fig. 16). The experiments were performed in an optical cryostat at temperatures of about 70 K. The DC I-V characteristics in Fig. 17 shows two steps at V = 0.1 V and V = 1.2 V, which are due to sequential resonant tunneling. At low bias, the current results from ground state to ground state tunneling. At V = 0.1 V, the detuning starts to be larger than the ground state width, and the tunnel current drops. The expected region of negative differential conductivity does not occur since domains are formed (Esaki and Chand, 1974; Choi et al., 1987; Grahn, 1995; Grahn et al., 1990, 1991). In the high-field domain, the conduction is supported through the ground state to first excited-state tunneling. The high-field domain increases for increasing bias until it

From Free-electron laser

';x

THz

Multi quantum well Superlattice

FIG. 16. Superlattice sample irradiated by radiation from the free-electron laser. The bonded wire acts as electrical top contact and as antenna for the THz radiation. The n+-doped substrate serves as the back contact (after Guimaraes et al., 1993).

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

157

Voltage (V) FIG. 17. Current-voltage characteristics of the superlattice sample measured without (solid line) and with 1.5 THz (dash-dotted line), 1.83 THz (dotted line), and 2.11 THz (dashed line) radiation. The arrows show the features attributed to photon-assisted tunneling (after Guimaraes et a[., 1993).

expands through the whole superlattice. This occurs at V = 1.2V, where the electric field becomes constant again throughout the superlattice. With external radiation, the I-V curves change and new steps can be observed. The results for three different frequencies show that the bias positions of the new steps move to lower voltages for increasing frequency. In Fig. 18, the frequency dependence of the induced steps is shown. For each frequency, the data are plotted for different intensities. The total applied DC voltage was divided by a factor of 100, which is the number of periods in the superlattice. A best straight-line fit is drawn through the data in Fig. 18. As a result we see that the data lie on straight lines with slopes -hie, which is a direct evidence for photon-assisted tunneling. From the extrapolation to zero frequency, we can identify the involved fundamental tunneling resonances. One line intersects the voltage axis at 11.8 meV, the other one at 30.6 meV. The calculated differences between the energy levels in the wells are E , - E , = 12.4 meV and E , - E , = 32.7 meV, respectively. Therefore,

158

KARLUNTERRAINER

Frequency (THz) FIG. 18. Frequency dependence of the voltage positions of the photon-assisted tunneling peaks. The voltage scale corresponds to the voltage drop across a single period of the superlattice assuming a constant field in the superlattice. Arrows point to the voltage offset between the ground state and the first two excited states (after Guimaraes et nl., 1993).

the first line was attributed to photon-assisted tunneling between the ground state and the first excited state of the next neighboring well by absorbing a photon. The second line is due to photon-assisted tunneling by the absorption of a photon between the ground state and the second excited state of the neighboring well. No photon-assisted tunneling peaks due to the emission of photons were observed. An unexpected result is that the voltage seems to drop homogeneously over the superlattice. This is surprising since the induced steps occur where domain formation is supported. The domain formation can be suppressed only when the additional current through the photon-assisted channels is large enough. This seems to be the case since the observed induced current in Fig. 17 is more than twice as high as the static current. At lower power levels, the induced current is indeed much lower and the domains are not suppressed completely. This explains the observed dependence of the bias position of the steps on the incident THz power. The observed features can be explained by the Tien-Gordon model for photon-assisted tunneling. Assuming a double-well structure with one well grounded and the other one driven by a time-dependent voltage V ( t ) = E,d cos(wt) results in an effective density of states in the excited well, which is distributed over sidebands separated by n-photon energies. The

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

159

photon-assisted channels become important when eE,djhw reaches the zeros of the Bessel functions J,(eE,d/hw). The rates of the n-photon-assisted tunneling channel should be proportional to J,'(eE,d/hw). Therefore a strong power dependence of the induced current is expected. However, no clear dependence on power could be extracted form the experiments since the coupling was inefficient and the photon-assisted features occurred only at the highest available power levels. In a subsequent work Keay et al. (1995) were able to improve the coupling and to increase the electric field in the superlattice sample. This enabled the investigation of the nonlinear behavior of the photon-assisted tunneling features. The samples used in this work were similar to those of Guimaraes et al. (1993), however, they had fewer periods. This was indented to increase the electric field E , inside the superlattice in comparison to the 100-period sample for the same intensity of the external radiation. The weakly coupled superlattice consisted of 10 33-nm GaAs wells and 11 4-nm Al,,,Ga,,,As barriers. The samples were epitaxially grown on semi-insulating GaAs substrate and the doping of the superlattice structure was 3 x 1015cm-3. The superlattice structure was embedded between two 300-nm-thick GaAs layers, which were doped to n = 2 x 1 O l 8 cm-, and served as contact layers. Between the contact layers and the superlattice a 50-nm GaAs spacer layer with a doping of n = 3 x lo1' cm-3 was grown. The samples were processed into 8-pm2 mesas and integrated with bow-tie antennas, as described in Section 111. The energies of the three lowest quantized states are calculated to be El = 4.2 meV, E, = 16.6 meV, and E , = 36.9 meV. The miniband width of the ground state is well below lmeV, and thus the coupling between the wells is weak. The measured I-V curve of the sample is shown in Fig. 19 for a temperature of about 10 K. The curve shows a strong increase of the current at about 0.15 V followed by sawtooth oscillations. Near zero voltage (0.02V) the current through the sample is due to ground state to ground state tunneling. For increasing voltage, the alignment of the ground states is lost and the current decreases. Above a voltage of O.lV, the current increases again because an alignment between the ground state of one well and an excited state in the neighboring well appears. The current increases until it reaches the maximum value that can be carried by ground state to first excited state tunneling. For higher voltages, the alignment between ground state and first excited state is destroyed and the current drops. However, no steady decrease of the current is observed until the voltage reaches the next resonances position. Instead, sawtooth oscillations of the current occur, which is explained by the formation of low- and high-field

160

KARLUNTERRAINER

0

0.1

0.2 0.3 0.4 0.5

0.6

Volts FIG. 19. The current-voltage characteristics of the bow-tie-coupled superlattice without (dotted line) and with (solid line) 3.42-THz radiation. Theoretical calculation (a) without domain formation and (b) including domain formation, and (c) experimental results (after Keay et al., 1995a).

domains within the superlattice. The first oscillation is explained by the break off of one quantum well forming a high-field domain. The high-field domain is characterized by the alignment of the ground state in one well to the second excited state in the neighboring well, which is connected with an increased overall tunneling probability. The next wells enter the high-field domain one by one when the voltage is increased further, thus forming the oscillatory structure. When the last well enters the high-field domain the electric field is again homogenous across the sample, defining the next

3

TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

161

resonant tunneling peak where for all wells, the ground state is aligned with the second excited state in the neighboring well. In the presence of external radiation, photon-assisted tunneling features appear. In Fig. 19, the result is shown for an external frequency of 3.42 THz. New steps appear in voltage below and above the first tunneling peak. An analysis shows that the first step is the two-photon absorption channel for tunneling from the ground state in one well into the second excited state in the neighboring well. The second step is the one-photon absorption channel for tunneling into the second excited state in the neighboring well. This step also shows the oscillatory dependence of the current, which demonstrates that the domain structure is maintained even under the presence of the THz field. The presence of domains causes transport to occur via photon-assisted tunneling in the low-field domain and by direct tunneling in the high-field domain. For a description of this behavior, a model for photon assisted tunneling was developed that includes domain formation based on a model introduced by Bonilla et al. (1994). The influence of the external field is included following Tien and Gordon by introducing virtual states separated form the ground state by f nAw and having a probability density given by the square of the nth Bessel function with the argument eE,d/hw. The results of this calculation are shown in Fig. 19b. The model explains the DC I-V curve and the irradiated I-V curves quite well. From a comparison of the calculated I-V curves with and without domains, we can see that the one-photon emission peak from the alignment to the first excited state is not present in the I-V curve, which includes domains. This is explained by the higher current of the ground state to second excited state tunneling than that of the ground state to first excited state resonance. As a consequence also the photon-assisted currents for the ground state to first excited state resonance are smaller and are dominated by the photon-assisted channels of the ground state to second excited state resonance. Figure 20 shows the frequency dependence of the voltage positions of the photon-assisted steps. The lowest line is the one-photon-absorption step for the ground state to first excited state resonance, which can only be observed up to 3 THz. The two upper lines are the mentioned two- and one-photonabsorption steps. The dotted lines are simple interpolations between the step locations without radiation and the expected zero voltage intercepts on the frequency axis, which is given by 2ho = E , - E , = 32.7 meV (3.95 THz) for the two-photon transitions, by hcc, = E , - E , = 32.7 meV (7.9 THz) for the one-photon transition, and by h a = E , - El = 12.4meV (3.0THz). The solid lines show the expected step positions from the preceding model. The intensity dependence of the strength of the photon-assisted channel is a good test for the validity of the Tien and Gordon model in the case of a

162

KARL UNTERRAINER 0.6,

,

I

I

,

I

,

h

-

0.4

-8

c 0.2

0 .c

8

0 J

0

n -0.2

aJ

85

-0.4

0 1 2 3 4 5 6 7 8

Frequency (THz) FIG. 20. Frequency dependence of the voltage positions of the photon-assisted steps: E , - E,, photon-assisted transition (solid diamonds); El - E,, two-photon-assisted transition (open circles); and E , - E,, one-photon-assisted transitions (full squares). The dotted lines are simple linear extrapolations. The solid lines are obtained from the model calculation including domain formation (after Keay et al., 1995a).

superlattice with domain formation. In Fig. 21, we can see the induced current through the sample irradiated by an external field at a frequency of 3.42 THz. The current is shown for three bias points corresponding to tunneling through the ground state (zero photon peak at 5meV), to the two-photon-absorption-assisted tunneling at 100 meV, and to the onephoton-absorption-assisted tunneling at 250 meV. The electric field axis is scaled so that the local maximum of the one-photon and two-photon process and the minimum for the zero photon process are a best fit to the corresponding maxima or minima of the squared nth Bessel functions. For the chosen electric field, the nonlinear dependence of the induced current can be quite well explained by the Tien and Gordon model. However, the prediction of the model including domains shows that no simple Bessel function behavior is expected due to the complex I-V characteristic. There is a qualitative agreement with the experimental findings, which becomes worse at higher electric fields. This work showed for the first time the expected nonlinear dependence of the photon-assisted processes. In addition, the validity of the Tien and Gordon model was demonstrated for an extended semiconductor superlattice. Domain formation in the presence of strong THz fields can be modeled by including photon-assisted levels in the calculation.

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

163

2.5 I

5

2-

v

F 1.52

a L

1-

-0 0)

2

0.5-

-0 S

-

0.

0

-0.5 5 10 15 20 Laser Field Strength (kV/cm)

FIG. 21. (a) Intensity dependence of the induced current measured at the photon-assisted step positions. (b) The calculated induced current from the Tien and Gordon model (after Keay et al., 1995a).

2. PHOTON-ASSISTED TUNNELING BETWEEN GROUND STATES (DYNAMIC ABSOLUTENEGATIVE CONDUCTANCE, STIMULATED LOCALIZATION, MULTIPHOTON EMISSION) In the previous section, we discussed the observation of multiphoton absorption-assisted tunneling in weakly coupled superlattices following the prediction of Tien and Gordon. In this section, we investigate the dynamic localization and absolute negative conductance in semiconductor quantum structures. These phenomena were predicted theoretically in the 1970s, as discussed in Section 11. These models are based on strongly coupled . superlattices, which form minibands. The main point of most models is the occurrence of Bloch oscillations in the presence of strong THz radiation, which coherently drives the electron beyond the miniband zone boundary.

164

KARLUNTERRAINER

For sequential resonant tunneling structures, however, this picture is not applicable since phase coherence is lost between the wells and only little theoretical work was done (Kazarinov et al., 1972). The experimental work for a more intensive study of photon-assisted transport in the weakly coupled superlattices was encouraged by the successful observation of photon-absorption-assisted tunneling in perfect agreement with theoretical predictions. However, the observation of stimulated photon-emission-assisted tunneling was not successful in the previous experiments, which was explained by the overlap of the assisted tunneling channels between the ground state to the first excited state and that of the ground state to second excited state tunneling. To resolve this problem a smaller quantum well width was chosen where the energy difference between the states was larger (Keay et al., 1995~).The width for GaAs well in these samples was 15 nm. The whole structure consisted of 10 such wells separated by 5-nm-thick Al,,,Ga,,,As barriers. The superlattice was grown on top of an undoped GaAs substrate after a 300-nm GaAs contact region doped at n = 2 x 10'8cm-3 and after a 50-nm-thick GaAs setback region. The superlattice and the setback were doped to n = 3 x l O ' ' ~ r n - ~ The . superlattice was capped with a setback and contact region identical to that for the bottom contact region. Since the bow-tie antenna coupling proved to be very efficient the samples were processed into small mesas (8 pm2 area) and integrated in the coplanar bow-tie antenna structures. Figure 22 shows the current-voltage characteristics without external radiation and with an external field at a frequency of 1.30THz at three

-2Ffco -3ho

.

0.6

r

Without THz Field

0.4 h

5 0.2 . v

E

s!

0-

L

Increasing THz field

3-0.2.

0

-0.2

-0.1

0

0.1

0.2

Volts FIG.22. Current-voltage characteristics of the 15 nm-5 nm superlattice without (thin solid line) and with 1.30-THz radiation at three different laser intensities (after Keay et al., 1995~).

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

165

different intensities. The I-V curve without external field shows an Ohmic region, where the current increases linearly with applied DC voltage. In this region, the current is due to sequential ground state to ground state tunneling. As the bias is increased more, the maximum current is reached and the ground state to ground state alignment becomes worse. Instead of a smooth decrease of the current, a high-field domain is formed in one well of the superlattice. This high-field domain is characterized by an alignment of the ground state in one well and the first excited state in the next well. In this high-field domain, current flow takes place through ground state to first excited state tunneling. As the bias is increased further, one well after another breaks off into the high-field domain, resulting in the observed sawtooth negative differential resistance structure in the I-V curve. The I-V curves changes dramatically when an external radiation is incident. The first effect is that the conductance at zero bias is suppressed. This indicates that the external field tends to localize the electrons. This effect is called dynamic localization (Ignatov and Romanov, 1976a; Dunlap and Kenkre, 1986; Holthaus, 1992). At higher DC bias the current is increased above the value for the DC I-V curve. At this bias an additional current can flow by stimulated emission of one photon from the ground state in one well to the ground state of the next neighbor via a one-photon sideband at -hole below the ground state of the initial well. The energy difference between the ground states is given by eVo/N, where N is the number of wells and V, is the DC bias. For the observation of photonassisted tunneling, the energy of the photons must equal the ground state energy difference:

hw

= eVo/N

At the next higher intensity a remarkable effect is observed, the conductance at low bias actually becomes negative. In this situation, the electrons tunnel against the applied DC field by absorbing a photon from the external field. The tunnel transition is through the ground state of one well to the ground state of the next well via a one-photon absorbing sideband at +hw/e above the ground state of the final well. When the DC bias is increased further, the conductance becomes positive again and the one-photonemission-assisted peak is observed again. At this step, the electrons can tunnel into the direction of the field by stimulated emission of a photon. At even higher DC bias, a two-photon-emission-assisted step is observed. At the highest intensity and at low bias, the current becomes positive again. The current at the one-photon-assisted step almost vanishes, in contrast to the two-photon step whose current has increased. At higher DC bias, a three-photon step can be observed where the tunneling is mediated by stimulated three-photon emission.

166

KARLUNTERRAINER

It is obvious from the experimental results that the strength of the photon-assisted channels and of the zero-bias conductance depends on the intensity of the external field in a very nonlinear way. The dependence on the laser field strength is shown in Fig. 23. For the one-, two-, and three-photon emission assisted step the step height is used and for the current at low bias, the conductance of the linear part of the I-V is used since no exact voltage position is known. For a further analysis, the results can be compared to the Tien-Gordon model assuming that sequential resonant tunneling is the dominant transport process. The Tien-Gordon model predicts the appearance of photon sidebands with probability amplitudes proportional to the squared Bessel functions J:(eE,d/ho), where II is the number of involved photons. Negative n corresponds to stimulated emission; positive n, to absorption of photons. Here J ; describes the dependence of the direct (zero-photon) current. From Section I we know that when e E , d / h o equals one of the zeros of J , , the phenomenon of dynamic localization is predicted that is the complete suppression of the conductance at zero bias. The intensity dependence of the photon-assisted steps is given by the higher Bessel functions. In Fig. 23, the electric field axis is scaled so that the minimum of the zero bias conductance and the maxima of the photon-assisted peaks align with the appropriate minimum and maximum of the squared Bessel functions. The arrows in Fig. 23 show the positions of the Bessel function extrema. The agreement with the experimen-

FIG.23. Intensity dependence of the induced current for one-, two-, and three-photonassisted tunneling and the zero-bias differential conductance (zero photons) for 1.30THz (after Keay et al., 199%).

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

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tally observed minimum and maxima is quite good, which shows that the Tien-Gordon model is still applicable. The appearance of absolute negative conductivity is not directly predicted by the Tien-Gordon model. The authors of the work explained this unexpected behavior by a broken symmetry between the emission and absorption channels due to linewidth broadening. This is shown in Fig. 24, where a broadened level in one well is coupled to the level separated by eE,d in the neighboring well. For a situation near the first zero of J i the channel for direct tunneling is suppressed and transport is only possible through photon assisted channels. In Fig. 23a, the applied field is smaller than holed and the one-photon-emission-assisted channel is inhibited, while the one-photon-absorption-assisted channel is allowed. In this situation, the current flows against the field direction. When the field is larger than Awled, the emission channel is allowed and the current can flow in the direction of the applied field. The authors developed a model to explain the observed features assuming levels with a Gaussian broadening and with a Boltzmann-like occupation. With that model, the authors could qualitatively explain their results. The absolute negative conductance is caused by two channels. The negative current at the positive bias side is generated by the one-photon-absorptionassisted tunneling peak from the negative bias side. The one-photon-absorption-assisted tunneling step for the positive side of the I-V curve occurs at the negative side causing a positive current for negative bias. Two other

\ FIG. 24. (a) Broadened levels in the presence of radiation at low DC bias. For eE,d < hw, emission is forbidden and absorption is allowed. (b) At higher bias, eE,d > hw, emission channels are allowed and absorption channels are inhibited. The vertical arrows represent photons with constant energy (after Keay et al., 1995~).

