Solid State Physics, Volume 55

SOLID STATE PHYSICS VOLUME 55

Founding Editors FREDERICK SEITZ DAVID TURNBULL

SOLID STATE PHYSICS Advances in Research and Applications

Editors HENRY EHRENREICH

FRANS SPAEPEN

Division of Engineering and Applied Sciences Harvard University Cambridge, Massachusetts

VOLUME 55

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Contents

........................................ PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTRILIUTORS

vii xi

Physics of Organic Electronic Devices

I . H . CAMPBELL AND D . L . SMITH I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Electronic Properties of n-Conjugated Organic Materials . . . . . . . . . . . . . 111. Metal/Organic Interface Electronic Structure . . . . . . . . . . . . . . . . . . . . . IV. Electrical Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Organic Diodes and Field-Effect Transistors . . . . . . . . . . . . . . . . . . . . . VI . Summarv and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . .

1 12 24 54 77 113 117

Charge Density Wave Formation in Nanocrystals

PHILIPKIM.JIAN ZHANG.AND CHARLESM. LIEBER I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Charge Density Waves in Bulk Transition Metal Dichalcogenides . . . . . . 111. Creation of CDW Nanocrystals by STM Tip . . . . . . . . . . . . . . .

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

120 126 132 142 148 156 157

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Martensitic Transitions and Shape-Memory Materials . . . . . . . . . . . . . . . 111. Precursor Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Lattice Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Phase Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII . Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Magnetic Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160 162 175 181 187 217 233 252 265 267

IV. Finite Size Effect and Fermi Surface Roughening . V . Nanocrystal Fabrication to Other TMD Systems . VI . Summary and Future Work . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

Vibrational Propertlesof Shape-Memory Alloys

ANTON PLANES AND LLU~S MA~OSA

V

vi

CONTENTS Grain Growth and Evolution of Other Cellular Structures CARLV . Thompson

I. I1. 111. IV . V. VI . VII . VIII . IX . X.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Phenomenological Model of Burke and Turnbull . . . . . . . . . . . . . . . Grain Growth in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean Field Models for Grain Growth . . . . . . . . . . . . . . . . . . . . . . . . Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiments on Two-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . Experiments on Foils and Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . Three-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

270 270 272 279 286 290 293 306 313 314

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

331

Contributors to Volume 55 Numbers in parentheses indicate the pages on which the authors’ contributions begin.

I. H. CAMPBELL (l), Los Alamos National Laboratory, Los Alamos, N M 87545

PHILIPKIM(1 19), Department of Physics, University of California, Berkeley, CA 94720 CHARLES M. LIEBER (1 19), Division of Engineering and Applied Sciences and Department of Chemistry and Chemical Biology, Harvard University, M A 02138 L L ~MAI~OSA S (159), Departament dEstructura i Constituents de la Matiria, Facultat de Fisica, Diagonal, 645, 08028 Barcelona, Catalonia (Spain) ANTONIPLANES (159), Departament dEstructura i Constituents de la Matiria, Facultat de Fisica, Diagonal, 645, 08028 Barcelona, Catalonia (Spain)

D. L. SMITH(l), Los Alamos National Laboratory, Los Alamos, N M 87545 CARLV . THOMPSON (269), Department of Materials Science and Engineering, M.I. T., Cambridge, M A 02139 JIAN ZHANG(1 19), Optikos Corporation, Cambridge, M A 02141

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Preface

The present volume deals with four diverse areas of considerable interest and importance: organic electronic device physics, charge density waves in nanocrystals, shape memory alloys, and grain growth of cellular structures. The article by Campbell and Smith presents a comprehensive survey of the basic physics underlying organic electronic devices, in particular, the most studied examples of light-emitting diodes (LEDs) and field-effect transistors. This exciting new area is rapidly unfolding in some ways, as the authors point out, analogously to the early development of inorganic semiconductor devices. Because of their unique properties permitting large area processing, mechanical flexibility, chemical sensing interactions, and biocompatibility, organic materials are beginning to have a wide range applications. The operating principles of organic LEDs are fundamentally distinct from those characterizing the conventional inorganic devices because carriers are injected at metallic contacts. The interface details, which are extensively discussed in this article, are therefore very significant. Moreover, the palette of organic materials is very diverse, and the search for new materials designed for specific applications presents new challenges. The article is both didactic and up to date, and therefore approachable by both newcomers to the field and active researchers. It is extensively referenced, thereby providing a literature source to important items such as the comprehensive review by N.C. Greenham and R.H. Friend, concerning semiconductor device physics of conjugated polymers, which appeared in Volume 49 of this Series. Kim, Zhang, and Lieber address the formation of charge density waves (CDWs) in 2D nanostructures, in particular, transition metal dichalcogenides (TMDs). In bulk form such layered materials have been known since the '70s to exhibit 2D CDW formation. However, when the crystal size approaches nanometer dimensions, electron quantum confinement effects become important. The quantization of allowed electron states in reciprocal space introduces Fermi surface roughening, which in turn is responsible for ix

X

PREFACE

irregularities of the observed CDW wavelengths. The understanding of such results, which are summarized and discussed in this review, is important because it leads to insights of how the electron-hole pairing is modified in mesoscopic systems. These insights shed light on the evolution of Fermi surface structures from the bulk to the nanoscale. The article also discusses the fabrication of nanocrystals using an STM tip for a variety of TMD crystals and the rich variety of phenomena such as STM tip-induced local phase transformations that are observed. It should be noted that Volume 44 of this Series contains a lengthy and detailed review of quantum transport in semiconductor nanostructures by Beenakker and van Houten. The shape-memory effect in certain metallic alloys is made possible by a reversible martensitic transformation. A general article about martensitic (i.e., displacive or shear) transformations by A.L. Roitburd appeared in Volume 33 of this Series. Shape-memory alloys have several technological applications, from safety valves to, most recently, micro-electromechanical systems (MEMS). The lattice dynamics of these alloys are the subject of the article by Planes and Mafiosa. The high-temperature phase of many shapememory alloys has the body-centered cubic (bcc) structure. Its phonon spectrum contains so-called “soft modes” of low energy and large vibrational entropy, which are thought to convey stability to the bcc phase at high temperature, as well as facilitate the transition towards the lowtemperature martensite phase. Most of the discussion is on Cu-based alloys, because they have been studied intensively experimentally and the electronic contribution to their entropy is negligible. Some discussion of the more recent ferromagnetic shape-memory alloys, in particular the Heusler alloys, is included as well. The article by Thompson reviews our understanding of the evolution of materials that are divided up into cells by internal surfaces, such as polycrystals or foams. The evolution is a type of coarsening, driven by a continuous decrease in the total interfacial area. In polycrystals the phenomenon is known as grain growth. Insights into this complex process, which was reviewed a first time in volume 50 of this Series by Weaire and McMurry, continue to develop. Much of the progress is driven by computer simulations, using a variety of complementary numerical approaches. For example, the front-tracking technique, which prescribes the motion of each segment of the boundary based on the local curvature, is the most effective one for elucidating two-dimensional grain growth, including the emergence of log-normal size distributions, the stagnation of growth, solute drag on the boundary motion, and abnormal (bi-modal) grain growth. As a result, our understanding of two-dimensional grain growth, which occurs in thin films and hence is technologically very important, is quite complete. By contrast,

PREFACE

xi

the understanding of grain growth in three dimensions is not nearly as advanced. Monte Carlo simulations of Potts models for polycrystals appear best suited for shedding light on this part of the problem.