168

KARLUNTERRAINER

groups developed models for the explanation of the experimental results (Plater0 and Aguado, 1997; Wagner, 1996). However, only qualitative agreement with the experiment was found. The observed absolute negative conductivity is important for the realization of an absolute frequency-to-voltage converter. Such a frequency-tovoltage converter was also predicted for a superlattice miniband by Dunlap et al. (1993). The main idea was that a system with negative conductivity connected to a capacitor should be unstable at zero bias. The capacitor would be charged by the absolute negative conductivity to a voltage at which the conductivity turns positive. This occurs at the position of the one-photon emission peak at V, = h w / e . N . Therefore, intense THz radiation will induce a constant voltage at the capacitor and the value of the voltage gives an absolute measure for the frequency. A closer look at the irradiated I-Vs in Fig. 22 shows another peak that does not shift with the frequency of the laser. This is the peak closest to the origin located at 20 mV. This peak is due to the direct tunneling and should also be observed in the unirradiated I-V. However, in the unirradiated I-V there is no observable feature at 20 mV. The maximum current for ground state to ground state tunneling is reached at about 50mV. In Fig. 25, the positions of the peaks are shown as a function of the photon energy. The dependence of the peaks can be quite well approximated by straight lines as expected for photon-assisted tunneling. The slopes of the lines are - 10, 10, 20, and 30, corresponding to one-photon absorption, one-photon emission, two-photon emission, and three-photon emission. All four lines intersect the abscissa at 20 mV. According to the standard model, the photon-assisted tunneling peaks are replicas of the tunneling peak of the unirradiated I-V. This is not the case for the described experiment. Zeuner et al. (1996) tried to explain this discrepancy by using an “instantaneous” I-V curve. They assumed that this instantaneous I-V curve determined the transport at THz frequencies and should be used in the Tucker and Feldmann (1985) formula. This means, on the other hand, that the instantaneous I-V should manifest itself in the irradiated I-V for sufficiently high intensities. Thus, the peak at 20 mV can be interpreted as the ground state to ground state tunneling peak of the instantaneous I-V. This would correspond to a ground-state level width of 2 meV assuming that the tunnel current decreases when the voltage drop per period becomes larger than the level width. This value is larger than the intrinsic width of the ground state. However, it is consistent with level broadening due to well-width fluctuations of one monolayer. Zeuner et al. (1996) argue that the unirradiated ILV is not the instantaneous I-V curve since domain formation is present. The formation of field domains is on the time scale of 0.1 to 1 ns and should therefore not be included in a situation where a THz field is present. Also, the charge built up, which is

3 TUNNELING r~ SEMICONDUCTOR QUANTUM STRUCTURES

169

300

emission 200

-

h

> E

1 photon

-100

0

absorption

I

I

I

5

10

15

hv ( meV ) FIG. 25. Frequency dependence of the photon-assisted peaks of the 15 nm-5 nm superlattice (after Zeuner et al., 1997).

necessary for domain formation, is too slow to follow the THz field. Sufficiently strong THz radiation should therefore suppress domain formation and the instantaneous current-voltage characteristic should be revealed. It was shown that a assuming a simple I-V without domains and a ground state to ground state tunneling peak at 20 mV together with Tucker’s expression can explain all the observed features in the irradiated I-vs. Zeuner et al. (1997) have also demonstrated the transition from the quantum response of photon-assisted tunneling to classical rectification. At 1.5 THz in Fig. 26, all photon-assisted channels can be clearly resolved and the observed peak position does not depend on the laser intensity. The positions are dependent only on the photon energy. At 0.6THz the experimental results are completely different. No additional photon-assisted peaks can be resolved. The main tunneling peak is shifting to higher bias

170

KARLUNTERRAINER

1.5 THz measured

I

I

I

FIG. 26. Current-voltage characteristics irradiated with ditferent intensities at frequencies of 2 and 0.6 THz. A transition from quantum behavior at high frequencies to classical rectification at low frequency is clearly seen (after Zeuner et al., 1997).

and decreasing in height for increasing laser intensity. The experimental results are reproduced within the Tucker model, showing that the quantum response vanishes when the variations of the nonlinearity in the I-V are in the range of the photon energy times the number of well N . h w / e . The limiting frequency for quantum response is around 0.5 THz, corresponding to a ground-state level width of 2 meV. A more quantitative explanation for the discrepancy between the unirradiated I-V curve and the photon-assisted tunneling peaks was found by Wacker et al. (1997). They developed a rigorous model for microscopic transport in a low-doped weakly coupled superlattice. For low-doped samples the low-field transport is influenced by the presence of impurity bands. This means tunneling can occur not only between free states but also between free states and impurity states. At low temperatures, tunneling takes place between impurity bands and free states. The maximum occurs at the energy where the bottom of the impurity band is aligned with the bandedge of the free-electron states in the neighboring well, which means that the

3 TUNNELING IN SEMICONDUCTOR QUANTUM STRUCTURES

171

voltage position of the low-field tunneling peak is not given by the free-ground-state-level width but by the width of the impurity band. This changes when the temperature is increased and the occupation of the impurity band becomes lower compared to the occupation in the free ground state. For this situation, the transport is dominated by free ground state to free ground state tunneling and the peak position is again given by the width of the ground state. This dependence of the peak position on temperature is used to explain the discrepancies between the unirradiated I-V and the photon-assisted replicas. At a low temperature and without external radiation, the I-V is dominated by the impurity band to free ground state tunneling resulting in a peak in the I-V at about 100 mV. When the external radiation is switched on, the electrons are excited from the impurity band into the free ground state. This corresponds to a higher effective carrier temperature at which the transport is determined by free ground state to ground state tunneling. Thus, for the irradiated samples the peak for the free ground state to free ground state tunneling at 20 mV is relevant for the position of the photonassisted peaks. Wacker et al. (1997) modeled the unirradiated I-V and found their theory reproduces the experimentally observed I-V at a temperature of 4 K quite well for low bias. At higher bias, the experimental I-V is better explained with simulations for a temperature of 35K, indicating that the electrons are heated up by the bias. For the simulation of the irradiated I-Vs an electron temperature of 35 K is used since the external radiation heats up the electrons already at zero bias. At this bias about 50% of the electrons are free. The calculated I-V for 35 K shows a double peak structure with one peak at 20mV (free ground state to free ground state tunneling) and a second peak at about 100mV (impurity band to free ground state tunneling). The first peak is absent in the unirradiated I-V and is observed only in the irradiated I-Vs. The simulation with external radiation using an effective carrier temperature of 35K is in very good agreement with the observed experimental results (Fig. 27). Small deviations at a higher bias are explained by domain formation. This shows that the Tien-Gordon model together with a microscopic model for the transport gives a very good description of the photon-assisted transport in sequential resonant tunneling superlattices. An interesting consequence of this work is that photon-assisted tunneling experiments can be used to obtain the instantaneous I-V of resonant tunneling structures. The linewidth broadening of the levels can be extracted from the position of the direct tunneling peak. Thus, different broadening mechanisms such as impurity scattering and interface roughness scattering can be investigated.

172

KARLUNTERRAINER 0.6

t

(a) hv = 6.3 meV

I

t

(C) CI = 2.4

hv=3.5 rneV

r 0

100 NFd ( m V )

200 0

..

. ... . , , , hv=4.0 rneV

-hvS.3 rnsV .....,... h v d . 3 rneV

100 NFd ( mV )

200

0.6

-%

0.4

0.2

v

-

0.0 a

-0.2 0

100

bias ( mV )

200 0

hvS.5 rneV hv=4.0meV hv5.3meV

100

200

bias ( m V )

FIG. 27. Current-voltage characteristics of an irradiated superlattice. (a) Calculations for hv = 6.3 meV and different field strength a = eE,d/hv, (b) experimental results for the same conditions, (c) calculation for a fixed field strength of a = 2.4 and different photon energies, and (d) experimental results for the same photon energies. The intensity was tuned to give maximum negative conductance (after Wacker et a[., 1997).

VII. Terahertz Transport in Superlattice Minibands For strongly coupled superlattices, minibands are formed, enabling coherent transport throughout several superlattice periods. The fundamental mode for transport in a miniband is a Bloch oscillation characterized by the Bloch frequency oB= eEd/h. For typical epitaxially grown superlattices, the period d is around 10 nm. For the observation of coherent Bloch oscillations the condition wB’z> 1 must be fulfilled. The scattering time z in superlattice depends on several parameters, such as doping, interface quality, and miniband width. From a design point of view, the miniband width is the

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most favorable parameter since it can be adjusted by the ratio between the barrier and well width. The strong scattering mechanism of optical phonon emission can be eliminated by making the miniband width smaller than the optical phonon energy hoop. The impurity scattering can be controlled by the doping concentration. For a superlattice homogeneously doped between 10’’ and 1016cm-3,we expect a scattering time of a few tens of picoseconds. This means that Bloch oscillation should be observable for wB> 1THz, which requires a field of 5 kV/cm that can be easily reached in practical superlattices. For the coherent interaction of an external radiation field with Bloch oscillation, many effects have been predicted since the 1970s. Since all the effects depend on the field strength eE,,d/ho, the THz frequency range seems to be the best choice for the observation of these nonlinear effects. A further motivation to work in the THz regime is the absence of reliable coherent radiation sources, which are desperately needed for a large field of spectroscopic applications. The theoretical calculations predict gain for the condition given earlier and when the superlattice is biased into the negative differential conductivity region. Fundamental research in the THz range can therefore help to develop a semiconductor radiation source and thus improve the technological coverage of this part of the electromagnetic spectrum. The existence of a negative differential conductivity region in semiconductor superlattices was observed by Sibille et al. (1989, 1990, 1993). However, no clear identification of the onset of the negative differential conductivity region was given. In optical experiments in undoped superlattices, the splitting for the miniband into a Wannier-Stark ladder was observed (Mendez et al., 1988; Voisin et al., 1988; Agullo-Rueda et al., 1989). The energy splitting was proved to be given by eEd. Mendez et al. observed that the optical interband transitions from electron states and hole states belonging to different wells vanished when the Stark splitting eEd became larger than the miniband width A. From that they conclude that the tunneling probability between the wells disappears and the superlattice becomes localized when the Stark splitting exceeds the miniband width. At this point, the vertical transport through the superlattice changes from a Bloch-type transport to a hopping transport. Bloch oscillations are only expected for a Stark splitting smaller than the miniband width eEd < A, where optical transitions from electron states to hole states from different wells are observed. In a transport experiment, negative differential conductivity should be observed when eEd > h/z, which should correspond to the low-field minimum for the observation of separate Wannier-Stark transitions. Sibille et al. (1992) performed a beautiful experiment where transport and the optical transitions were measured at the same time. The onset of

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negative differential conductivity was observed not at the bias where separate Wannier-Stark transitions could be distinguished, but at a higher voltage where only transitions from next neighboring wells remained. This makes the windows for observing Bloch oscillation rather narrow. In time-resolved optical experiments, coherent oscillations of the polarization were observed first in degenerate four-wave mixing experiments and later also by detecting the emitted THz radiation (Feldmann et al., 1992; Waschke et al., 1993; Leo, 1998; Rossi, 1998). In a 1997 experiment, the group of K. Leo showed that the oscillating polarization is indeed connected to a real displacement of the electric charge over several superlattice periods as expected for Bloch oscillation (Lyssenko et al., 1997). (These experiments are discussed in Chapter 8.) These beautiful optical experiments do not answer the open questions of whether Bloch oscillations can be excited electrically. It is somewhat strange that the Bloch oscillation, which is the normal mode of carrier dynamics in a superlattice, has so far escaped observation in a pure electrical experiment. The observation of a unique signature of Bloch oscillation in an electrical experiment must be the first step before an attempt is made for the realization of an electrically driven Bloch oscillator. The work of Sibille's group has shown that negative differential conductance can be achieved in a semiconductor superlattice even at high current densities and at room temperature. However, it was also reported that charge built up leads to an electric field that changes throughout the superlattice, which would prevent the observation of a clear resonance at one specific Bloch frequency. The results on photon-assisted tunneling in sequential resonant tunneling showed that the instantaneous current-voltage characteristics can be obtained from measurements of the THz response. Consequently, it should be possible to observe signatures of Bloch oscillation in the current-voltage characteristic of THz-driven superlattices. The first study on the THz response of a miniband superlattice was reported by a group of the University of Regensburg (Ignatov et al., 1994). THz radiation was coupled to the superlattice sample using a corner cube reflector and a whisker antenna similar to the original approach of the UCSB group. The superlattice sample consisted of 80 periods with 5.4-nm-thick GaAs wells and 1.I-nm-thick Al,.,Ga,,,As barriers. The superlattice was doped with Si to lo1' cmP3 and was grown on nf-GaAs substrate with a doping concentration of 2 x 1 0 " ~ m - ~ .Between the GaAs substrate and the superlattice a layer with gradual composition was grown to avoid abrupt heterojunctions between the GaAs and the superlattice. The calculated miniband width of the structure is A = 65 meV. The sample had a cross section of 10 x 10 pm and Au-Ge-Ni contacts were evaporated to the top

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contact and to the substrate. The top contact is connected to a 360-pm-long gold wire antenna, which is mounted in a grounded corner cube for optimum coupling. The DC bias is applied to the substrate contact through an inductive resistance. The THz-radiation-induced current signal is measured with a transient recorder as the voltage signal at its 50-R impedance input, which was decoupled from the DC bias circuit by a capacitor. As a THz source, a high-pressure CO, laser was used to excite a NH, gas laser, which generates THz radiation through stimulated Raman scattering. The CO, laser was self-mode-locking, giving pulse trains with 1-ns-long pulses separated by 24 ns. The system was tuned to emit radiation at 3.3 THz with a pulse energy of about 2 pJ. The repetition rate for the pulse trains is about 1 Hz. The THz radiation is focused onto the antenna with the electric field polarization parallel to the antenna. The current-voltage characteristic of the superlattice sample showed a nonlinear increase of the current around zero bias, followed by a linear increase and a maximum at about U = 2 V (see Fig. 28). The negative differential resistance region expanded up to U = 4 V. The nonlinear behavior was attributed to nonohmic contacts. According to the contact and series resistance of the ’n layers a large voltage drop of 1.5V at U = 2 V bias was estimated. Therefore, the voltage across the superlattice was only 500 mV and the DC electric field at the current maximum was E = 10 kV/ cm. We estimate a scattering time of z = 0.7 ps by assuming that the onset of negative differential conductivity is caused by electrons reaching the miniband zone boundary and performing Bloch oscillation.

Uapp,

(V)

-

FIG. 28. Current-voltage characteristic of the 5.4 nm-1.1nm superlattice and the induced current change 61 by the 3.3-THz radiation (after Ignatov et al., 1994).

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The THz response was measured around the current peak at U = 2 V. The observed signal was due to an induced reduction of the current through the superlattice. The magnitude of the signal was about 10% of the DC current. The signal height showed a linear dependence on the laser intensity. In Fig. 28, the THz response is shown as a function of the bias. For increasing bias, the signal corresponds also to a current reduction with decreasing amplitude. The signal vanishes at about U = 4 V. At a low bias ( U < 1V), an increased current was induced by the THz pulses, which was attributed to rectification of the THz radiation due to the nonlinear I-V curve caused by the nonohmic contacts. In subsequent experiments, different superlattices with 3.63-nm GaAs wells and 1.17-nm barriers with AuGeNi top and bottom contacts were used. The response of these superlattices was investigated using microwave radiation at 78, 90, and 450 GHz in addition to the 3.9-THz radiation from the gas laser system (Schomburg et al., 1996; Winner1 et al., 1997). The results for 90 GHz and 3.9 THz are shown in Fig. 29. For 90 GHz classical rectification is observed since the peak of the signal is found at the position of the maximum curvature of the DC I-V. The maximum response for 3.9 THz is observed at the voltage position where the maximum current in the DC I-V is observed. The authors explained their results with an interaction of theTHz field with Bloch oscillation of electrons in the superlattice. According to the theory described in Section 11, we would expect a decrease of the current at the peak with a complete suppression of the current when the field strength eE,d/Aw is equal to the first zero of J , and an increase of the current above that value. The observed reduction of the current and its monotonic dependence on laser intensity indicates that the THz electric field strength is too small for the observation of nonlinear effects. However, for 3.9THz even at this field strength, photon-assisted peaks should be observed in the negative differential conductivity region. Two possible causes of the absence of such peaks are the presence of a very inhomogeneous electric field inside the superlattice, which could be plausible from nonohmic contacts or dynamic domain formation, or that the actual scattering time is much shorter or is not constant so that the condition for W ’ T > 1 is not fulfilled for the whole voltage range. The authors extracted a scattering time of 100 fs from the DC I-V and from the dynamical behavior at small bias. For the situation w.7 < 1, only classical rectification is expected, which does not show quantum effects such as photon-assisted tunneling peaks. The preceding results of the Regensburg group have shown that THz radiation directly interacts with the electrons in a superlattice and influences the miniband transport. The response is fast and not due to heating, which makes it attractive for detector applications.

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UIU, FIG. 29. Measured (dashed line) and calculated (solid line) current-voltage characteristics and field-induced change: (a) without radiation, (b) for 90 GHz radation (calculation for WT = 0.07), and (c) for 3.9-THz radiation at different power levels (calculation for WT = 3) (after Winner1 et al., 1997).

In another work by the UCSB group it was shown that DC-currentdriven Bloch oscillation couples to external radiation either by emission or absorption of THz photons. They explored this phenomenon by investigating the inverse Bloch oscillator efect, which senses changes in the DC conductivity under the influence of an external THz field (Unterrainer et al., 1996b).

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The samples used in this study are GaAs-Al,Ga, -,As (x = 0.3) superlattices grown on a semi-insulating GaAs substrate by molecular beam epitaxy. The superlattice structure consists of 40 periods of 80-A-wide GaAs wells and 20 A-thick AlGaAs barriers. The superlattice is homogeneously Si doped with a concentration of n = 3 x 10'5cm-3. The superlattice is embedded in 3000 A-thick GaAs layers with a carrier concentration of n = 2 x 1018cm-3, which serve as contact regions. The highly doped contact layers are separated from the superlattice by 80-A GaAs lightly doped setback layers. A band-structure calculation in the envelope function approximation results in a width for the lowest miniband A = 22 meV (Bastard, 1981). This means that electrons moving in the lowest miniband are not scattered by optical phonons at low temperatures. The effective mass of electrons at the bottom of the miniband is mSL= 0.07 m,. The second miniband is separated from the first one by about 100 meV. Thus, for low applied Bias (< 200 mV) we do not have to consider tunneling to the second miniband. Superlattice mesas with an area of 8pm2 were integrated with bow-tie antennas as described in Section 111. The experiments were performed at 10 K in a temperature-controlled flow cryostat with Z-cut quartz windows. The conductance of the superlattice was measured during the microsecond long pulses of THz radiation provided by the UCSB free electron lasers. The current voltage characteristics of the superlattice device without THz radiation (DC I-V) can be seen in Fig. 30 (curve at the top). The current is linear for bias voltages below 20 mV. The negative differential conductivity (NDC) region begins at a bias of 20mV, which corresponds to a critical electric field of 500 V/cm. Assuming that the onset of the NDC region is due to Esaki-Tsu type localization when uB.t> 1, we find a scattering time z = 1.3 ps. The maximum current density is about 100A/cmz. The I-V curve shows a small asymmetry: For positive bias (injection from the top contact) we find a more pronounced NDC than for negative bias where the bottom contact is the emitter. This asymmetry is present in all devices and we think is due to the different geometry of top and bottom contacts or due to inhomogeneity in the doping. From the preceding scattering time of z = 1.3ps, we would expect to see coherent effects for a frequency of a few 100 GHz. The irradiated I-Vs in Fig. 30 for 0.24THz show only classical rectification effects and no quantum effects can be observed. The curves are shown for increasing AC field strength (the curves are displaced downward with increasing intensity for clarity). The conductivity decreases with increasing intensity and the main peak shifts to higher voltages. This is an indication that the scattering time extracted from the maximum of the D C I-V is not correct.

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1 v=0.24THz

0

0.05

0.1

DC Bias (V) FIG. 30. Current-voltage characteristics of the 8 nm-2 nm superlattice without (curve at the top) and with 0.24-THz radiation. With increasing intensity, the conductivity decreases and the main peak shifts to higher voltages. No quantum effects are observed at this frequency.

Figure 31a, shows the influence of an external THz electric field at a frequency of 0.6 THz on the superlattice current. At low intensities, an additional peak emerges in the NDC region. We attribute the first additional peak to a resonance of the external laser field with the Bloch oscillation wB= a.When the intensity is increased further, the first peak starts to decrease and a second peak at about twice the voltage of the first peak is observed and assigned to a two-photon resonance. At the highest intensities, we observe a four-photon resonance. The initial current peak of the DC I-V decreases with increasing intensity indicating the onset of dynamic localization. At very high intensities, a small bump at the original position of the peak recovers. The position of the peaks do not change with intensity of the FEL. We observe quantum effects for frequencies larger than 480 GHz, which corresponds to a scattering time of z = 0.33 ps, approximately 3-4 times shorter than the previous estimate. This value is consistent with earlier cyclotron resonance measurements (magnetic field perpendicular to the growth direction) (Duffield et al., 1986). Figure 31b shows the results for a laser frequency of 1.5 THz. The peaks are shifted to higher voltages and are much more pronounced. Only the fundamental and the second harmonic are observed, since, for a given Em, eE,d/ho is smaller at higher frequencies. In addition, we observe a sup-

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0

,

0.05 0.1 0.15 DC Bias (V)

1

0.2 0

0.05

0.1

0.15

0.2

DC Bias (V)

FIG. 31. DC current-voltage curve for increasing FEL intensity (the curves are shifted downward for increasing laser intensity). The FEL frequency was fixed to 0.6 THz (a) and to 1.5THz (b). In the NDC region, additional features occur attributed to resonances a t the Bloch frequency and its subharmonics.

pression of the current value in between the peaks. The peaks show a clear asymmetry with a steeper slope on the high-voltage side. This asymmetry is different from the shape of the peak of original DC I-V, which shows a steeper slope at the low voltage side. Figure 32 shows the peak positions as a function of FEL frequency. The relationship is linear and the slopes of the Nth harmonic are N times the slope of the one-photon resonance. The magnitude of the slope is larger than expected from a voltage drop across the whole superlattice. The most reasonable explanation is that a high electric field domain is formed that extends over approximately one-third of the superlattice. For this stable situation, the electric field in the low-field domain is below the critical field for localization and puts this part of the superlattice in the high conductive miniband transport regime. The formation of a high-field domain could also explain the discrepancies between the values for the scattering times deduced from the DC I-V and from the THz measurements. If the onset of the NDC is more likely caused by domain formation, localization over a fraction of the superlattice, and not by the onset of localization over the entire superlattice, the value for the scattering time that we derived from the assumption of uniform localization is incorrect. Figure 33 shows the intensity dependence of the current at the different resonances at 0.6THz. In addition, the predicted current from Eq. (32) is

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FIG. 32. DC bias positions of the induced current peaks versus FEL frequency (after Unterrainer et al., 1996).

0

2

4 6 edE Ihv

8

10

FIG. 33. Data points show the intensity dependence of the current for constant DC bias at the different resonance positions at 0.6 THz. The upper curve shows the behavior at a bias of 45 mV, where the one-photon resonance occurs; the other curves show the behavior at the bias positions of the higher resonances. The lines show the predicted current at these resonances as a function of AC field strength (after Unterrainer et al., 1996).

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plotted. The absolute value of the electric field is obtained from a fit to the maximum of the current of the one-photon resonances. The coincidence of the first maxima for the higher photon resonances is very good. Thus, we can use the fit to the maxima to calibrate our electric field in the sample. A discrepancy exists for the predicted smaller oscillations at higher intensities, which do not show up in the experimental data. Multiphoton resonances with Bloch oscillation in a superlattice in a DC electric field have been observed. These results show clearly that the external radiation couples to Bloch oscillations, contrary to theoretical suggestions that THz radiation would not couple to a uniform Wannier-Stark ladder. This result is intimately related to dissipation and line broadening of the otherwise identical states in the ladder: absorption appears above the Wannier-Stark splitting (CD, < w ) and gain below (w, > w). The effect is an analogy to Shapiro steps in S-I-S junctions that support the AC Josephson effect.

VIII. Photon-Assisted Tunneling and Terahertz Amplification We can discuss the THz amplification of resonant tunneling diodes from the results of Drexler (1995). The current of the peak at the higher voltage side of the main tunneling peak is due to stimulated emission of photons. From the value of the current density of the one-photon-stimulated emission peak at the lowest intensity we can estimate the power emitted from the sample. The current density of the main tunneling peak is about 70 Ajcm' and that of the stimulated photon emission peak for a frequency of 1 THz has about the same current density. The current of this photon assisted peak is only mediated by the emission of photons. Thus, the stimulated photon emission rate can be estimated to be j / e = 70j1.6 x = 4.3 x 1O2's-' cm-'. The emitted power density for a photon energy of 4.1 meV is thus about 0.3 W/cm2. This is not a very high value compared to the estimated input power density of for the bow-tie device. 50kW/cm2, which results in a low gain of Since the current density of the measured RTD is quite low compared to that of RTDs designed for high current densities, there is room for further improvement. However, higher current densities should be realized only when considering that the tunneling peaks do not broaden significantly. In addition, the device geometry must be changed to ensure a larger interaction volume for the radiation. In the experiments with miniband superlattices, we attribute a positive photocurrent in the NDC to stimulated emission of photons (electrons move downward in the Wannier-Stark ladder). From the value of the photocur-

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rent we can estimate the power transfer from the DC electric field into the photon field to be about 50 nW (0.56 W/cm2 in the mesa). The intensity of the AC field inside the superlattice at the maximum of the one-photon resonance is 42 kW/cmz, which leads to a total THz amplification coefficient of our superlattice mesa of 1.3 x 10W5.Assuming a 50-0 impedance of the bow-tie structure we estimate a negative THz conductance of 1.3 x lop4 (Qcrn)-’, which compares to the theoretical value of 5 x lop3(Qcm)-’. The experimentally determined THz conductance is more than one order of magnitude smaller than the theoretical prediction. The values for the amplification are quite small; however, these results show that the theoretical predictions for the nonlinear behavior of semiconductor superlattices are correct. The observation of this parametric gain of a THz-driven superlattice proves that the assumptions we had to make in the calculations of the nonlinear behavior are valid. In consequence, the prediction of negative AC conductivity in a superlattice without a driving AC field seems to be quite realistic. However, the direct experimental proof of small signal gain in a biased superlattices remains missing,

IX. Summary and Outlook The experiments on photon-assisted tunneling in semiconductor quantum structures have produced new results for the nonlinear interaction with external AC fields. Some of these effects had already been predicted in the 1970s and their relevance for realistic semiconductor quantum structures has been quite disputed. The results for the resonant tunneling diodes and for the sequential resonant tunneling superlattices show in a very impressive way by the observation of photon-assisted side peaks and by the occurrence of dynamic localization that these models are correct. The observation of absolute negative conductance was not predicted for such systems. More refined models show that absolute negative conductance is a consequence of dynamic localization. Furthermore, the results from the resonant tunneling diode showed that the occupation of the wells influences the photon-assisted tunneling process. This -together with a better theory for photon-assisted tunneling- could lead to quantitative determination of the occupation in the wells from photon-assisted tunneling measurements. For sequential resonant tunneling structures, it was shown that photon-assisted tunneling experiments reveal the “instantaneous” current-voltage characteristics, which is very important in these structures where, under DC bias, domain formation dominates the current-voltage characteristics. The experiments on miniband superlattices have also proven the theoreti-

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cal predictions for the nonlinear interaction with external AC fields. Furthermore, the successful experiments in superlattices showed that in a superlattice supplied with a constant current, Bloch oscillation couples to an external AC field. With the observation of photon-assisted peaks in the negative differential conductivity region, the analogy to Shapiro steps in AC Josephson junctions is established. The suppression of the low-field conductance by an external AC field through the dynamic localization is explained by the semiclassical model and by the full quantum mechanical approach. This effect is also important for detector applications for THz electronics. The dynamic localization is very fast since it is an instantaneous effect and does not involve carrier recombination. Resonant tunneling diodes also have the potential to be used in detector applications using the photonabsorption-assisted tunneling channel. The observed nonlinear characteristics of superlattices are important for modulators and frequency converters (mixers, harmonic generators). These applications could be equally important as the most discussed possible application of a superlattice as a tunable THz source.

ACKNOWLEDGMENTS The author would like to thank the staff at the Center for Free-Electron Laser studies: J. R. Allen, D. Enyeart, J. P. Kaminski, G. Ramian and D. White, S. J. Allen, B. J. Keay, M. C . Wanke, H. Drexler, S. Zeuner, and E. Schomburg for their collaboration, support, and for supplying their material for this review, and M. Helm for discussions and for reading the manuscript. Most of the samples were grown by K. L. Campman, K. D. Maranowski, A. C. Gossard, D. Leonard, and G. Medeiros-Ribeiro.

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Lam, J. F., B. D. Guenther, and D. D. Skatrud. (1990). Appl. Phys. Lett. 56, 773. Lebwohl, P. A., and R. Tsu. (1970). J . Appl. Phys. 41, 2664. Leo, K. (1998). Semicond. Sci. and Technol. 13, 249. Lyssenko, V. G., G. Valusis, F. Loser, T. Hasche et al. (1997). Phys. Rev. Lett. 79, 301. Mendez, E. E., F. Agullo-Rueda, and J. M. Hong. (1988). Phys. Rev. Lett. 60, 2426. Pavlovich, V. V., and E. M. Epshtein. (1976). Sou. Phys. Semicond. 10, 1196. Platero, G., and R. Aguado. (1997). Appl. Phys. Lett. 70, 3546. Ramian, G. R. (1992). Nucl. Instr. Meth. A 318, 225. Reed, M. A., W. R. Frensley, W. M. Duncan, R. J. Matyi, A. C. Seabaugh, and H. L. Tsai. (1989). Appl. Phys. Lett. 54, 1256. Richards, P. L., and T. M. Shen. (1980). IEEE Trans. Electron Devices 27, 1909. Romanov, Yu. A. (1972). Opt. Spektrosk. 33, 917. Rosencher, E., A. Fiore, B. Vinter, V. Berger, Ph. Bois, and J. Nagle. (1996). Science 271, 168. Ross], F. (1998). Semicond. Sci. Technol. 13, 147. Schomburg, E., et al. (1996). Appl. Phys. Lett. 68, 1096. Schomburg, E., et al. (1997). Appl. Phys. Lett. 71, 401. Shik, A. Ya. (1975). Sou. Phys. Semicond. 8, 1195. Sibille, A., J. F. Palmier, C. Minot, and F. Mollot. (1989). Appl. Phys. Lett. 54, 165. Sibille, A,, J. F. Palmier, H. Wang, and F. Mollot. (1990). Phys. Rev. Lett. 64, 52. Sibille, A,, J. F. Palmier, and F. Mollot. (1992). Appl. Phys. Lett. 60, 457. Sibille, A,, J. F. Palmier, M. Hadjazi, H. Wang, G. Etemadi, E. Dutisseuil, and F. Mollot. (1993). Superlat. Microstruct. 13, 247. Tien, P. K., and J. P. Gordon. (1963). Phys. Rev. 129, 647. Tsu, R., and L. Esaki. (1971). Appl. Phys. Lett. 19. 246. Tucker, J. R., and M. J. Feldman. (1985). Rev. Mod. Phys. 57, 1055. Unterrainer, K., B. J. Keay, M. C. Wanke, S. J. Allen, D. Leonard, G. Medeiros-Ribeiro, U. Bhattacharya, and M. J. W. Rodwell. (1996a). In Hot Carriers in Semiconductors, ed. K. Hess et al., (Plenum Press, New York), p. 135. Unterrainer, K., B. J. Keay, M. C. Wanke, S. J. Allen, D. Leonard, G. Medeiros-Ribeiro, U. Bhattacharya, and M. J. W. Rodwell. (1996b). Phys. Rev. Lett. 76, 2973. Voisin, P., J. Bleuse, C. Bouche, S. Gaillard, C. Alibert, and A. Regreny. (1988). Phys. Rev. Lett. 61, 1357. Wacker, A., A. P. Jauho, S. Zeuner, and S. J. Allen. (1997). Phys. Rev. B 56, 13268. Wagner, M. (1996). Phys. Rev. Lett. 76, 4010. Wanke, M. C., A. G. Markelz, K. Unterrainer, S. J. Allen, and R. Bhatt. (1996). In Hot Carriers in Semiconductors, ed. K. Hess et al. (Plenum Press, New York), p. 161. Wannier, G. H. (1962). Rev. Mod. Phys. 34, 645. Waschke, C., H. G. Roskos, R. Schwedler, K. Leo, and H. Kurz. (1993). Phys. Rev. Lett. 70, 3319. Weisbuch, C., and B. Vinter. (1991). In Quantum Semiconductor Structures: Fundamentals and Applications (Academic Press, Boston). West, L. C., and S. J. Eglash. (1985). Appl. Phys. Lett. 46, 1156. Winnerl, S., et al. (1997). Phys. Rev. B 56, 10303. Wyss, R., C. C. Eugster, J. A. de Alamo, and Q. Hu. (1993). Appl. Phys. Lett. 63, 1522. Zak, J. (1993). Phys. Rev. Lett. 71, 2623. Zeuner, S., B. J. Keay, S. J. Allen, K. D. Maranowslu, A. C. Gossard, U. Bhattacharya, and M. J. W. Rodwell. (1996). Phys. Rev. B 53, R1717. Zeuner, S., B. J. Keay, S. J. Allen, K. D. Maranowski, A. C. Gossard, U. Bhattacharya, and M. J. W. Rodwell. (1997). Superlatt. Microstr. 22, 149.

S E M I C O N D U C I O R S A N D SEMIMETALS, VOL 66

CHAPTER 4

Optically Excited Bloch Oscillations -Fundamentals and Application Perspectives P. Haring Bolivar, T Dekorsy, and H. Kurz INSTlTUT

HALBLEI~ERTECHNIK 11

RWTH AACHEN,GERMANY

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. HISTORICAL BACKGROUND BLWH OSCILLATIONS IN THE SEMICLASSICAL MODEL . . . . . . . . . . . . . . . . . . . . . . . . 111. WANNIER-STARK DESCRIPTION OF BLOCHOSCILLATIONS . . . . . . . . . . IV. TIME-RESOLVED INVESTIGATION OF BLOCH OSCILLATIONS . . . . . . . . . . V. BLOCHOSCILLATIONS AS A MODELSYSTEM FOR COHERENT CARRIER DYNAMICS IN SEMICONDUCTORS . . . . . . . . . . . . . . . . . . . . . VI. APPLICATIONOF BLOCHOSCILLATIONS AS A COHERENT SOURCE OF TUNABLE TERAHERTZ RADIATION . . . . . . . . . . . . . . . . . . . . . . . . VII. SUMMARY.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187

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REFERENCES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

188 192 197 203

210 214 215

I. Introduction One of the most intriguing phenomena in solid state physics is the behavior of an electronic charge in the periodic potential of a crystal lattice under the influence of a constant electric field. Counterintuitively and opposing our everyday experience from the well-known Ohmic law, in an ideal scattering free system a charge carrier will not follow uniformly the electric field and give rise to a constant electrical current, but will perform a periodic oscillatory motion in real space-it will perform Bloch osciflations.

Despite that the theoretical concept of Bloch oscillations was postulated more than 70 years ago by F. Bloch (1928) and C. Zener (1934), only recent experiments exploiting the high time resolution achievable with femtosecond lasers and the high quality of modern molecular-beam-epitaxy grown heterostructures have enabled the observation of this fundamental phenomenon (Feldmann et al., 1992a; Leo et al., 1992). This chapter contains a 187 Copyright ;( 2000 by Academic Press All rights of reproduction in any form reserved ISBN 0-12.752175-5 ISSN 0080-8784’00 $3000

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general overview of fundamental aspects of Bloch oscillations and presents various experimental work performed to investigate these oscillations and derive their characteristic dependencies. The study of Bloch oscillations has not only opened the path for prospective applications especially as tunable sources of electromagnetic radiation in the terahertz range, but has also enabled us to gain deep insight into fundamental properties of ultrafast coherent charge carrier dynamics in semiconductors. The outline of this chapter is as follows: the first section contains an overview of the historical background of Bloch oscillations and their theoretical description in the semiclassical model. Section I11 introduces the Wannier-Stark description of Bloch oscillations and discusses difficulties for the correct theoretical modeling of impulsively, optically excited coherent charge carrier dynamics in semiconductor superlattices. The third section describes different experimental approaches that have enabled the time-resolved observation of this fundamental phenomenon. The first two sections focus on the investigation of Bloch oscillations by terahertz emission experiments: both fundamental aspects (Section V) and application perspectives (Section VI) are discussed.

11. Historical Background -Bloch Oscillations in the Semiclassical Model

In 1928 Felix Bloch (1928) analyzed theoretically the behavior of an electronic charge e in a periodic potential under the influence of a static electric field F. Two equivalent models can be adopted to describe this problem: It can either be solved by the solution of Schrodinger’s equation including the electric field explicitly, which corresponds to a description in a Wunnier-Stark-Basis in real space and which is described later. A common alternative ansatz, the semiclassical model, however, enables us to gain a more illustrative picture of Bloch oscillations in a k-space formulation. The basic idea behind this semiclassical description is a formal transformation of the Schrodinger equation (Haug, 1964; Ashcroft and Mermin, 1976; Krieger and Iafrate, 1985): hk

=

-eF

where h is Planck’s constant, k is the momentum, e is the elementary charge, and F is the electric field. This mathematical transformation enables the elimination of the electric field in the time dependent Schrodinger equation describing the system. The problem is thus reduced to a description of Bloch electrons (ie., to a description of the system with the wave functions of the

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field-free case) by introducing a time-dependent k-vector. Besides this entirely mathematical point of view, Eq. (1) can also be interpreted as an equation of motion for the momentum k. It is therefore also known as the uccelerution theorem, as any charge carrier, or more precisely, any wave packet with a narrow k-space distribution (a Houston state; Houston, 1940), will move in k-space with a constant velocity k given by Eq. (1). Since the potential of the crystal E(k) in which the wave packet moves is periodic (period 2n/d), where d is the real-space lattice constant, the wave packet will return to its original state after a finite time. In other words, due to the periodicity of E(k), the wave packet cannot gain an arbitrary high k-vector, but will be Bragg-reflected at the end of the Brillouin zone at k = n/d. In the reduced Brillouin zone, it will thus continue its motion at the left end of the Brillouin zone at k = - n / d (see, e.g., Fig. 1). Formally speaking, the momentum is only a good quantum number modulo the reciprocal wave vector of the periodic potential defined by 2 4 d . In the absence of scattering it will thus return to its original initial state with a frequency VBloch (period TBloch = l/VBl,,ch) given by

VBloch

=

~

k - eFd __ 2n/d - h

Equation (2) already demonstrates that any electronic wave packet in any periodic potential will perform a periodic motion in k-space with a characteristic frequency vBloch that is linearly dependent on the static electric field F. Note that the Bloch frequency is independent on the specific bandstructure of the material; that is, as it is independent of the electronic mass, it behaves utterly differently in comparison to classic behavior. This frequency

FIG. 1. Semiclassical picture of Bloch oscillations: (a) dispersion relation E ( k ) and realspace group velocity u and (b) associated real-space oscillation.

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dependence [Eq. (2)] is hence one of the primary characteristics of Bloch oscillations. To calculate the motion in real space of electrons performing Bloch oscillations, one can derive the real-space group velocity u of a wave packet from the dispersion relation E(k) since 0

i a h ak

= - -E(k)

(3)

As an example for the real-space dynamics of Bloch oscillations, one can take an ideal one-dimensional Kronig-Penney model for the crystal as depicted in Fig. 1 (spatial coordinate z, field F in the - z direction), which delivers E(k) = A/2-(1 - cos(kd)), where A represents the width of the considered electronic band. Assuming an ideally narrow electronic wave packet with an initial wave vector k = k , at position z = z , for t = 0 one can easily calculate from Eqs. (1) and (3) the velocity v(t), which yields after time integration the position z(t) of the wave packet. For this simplified ideal system one gets

which demonstrates that the periodic motion in k-space can directly be associated with an oscillation in real space. The phase of the oscillation will then depend on the initial condition k,. An interesting property of this oscillation is that the oscillation amplitude

LZ.- A

2eF

is inverse proportional to the applied field. This amplitude L will later

reappear as a localization length in the quantum mechanical description. From Eq. (4)one can also derive directly the velocity of the wave packet. Interestingly, despite the absence of scattering in this model, the maximum velocity a carrier can reach vmax = Ad/2h is independent of the field. This characteristic behavior is in pronounced contrast to what one would expect for the acceleration of a free carrier in an electric field and thus underlines the Bloch character of the motion of charge carriers in a periodic potential. As already mentioned, the phase of a particle performing Bloch oscillations depends critically on the initial conditions. Hence, when not only one but an ensemble of carriers is excited coherently with an ultrashort laser

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pulse, the initial distribution in k-space can strongly modify the motion of the total charge distribution. Interference effects can then influence the total dynamics drastically. Well-known special cases are: (i) the excitation of carriers in the center of the band at E(k) = A/2 (at k , = +_n/2d)of a onedimensional system, which leads to the characteristic breathing mode, where the center of mass of the ensemble remains constant, but the wave packet expands and contracts with the Bloch frequency, or (ii) the homogeneous excitation of carriers over the entire Brillouin zone, which leads to no net carrier motion at all. Nevertheless, for the excitation of an ensemble of carriers close to the bandedge (at k, z 0), the behavior of the ensemble will closely follow the oscillation of a single particle as described by Eq. (4). It is important to note that the semiclassical picture of Bloch oscillations is a mathematically exact transformation which allows to employ the well-known field-free band structure and wave functions for the description of the dynamics of wave packets in the system. The problem is that all time dependencies are transformed by the acceleration theorem, which makes the correct description of initial conditions particularly difficult. Although the preceding calculation of the real-space dynamics [Eq. (4)] delivers the correct general dependencies for Bloch oscillations, the adopted energetically and temporally sharp initial conditions do not fulfill the uncertainty principle. A true quantum mechanical calculation must be performed to take into account initial conditions. This can then include effects such as, for example that a photoexcited carrier distribution will already be accelerated and thus redistributed in k-space during the finite temporal duration of the exciting laser pulse. Zener tunneling (Zener, 1934) -that is, the coupling of the carriers to other bands is also frequently neglected in the semiclassical approach to simplify the modeling. In the sample structures described later, however, Zener tunneling is found to be negligible. The entire concept of the Bloch oscillation of an electron in an electronic band under the influence of an electric field seems to contradict our everyday experience and intuition, and in fact although crystalline materials with a high degree of perfection like semiconductors exist, Bloch oscillations are extremely difficult to observe. The reason is that in most material systems, scattering of the carrier momentum occurs on an ultrashort timescale -in typical semiconductors scattering times are in the subpicosecond range. To observe at least one Bloch oscillation before scattering has affected the motion of the carrier, one has thus to tune the Bloch frequency to above several terahertz (THz). For typical semiconductors with lattice constants of the order of a few angstrom this would require [see Eq. (2)] fields beyond or very close to typical break-down fields. This is one of the main factors that has prevented the observation of Bloch oscillations. In 1970 L. Esaki and R. Tsu made a proposal that allowed circumvention ~

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of this problem. The idea is to grow by molecular beam epitaxy (MBE)the growth technique pioneered by Cho and Arthur (1975) -artificial crystal structures with a much higher lattice period than in natural crystalline semiconductors by growing alternating layered sequences of semiconductors with different bandgaps. These superlattices now enable us to design artificial periodic potentials with tunable miniband widths and larger spatial periods in the tens of nanometers range. With these parameters Bloch oscillation periods below typical scattering times can be attained already for fields on the order of several kilovolts per centimeter, which can easily be applied in p-i-n diode structures. Superlattices have therefore opened the possibility for observing Bloch oscillations experimentally for the first time, and, in fact, the great majority of experiments described later were performed in these heterostructure systems.