HENRYEHFWNREICH FUNS SPAEPEN

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SOLID STATE PHYSICS VOLUME 55

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SOLID STATE PHYSICS. VOL. 55

Physics of Organic Electronic Devices I . H . CAMPBELL AND D . L. SMITH Los Alamos National Laboratory. Los Alamos. New Mexico

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Organic Electronic Materials . . . . . . . . . . . . . . . . . . . . . . 2. Organic Electronic Devices . . . . . . . . . . . . . . . . . . . . . . 3. Scope of This Article . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Electronic Properties of a-Conjugated Organic Materials . . . . . . . . . . 4. Elementary Picture of the Electronic Structure . . . . . . . . . . . . . 5 . Electron-Ion and Electron-Electron Interactions . . . . . . . . . . . . 6. Solid State Properties . . . . . . . . . . . . . . . . . . . . . . . . . I11. Metal/Organic Interface Electronic Structure . . . . . . . . . . . . . . . . 7. Built-in Potentials in Metal/Organic/Metal Structures . . . . . . . . . 8. Built-in Potential and Schottky Energy Barrier Measurement . . . . . . Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Built-in Potential and Schottky Energy Barrier Measurement . . . . . . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Solid State and Molecular Properties . . . . . . . . . . . . . . . . . . 11. Manipulating Schottky Energy Barriers Using Dipole Layers . . . . . . IV. Electrical Transport Properties . . . . . . . . . . . . . . . . . . . . . . . 12. Time-of-Flight Mobility Measurements . . . . . . . . . . . . . . . . . 13. Mobility from Single-Carrier SCL Diode I-V Characteristics . . . . . . 14. Mobility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . V . Organic Diodes and Field-Effect Transistors . . . . . . . . . . . . . . . . . 15. Organic Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16. Field-Effect Transistors . . . . . . . . . . . . . . . . . . . . . . . . VI . Summary and Future Directions . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

1 Introduction

1 4 8 11 12 13 16 21 24 26

29 32 45 49 54 57 63 66 77 79 108 113 117

.

This article discusses the basic device physics that governs the operation of organic electronic devices. Organic electronic devices are a new class of solid state electrical devices that have been the subject of intense research in the last decade. The two most widely studied devices are light-emitting diodes (LEDs) and field-effect transistors (FETs). These organic devices are attract1 ISBN 0-12-607755-X ISSN 0081-1947/01 $35.00

Copyright 02001 by Academic Press All rights of reproduction in any form meNed.

2

I. H. CAMPBELL A N D D. L. SMITH

ing considerable interest because they have processing and performance advantages for low-cost and/or large-area applications. An organic lightemitting diode consists of a thin-film of a luminescent organic material contacted by metal electrodes on the top and bottom of the film. One electrode serves as an electron injecting contact and the other as a hole injecting contact. When a sufficient voltage bias is applied to the metal contacts, electrons and holes are injected into the organic material. The injected electrons and holes recombine in the organic material, emitting light. Organic field-effect transistors are lateral devices consisting of a conducting gate contact, a gate insulator, source and drain electrodes electrically isolated from the gate contact by the gate insulator, and an organic film in contact with the source and drain electrodes. When a bias is applied between the gate and source-drain electrodes, charge is injected into the organic film from the source and drain contacts. With an additional bias applied between the source and drain contacts a current flows laterally between these two contacts. The source to drain current is modulated by the gate voltage producing the field-effect transistor action. Organic electronic devices use undoped, insulating organic materials as the light-emitting and charge-transporting layers. The charge carriers in the devices are injected from the contacts. Electronic devices based on doped organic materials have not been developed in a manner analogous to doped inorganic semiconductor devices. To date, it has proven difficult to obtain robust n- and p-type doping of organic materials suitable for electronic device development. Doped, conducting organic materials, such as aciddoped polymers, have electrical properties similar to low conductivity metals. They have been used in organic electronic devices as a metallic contact. This article considers organic electronic devices that use thin films of undoped, conjugated organic materials for the active layer. These devices are the focus of both current scientific research and commercial development. References [l-41 present recent reviews of the electronic properties and Refs. [S-91 A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su, Reviews of Modern Physics 60,781 (1988). N . C. Greenharn and R. H. Friend, in Solid State Physics (H. Ehrenreich and F. Spaepen, eds.), Vol. 49, Academic Press, New York (1995). C. E. Swenberg and M. Pope, Electronic Processes of Organic Crystals and Polymers, Oxford University Press, Oxford (1999). P. M. Borsenberger and D. S. Weiss, Organic Photoreceptorsfor Xerography, Marcel Dekker, New York (1998). L. J. Rothberg and A. J. Lovinger, J. Materials Research 11, 3174 (1996). P. E. Burrows, G. Gu, V. Bulovic, Z. Shen, S. R. Forrest, and M. E. Thompson, ZEEE Trans. on Electron Devices 44, 1188 (1997). C. H. Chen, J. Shi, and C. W. Tang, Macromolecular Symposia 125, 1 (1998). Semiconducting Polymers: Chemistry, Physics and Engineering, (G. Hadziioannou and P. Van Hutten, eds.), John Wiley & Sons, New York (2000).