III. Wannier-Stark Description of Bloch Oscillations

As elucidated previously, Bloch oscillations can be described theoretically also in the Wannier-Stark picture, which we use now to derive the dynamics of Bloch oscillations in superlattices after photoexcitation with femtosecond laser pulses. By using laser pulses with a duration shorter than the Bloch oscillation phase, one can easily guarantee that photoexcited carriers performing Bloch oscillations have a narrow initial distribution in k-space, which is essential to avoid destructive interference. In the Wannier-Stark picture the basis for the description of the dynamics of the system are the eigenstates of the Hamiltonian including the electrical field. A complete quantum mechanical calculation is necessary to derive these states correctly at each field position. This demands a greater effort, but in contrast to the semiclassical approach, this description lacks the difficulty for the description of initial conditions, which makes the modeling of time-resolved optically excited experiments often simpler. To demonstrate the dynamics of Bloch oscillations in a superlattice in the Wannier-Stark picture, a tight-binding approach enables us to derive the essential characteristics. This approach is analogous to assuming a cosineshaped E(k) dispersion, as discussed previously in the semiclassical approach, and constitutes a very good model for a semiconductor superlattice with square well potentials. In this approach, the Wannier-Stark (WS) states 'Fnare determined by (Wannier, 1960)

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193

if nearest-neighbor coupling is taken into account and coupling to other bands (Zener tunneling) is neglected (Bleuse et al., 1988). Here @(z) is the wave function of an isolated well and .Ii is the Bessel function of order i. The argument of the Bessel function corresponds to a localization length L = A/2eF and enables us to distinguish three different field regimes, as depicted in Fig. 2. Note that L is exactly also the amplitude of the Bloch oscillations in the semiclassical picture [see Eq. ( 5 ) ] . For low fields ( F s 0), the Bessel functions are delocalized over the superlattice and give rise to the miniband states. For very high fields (eFd > A) the localization length is smaller than the superlattice period, and hence the system behaves only like a group of uncoupled single quantum wells. The interesting regime is the intermediate regime, where wave functions extend over several lattice periods. It is easily derived, from the translational invariance against simultaneous shift of energy by eFd and space by d, that the energy levels and consequently also the optical transitions in the miniband will split up into a series

-

valence band

FIG. 2. Scheme of the eigenstates in a superlattice in diverse field regimes: (a) small-field, (b) Wannier-Stark ladder (WSL), and (c) high-field regime.

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P. HARINGBOLIVARET AL.

of discrete levels, the WSL E

=E,

+ neFd

n

= 0,f 1,

f 2 , . ..

(7)

where eFd corresponds to the potential drop over one superlattice period due to the field, and E , is the energy of the transition in the isolated wells. A calculated typical field dependency of the linear absorption of the WSL in an one-dimensional A = 18 meV Al,Ga, -,As-GaAs superlattice is plotted in Fig. 3a as an example. Here E = 0 is set at the energy of the n = 0 transition. The thickness of the lines indicates the oscillator strength of the transition. In this intermediate field range, the wave functions of neighboring wells overlap, as indicated by the finite oscillator strength of the n # 0 transitions. Hence quantum mechanical superpositions (wave packets) of several WS wave functions can be constructed with femtosecond laser pulses exciting transitions from hole to several electron states. The wave packets will oscillate with the energy difference of the constituting WS wave functions. Hence, in analogy to the semiclassical description, a coherent superposition of neighboring WS states (e.g., Yo and Y - will perform Bloch oscillations in real space, as depicted in Fig. 4. Clearly also in this case of a WS description, the characteristic frequency dependence VBloch = eFd/h can directly be derived from Eq. (7). Other typical situations can easily be constructed by a superposition of WS states, such as, for example, a breathing mode by a superposition of the WS Yo and Y t: states (Sudzius et al., 1998).

$'.'""P

20 10

0

10 20 sFd [me4

30

0

5

10 15 eFd [mew

20

FIG.3 Numerical calculation of a fan chart of the WSL as a function of the electric field (a) single particle calculation and (b) calculation including the electron-hole Coulomb interaction (courtesy G. Bartels, 1998).

195

4 OPTICALLY EXCITEDBLOCHOSCILLATIONS

U

10 nm i =OfS

t = 300 fs

t = 600 fs FIG. 4. Numerically calculated electronic wave functions in a semiconductor superlattice with 97-A Al,,,Ga,,,As wells and 17-A GaAs barriers. The upper part of the figure illustrates the Yo and Y - WSL wave functions in the periodic potential of the superlattice at 6 kV/cm. The lower part of the figure depicts the dynamics of the electronic probability density as a function of time. The dark grey line depicts the location of the electronic center of mass as a function of time.

This ideal case for the WSL contains the basic ingredients for the description of Bloch oscillations in semiconductor superlattices. A first important modification is nevertheless introduced by the Coulomb interaction between the electronic and hole WS states (Dignam and Sipe, 1990, 1991). As an example, Fig. 4b contains numerical calculations for a onedimensional superlattice including this interaction. In comparison to the single-particle calculation that clearly exhibits the equally spaced WS translations described by Eq. (7), the inclusion of Coulomb effects modifies the ladder. At low fields, the modification can be quite substantial and a characteristic series of anticrossings between the WSL states become visible. At high fields, the influence of the Coulomb interaction is less and the splitting between the WS states becomes again nearly proportional to the applied field. The influence of the Coulomb interaction on Bloch oscillations is discussed later in Section V. Another difficulty arises from the three-

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P. HARINGBOLIVARET AL.

dimensional character of real superlattice samples, which leads to a continuum of states in the k , and k, directions (i.e., the momentum perpendicular to the growth direction). As a consequence, a large number of states must be taken into account to describe Bloch oscillations in semiconductor superlattices and to account for interference effects at higher excitation energies. One should be aware that the correct theoretical description of the coherent dynamics of Bloch oscillations after excitation by the electromagnetic field is even far more demanding than described here. The problem of this elementary formulation is that the dynamics of the system are calculated within a fixed set of wave functions calculated as stationary solutions of the system. Nevertheless, the timescale of the dynamics that are analyzed with ultrashort laser pulses reaches far into a time regime, where Heisenberg’s uncertainty relation yields energy uncertainties ( h = 6.5 meV. 100 fs) of the order of the typical energetic spacings in the experiments. Static solutions are therefore not valid solutions for describing the dynamics of the system on this ultrashort timescale. The calculation of the dynamical wave functions is therefore necessary, which is extraordinarily difficult for real semiconductors. In simple words, it is necessary to attain the time-dependent wave functions for the interacting N electron system that determine the macroscopic observables. As N varies with time, the whole basis of wave functions of the system must be recalculated for each time step. A less demanding formulation, in the second quantization (Heisenberg picture) is therefore more appropriate and the dynamics of the system are therefore frequency expressed in terms of electronic creation (2f) and annihilation (2) operators in a density matrix formalism. In semiconductors, it is reasonable to introduce also operators for the holes (it and 2). Another difficulty arises due to the Coulomb interaction of the N-particle system, which gives rise to an infinite hierarchy of different orders of correlation functions (including higher-order correlations) describing the interaction between all N carriers. This represents an insurmountable difficulty and approximations are necessary to deal with the resulting infinite set of coupled equations of motion. A very successful and widespread approach is based on a reduction of all higher order correlations via a Hartree-Fock decoupling scheme into products of two-point correlations (e.g., (2!j]jkt,)-+( a j c i ) * (jk2,))and solving the equations of motion at a two-point level. This ansatz was originally developed in a real-space representation (Huhn and Stahl, 1984; Stahl and Balslev, 1987), but is more widely employed in its k-space formulation (Schmitt-Rink and Chemla, 1986; Haug and Koch, 1993). The approach is known as the semiconductor bloch equations (SBE). The SBE have extensively and very successfully been applied to a multitude of problems, specifically also for diverse observations of the coherent dynamics of Bloch oscillations (for a review see, e.g., Rossi,

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1998; and Meier et al., 1998) and calculations now include even microscopic descriptions of scattering mechanisms, avoiding simplifications related to the introduction of phenomenological relaxation constants. Despite its wide acceptance and success, the SBE approach is nevertheless based on the Hartree-Fock decoupling (random phase approximation) of higher-order correlations, which is rigorously only valid for a Gaussian ensemble in equilibrium, and can thus be problematic for ultrafast dynamical processes which are normally performed far from equilibrium. An alternative approach was therefore derived, the dynamics controlled truncation (DCT) scheme (Axt and Stahl, 1994). This approach incorporates systematic truncation concepts to define a finite set of density matrices (including higher-order correlations) that must be taken into account as independent dynamic variables to describe experimental observables exactly within a prescribed order of the electromagnetic field. Under many experimental situations, the inclusion of higher correlations beyond the HartreeFock approximation yields only minor corrections, but shown in Section V, severe deviations are possible when the experimental observables are dominated by density-like quantities that are strongly influenced by Coulomb interaction. There are many more approaches for the description of ultrafast processes in semiconductors. A very popular formalism is based on the nonequilibrium Green functions (Miiller et al., 1987). This overview is intended only to emphasize that the theoretical description of coherent dynamics in semiconductors is still a subject of intense research and that ultrafast experiments, as the observation of Bloch oscillations dynamics described later, are important aids for the further development of appropriate models.

IV. Time-Resolved Investigation of Bloch Oscillations One of the ingredients that opened the path for the experimental observation of Bloch oscillations was the impressive advance in MBE growth techniques for semiconductor heterostructures. These well-developed techniques enabled the growth of high-quality Al,Ga, -,As-GaAs superlattices exhibiting sharp and homogeneous optical transitions. The initial experimental step relevant for the observation of Bloch oscillations was the demonstration of a WSL in cw experiments. The first observation was made in 1988 by photocurrent spectroscopy at low temperatures (Mendez et al., 1988; Voisin et al., 1988) and later even at room temperature (Agullo-Rueda et al., 1989; Mendez et al., 1990; Kawashima et al., 1991). Closely related to the experimental demonstration of Bloch oscillations was the observation of

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negative differential conductance in superlattices by Sibille (Sibille et al., 1990, 1992), which is closely related to the concept of a localization length decreasing with the electric field. Parallel to the progress in the MBE growth techniques, a key aspect that made the observation of Bloch oscillations feasible was the development of reliable femtosecond lasers sources. With the upcoming of Kerr-Lens mode-locked titan:sapphire lasers in 1991 (Spence et al., 1991), a flexible and stable source of tunable ultrafast laser pulses with a duration of few tens of femtoseconds was developed that enables us to perform sensitive optical correlation experiments with a temporal resolution given only by the duration of the laser pulses. Hence, ultrafast electronic processes can be analyzed directly in the time regime. Bloch oscillations could then be observed experimentally for the first time in Al,Ga,-,As-GaAs superlattices in 1992 (Feldmann et al., 1992; Leo et al., 1992) by transient four-wave mixing (FWM), which is an elegant experimental technique for detecting coherent interband dynamics (Yajima and Taira, 1979). A typical FWM setup is depicted in Fig. 5. These experiments followed a theoretical prediction that Bloch oscillations should be observable as a modulation of the third-order nonlinear optical polarization (von Plessen and Thomas, 1992). Data of these first FWM measurements (Haring Bolivar et al., 1993) are represented in Fig. 6, where the characteristic VBloch F dependency could clearly be demonstrated in an AI,,,Ga,,,AsGaAs superlattice with a combined miniband width of A = 18 meV. In this

-

’ I I

Z

’ I t

FIG.5. Scheme of a four-wave mixing experimental setup. The third-order nonlinear polarization P::!cr generated by both laser pulses in the material, leads to the diffraction of a signal which is proportional to the coherent interband dynamics in the material.

4 OPTICALLY EXCITED BLOCHOSCILLATIONS

199

? 3.

0

C

$

-1

0

1 2 time delay (ps)

3

FIG. 6 . Data of one of the first four wave mixing measurements of Bloch oscillations for different applied biases (T = 10 K) (Haring Bolivar et al., 1993). The inset shows the extracted Bloch oscillation frequencies as a function of the applied field.

experimental technique, the optical interband coherence of the system is probed by two laser pulses via the third-order polarization. The coherent superposition of optical transitions to different WSL states leads then to quantum interference effects (known as quantum beats from atomic spectroscopy) manifested as a modulation of the transients. The beats observed in the experiments reflect the coherent superposition of WSL states, which corresponds to wave packets performing Bloch oscillations. The first experiments showed already a tunability of these oscillations between 500 GHz and 2.5 THz. Later, measurements with a higher time resolution quickly demonstated a higher tuning range from 300 GHz to 5 THz (Leisching et al., 1994), where the upper frequency was limited only by the employed laser pulsewidth. Further FWM experiments were performed to study the dephasing behavior of Bloch oscillations in diverse structures with different miniband widths, to investigate excitation parameter, electric field and sample temperature dependencies. Even the appearance of higher harmonics or the influence of many-particle effects was analyzed by this experimental technique. For a further review of FWM observations, see Feldmann (1992b) and Leisching et al. (1994) and references therein. Although the coherent superposition of WSL states demonstrated by the FWM experiments corresponds directly to a Bloch oscillation in real space,

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as demonstrated mathematically in Bastard and Fereirra (1989), this first experimental observation of Bloch oscillations remained debated. The point of discussion was that as the FWM technique probes only interband contributions, no direct information regarding the real-space (intraband) dynamics of the coherent oscillation within the minibands is attainable. The discussion was nevertheless quickly resolved when THz emission spectroscopy measurements directly probing the coherent dynamics within the minibands (the intraband dynamics) proved in 1993 that Bloch oscillations take place in real space (Waschke et al., 1993). The advantage of the time-resolved THz emission spectroscopy technique (see experimental setup in Fig. 7) is that as the emitted electric field is directly proportional to the acceleration of charge carriers within the sample. By monitoring the time evolution of the emitted radiation one can thus detect the spatial motion of the charge carrier ensemble directly in the time domain. Making use of this advantage, a wide variety of THz experiments on Bloch oscillations have been performed. The next sections of this chapter focuses on a more detailed discussion of these results, but before moving on to the THz experiments, further experimental approaches are referred to in this section. Another experimental technique was later established for analyzing Bloch oscillations. This technique, depicted in Fig. 8, is based on a pump-probe scheme where the transient polarization changes of the probe pulse according to the electrooptic effect are detected as a function of time delay (Dekorsy et al., 1994). By employing this transmittive electrooptic sampling

FIG. 7. Optical setup for time-resolved terahertz (THz) emission spectroscopy, which enables us to monitor the domain of the emitted electric field in amplitude and phase directly in time. The emitted radiation is collected with parabolic mirrors and detected with a photoconductive THz antenna gated by the second time-delayed laser pulse.

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FIG. 8. Experimental setup for transmittive electrooptic sampling. The general idea is to monitor the change in the optical polarization of the probe pulse in a differential technique. The TEOS signal is then directly proportional to the internal electric field in the sample Pi:/ra, due to the linear electrooptic or Pockels effect. The signal is nevertheless also influenced by ) resonantly probed experiments (Lovenich et al., third order correction terms ( - P ~ ~ ! , ,for 1997).

(TEOS) technique with shorter laser pulses, Bloch oscillations could then be observed within a wider tuning range from 250GHz to 8THz in a A = 37 meV superlattice at 10 K. These high frequencies correspond to oscillation periods below typical scattering times at room temperature, and indeed TEOS experiments in this A = 37 meV sample enabled the first observation of Bloch oscillations at room temperature (Dekorsy et al., 1995). Figure 9 depicts the corresponding oscillatory traces and Fourier transforms of the data. The highest observed frequencies are in the vicinity of the optical phonon resonance in GaAs. The effects of resonance conditions of Bloch oscillations and optical phonons are an intriguing subject concerning the influence on the electronic dephasing time, which is under present investigation (Dekorsy et al., 1998). The dephasing time in the subpicosecond range a t room temperature stems from the strong population of optical phonons, which leads to an ultrafast randomization of the carrier momentum via LO phonon absorption. This process must be distinguished from the phonon emission process

P. HARINGBOLIVARET AL.

202

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2

<

Y

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1.5

-a

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$ I-

0,5

0,o 0,O

0,5 1,0 time delay (ps)

1,5

0

5

10

15

frequency (THz)

FIG.9. Transmittive electrooptic sampling transients of Bloch oscillations at 300 K. The left figure depicts the observed transients for different applied electric fields; the right, the corresponding Fourier transforms (transients and Fourier transforms are shifted vertically for clarity).

under excitation with optical excess energy, which has a markedly different influence on the coherence of the system (see Section V). Although TEOS is a powerful experimental technique concerning the temporal resolution, the interpretation of the observed amplitudes has to take into account that both interband and intraband polarizations contribute to the detected optical anisotropy (Lovenich et al., 1997). Thus real-space dynamics cannot easily be derived from the data. We show later how this important information, and especially the real-space amplitude of Bloch oscillations, can be attained directly from THz emission experiments. The majority of investigations on Bloch oscillations have concentrated on the Al,Ga, -,As-GaAs materials system, which can be grown by MBE with excellent quality. Bloch oscillations can now also be observed at low lattice temperatures in an In, -,Ga,As,P, -=-Inl -,Ga,As,P, -= superlattices grown by low-pressure metal-organic vapor phase epitaxy. The experiments were based on pump-probe spectroscopy employing an optical parametric oscillator delivering pulses of 100 fs duration around 1.55 pm (Cho et al., 1996). The dephasing times observed in this material composition were in the subpicosecond range at low lattice temperatures (10 K). In contrast the longest dephasing time observed in a AlGaAs-GaAs (A = 19 meV) have been about 3 ps (Dekorsy et al., 1994) reflecting the influence of the material quality. In addition the InGaAsP-InGaAsP composition gives a shallow

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confinement for electrons of only 80 meV compared to approx. 270 meV in the investigated AlGaAs-GaAs systems. Due to the faster scattering in the quarternary material and the fast sweep-out of carriers out of the shallow superlattice within the dephasing time, a transient chirp imposed on the Bloch oscillations could be observed (Cho et al., 1997). A detailed analysis of the chirp factor of Bloch oscillations allows us to observe the field screening in the superlattice on the subpicosecond time scale and thus to determine the carrier mobilities (Forst et al., 1999). These experiments hence establish a direct time resolved observation of the transition from coherent (Bloch oscillation dominated) to the incoherent (Drude-like) transport regime. Recently, the transition regime between coherent and incoherent transport has been studied in GaAs- AlCaAs superlattices by hot electron microscopy, too (Rauch et al., 1998). Beside the observation of Bloch oscillations in the solid state discussed here, recently Bloch oscillations of laser-cooled atoms in the periodic potential of a standing light field could be observed experimentally (Ben Dahan et al., 1996; Wilkinson et al., 1996). Here the constant force is produced by a time dependent frequency difference of two laser beams. This observation demonstrates that the phenomenon of Bloch oscillations can be generalized to any quantum mechanical particle in a periodic potential under the influence of a constant force. Other interesting experiments include the observation of the inverse effect of the emission of THz radiation from Bloch oscillations, which was recently observed using the intense THz radiation from a free electron laser (Unterrainer et a/., 1996). Further observations of the coherent intraband dynamics of Bloch oscillations will be presented in the next two sections. Section V will present fundamental experiments of Bloch oscillation dynamics in superlattices, which have aided the development of general theoretical concepts to describe coherent electronic dynamics in semiconductors. Section VI describes the application of Bloch Oscillations as tunable sources of electromagnetic radiation in the THz range and the measurement of the Bloch oscillation amplitude.

V. Bloch Oscillations as a Model System for Coherent Carrier Dynamics in Semiconductors

One important aspect in the analysis of Bloch oscillations in semiconductor superlattices is the deduction of fundamental properties of coherent carrier dynamics in semiconductors. The ongoing investigation and development of new concepts and theoretical approaches for the general descrip-

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tion of coherent charge carrier dynamics has been aided substantially by diverse observations of the behavior of Bloch oscillations. Especially questions regarding the microscopic analysis of dephasing processes of coherent intraband dynamics have benefited from the study of Bloch oscillations. One interesting new notion is the conservation of intraband coherence after scattering processes. In general, it is assumed that scattering of carriers by the emission of phonons (i.e., energy relaxation) destroys phase coherence. Doubts have arisen as a consequence of the observation that the phase coherence of wave packets excited with noticeable excess energies was destroyed more slowly than the corresponding LO-phonon emission times (Some and Nurmikko, 1994). The same observations were made for Bloch oscillations (Roskos, 1994) and led to the conclusion that intraband coherence can be preserved even after energy relaxation (Wolter et al., 1997). A more detailed analysis of the THz emission together with a theoretical modeling of the microscopic relaxation processes (Wolter et al., 1998) has led to the conclusion that for high enough excitation energies, the relaxation process is even an indispensable prerequisite for the development of a coherent carrier motion and consequently the emission of THz radiation. This conclusion is founded on several experimental observations. A first step was made in two-color pump-probe experiments (Wolter et a/., 1997), which were corroborated by an elaborate three-dimensional microscopic modeling of carrier relaxation (Meier et al., 1998). These experiments demonstrated that relaxation by LO-phonon emission within a superlattice miniband takes place within less than 200fs. This time is clearly much shorter than the THz dephasing rates, typically of the order of 1 ps. Further observations of the excitation energy dependence of THz emission within the continuum of the first miniband transition demonstrated surprising effects (Wolter et al., 1998) (see Fig. 10): (i) the dephasing time of the THz transients remains constant, even when the onset for LO-phonon emission is crossed (ii) the signal decreases for increasing carrier excess energies, but increases again above the LO-emission threshold; and (iii) a phase jump is observed at threshold. The constant dephasing time demonstrates that the intraband coherence survives the energy relaxation process. In the semiclassical picture, the coherence conservation is equivalent to retaining a localized distribution of carriers in k-space. More interestingly, for an excitation above the upper edge of the miniband (i.e., with an excess energy of A), one would not expect any THz signal at all without energy relaxation, for reasons similar to the absence of current in a completely filled band (see semiclassic predictions in Section 11). The observation of a THz emission at high excess energies hence not only demonstrates that the energy relaxation process leads to a redistribution of the initially excited carriers while preserving at least partially the intraband coherence, but is even mandatory

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time delay (ps) FIG. 10. Excitation energy dependence of the THz emission from an AI,Ga, -.As superlattice with an 18 meV miniband width.

for the development of a macroscopic coherent intraband polarization. The redistribution of the carriers in k-space is obvious also from the phase jump of the THz emission at the LO-emission onset. A microscopic theoretical analysis of the three-dimensional system (Wolter et al., 1998) enables us to distinguish two distinct components responsible for the relaxation-induced macroscopic intraband signal: one contribution arises from the redistributed carriers after energy relaxation, the other one is interestingly the remaining intraband hole formed by the leaving of carriers from the originally excited homogeneous distribution. This analysis of Bloch oscillations illustrates how the study of coherent carrier dynamics in semiconductors remains an innovative field of research, delivering important fundamental information of general aspects of ultrafast carrier dynamics. This investigation introduces, for instance, the new notion that a scattering process not only does not necessarily destroy the coherence of a system, but can even be a mechanism for creating a net coherent carrier motion. Another aspect of coherence conservation after scattering has given rise to considerable attention in the recent past. The idea is to employ luminescence upconversion experiments (see Fig. 11) to determine electronic dynamics at the initial stages of momentum relaxation in semiconductors (Wang et al., 1995): Before momentum is scattered, carrier recombination in

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FIG.11. Experimental setup for detecting the resonantly excited secondary emission of light. The emitted radiation is time resolved by generating a sum frequency signal (upconversion) with a second optical pulse from an optic parametric oscillator (OPO).

optically excited semiconductors emits photons only in specific diffraction directions (in reflection or transmission directions with respect to the optical excitation). By monitoring the luminescence with a very high time resolution in an “off-resonant’’ direction one can thus directly monitor the initial stages of scattering of the optically induced coherence in momentum space. This effect is nevertheless superposed to Rayleigh scattering of the optically excited polarization. The terminus luminescence is therefore not appropriate and the term resonant secondary emission (RSE) was thus proposed by Birkedal and Shah (1998). First experiments were performed in singlequantum-well structures (Wang et al., 1995; Haacke et al., 1997). The coherent signatures on the RSE signals were then interpreted either in terms of coherence preserving momentum scattering (Wang et nl., 1995) or of Rayleigh (Haacke et al., 1997) contributions. We performed analogous measurements on Bloch oscillations in A1,Ga, -,As-GaAs superlattices (Niisse et al., 1999). In this system, one can measure the influence of momentum scattering on real-space intraband oscillations. The measurements also enable us to monitor the influence of an applied field on the momentum relaxation of charge carriers for the first time. Typical transients of the resonant secondary emission of a superlattice as a function of the applied electric bias are plotted in Fig. 12. A detailed analysis of the influence of carrier density, the polarization dependencies, and the varying inhomogeneous contributions and the systematic analysis

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207

104 103

102 10'

100

8

k

-0.5 0.0

0.5

1.0

1.5 2.0 2.5 time delay (ps)

FIG 12 Resonant secondary emission from a A (Ndsse et al., 1999).

=

3.0 3.5

18 meV AI,Ga, -,As superlattice at 10 K

of the influence of the detection direction have enabled us to clear up previous experimental interpretations (Wang et al., 1995; Haacke et al., 1997) and to demonstrate that the experimental conditions (especially the detection direction) determine which of the two contributions dominates the RSE. Under appropriate experimental conditions, the observed signals can exclusively reflect the recombination of carriers having experienced momentum scattering. In contrast to previous interpretations, we nevertheless interpret these coherent contributions as arising from spatially oscillating electrons that have conserved their intraband coherence, and that the momentum scattering process has destroyed interband coherence only in this initial stage. The spatially varying overlap of the intraband coherent electrons with localized holes can give rise to a modulation of the signal, independently of whether the optical coherence (the phase relation to the holes) has dephased. In our point of view, coherent RSE signatures thus demonstrate only the conservation of intraband coherence of a spatially oscillating wave packet after momentum scattering. For our experimental situation, the momentum scattering is found to be dominated by carrier-carrier scattering. Interestingly, these scattering rates are

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found to be influenced severely by the applied electric field. The further refinement of this analytic method constitutes a valuable tool for microscopically analyzing momentum scattering of electrons and to aid the development of more elaborate models for the description of carrier scattering in semiconductors. A last aspect to be considered here is the influence of the Coulomb interaction on coherent intraband dynamics, which has evolved into a point of intense theoretical discussion. This is an example on how the detailed analysis of Bloch oscillations has enabled us to derive important general features of coherent charge carrier dynamics in semiconductors. In this context, THz experiments on Bloch oscillations in a narrow-miniband superlattice demonstrated the dominance of excitonic contributions to intraband dynamics even a f e r the dephasing of the interband polarization (Haring Bolivar et al., 1997). This observation contradicted theoretical predictions with the widespread SBE model (Meier et al., 1994) and agrees with alternative DCT calculations (Axt et al., 1996), which explain that the deviations of the SBE predictions are an inherent consequence of the Hartree-Fock approximation. The general conclusion from this discussion is that the widely used Hartree-Fock (random-phase) approximation leads to an uncontrolled description of Coulomb effects on intraband semiconductor dynamics already at a x ' ~ 'level. The prolem was shown to arise from the substitution of a fourth-order correlation, the density N i j k l ,which determines Coulomb contributions to intraband dynamics, by a product of two interband polarizations Yg ykl. In calculations beyond the Hartree-Fock approximation, where as Nijk,dephases synchronously with the intraband coherence (as can be derived rigorously from the contraction theorem, as shown by Victor et al., 1995), it is automatically guaranteed that intraband dynamics are always influenced by the Coulomb interaction. The HartreeFock substitution of N i j k l by an interband quantity then destroys this delicate balance, and Coulomb effects can now exist only as long as the optical (interband) polarization yr has not dephased. After this time, artificial Coulomb-free dynamics arise. This is the reason why, depending sensibly on the relation of inter- and intraband dephasing, either Coulombrenormalized or Coulomb-free situations for intraband dynamics are predicted by SBE (Binder et al., 1994). As the physical meaning of the dynamic variables describing the Coulomb interaction is changed by the HartreeFock approximation even before any damping is introduced, this intrinsic problem cannot be overcome by an elaborate microscopic modeling of dephasing or by full three-dimensional calculations. To compare theoretical predictions with experimental observation it is thus essential to compare exactly not only what frequencies are observed (Coulomb-free of Coulomb-renormalized), but more importantly how long

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these prevail. Meier et al. (1998) argued by presenting Coulomb-renormalized features (see their Fig. 8) that SBE calculations could explain the experimental observations (H aring Bolivar et al., 1997). Nevertheless, these calculations presented spectrally too broad Coulomb-renormalized features, which indicates that these contributions dephase too rapidly in comparison with experiment. Apart from that, the calculations were performed with erroneous parameters, which do not enable any distinction at all. As demonstrated in Fig. 13, comparing SBE and DCT predictions for the observed contributions for the experimental situation (Haring Bolivar et al., 1997) and for the longer excitation pulses specified in Meier et al. (1998). Under our experimental conditions, a distinction is possible and the Hartree-Fock artifacts are evident: the rapidly decaying Coulomb-renormalized contributions in SBE deliver a very broad peak at the excitonic frequencies, as well as the artifically emerging Coulomb-free contribution. Clearly, too-long pulses no longer enable a clear distinction as the excitation pulse duration is longer than the dephasing time of the SBE Coulomb contributions, these appear narrower; as it is even longer than the oscillation period of the Coulomb-free contributions these are even artifically spectrally cut off. The experimental demonstration of SBE artifacts thus maintains its validity (Haring Bolivar et a/., 1997). We emphasize that the SBE approach nevertheless continues to be a powerful tool for modeling coherent carrier

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dynamics in semiconductors, especially as calculations beyond the HartreeFock approximation can become extraordinarily demanding in terms of computing power. In the future, it is nevertheless important to consider that the underlying Hartree-Fock approximation can lead to an inappropriate description of Coulomb effects. One interesting way to minimize the effect of the Hartree-Fock approximation is described in (Axt et al., 1996) proposing a temporal rescaling of dynamic variables, the tirne-corrected coherent limes (TCCL). This last fundamental aspect of the analysis of coherent dynamics in semiconductor superlattices shows directly how the wide range of information attainable by the time-resolved study of Bloch oscillations can directly aid the refinement of general theoretical concepts for the description of coherent carrier dynamics. This is extremely important, as the further development of these concepts is necessary for us to be able to design, model, and optimize future ultrafast optoelectronic devices.

VI. Application of Bloch Oscillations as a Coherent Source of Tunable Terahertz Radiation One of the main prospective applications for Bloch oscillations is the development of a tunable sources of electromagnetic radiation in the THz frequency range. A wide variety of applications for THz spectroscopy and imaging have been demonstrated (for an overview see, e.g., Nuss and Orenstein, 1998, and references therein), but no efficient flexible sources exist to date. The basic idea behind most THz emitters is that any charge carrier acceleration will lead to the emission of an electromagnetic field ETHz, according to Faraday’s induction law,

a

ETHz - j ( t ) N

at

a

= -ni(t)

at

where j ( t ) is the current density associated with the motion of the charge carriers. Hence, the oscillatory motion of Bloch oscillations will emit an electromagnetic field. The attractive characteristic of a Bloch oscillation based THz emitter is that the emission frequency can easily be tuned electrically by adjusting the static electric field. For fundamental investigations, it is important to underline that due to the simple linear relation between field and carrier position in Eq. (S), by monitoring the timeresolved THz emission from an ensemble of carriers one can derive its motion directly in real space. Consequently, this technique allowed the first

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demonstration that Bloch oscillations are associated with an oscillation in real space (Waschke et al., 1993). The typical THz emission transients depicted in Fig. 14 exhibit the clear 1/F dependency of the Bloch oscillation period and demonstrate the widely tunable THz emission frequency. The observation of this effect thus constitutes an impressive verification of the original Esaki-Tsu proposal for the realization of a Bloch oscillator. Bloch oscillations now represent an attractive source of tunable electromagnetic radiation in the THz range-a tunability from several 100GHz to more than 10THz has been demonstrated, which is equivalent to an impressive wavelength range from 1mm to 30 pm. The application of Bloch oscillations as a THz source has nevertheless been hampered by the low conversion efficiency of the generation process, which is typically below 10- The main reason for the low efficiency is that scattering processes with impurities, phonons, inhomogeneities, or other carriers destroy the phase of the coherent oscillation of the ensemble before a considerable amount of the energy in the system can be emitted as a coherent THz radiation (Roskos, 1994). Many efforts have been made to increase the emitted power levels and an impressive possibility was predicted by Victor et al. (1994) and confirmed experimentally later by Martini et al., (1996). The idea is to increase the emission rate of the system by exploiting the superradiant characteristics of the coherent ensemble in order to be able

’.

-1

0

1

2

3

4

5

time delay (ps) FIG.14. Typical transients for the THz emission from Bloch oscillations in a semiconductor superlattice. at 10K for excitation within the first miniband.

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to extract a greater amount of energy from the ensemble before dephasing destroys the coherence of the system. Power-dependent measurements indeed presented evidence for the quadratic scaling of the emitted power levels, as expected for a superradiant emission, underlining the cooperative character of the THz emission from Bloch oscillations. Data of these measurements are presented in Fig. 15. The emitted power levels were increased by more than one order of magnitude. Nevertheless, at higher excitation densities, the efficiency enhancement saturated due to an increase in the dephasing rate reflecting the increase of carrier-carrier scattering (Wolter et at., 1999). The measurements of the superradiance of Bloch oscillations were performed with a calibrated THz setup to quantify the concrete emitted power levels. This absolute calibration had the additional advantage of enabling the measurement of the absolute microscopic Bloch oscillation amplitude for the first time. As the THz emission is directly proportional to the acceleration of the charge carriers, by integrating twice over the measured data one can directly monitor the real-space dynamics of the emitting carrier ensemble. The only requirements to attain the absolute spatial dynamics are the calibration of the emitted radiation, taking into account internal transmission losses, and the inclusion of the collective character of the emitted radiation. By this method, the absolute Bloch oscillation amplitude could be detected (Martini et al., 1996). As shown in Fig. 16, the resulting amplitudes are smaller than the simple semiclassical estimate but agree well with a more precise quantum mechanical calculation. Basically, the difference between the semiclassical limit and the real amplitudes arises from both

0 FIG. 15.

density.

20 40 60 input optical power (mW)

Quadratic dependency of the emitted THz power as a function of the excitation

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300 n

9

3

03

4

5

7

6

8

electric field (kV/cm) FIG. 16. Bloch oscillation amplitude as a function of the applied bias: measured values (circles), semiclassic estimate (straight line), and quantum mechanical calculation of the superposition of the WS, and WS- states (dashed line).

the attractive interaction of the electrons with the holes and the finite number of WSL states excited by the laser pulses. An alternative experimental approach later confirmed the THz results (Lyssenko et al., 1997). Here the approach was to derive the transient change of the internal field in the superlattice induced by the screening of the external field due to the oscillating electrons from the energetic shift of spectrally resolved FWM transients. The analysis of this frequency shift hence enables us to monitor the intraband polarization of the ensemble. One of our main interests in the analysis of the THz emission from Bloch oscillations is the development of new inversionless amplification mechanisms to achieve higher conversion efficiencies. New approaches include the enhancement of the THz emission by superposing a coherent THz background radiation EB, during the emission process. If properly synchronized, the emitted field of the Bloch oscillator E,, will superpose coherently and the emitted power will increase by a gain term P 2E,,EB0. In some way, this effect can be understood as a parametric amplification of THz emission. First studies of this amplification mechanism applied to surface field emitters have demonstrated a drastic increase of the conversion efficiency. The amplification is now subject of active research (Martini et al., 1998a) and is currently even exploited in resonator configurations, where the emission from the THz emitter itself is used as a THz background field to enhance the THz emission at a later time. The achievable efficiency increase is currently restricted only by the quality of a THz resonator. Despite the

-

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current bad quality of our initial resonator setup with losses of the order of 60% per round-trip, an efficiency increase by almost an order of magnitude has already been demonstrated (Martini et a]., 1998b, 1998~).The special attractivity of this amplification mechanism stems from the fact that it is based solely on the coherent properties of the emitting ensemble. It can thus be applied to any coherent source of THz radiation. Additionally, in contrast to conventional amplification by stimulated emission, no population inversion is requied. This is a great advantage in view of the difficulty in achieving inversion in the THz frequency range, as the typical THz transition energies are comparable to energy level linewidths and are smaller than the LO-phonon energy, which leads to ultrafast relaxation dynamics. This amplification mechanism is currently a promising field of research in view of the broad applicability of Bloch oscillations as a tunable source of electromagnetic radiation in the THz frequency range. Another access toward the application of semiconductor superlattices as sources of THz radiation is derived from amplification by stimulated emission. The Scamarcio et al. (1998) demonstration of the emission of far-infrared radiation (A = 6 pm) from inter-WSL states indicates the feasibility for the application of electrically driven superlattices as tunable sources of THz radiation. One day, Bloch oscillations might present the active medium for the tunable emission of THz radiation in structures similar to the quantum cascade laser (Capasso, 1987). Nevertheless, a great number of problems remain to be overcome. Presumably, resonant tunneling structures injecting energetically narrow distributions of electrons into superlattice structures will be important for us to be able to resolve some of the difficulties in generating a population inversion (Beltram et al., 1990 Rauch et a/., 1997).

VII. Summary Altogether, the study of Bloch oscillations continues to be an interesting and fascinating field of research. The growing interest in the potential applications of coherent dynamics is one of the driving forces behind this active field of semiconductor physics and has especially promoted the role of Bloch oscillations as a source of tunable electromagnetic radiation in the THz range. The ultrafast optical experiments performed on Bloch oscillations both fulfill the role of an analytic tool to determine and expand the potential and the limiting factors of coherent phenomena, but they are also an important approach to aid in the development of the still incomplete theoretical instrumentarium to describe coherent electronic dynamics in semiconductor systems.

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ACKNOWLEDGMENTS We would like to acknowledge many people who are contributing actively to this research work. We appreciate the collaboration of K. Leo, H. G. Roskos, P. Leisching, C. Waschke, F. Wolter, R. Martini, S. Nusse, and G. C. Cho and the constant theoretical support by A. Stahl, M. Axt, K. Victor, and G. Bartels. Excellent samples were supplied by K. Kohler, R. Hey, and H. T. Grahn. We appreciate the financial support by the Deutsche Forschungsgemeinschaft, by the Volkswagen Stiftung, by the European Community under the TMR project Interact, by the European Research Office of the U.S. Army, and by the Bundesministerium fur Bildung, Wissenschaft, Forschung und Technologie.