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PHYSICS OF ORGANIC ELECTRONIC DEVICES

3

present recent reviews of device applications of organic electronic materials. Organic electronic devices have improved dramatically since their invention, and a wide range of products are now available or in development.’’- l 3 Organic LEDs have been reported with luminous efficiencies of 20-30 lm/W and with external quantum efficiencies of 7-8%.14-” These efficiencies compare favorably with other emissive display technologies which typically have luminous efficiencies of about 5 lm/W’ *. An external quantum efficiency of 7% corresponds to an internal quantum efficiency of about 35%. In organic LEDs the injected electrons and holes form excitons, which then recombine. Both singlet and triplet excitons are formed but, usually, only the singlet excitons recombine radiatively. The branching ratio for singlet/triplet formation is not known, but if governed by spin statistics alone then three triplets would be formed for every singlet. Therefore, statistics alone would argue that the maximum internal quantum efficiency would be 25%. The reported 35% internal quantum efficiency implies that spin statistics alone do not limit the maximum internal quantum efficiency. Recent experiments using phosphorescent organic molecules, in which both the singlet and triplet excitons can recombine radiatively, suggest that it may be possible to produce organic LEDs with close to 100% internal quantum e f f i ~ i e n c y . ” ~ ’ ~Th - ~e~development of organic field-effect transistors has progressed more slowly than that of organic LEDs. However, organic FETs are now beginning to achieve performance levels suitable for circuit appliG. Horowitz, Advanced Materials 10, 365 (1998). J. R. Sheats, H. Antoniadis, M. Hueschen, W. Leonard, J. Miller, R. Moon, D. Roitman, and A. Stocking, Science 273, 884 (1996). D. B. Roitman, H. Antoniadis, J. Sheats, and F. Pourmirzaie, Laser Focus World 34, 163 (1998). l 2 P. E. Burrows, G. Gu, V. Bulovic, Z . Shen, S. R. Forrest, and M. E. Thompson, IEEE Trans. Elec. Device 44, 1188 (1997). l 3 R. H. Friend, R. W. Gymer, A. B. Holmes, J. H. Burroughes, R. N. Marks, C. Taliani, D. D. C. Bradley, D. A. DosSantos, J. L. Bredas, and M. Logdlund, Nature 397, 121 (1999). l4 Results from many corporations. See, for example, Cambridge Display Technology (www.cdtltd.co.uk) or Uniax Corporation (www.uniax.com). J. Kido and Y. Iizumi, Appl. Phys. Leu. 73, 2721 (1998). l 6 S. E. Shaheen, G. E. Jabbour, B. Kippelen, N. Peyghambarian, J. D. Anderson, S. R. Marder, N. R. Armstrong, E. Bellmann, and R. H. Grubbs, Appl. Phys. Lett. 74, 3212 (1999). l 7 M. A. Baldo, S. Lamansky, P. E. Burrows, M. E. Thompson, and S. R. Forrest, Appl. Phys. Lett. 75, 4 (1999). S. Matsumoto, in Electronic Display Devices ( S . Matsumoto, ed.), Chapter 1, John Wiley & Sons, New York (1990). M. A. Baldo, D. F. OBrien, Y. You, A. Shoustikov, S. Sibley, M. E. Thompson, and S. R. Forrest, Nature 395, 6698 (1998). 2o D. F. OBrien, M. A. Baldo, M. E. Thompson, and S . R. Forrest, Appl. Phys. Lett. 74, 442 (1999). 2 1 R. C. Kwong, S. Sibley, T. Dubovoy, M. Baldo, S. R. Forrest, and M. E. Thompson, Chemistry of Materials 11, 3709 (1999).