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SEMICONDUCTORS AND SEMLMETALS. VOL. 66

Index Note: Page numbers in italics refer to a figure on that page. definition, 128, 172 description, 187 dynamics and stationary solution, 196 dynamics controlled truncation (DCT). 197 electric experiments, 174- 176 interband and intraband contributions,

A ,y'*'.

see Nonlinear susceptibilities

xf3'.see Nonlinear susceptibilities; Secondorder nonlinear susceptibility

A

200

interference effects, 191 inverse effect of THz radiation emission, 203 and Kronig-Penney model of superlattices, 190 in laser-cooled atoms, 203 luminescence upconversion experiments, 205 particle phase, 190 real-space dynamics, 190- 191 SBE approach to dynamics, 196-197 semiclassical model, 188- 192 as source of tunable terahertz radiation, 210-214 time-resolved investigation, 197-203 tunability, 199 Wannier-Stark description, 192- 197 Bloch oscillation observation, 191- 192 femto laser sources, 198 and four-wave mixing (FWM), 198 at room temperature, 201 semiconductor material, 202 time-resolved THz emission spectroscopy, 200 transmittive electrooptic sampling (TEOS), 200-201 Bragg confinement, and vertical QC laser, 31 Bragg reflectors, 189

Absorption coefficient as function of applied electric field, 100 relationship to absorbance, 94 Absorption spectra, QC laser with diagonal transition, 14- 16 AC Josephson effect, 146-147 Acceleration theorem, 189 Anticrossing QC laser with diagonal transition, 18-20 between Wannier-Stark ladder states, 195 Applied electric field tunability, 86

B Band structure engineering and quantum cascade lasers, 5 and semiconductor structures, 85 Bloch frequency, 189 Bloch oscillation acceleration theorem, 189 in biased superlattice, 136- 137 chirp factor, 203 and coherent carrier dynamic properties, 201-210 Coulomb interaction of N-particle system, 196

219

220

INDEX

Breathing mode, 191, 194 Brillouin zone, 189

C

Carrier homogeneous excitation over Brillouin zone, 191 maximum velocity, 190 Carrier dynamics. see Coherent carrier dynamics Coherent carrier dynamics Bloch oscillation as model, 203-210 dynamics controlled truncation (DCT) model, 197 semiconductor bloch equations (SBE) model, 196 time-corrected coherent limes (TCCL), 210 Coherent transport, in minibands, 134-138 Conduction band diagram double quantum well, 129 QC Iaser with diagonal transition, 21 Coulomb effects artificial Coulomb-free dynamics, 208 on intraband semiconductor dynamics, 208 Coupled quantum wells high order nonlinear optical susceptibilities, 87-89 and nonlinear optical properties, 86 nonlinear optical properties, 90-93

device processing, 27 doping profile, 16-18 laser characteristics, 27-29 photocurrent, 14-16 waveguide, 24-27 Diagonal transition Q C laser vs vertical transition QC laser, 60 wavelength = 8.4 pm, 56 Difference frequency mixing (DFM). see Third harmonic generation (THG) optimization in coupled quantum wells, 90-93 Distributed feedback Q C lasers. see DFB QC lasers Dynamics controlled truncation (DCT), 197 comparison to semiconductor bloch equations, 209

E Electroluminescence, vertical vs diagonal QC laser, 33 Electronic states, quantum cascade lasers, 6

F Four-wave mixing (FWM), and Bloch oscillation observation, 198 Funnel injector, 44-52

G

D DCT. see Dynamics controlled truncation (DCT) DFB QC lasers, 68-69 applications, 77-80 index coupled, 73-77 loss-coupled, 69-72 DFM. see Difference frequency mixing Diagonal transition cascade laser, 15-29 absorption spectra, 14-16 anticrossing, 18-20 band structure, 20-24 cross section, 25

Gain, in intersubband transitions, 2 Green functions. 197

H Hartree-Fock method and Coulomb effects on intraband semiconductor dynamics, 208 minimizing approximation, 2 10 and SBE approach to Bloch oscillation dynamics, 197 Homogeneous broadening, 102

221

INDEX I Infrared detectors. see also QWIP Infrared imaging systems. see QWIP Intersubband absorption condensation from electron-electron interaction, 22 and Stark effect in coupled QW, 93-102 Intersubband absorption spectra. see Absorption spectra Intersubband transitions description, 2 and laser development, 4-5 Intraband coherence conservation, 204-205 influence of Coulomb interaction, 208 Intraband oscillations and momentum scattering, 206-207

and Bloch oscillation observation, 192, 197 and quantum cascade lasers, 5 and semiconductor structures, 85 Multi-quantum well (MQW), 3 Multiphoton electron emission, 116-121, 163- 172

N

Nonlinear optical properties. 86 Nonlinear susceptibilities, 86-87 coupled quantum wells, 90-93 far-infrared generation, 109-1 12 high-order derivation in coupled QW, 87-89 large, 86-87 Nonparabolicity. 7

K

0 Kronig-Penney model of superlattices, 190 Optican phonon scattering rate, 7-8 Oscillator strength, 20

L Lasers based on intersubband transitions, 4-5 Line broadening, 23 LO phonon absorption, 201 LO phonon emission, 204 Long wavelength QC laser, 52-53 based on vertical transition, 56-61 plasmon enhanced waveguide, 53-55 room temperature design, 61-65

M MBE. see Molecular beam epitaxy Minibands coherent transport, 134-138 relaxation by LO phonon emission, 204 transport in external AC fields, 140- 144 tunable widths and Bloch oscillation observation, 192 Wannier-Stark ladder, 138-139 Molecular beam epitaxy

P Photocurrent as function of laser polarization, 119 in multiphoton emission regime, 119 QC laser with diagonal transition, 14- 16 Photon-assisted tunneling description, 129 experimental methods, 147- 149 between ground and excited states, 155-162 between ground states, 163-172 nonperturbative limit, 130-134 perturbation limit, 130 in resonant tunneling diodes, 149-155 states, 170 and terahertz amplification, 182- 183 Tien-Gordon model, 130- 134 in weakly coupled superlattice, 155-172 Photon emission, and THz radiation source development, 142-143 Plasmon waveguides, 53-55

222

INDEX

and semiconductor lasers, 65-80 Population inversion evidence for, 152 and laser operation, 2 QC laser with diagonal transition, 23,29 and quantum cascade laser design, 7 Power efficiency, quantum cascade lasers, 3

Q Quantum beats, 199 Quantum cascade laser design philosophy, I0 DFB (distributed feedback) design, 68-69 diagonal transition design. see Diagonal transition cascade laser difference from diode lasers, 2 emitter structure, 15 energy band diagram, 10-1 1 geometry, 2-3 injector doping, 13 long wavelength design. see Long wavelength QC laser optical constants, 12- 13 plasmon enhanced waveguide, 53-55 population inversion and instability, 13 power efficiency, 3 room temperature performance. 5 schematic diagram of tunneling/ relaxation processes, 35-38 semiconductor material, 11 stability requirements, 13 theory, 6-9 vertical transition design. see Vertical transition cascade laser Quantum well infrared photodetectors. see QWIP Quantum well, multiphoton electron emission, 116-121

R Radiative efficiency, quantum cascade lasers, 8 Resonant secondary emission (RSE), 206-208

S SBE. see Semiconductor bloch equations Scattering processes in semiconductors, 204-205 Schrodinger equation, adaptation to bloch oscillation study, 188 Second-harmonic generation (SHG), 86 in coupled QW, 102-107 optimization in coupled quantum wells, 90-93 selection rules, 105-106 Stark effect, 99 Second-harmonic power derivation, 105 as function of pump power, 107, 108 Second-order nonlinear susceptibility electric field-induced enhancement, 106 as function of bias, 109 resonant Stark tuning, 102-107 Semiconductor bloch equations, 196 comparison to dynamics controlled truncation, 209 Semiconductor lasers, based on surface plasmon waveguides, 65-80 Semiconductors absolute negative conductance, 167-172 dynamic localization, 163-172 stimulated multiphoton emission, 163172 ultrafast process descriptions, 196-197 SHG. see Second-harmonic generation Spontaneous terahertz radiation, 128 and stimulated photon emission, 142- 143 Stark effect and confined state energy, 97 and intersubband absorption in coupled QW, 93-102 Step quantum wells, local to global state transitions, 98 Superlattices and bloch oscillation observation, 192 current and applied bias, 142-144 eigenstates in diverse field regimes, 193 in external AC fields, 144-147 numerically calculated electronic wave functions, 195 as oscillator, 138 photon-assisted peaks from AC field, 140- 142

223

INDEX

T

V

Terahertz amplification in resonant tunneling diodes, 468-183 Terahertz radiation Bloch oscillations as source, 210-214 spontaneous. see Spontaneous terahertz radiation Terahertz transport in minibands, 172- 182 THG. see Third harmonic generation

Vertical transition cascade laser, 29-39 vs diagonal transition QC laser, 60 high frequency response, 38-39 injection efficiency rate equations, 38 optimization, 40-44 quantum design, 30-35 spectral characteristics, 41-44 threshold current rate equations, 35-38 Vertical transition cascade laser with funnel injector, 44-52 continuous wave operation, 50-52 pulsed operation, 46-48 spectral properties, 52-53 threshold current density, 48-49

(THG) Third harmonic generation (THG), 86 dependence on pump wavelength, 114-115 optimization in coupled quantum wells, 90-93 Stark effect, 99 and triply resonant nonlinear susceptibility, 112-116 via continuum resonance, 121-122 Third harmonic power cryogenic enhancement, 115-1 16 equation, 114 as function of applied bias, 122 THz. see Spontaneous terahertz radiation Tien-Gordon model, 130- 134 comparison to experimental model, 152155 and photon-assisted tunneling in weakly coupled superlattices, 158 test of validity, 161-162 Time-resolved THz emission spectroscopy, 200 Transmittive electrooptic sampling (TEOS), 200-201 Tunnel current, 129 external modification, 132-134

W Wall pug efficiency, 3 Wannier-Stark ladder, 138- 139 ideal case, 195 and stimulated photon emission, 143 Wave functions numerically calculated for superlattices, 195 quantum cascade lasers, 6 Wave packets, description, 194 WSL. see Wannier-Stark ladder

2 Zener tunneling, 191, 193

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Contents of Volumes in This Series

Volume 1 Physics of 111-V Compounds C . Hilsum, Some Key Features of 111-V Compounds F . Bassani, Methods of Band Calculations Applicable to 111-V Compounds E . 0. Kane, The k-p Method !l L. Bonch-Brueuich, Effect of Heavy Doping on the Semiconductor Band Structure D. Long, Energy Band Structures of Mixed Crystals of 111-V Compounds L. M . Rofh and P. N. Argyres, Magnetic Quantum Effects S. M . Puri and T. H . Geballe, Thermomagnetic Effects in the Quantum Region W. M. Becker, Band Characteristics near Principal Minima from Magnetoresistance E. H. Putley, Freeze-Out Effects, Hot Electron Effects, and Submillimeter Photoconductivity in InSb H . Weiss, Magnetoresistance B. Ancker-Johnson, Plasma in Semiconductors and Semimetals

Volume 2 Physics of 111-V Compounds M. G. Holland, Thermal Conductivity S. I. Novkova, Thermal Expansion U. Piesbergen, Heat Capacity and Debye Temperatures G. Giesecke, Lattice Constants J. R Drabble, Elastic Properties A . U. Mac Rae and G. W Gobeli, Low Energy Electron Diffraction Studies R. Lee Mieher, Nuclear Magnetic Resonance B. Goldstein, Electron Paramagnetic Resonance T. S. Moss, Photoconduction in 111-V Compounds E. Antoncik and J. Tauc, Quantum Efficiency of the Internal Photoelectric Effect in InSb G. W. Gobeli and I. G. Allen, Photoelectric Threshold and Work Function P. S. Pershan, Nonlinear Optics in 111-V Compounds M . Gershenzon, Radiative Recombination in the 111-V Compounds F. Stern, Stimulated Emission in Semiconductors

225

226

CONTENTS OF VOLUMES IN THISSERIES

Volume 3 Optical of Properties 111-V Compounds M. Hass, Lattice Reflection W: G. Spitzer, Multiphonon Lattice Absorption D. L. Stierwalt and R. F. Potter, Emittance Studies H. R. Philipp and H. Ehrenveich, Ultraviolet Optical Properties M. Cardona, Optical Absorption above the Fundamental Edge E. J. Johnson, Absorption near the Fundamental Edge J. 0. Dimmock, Introduction to the Theory of Exciton States in Semiconductors B. Lax and J. G. Mavroides, Interband Magnetooptical Effects H. Y. Fan, Effects of Free Carries on Optical Properties E. D.Palik and G. B. Wright, Free-Carrier Magnetooptical Effects R H. Bube, Photoelectronic Analysis B. 0. Seraphin and H. E. Bennett, Optical Constants

Volume 4 Physics of 111-V Compounds N. A. Goryunova, A. S. Borschevskii, and D N. Tretiakov, Hardness A? N. Sirota, Heats of Formation and Temperatures and Heats of Fusion of Compounds A*"Bv D. L. Kendall, Diffusion A. G. Chynoweth, Charge Multiplication Phenomena R W. Keyes, The Effects of Hydrostatic Pressure on the Properties of 111-V Semiconductors L. W Aukerman, Radiation Effects N. A. Goryunova, F. P. Kesamanly, and D. N. Nasledov, Phenomena in Solid Solutions R. T. Bate, Electrical Properties of Nonuniform Crystals

Volume 5 Infrared Detectors H. Levinstein, Characterization of Infrared Detectors P. W Kruse, Indium Antimonide Photoconductive and Photoelectromagnetic Detectors M. B. Prince, Narrowband Self-Filtering Detectors I. Melngalis and T. C. Harman, Single-Crystal Lead-Tin Chalcogenides D. Long and J. L. Schmidt, Mercury-Cadmium Telluride and Closely Related Alloys E. H. Putley, The Pyroelectric Detector N. B. Stevens, Radiation Thermopiles R. J. Keyes and T. M. Quist, Low Level Coherent and Incoherent Detection in the Infrared M . C. Teich, Coherent Detection in the Infrared F. R. Arums, E. W. S a d , B. J. Peyton, and F. P. Pace, Infrared Heterodyne Detection with Gigahertz IF Response H. S. Sommers3Jr., Macrowave-Based Photoconductive Detector R Sehr and R. Zuleeg, Imaging and Display

Volume 6 Injection Phenomena M. A. Lamperr and R. B. Schilling, Current Injection in Solids: The Regional Approximation Method R Williams, Injection by Internal Photoemission A. M. Barnett, Current Filament Formation

CONTENTS OF VOLUMES IN THISSERIES

227

R. Baron and J. K Mayer, Double Injection in Semiconductors W. Ruppel, The Photoconductor-Metal Contact

Volume 7 Application and Devices Part A J. A. Copeland and S. Knight, Applications Utilizing Bulk Negative Resistance F. A . Padovuni, The Voltage-Current Characteristics of Metal-Semiconductor Contacts P. L. Hower, W. W. Hooper, B. R. Cairns. R. D. Fairman, and D. A . Tremere, The GaAs Field-Effect Transistor M. H. White, MOS Transistors G. R. Antell, Gallium Arsenide Transistors T. L. Tanslqy, Heterojunction Properties

Part B T. Misawa, IMPATT Diodes H. C. Okean, Tunnel Diodes R. B. Campbell and Hung-Chi Chang, Silicon Junction Carbide Devices R E. Enstrom, H. Kressel, and L. Krassner, High-Temperature Power Rectifiers of GaAs, -*PX

Volume 8 Transport and Optical Phenomena R. J. Stirn, Band Structure and Galvanomagnetic Effects in 111-V Compounds with Indirect Band Gaps R. W. Ure, Jr., Thermoelectric Effects in 111-V Compounds H. Piller, Faraday Rotation H. Barry Bebb and E. W. WiNiam.r, Photoluminescence I: Theory E. W. Williams and H. Barry Bebh, Photoluminescence 11: Gallium Arsenide

Volume 9 Modulation Techniques B. 0. Seraphin, Electroreflectance R. L. Aggarwal, Modulated Interband Magnetooptics D. F. Blossey and Paul Handler, Electroabsorption B. Batz, Thermal and Wavelength Modulation Spectroscopy I. Balslev, Piezopptical Effects D. E. Aspnes and N. Bottka, Electric-Field Effects on the Dielectric Function of Semiconductors and Insulators

Volume 10 Transport Phenomena R. L. Rho& Low-Field Electron Transport J. D. Wiley, Mobility of Holes in 111-V Compounds C. M. Woye and G. E. Stillman, Apparent Mobility Enhancement in Inhomogeneous Crystals R. L. Petersen, The Magnetophonon Effect

228

CONTENTS OF VOLUMES IN THISSERIES

Volume 11 Solar Cells H. J. Hovel, Introduction; Carrier Collection, Spectral Response, and Photocurrent; Solar Cell Electrical Characteristics; Efficiency; Thickness; Other Solar Cell Devices; Radiation Effects; Temperature and Intensity; Solar Cell Technology

Volume 12 Infrared Detectors (11) W. L. Eiseman, J. D. Merriam, and R. F. Potter, Operational Characteristics of Infrared Photodetectors P. R. Bratt, Impurity Germanium and Silicon Infrared Detectors E. H. Putley, InSb Submillimeter Photoconductive Detectors G. E. Stillman. C. M. Woljie, and J. 0. Dimmock, Far-Infrared Photoconductivity in High Purity GaAs G. E. Stillman and C. M. Wove, Avalanche Photodiodes P. L. Richards, The Josephson Junction as a Detector of Microwave and Far-Infrared Radiation E. H. Pulley, The Pyroelectric Detector- An Update

Volume 13 Cadmium Telluride X Zanio, Materials Preparations; Physics; Defects; Applications

Volume 14 Lasers, Junctions, Transport N. Holonyak, Jr. and M. H. Lee, Photopumped 111-V Semiconductor Lasers H. Kressel and J. K. Butler, Heterojunction Laser Diodes A Van der Ziel, Space-Charge-Limited Solid-state Diodes P. J. Price, Monte Carlo Calculation of Electron Transport in Solids

Volume 15 Contacts, Junctions, Emitters B. L. Sharma, Ohmic Contacts to IIILV Compounds Semiconductors A. Nussbaum, The Theory of Semiconducting Junctions J. S. Escher, NEA Semiconductor Photoemitters

Volume 16 Defects, (HgCd)Se, (HgCd)Te H. Kressel, The Effect of Crystal Defects on Optoelectronic Devices C. R. Whitsett, J. G. Broerman, and C. J. Summers, Crystal Growth and Properties of Hg, _xCd,Se alloys M. H. Weiler, Magnetooptical Properties of Hg, _xCd,Te Alloys P. W. Kruse and J. G. Ready, Nonlinear Optical Effects in Hg, _xCd,Te

Volume 17 CW Processing of Silicon and Other Semiconductors J. F. Gibbons, Beam Processing of Silicon A. Lietoila, R. B. Gold, J. F. Gibbons, and L. A. Christel, Temperature Distributions and Solid Phase Reaction Rates Produced by Scanning CW Beams

CONTENTS OF VOLUMESIN THISSERIES

229

A. Leitoila and J. F Gibbons, Applications of CW Beam Processing to Ion Implanted Crystalline Silicon N. M. Johnson, Electronic Defects in CW Transient Thermal Processed Silicon K. F Lee, T. J. Stultz, and J. F Gibbons, Beam Recrystallized Polycrystalline Silicon: Properties, Applications, and Techniques T. Shibata, A. Wakita, T. W. Sigmon, and J. F. Gibbons, Metal-Silicon Reactions and Silicide Y. I. Nissim and J. F Gibbons, CW Beam Processing of Gallium Arsenide

Volume 18 Mercury Cadmium Telluride P. W Kruse, The Emergence of (Hgl _,Cd,)Te as a Modern Infrared Sensitive Material H. E. Hirsch, S. C. Liang, and A. G. White, Preparation of High-Purity Cadmium, Mercury, and Tellurium W. F. H. Micklerhuwite, The Crystal Growth of Cadmium Mercury Telluride P. E. Petersen, Auger Recombination in Mercury Cadmium Telluride R. M. Broudy und V. J. Mazurczyck, (HgCd)Te Photoconductive Detectors M. B. Reine, A . K. Soad, and T. J. Tredwell, Photovoltaic Infrared Detectors M. A . Kinch, Metal-Insulator-Semiconductor Infrared Detectors

Volume 19 Deep Levels, GaAs, Alloys, Photochemistry G. F. Neurnark and K. Kosai, Deep Levels in Wide Band-Gap IIILV Semiconductors D. C. Look, The Electrical and Photoelectronic Properties of Setn-Insulating GaAs R. F Brebrick, Ching-Hua Su, and Pok-Kai Liao, Associated Solution Model for Ga-In-Sb and Hg-Cd-Te Y Ya. Gurevich and Y. V. Pleskon. Photoelectrochemistry of Semiconductors

Volume 20 Semi-Insulating GaAs R. N. Thomas, H. M. Hobgood, G. W. Eldridge. D. L. Barrett, T. T. Braggins. L. B. Ta, and S. K. Wang, High-Purity LEC Growth and Direct Implantation of GaAs for Monolithic Microwave Circuits C. A . Stolte, Ion Implantation and Materials for GaAs Integrated Circuits C. G. Kirkpatrick, R. T. Chen, D. E. Hobnes, P. M. Asbeck, K. R. Ellioti, R. D. Fairman, and J. R. Oliver, LEC GaAs for Integrated Circuit Applications J. S . Blakemore and S. Rahimi, Models for Mid-Gap Centers in Gallium Arsenide