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I. H. CAMPBELL AND D. L. SMITH

cations.23- 3 2 Transistors with field-effect mobilities of about 1 cmZ/vs, comparable to amorphous silicon, have been rep~rted.~'High mobility n-channel organic FETs and complementary n- and p-channel organic circuits have been d e ~ e l o p e d .The ~ ~ use of high dielectric constant gate insulators has produced transistors with operating voltages below 5 V that are compatible with conventional inorganic circuitry.32 1. ORGANIC ELECTRONIC MATERIALS The organic materials used in this new class of electronic devices are n-conjugated materials, either small molecules or polymers. They have energy gaps ranging from about 1.5 eV to 3.5 eV,2*3are undoped, and therefore have essentially no free carriers at room temperature. Figure I. 1 shows the chemical structure of three representative materials: the small molecule tris-(8-hydroxyquinolate)-aluminum [Alq], the polymer poly(pphenyelene vinylene) [PPV], and the small molecule pentacene. Tris-(8hydroxyquino1ate)-aluminum was used in the first organic light-emitting diode^.^^,^^ Poly (p-phenylene vinylene) was the active material used in the first polymer light-emitting diodes.35 It is an insoluble polymer, i.e. it does not dissolve in organic solvents, and is prepared by thermal conversion of

22

V. Cleave, G. Yahioglu, P. LeBarny, R. H. Friend, and N. Tessler, Advanced Materials 11,

285 (1999).

H. Sirringhaus, N. Tessler, and R. H. Friend, Science 280, 1741 (1998). H. Sirringhaus, P. J. Brown, R. H. Friend, M. M. Nielsen, K. Bechgaard, B. M. W. LangeveldVoss, A. J. H. Spiering, R. A. J. Jannssen, and E. W. Meijer, Nature 401, 685 (1999). 2 5 Z. Bao, A. Dodabalapur, and A. J. Lovinger, Appl. Phys. Lett. 69,4108 (1996). 2 6 Z. Bao, A. J. Lovinger, and A. Dodabalapur, Appl. Phys. Lett. 69, 3066 (1996). 2 7 A. Dodabalapur, Z. Bao, A. Makhija, J. G. Laquindanum, V. R. Raju, Y. Feng, H. E. Katz, and J. Rogers, Appl. Phys. Lett. 73, 142 (1998). R. Hajlaoui, G. Horowitz, F. Garnier, A. ArceBrouchet, L. Laigre, A. ElKassmi, F. Demanze, and F. Kouki, Advanced Materials 9, 389 (1997). 29 C. J. Drury, C. M. J. Mutsaers, C. M. Hart, M. Matters, and D. M. deleeuw, Appl. Phys. Lett. 73, 108 (1998). 30 S. F. Nelson, Y.-Y. Lin, D. J. Glundlach, and T. N. Jackson, Appl. Phys. Lett. 72, 1854 23 24

(1998).