Volume 21 Hydrogenated Amorphous Silicon Part A J. I. Pankove. Introduction M. Hirose, Glow Discharge; Chemical Vapor Deposition Y Uchida, di Glow Discharge T. D. Moustakas, Sputtering I. Yarnada, Ionized-Cluster Beam Deposition B. A. Scott, Homogeneous Chemical Vapor Deposition

230

CONTENTS OF VOLUMES IN THISSERIES

F. J. Kumpus, Chemical Reactions in Plasma Deposition P. A . Longewuy, Plasma Kinetics H. A. Weukliem, Diagnostics of Mane Glow Discharges Using Probes and Mass Spectroscopy L. Gluttmun, Relation between the Atomic and the Electronic Structures A. Chenevus-Puule, Experiment Determination of Structure S. Minomuru, Pressure Effects on the Local Atomic Structure D.Adler, Defects and Density of Localized States

Part B J. I. Punkove, Introduction G. D. Cody, The Optical Absorption Edge of a-Si:H N. M . Amer und W. B. Jackson, Optical Properties of Defect States in a-Si: H P. J. Zunzucchi, The Vibrational Spectra of a-Si: H Y. Humukuwu, Electroreflectance and Electroabsorption J. S. Lunnin, Raman Scattering of Amorphous Si, Ge, and Their Alloys R. A. Street, Luminescence in a-Si: H R. S. Crundull, Photoconductivity J. Tuuc, Time-Resolved Spectroscopy of Electronic Relaxation Processes P. E. Vunier, IR-Induced Quenching and Enhancement of Photoconductivity and Photo luminescence H. Schude, Irradiation-Induced Metastable Effects L. Ley, Photoelectron Emission Studies

Part C J. I. Punkove, Introduction J. D. Cohen, Density of States from Junction Measurements in Hydrogenated Amorphous Silicon P. C. Taylor, Magnetic Resonance Measurements in a&: H K. Moriguki, Optically Detected Magnetic Resonance J. Dresner, Carrier Mobility in a-SL H T. Tiedje, Information about band-Tail States from Time-of-Flight Experiments A. R. Moore, Diffusion Length in Undoped a-Si: H W. Beyer and J. OverhoA Doping Effects in a-Si: H H. Fritzche, Electronic Properties of Surfaces in a-Si:H C. R. Wronski, The Staebler-Wronski Effect R. J. Nemunich, Schottky Barriers on a-Si: H B. Abeles und T. Tiedje, Amorphous Semiconductor Superlattices

Part D J. I. Punkove, Introduction D. E. Curlson, Solar Cells G. A. Swurtz, Closed-Form Solution of 1-V Characteristic for a a-Si:H Solar Cells I. Shimizu, Electrophotography S. Zshioku, Image Pickup Tubes

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P. G. LeComber and W. E. Spear, The Development of the a-Si: H Field-Effect Transistor and Its Possible Applications D. G. Ast, a-Si: H FET-Addressed LCD Panel S. Kaneko, Solid-state Image Sensor M. Matsumura, Charge-Coupled Devices M. A. Bosch, Optical Recording A. D’Amico and G. Fortunato, Ambient Sensors H. Kukimoto, Amorphous Light-Emitting Devices R. J. Phelan, Jr., Fast Detectors and Modulators J. I. Pankove, Hybrid Structures P. G. LeComber, A. E. Owen, W. E. Spear, J. Hajto, and W K. Choi, Electronic Switching in Amorphous Silicon Junction Devices

Volume 22 Lightwave Communications Technology Part A K. Nakajimn, The Liquid-Phase Epitaxial Growth of InGaAsP W. T. Tsang, Molecular Beam Epitaxy for 111-V Compound Semiconductors G. B. Stringfellow, Organometallic Vapor-Phase Epitaxial Growth of 111-V Semiconductors G. Beuchet, Halide and Chloride Transport Vapor-Phase Deposition of InGaAsP and GaAs M . Razeghi, Low-Pressure Metallo-Organic Chemical Vapor Deposition of GaJn, -xAsP, - y Alloys P. M. Petroff, Defects in IIILV Compound Semiconductors

Part B J. P. van der Ziel, Mode Locking of Semiconductor Lasers K Y. Lau and A. Yariv, High-Frequency Current Modulation of Semiconductor Injection Lasers C. H. Henry, Special Properties of Semiconductor Lasers Y. Suematsu, K. Kishino, S. Arai, and F. Koyama, Dynamic Single-Mode Semiconductor Lasers with a Distributed Reflector W. T. Tsang, The Cleaved-Coupled-Cavity (C3) Laser

Part C R. J. Nelson and N. X Dutta, Review of InGaAsP InP Laser Structures and Comparison of

Their Performance N. Chinone and M. Nakamura, Mode-Stabilized Semiconductor Lasers for 0.7-0.8- and 1.1- 1.6-pm Regions Y. Horikoshi, Semiconductor Lasers with Wavelengths Exceeding 2 pm B. A. Dean and M . Dixon, The Functional Reliability of Semiconductor Lasers as Optical Transmitters R. H. Saul, T. P. Lee, and C. A. Burus, Light-Emitting Device Design C. L. Zipfel, Light-Emitting Diode-Reliability T. P. Lee and T. Li, LED-Based Multimode Lightwave Systems K. Ognwa, Semiconductor Noise-Mode Partition Noise

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Part D F. Capasso, The Physics of Avalanche Photodiodes T. P. Pearsall and M. A. Pollack, Compound Semiconductor Photodiodes T. Kaneda, Silicon and Germanium Avalanche Photodiodes S. R. Forrest, Sensitivity of Avalanche Photodetector Receivers for High-Bit-Rate LongWavelength Optical Communication Systems J. C. Campbell, Phototransistors for Lightwave Communications

Part E S. Wang, Principles and Characteristics of Integrable Active and Passive Optical Devices S. Margalit and A . Yariv, Integrated Electronic and Photonic Devices T. Mukai, Y. Yamamoto, and T. Kimura, Optical Amplification by Semiconductor Lasers

Volume 23 Pulsed Laser Processing of Semiconductors R. F. Wood, C. W. White, and R. T. Yotmg, Laser Processing of Semiconductors: An Overview C. W White, Segregation, Solute Trapping, and Supersaturated Alloys G. E. Jellison, Jr., Optical and Electrical Properties of Pulsed Laser-Annealed Silicon R. F. Wood and G.E. Jellison, Jr., Melting Model of Pulsed Laser Processing R. F. Wood and F. W. Young, Jr., Nonequilibrium Solidification Following Pulsed Laser Melting D.H. Lowndes und G. E. Jellison. Jr., Time-Resolved Measurement During Pulsed Laser Irradiation of Silicon D.M. Zebner, Surface Studies of Pulsed Laser Irradiated Semiconductors D. H. Lowndes, Pulsed Beam Processing of Gallium Arsenide R. B. James, Pulsed CO, Laser Annealing of Semiconductors R. T. Youag and R. F. Wood, Applications of Pulsed Laser Processing

Volume 24 Applications of Multiquantum Wells, Selective Doping, and Superlattices C. Weisbuch, Fundamental Properties of 111-V Semiconductor Two-Dimensional Quantized Structures: The Basis for Optical and Electronic Device Applications H. Morkoc and H. Unlu, Factors Affecting the Performance of (Al,Ga)As/GaAs and (Al, Ga)As/InGaAs Modulation-Doped Field-Effect Transistors: Microwave and Digital Applications N. 7: Linh, Two-Dimensional Electron Gas FETs: Microwave Applications M. Abe ef a/., Ultra-High-speed HEMT Integrated Circuits D. S. Chemla, D. A. B. Miller, und P. W. Smith, Nonlinear Optical Properties of Multiple Quantum Well Structures for Optical Signal Processing F. Cupasso, Graded-Gap and Superlattice Devices by Band-Gap Engineering W: T. Tsang, Quantum Confinement Heterostructure Semiconductor Lasers G. C. Osbourn el al., Principles and Applications of Semiconductor Strained-Layer Superlattices

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Volume 25 Diluted Magnetic Semiconductors W. Giriat and J. R Furdyna, Crystal Structure, Composition, and Materials Preparation of Diluted Magnetic Semiconductors W. M. Becker, Band Structure and Optical Properties of Wide-Gap Ay-xMn,B,, Alloys at Zero Magnetic Field S. Osevof and P. H. Keesom, Magnetic Properties: Macroscopic Studies T. Giebultowicz and T. M. Holden, Neutron Scattering Studies of the Magnetic Structure and Dynamics of Diluted Magnetic Semiconductors J. Kossut, Band Structure and Quantum Transport Phenomena in Narrow-Gap Diluted Magnetic Semiconductors C. Riquaux, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. A. Gaj, Magnetooptical Properties of Large-Gap Diluted Magnetic Semiconductors J. Mycielski, Shallow Acceptors in Diluted Magnetic Semiconductors: Splitting, Boil-off, Giant Negative Magnetoresistance A . K. Ramadas and R. Rodriquez, Raman Scattering in Diluted Magnetic Semiconductors P. A. Wo@ Theory of Bomd Magnetic Polarons in Semimagnetic Semiconductors

Volume 26

111-V Compound Semiconductorsand Semiconductor Properties of Superionic Materials

Z. Yuanxi, 111-V Compounds H. V. Winston, A. T Hunter, H. Kimura, and R. E. Lee, InAs-Alloyed GaAs Substrates for Direct Implantation P.K Bhattacharya and S. Dhar, Deep Levels in IIILV Compound Semiconductors Grown by MBE Y. Ya. Gurevich and A. K . Ivanov-Shits, Semiconductor Properties of Supersonic Materials

Volume 27 High Conducting Quasi-One-DimensionalOrganic Crystals E. M. Conwell, Introduction to Highly Conducting Quasi-One-Dimensional Organic Crystals I. A. Howard, A Reference Guide to the Conducting Quasi-One-Dimensional Organic Molecular Crystals J. P. Pouquet, Structural Instabilities E. M. Conwell, Transport Properties C. S. Jaacobsen, Optical Properties J. C. Scott, Magnetic Properties L. Zuppiroli, Irradiation Effects: Perfect Crystals and Real Crystals

Volume 28 Measurement of High-speed Signals in Solid State Devices J. Frey and D. Ioannou, Materials and Devices for High-speed and Optoelectronic Applications H. Schumacher and E. Strid, Electronic Wafer Probing Techniques D. H. Auston, Picosecond Photoconductivity: High-speed Measurements of Devices and Materials J. A . Valdmanis, Electro-Optic Measurement Techniques for Picosecond Materials, Devices, and Integrated Circuits. J. M. Wiesenfeidand R. K, Jain, Direct Optical Probing of Integrated Circuits and High-speed Devices G. Plows, Electron-Beam Probing A. M. Weiner and R B. Marcus, Photoemissive Probing

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Volume 29 Very High Speed Integrated Circuits: Gallium Arsenide LSI M. Kuzuhara and T. Nazaki, Active Layer Formation by Ion Implantation

H. Hasimoto, Focused Ion Beam Implantation Technology T. Nozaki and A. Higashisaka, Device Fabrication Process Technology M. Zno and T. Takada, GaAs LSI Circuit Design M. Hirayama, M. Ohmori, and K. Yamasaki, GaAs LSI Fabrication and Performance

Volume 30 Very High Speed Integrated Circuits: Heterostructure H. Watanabe, T. Mizutani, and A. Usui, Fundamentals of Epitaxial Growth and Atomic Layer Epitaxy S. Hiyamizu, Characteristics of Two-Dimensional Electron Gas in 111-V Compound Heterostructures Grown by MBE T. Nakanisi, Metalorganic Vapor Phase Epitaxy for High-Quality Active Layers T. Nimura, High Electron Mobility Transistor and LSI Applications T. Sugeta and T. Ishibashi, Hetero-Bipolar Transistor and LSI Application H. Matsueda, T. Tanaka, and M. Nakamura, Optoelectronic Integrated Circuits

Volume 3 1 Indium Phosphide: Crystal Growth and Characterization J. P. Farges, Growth of Discoloration-free InP M. J. McCollum and G. E. Stillman, High Purity InP Grown by Hydride Vapor Phase Epitaxy T. Inada and T. Fukuda, Direct Synthesis and Growth of Indium Phosphide by the Liquid Phosphorous Encapsulated Czochralski Method 0. Oda, K. Katagiri, K. Shinohara, S. Katsura, Y. Takahashi, K Kainosho, K. Kohiro, and R. Hirano, InP Crystal Growth, Substrate Preparation and Evaluation K. Tada, M. Tatsumi, M. Morioka, T. Araki, and T. Kawase, InP Substrates: Production and Quality Control M. Razeghi, LP-MOCVD Growth, Characterization, and Application of InP Material T. A. Kennedy and P. J. Lin-Chung, Stoichiometric Defects in InP

Volme 32 Strained-Layer Superlattices Physics T. P. Pearsall, Strained-Layer Superlattices

F. H. Pollack, Effects of Homogeneous Strain on the Electronic and Vibrational Levels in Semiconductors J. Y. Marzin, J. M. Gerard, P. Voisin, and J. A. Brum, Optical Studies o f Strained IIlLV Heterolayers R. People and S. A. Jackson, Structurally Induced States from Strain and Confinement M. Jaros, Microscopic Phenomena in Ordered Superlattices

Volume 33 Strained-Layer Superlattices: Materials Science and Technology R. Hull and J. C. Bean, Principles and Concepts of Strained-Layer Epitaxy W. J. SchaJ P. J. Tasker, M. C. Foisy, and L. F. Eastman, Device Applications of Strained-Layer Epitaxy

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S. T. Picraux, B. L. Doyle, and J Y . Tsao, Structure and Characterization of Strained-Layer

Superlattices E. Kusper and F. Schafer, Group IV Compounds D. L. Marfin, Molecular Beam Epitaxy of IV-VI Compounds Heterojunction R. L. Gunshor, L. A. Kolodziejski, A . V. Nurmikko. and N. Otsuka, Molecular Beam Epitaxy of 11-VI Semiconductor Microstructures

Volume 34 Hydrogen in Semiconductors J. I. Pankove and N. M. Johnson, Introduction to Hydrogen in Semiconductors C. H. Seager, Hydrogenation Methods J. I. Pankove, Hydrogenation of Defects in Crystalline Silicon J. W. Corbert, P. Deak, U . !t Desnica, and S . J . Pearton, Hydrogen Passivation of Damage Centers in Semiconductors S. J. Pearton, Neutralization of Deep Levels in Silicon J. I. Pankove, Neutralization of Shallow Acceptors in Silicon N. M. Johnson, Neutralization of Donor Dopants and Formation of Hydrogen-Induced Defects in n-Type Silicon M. Stavola and S.J. Pearton, Vibrational Spectroscopy of Hydrogen-Related Defects in Silicon A . D. Marwick, Hydrogen in Semiconductors: Ion Beam Techniques C. Herring and N . M . Johnson, Hydrogen Migration and Solubility in Silicon E. E. Huller, Hydrogen-Related Phenomena in Crystalline Germanium J. Kakalios, Hydrogen Diffusion in Amorphous Silicon J. Chevalier, B. Clerjaud, and B. Prijot, Neutralization o f Defects and Dopants in 111-V Semiconductors G. G. DeLeo and W. B. Fowler, Computational Studies of Hydrogen-Containing Complexes in Semiconductors R. F. Kiefl atid T. L. Esrk, Muonium in Semiconductors C. G. Van de Walk, Theory of Isolated Interstitial Hydrogen and Muonium in Crystalline Semiconductors

Volume 35 Nanostructured Systems M. Reed, Introduction H. van Houten, C. W. J. Beenakker, and B. J. van Wees, Quantum Point Contacts G. Timp, When Does a Wire Become an Electron Waveguide? M. Biittiker, The Quantum Hall Effects in Open Conductors W. Hansen, J. P. Korthaus, and U Merkr, Electrons in Laterally Periodic Nanostructures

Volume 36 The Spectroscopy of Semiconductors D. Heiman, Spectroscopy of Semiconductors at Low Temperatures and High Magnetic Fields A . V. Nurmikko, Transient Spectroscopy by Ultrashort Laser Pulse Techniques A. K Ramdas and S. Rodriguez, Piezospectroscopy of Semiconductors 0. J. Glembocki and B. V. Shanabrook, Photoreflectance Spectroscopy of Microstructures D. G. Seiler, C. L. Littler, and M . H. Wiler, One- and Two-Photon Magneto-Optical Spectroscopy of InSb and Hg, _xCd,Te

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CONTENTS OF VOLUMESIN THISSERIES

Volume 37 The Mechanical Properties of Semiconductors A,-B. Chen, A . Sher and W. T. Yost, Elastic Constants and Related Properties of Semiconductor Compounds and Their Alloys D. R. Clarke, Fracture of Silicon and Other Semiconductors H. Sietho$, The Plasticity of Elemental and Compound Semiconductors S. Guruswamy, K. T. Faber and J. P. Hirth, Mechanical Behavior of Compound Semiconductors S. Mahajun, Deformation Behavior of Compound Semiconductors J. P. Hirth, Injection of Dislocations into Strained Multilayer Structures D. Kendall, C. B. Fleddermann, and K. J. Malloy, Critical Technologies for the Micromachining of Silicon I. Matsuba and K. Mokuyu, Processing and Semiconductor Thermoekdstic Behavior

Volume 38 Imperfections in IIUV Materials U. Scherz and M . Schefler, Density-Functional Theory of sp-Bonded Defects in III/V Semiconductors M. Kaminska und E. R. Weber, El2 Defect in GaAs D.C . Look, Defects Relevant for Compensation in Semi-Insulating GaAs R. C. Newman, Local Vibrational Mode Spectroscopy of Defects in III/V Compounds A. M. Hennel, Transition Metals in IlIjV Compounds K. J. Malloy and K. Khachaturyan, DX and Related Defects in Semiconductors V. Swaminuthan and A. S. Jordan, Dislocations in III/V Compounds K. W. Nauka, Deep Level Defects in the Epitaxial III/V Materials

Volume 39 Minority Carriers in 111-V Semiconductors: Physics and Applications N. K. Dutta, Radiative Transitions in GaAs and Other 111-V Compounds R. K. Ahrenkiel, Minority-Carrier Lifetime in 111- V Semiconductors T. Furuta, High Field Minority Electron Transport in p-GaAs M . S. Lundstrom, Minority-Carrier Transport in IIILV Semiconductors

R. A. Abram, Effects of Heavy Doping and High Excitation on the Band Structure of GaAs D. Yevick and W. Bardyszewski, An lntroduction to Non-Equilibrium Many-Body Analyses of Optical Processes in 111-V Semiconductors

Volume 40 Epitaxial Microstructures E. F. Schubert, Delta-Doping of Semiconductors: Electronic, Optical, and Structural Properties of Materials and Devices A. Gossard, M . Sundaram, and P. Hopkins, Wide Graded Potential Wells P. Petro$, Direct Growth of Nanometer-Size Quantum Wire Superlattices E. Kapon, Lateral Patterning of Quantum Well Heterostructures by Growth of Nonplanar Substrates H. Temkin, D. Gershoni, and M. Punish, Optical Properties of G al -,In,As/InP Quantum Wells

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Volume 41 High Speed Heterostructure Devices F. Capasso, F. Beltram, S. Sen, A. Pahlevi, and A. Y. Cho, Quantum Electron Devices: Physics and Applications P. Solomon, D. J. Frank, S. L. Wright, and F. Canora, GaAs-Gate Semiconductor-InsulatorSemiconductor FET M. H. Hashemi and U.K. Mishra, Unipolar InP-Based Transistors R Kiehl, Complementary Heterostructure FET Integrated Circuits T. Ishibashi, GaAs-Based and InP-Based Heterostructure Bipolar Transistors H. C. Liu and T. C. L. G. Sollner, High-Frequency-Tunneling Devices H. Ohnishi, T. More, M. Takatsu, K. Imumura, and N . Yokoyama, Resonant-Tunneling Hot-Electron Transistors and Circuits

Volume 42 Oxygen in Silicon F. Shzmura, Introduction to Oxygen in Silicon W. Lin, The Incorporation of Oxygen into Silicon Crystals T J. Schaffner and D. K. Sehroder, Characterization Techniques for Oxygen in Silicon W M. Bullis, Oxygen Concentration Measurement S. M. Hu, Intrinsic Point Defects in Silicon B. Pajot, Some Atomic Configurations of Oxygen J. Michel and L. C. Kimerling, Electical Properties of Oxygen in Silicon R. C. Newman and R. Jones, Diffusion of Oxygen in Silicon T. Y. Tan and W. J. Taylor, Mechanisms of Oxygen Precipitation: Some Quantitative Aspects M. Sehrems, Simulation of Oxygen Precipitation K Simino and I. Yonenagu, Oxygen Effect on Mechanical Properties W. Bergholz, Grown-in and Process-Induced Effects F. Shimura, Intrinsic/Internal Gettering H. Tsuya, Oxygen Effect on Electronic Device Performance

Volume 43 Semiconductors for Room Temperature Nuclear Detector Applications 8. James and T. E. Schlesinger, Introduction and Overview S. Darken and C. E. Cox, High-Purity Germanium Detectors Burger, D. Nason, L. Van den Berg, and M. Schieber, Growth of Mercuric Iodide J . Bao, T. E. Schlesinger, and R. B. James, Electrical Properties of Mercuric Iodide X. J. Bao, R. B. James, and T. E. Schlesinger, Optical Properties of Red Mercuric Iodide M. Huge-Ali and P. Sijfert, Growth Methods of CdTe Nuclear Detector Materials M. Huge-Ali and P Siffert, Characterization of CdTe Nuclear Detector Materials M. Huge-Ali and P. Szffert, CdTe Nuclear Detectors and Applications R. B. James, T. E. Schlesinger, J. Lund, and M. Sehieber, Cd, -xZn,Te Spectrometers for Gamma and X-Ray Applications D. S. MeGregor, J. E. Kammeraad, Gallium Arsenide Radiation Detectors and Spectrometers J. C. Lund, F. Olschner, and A. Burger, Lead Iodide M. R. Squillante, and K. S. Shah, Other Materials: Status and Prospects V. M. Gerrish, Characterization and Quantification of Detector Performance J. S. Iwanezyk and B. E. Part, Electronics for X-ray and Gamma Ray Spectrometers M. Schieber, R. B. James, and T. E. Schiesinger, Summary and Remaining Issues for Room Temperature Radiation Spectrometers

R. L. A. X.

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CONTENTS OF VOLUMESIN THISSERIES

Volume 44 11-IV BluelGreen Light Emitters: Device Physics and Epitaxial Growth J. Han and R L. Gunshor, MBE Growth and Electrical Properties of Wide Bandgap

ZnSe-based 11-VI Semiconductors S. Fujita and S. Fujita, Growth and Characterization of ZnSe-based 11-VI Semiconductors by MOVPE E. Ho and L. A. Kolodziejski, Gaseous Source UHV Epitaxy Technologies for Wide Bandgap 11-VI Semiconductors C. G. Van de Wulle, Doping of Wide-Band-Gap 11-VI Compounds-Theory R. Cingokmi, Optical Properties of Excitons in ZnSe-Based Quantum Well Heterostructures A. Ishibashi and A. V. Nurmikko, 11-VI Diode Lasers: A Current View of Device Performance and Issues S. Guha and J. Petruzello, Defects and Degradation in Wide-Gap 11-VI-based Structures and Light Emitting Devices

Volume 45 Effect of Disorder and Defects in Ion-Implanted Semiconductors: Electrical and Physiochemical Characterization H. Ryssel, Ion Implantation into Semiconductors: Historical Perspectives You-Nian Wung and Teng-Cai Ma. Electronic Stopping Power for Energetic Ions in Solids S. T.Nakagawa. Solid Effect on the Electronic Stopping of Crystalline Target and Application to Range Estimation G. Muller, S. Kalbitzer and G. N. Greaves, Ion Beams in Amorphous Semiconductor Research J. Boussey-Said, Sheet and Spreading Resistance Analysis of Ion Implanted and Annealed Semiconductors M. L. Polignano and G. Queirofo, Studies of the Stripping Hall Effect in Ion-Implanted Silicon J. Stoemenos, Transmission Electron Microscopy Analyses R Nipoti and M . Servidori, Rutherford Backscattering Studies of Ion Implanted Semiconductors P. Zaumseil, X-ray Diffraction Techniques

Volume 46 Effect of Disorder and Defects in Ion-Implanted Semiconductors Optical and Photothermal Characterization M. Fried, T. Lohner and J. Gyulai, Ellipsometric Analysis A. Seas and C. Christofdes, Transmission and Reflection Spectroscopy on Ion Implanted Semiconductors A. Othonos and C. Christofdes, Photoluminescence and Raman Scattering of Ion Implanted Semiconductors. Influence of Annealing C. Christofdes, Photomodulated Thermoreflectance Investigation of Implanted Wafers. Annealing Kinetics of Defects U.Zammit, Photothermal Deflection Spectroscopy Characterization of Ion-Implanted and Annealed Silicon Films A. Mandelis, A. Budiman and M. Vargas, Photothermal Deep-Level Transient Spectroscopy of Impurities and Defects in Semiconductors R. Kalish and S. Charbonneau, Ion Implantation into Quantum-Well Structures A. M. Mymnikov and N. N. Gerasimenko, Ion Implantation and Thermal Annealing of 111-V Compound Semiconducting Systems: Some Problems of 111-V Narrow Gap Semiconductors

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Volume 47 Uncooled Infrared Imaging Arrays and Systems R. G. Buser and M . P. Tompsett, Historical Overview P. W Kruse, Principles of Uncooled Infrared Focal Plane Arrays R. A . Wood, Monolithic Silicon Microbolometer Arrays C. M. Hanson, Hybrid Pyroelectric-Ferroelectric Bolometer Arrays D.L. Polla and J . R. Choi, Monolithic Pyroelectric Bolometer Arrays N . Teranishi, Thermoelectric Uncooled Infrared Focal Plane Arrays M . F . Tompsett, Pyroelectric Vidicon 7: W Kenny, Tunneling Infrared Sensors J . R. Vig, R. L. Filler and I:Kim, Application of Quartz Microresonators to Uncooled Infrared Imaging Arrays P. W Kruse, Application of Uncooled Monolithic Thermoelectric Linear Arrays to Imaging Radiometers

Volume 48 High Brightness Light Emitting Diodes G. B. Stringfellow, Materials Issues in High-Brightness Light-Emitting Diodes M . G. Craford, Overview of Device issues in High-Brightness Light-Emitting Diodes F. M . Steranka, AlGaAs Red Light Emitting Diodes C. H . Chen, S. A. Stockman, M . J . Peariasky, and C. P. Kuo, OMVPE Growth of AlGaInP for High Efficiency Visible Light-Emitting Diodes F. A. Kish and R. M . Fletcher, AlGaInP Light-Emitting Diodes M . W Hodapp, Applications for High Brightness Light-Emitting Diodes I . Akasaki and H . Ammo, Organometallic Vapor Epitaxy of GaN for High Brightness Blue Light Emitting Diodes S. Nakamura, Group 111-V Nitride Based Ultraviolet-Blue-Green-Yellow Light-Emitting Diodes and Laser Diodes

Volume 49 Light Emission in Silicon: from Physics to Devices D. J. Lockwood, Light Emission in Silicon G. Abstreirer, Band Gaps and Light Emission in Si/SiGe Atomic Layer Structures T. G. Brown and D. G. Hall, Radiative Isoelectronic Impurities in Silicon and Silicon-Germanium Alloys and Superlattices J. Michel, L. V. C. Assali, M. T. Morse, andL. C. Kimerling, Erbium in Silicon Y Kanemitsu, Silicon and Germanium Nanoparticles P. M. Fauchet, Porous Silicon: Photoluminescence and Electroluminescent Devices C. Delerue, G. Allun, and M. Lannoo, Theory of Radiative and Nonradiative Processes in Silicon Nanocrystallites L. Brus, Silicon Polymers and Nanocrystals

Volume SO Gallium Nitride (GaN) J. I Pankove and T. D. Moustakas, Introduction S. P. DenBaars and S. Krller, Metalorganic Chemical Vapor Deposition (MOCVD) of Group 111 Nitrides W. A. Bryden and T. J. Kutenmacher, Growth of Group IIILA Nitrides by Reactive Sputtering N. Newman, Thermochemistry of IIILN Semiconductors S. J. Pearton and R. J. Shul, Etching of 111 Nitrides

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S. M. Bedair, Indium-based Nitride Compounds A. Trampert, 0. Brandt, and K H. Ploog, Crystal Structure of Group 111 Nitrides H. Morkoc, 1;. Hamdani, and A. Salvador, Electronic and Optical Properties of 111-V Nitride based Quantum Wells and Superlattices X Doverspike and J. I. Pankove, Doping in the 111-Nitrides T. Suski and P. Perlin, High Pressure Studies of Defects and Impurities in Gallium Nitride B. Monemar, Optical Properties o f GaN W. R. L. Lambrecht, Band Structure of the Group 111 Nitrides N. E. Christensen and P. Perlin. Phonons and Phase Transitions in GaN S. Nakamura, Applications of LEDs and LDs I. Akasaki and H. Amano, Lasers J. A. Cooper, Jr., Nonvolatile Random Access Memories in Wide Bandgap Semiconductors

Volume 51A Identification of Defects in Semiconductors G. D. Watkins, EPR and ENDOR Studies of Defects in Semiconductors J. -M. Spaeth, Magneto-Optical and Electrical Detection of Paramagnetic Resonance in Semiconductors T. A. Kennedy and E. R. Glaser, Magnetic Resonance of Epitaxial Layers Detected by Photoluminescence X H. Chow, B. Hitti, and R F. KieJ, pSR on Muonium in Semiconductors and Its Relation to Hydrogen K. Saarinen, P. Hautojarvi, and C. Corbel, Positron Annihilation Spectroscopy of Defects in Semiconductors R Jones and P. R Briddon, The A b Initio Cluster Method and the Dynamics of Defects in Semiconductors

Volume 51B Identification of Defects in Semiconductors G. Davies, Optical Measurements of Point Defects P. M . Mooney, Defect Identification Using Capacitance Spectroscopy M. Stavola, Vibrational Spectroscopy of Light Element Impurities in Semiconductors P. Schwander, W. D. Rau, C. Kisielowski, M . Gribelyuk, and A. Ourmazd, Defect Processes in Semiconductors Studied at the Atomic Level by Transmission Electron Microscopy N. D. Jager and E. R Weber, Scanning Tunneling Microscopy of Defects in Semiconductors

Volume 52 SIC Materials and Devices K. Jurrendahl and R F. Davis, Materials Properties and Characterization of Sic V. A. Dmitriev and M. G. Spencer, SIC Fabrication Technology: Growth and Doping V. Saxena and A. J. Sreckl, Building Blocks for Sic Devices: Ohmic Contacts, Schottky Contacts, and p-n Junctions M. S. Shur, Sic Transistors C. D. Brandt, R C. Clarke, R R Siergiej, J. B. Casady, A. W. Morse. S. Sriram, and A. K Agarwal, Sic for Applications in High-Power Electronics R. J. Trew. Sic Microwave Devices

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J. Edmond, H. Kong, G Negley, M. Leonard K Doverspike, W. Weeks, A. Suvorov. D. Waltz, and C. Carter, Jr., Sic-Based UV Photodiodes and Light-Emitting Diodes H. Morkoc, Beyond Silicon Carbide! 111-V Nitride-Based Heterostructures and Devices

Volume 53 Cumulative Subject and Author Index Including Tables of Contents for Volume 1-50

Volume 54 High Pressure in Semiconductor Physics I W. Paul, High Pressure in Semiconductor Physics: A Historical Overview N. E. Christensen, Electronic Structure Calculations for Semiconductors under Pressure R. J. Neimes and M. I. McMahon, Structural Transitions in the Group IV, 111-V and 11-VI Semiconductors Under Pressure A. R. Goni and K. Syassen, Optical Properties of Semiconductors Under Pressure P. Trautman, M. Baj, and J. M. Baranowski, Hydrostatic Pressure and Uniaxial Stress in Investigations of the EL2 Defect in GaAs M . Li and P. Y. Yu, High-pressure Study of DX Centers Using Capacitance Techniques T. Suski, Spatial Correlations o f Impurity Charges in Doped Semiconductors N. Kuroda, Pressure Effects on the Electronic Properties of Diluted Magnetic Semiconductors

Volume 55 High Pressure in Semiconductor Physics I1 D. K. Maude and J. C. Portal, Parallel Transport in Low-Dimensional Semiconductor Structures P. C. Klipstein, Tunneling Under Pressure: High-pressure Studies of Vertical Transport in Semiconductor Heterostructures E. Anastassakis and M . Cardona, Phonons, Strains, and Pressure in Semiconductors F. H. Pollak, Effects of External Uniaxial Stress on the Optical Properties of Semiconductors and Semiconductor Microstructures A. R. A d a m , M. Silver, ond J. Allam, Semiconductor Optoelectronic Devices S. Porowski and I. Grzegory, The Application o f High Nitrogen Pressure in the Physics and Technology of 111-NCompounds M. YousuJ Diamond Anvil Cells in High Pressure Studies of Semiconductors

Volume 56

Germanium Silicon: Physics and Materials

J. C. Bean, Growth Techniques and Procedures D. E. Savage, F. Liu, V. Zielasek, and M. G. Lagally, Fundamental Crystal Growth Mechanisms R Hull, Misfit Strain Accommodation in SiGe Heterostructures M. J. Shaw and M. Jaros, Fundamental Physics o f Strained Layer GeSi: Quo Vadis? F. Cerdeira, Optical Properties S. A. Ringel and P. N. Grillot, Electronic Properties and Deep Levels in Germanium-Silicon J. C. Campbell, Optoelectronics in Silicon and Germanium Silicon K. Eberl, K. Brunner, and 0. G. Schmidt, Si,-,C, and Si,~,-,,Ge,C, Alloy Layers

242

CONTENTS OF VOLUMES IN THISSERIES

Volume 57 Gallium Nitride (GaN) IS R. J. Molnar, Hydride Vapor Phase Epitaxial Growth of 111-V Nitrides T. D.Moustakas, Growth of 111-V Nitrides by Molecular Beam Epitaxy Z. Liliental- Weber, Defects in Bulk GaN and Homoepitaxial Layers C. G. Van de Walk and N. M. Johnson, Hydrogen in 111-V Nitrides W Gotz and N. M. Johnson, Characterization of Dopants and Deep Level Defects in Gallium Nitride B. Gil, Stress Effects on Optical Properties C. Kisielowski, Strain in GaN Thin Films and Heterostructures J. A. Miragliotfa and D. X Wickenden, Nonlinear Optical Properties of Gallium Nitride B. K. Meyer, Magnetic Resonance Investigations on Group 111-Nitrides M. S. Shur and M. AsifKhan, GaN and AlGaN Ultraviolet Detectors C. H. Qiu, J. I. Punkove, and C. Rossington, 111-V Nitride-Based X-ray Detectors

Volume 58 Nonlinear Optics in Semiconductors I A. Kost, Resonant Optical Nonlinearities in Semiconductors E. Garmire, Optical Nonlinearities in Semiconductors Enhanced by Carrier Transport D. S. Chemla, Ultrafast Transient Nonlinear Optical Processes in Semiconductors M. Sheik-Buhae and E. W. Van Strylond, Optical Nonlinearities in the Transparency Region of Bulk Semiconductors J. E. Millerd, M. Ziari, and A. Purtovi, Photorefractivity in Semiconductors

Volume 59 Nonlinear Optics in Semiconductors I1 J. B. Khurgin, Second Order Nonlinearities and Optical Rectification X L. Hall, E. R. Thoen, and E. P. Ippen, Nonlinearities in Active Media E. Hanamura, Optical Responses of Quantum Wires/Dots and Microcavities U. Keller, Semiconductor Nonlinearities for Solid-state Laser Modelocking and Q-Switching A. Miller, Transient Grating Studies of Carrier Diffusion and Mobility in Semiconductors

Volume 60 Self-Assembled InGaAslGaAs Quantum Dots Mitsuru Suguwara, Theoretical Bases of the Optical Properties of Semiconductor Quantum Nano-Structures Yoshiaki Nakata, Yoshihiro Sugiyama, and Mitsuru Sugawara, Molecular Beam Epitaxial Growth of Self-Assembled InAs/GaAs Quantum Dots Kohki Mukai, Mitsuru Sugawara, Mitsuru Egawa, and Nobuyuki Ohtsuka, Metalorganic Vapor Phase Epitaxial Growth of Self-Assembled InGaAs/GaAs Quantum Dots Emitting at 1.3 pm Kohki Mukai and Mifsuru Sugawara, Optical Characterization of Quantum Dots Kohki Mukai and Mitsuru Sugawara, The Photon Bottleneck Effect in Quantum Dots Hajjme Shoji, Self-Assembled Quantum Dot Lasers Hiroshi Ishikawa, Applications of Quantum Dot to Optical Devices Mitsuru Sugawara, Kohki Mukai, Hiroshi Ishikawa, Koji Otsubo, and Yoshiaki Nakata, The Latest News

CONTENTS OF VOLUMES I N THISSERIES

243

Volume 61 Hydrogen in Semiconductors I1 Norbert H. Nickel, Introduction to Hydrogen in Semiconductors I1 Noble M . Johnson und Chris G. Vun de Wulle, Isolated Monatomic Hydrogen in Silicon Yurzj V. Gorelkinskii, Electron Paramagnetic Resonance Studies of Hydrogen and HydrogenRelated Defects in Crystalline Silicon Norbert H. Nickel, Hydrogen in Polycrystalline Silicon Wo/fhard Beyer, Hydrogen Phenomena in Hydrogenated Amorphous Silicon Chris G. Van de Wulle, Hydrogen Interactions with Polycrystalline and Amorphous Silicon- Theory Karen M. McNumura Rutledge, Hydrogen in Polycrystalline CVD Diamond Roger L. Lichti, Dynamics of Muonium Diffusion, Site Changes and Charge-State Transitions Mutthew D.McCluskey and Eugene E. Huller, Hydrogen in 111-V and 11-VI Semiconductors S. J . Peurton und J. W. Lee, The Properties of Hydrogen in GaN and Related Alloys Jorg Neugebuuer und Chris G. Van lit> Wolle. Theory of Hydrogen in C a N

Volume 62 Hydrogen in Semiconductors I1 Manfred Helm, The Basic Physics of Intersubband Transitions Jerome Faist, Curlo Sirtori, Federico Cupusso, Loren N. Pfeirer, Ken W. West, Deborah L. Sivco. und Alfred Y. Choo, Quantum Interference Effects in Intersuhband Transitions H. C. Liu, Quantum Well Infrared Photodetector Physics and Novel Devices S. D. Gunapulu und S. V. Bunduru, Quantum Well Infrared Photodetector (QWIP) Focal Plane Arrays

Volume 63 Chemical Mechanical Polishing in Si Processing Frank B. Kuufman, Introduction Thomas Bibby und Kurey Holland, Equipment John P. Bare, Facilitization Duune S. Boning ond Okumu Oumu, Modeling and Simulation Shin Hwa Li,Bruce Tredinnick, and Me1 Hofman. Consumables I: Slurry Lee M. Cook, C M P Consumables TI: Pad Frangois Turd$ Post-CMP Clean Shin Hwa Li, Tura Chliu~par,und Frederic Robert, C M P Metrology Shin Hwu Li, Visun Buclzu, and Kyle Wooldridge, Applications and CMP-Related Process Problems

Volume 64 ElectroluminescenceI M. G. Cruford, S. A. Stockman, M . J . P w m k y , and F. A. Kish, Visible Light-Emitting Diodes H. Chui, N. F. Gardner, P. N. GriNor, J. W. Huang, M . R. Krames, and S. A . Murunowski, High-Efficiency AlGaInP Light-Emitting Diodes R. S. Kern, W. Gotz, C. H. Chen, H . Liu, R. M. Fletcher, and C. P. Kuo, High-Brightness Nitride-Based Visible-Light-Emitting Diodes Yoshihuru Suto, Organic L E D System Considerations V. BuloviC, P. E. Burrows, und S. R. Forrest. Molecular Organic Light-Emitting Devices

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CONTENTS OF VOLUMES IN THISSERIES

Volume 65 Electroluminescence I1 V. BuloviC and S. R Forrest, Polymeric and Molecular Organic Light Emitting Devices: A Comparison Regina Mueller-Mach and Gerd 0. Mueller, Thin Film Electroluminescence Markku Leskela, Wei-Min Li,and Mikko Ritala, Materials in Thin Film Electroluminescent Devices Kristiaan Neyts, Microcavities for Electroluminescent Devices

ISBN 0-32-752375-5