Y.-Y.Lin, A. Dodabalapur, R. Sarpeshkar, Z. Bao, W. Li, K. Baldwin, V. R. Raju, and H. E. Katz, Appl. Phys. Lett. 74,2714 (1999). 32 C. D. Dimitrakopoulos, S. Purushothaman, J. Kymissis, A. Callegari, and J. M. Shaw, Science 283, 822 (1999). 33 C. W. Tang and S. A. Van Slyke, Appl. Phys. Lett. 51, 913 (1987). 34 C. W. Tang, S. A. Van Slyke, and C. H. Chen, J. Appl. Phys. 65, 3610 (1989). 3 5 J. H. Burroughes, D. D. C. Bradley, A. R. Brown, R. N. Marks, K. Mackay, R. H. Friend, P. L. Burns, and A. B. Holmes, Nature 347,539 (1990).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

5

FIG. 1.1. Chemical structures of Alq (left), PPV (center) and pentacene (right).

a precursor polymer. Soluble derivatives of PPV, such as poly [2-methoxy, 5-(2’-ethyl-hexyloxy)-I,Cphenylene vinylene] (MEH-PPV),36 which can be spin cast from organic solutions, are now more widely used. Pentacene is widely used in organic thin-film transistors.30932These materials all have large conjugated units, i.e. regions with resonant single and double bonds, which determine their conduction and valence energy levels and thus their energy gaps. The main differences between the small molecule and polymer materials are their processing methods and mechanical properties. Thin films of small molecules, such as Alq and pentacene, are usually prepared by vacuum evaporation, whereas thin polymer films are formed by solution processing methods such as spin casting. In both cases, the resulting films are amorphous or small grain polycrystalline (5-20 nm crystallites) and are highly d i ~ o r d e r e d . ~Figure .~~.~ 1.2~ shows the crystal structure of PPV determined from x-ray diffraction measurement^.^' These organic films have densities of W/cm-K, and about 1 g/cm3, thermal conductivities of about 1 x specific heats of about 1 J/g-K.39,40Polymer films have better mechanical integrity than films of small molecules. The polymer chain is held together by strong covalent bonds and, in a polymer film, the chains are typically entangled, which further increases the mechanical strength of the film. By D. Braun and A. J. Heeger, Appl. P h p . Lett. 58, 1982 (1991). D. J. Gundlach, Y.-Y. Lin, T. N . Jackson, S. F. Nelson, and D. G. Schlom, IEEE Electron Device Letters 18,87 (1997). 38 D. Chen, M. J. Winokur, M. A. Masse, and F. E. Karasz, Phys. Rev. B41,6759 (1990). 39 Y. Agari, M. Shimada, and A. Ueda, Polymer 38, 2649 (1997). 40 C. H. M. Maree, R. A. Weller, L. C. Feldman, K. Pakbaz, and H. W. H. Lee, J. Appl. Phys. 84,4013(1998). 36

37

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I. H. CAMPBELL AND D. L. SMITH

a = 8.07 A b = 6.05 A

c=6.6A

A

FIG. 1.2. Crystal structure parameters for PPV (courtesy of M. J. Winokur).

contrast, there is only a weak Van der Waals attraction between molecules in a molecular film. The electrical and optical properties of thin films of small molecules and polymers are generally similar. From an electronic structure point of view, the molecular films can be considered as a collection of distinct molecular sites. For the polymer films, the extended polymer chain is broken into independent sites by a combination of structural and chemical defects. In these organic films, electronic conduction occurs by hopping from one localized site to another. Hopping conduction is an essential aspect of the electrical transport in these materials. The optical absorption and electroluminescence spectra of a 50-nm MEH-PPV film are shown in Fig. 1.3. The absorption coefficient at the peak of the spectrum is about 2 x lo5 cm-1.41 The absorption band is about 0.5 eV wide and the luminescence band is about 0.2 eV wide. The peak of the emission spectrum is red shifted from the peak of the absorption spectrum. Phonon replicas are apparent in both the absorption and emission spectra. The spectral width and the energy shift between the two spectra are due to a combination of structural relaxation and site energy disorder effects in the solid film. Because the absorption and emission spectra do not overlap appreciably, re-absorption of the emitted light is not significant in organic light-emitting diodes. The electroluminescence and photoluminescence spectra (not shown) are essentially identical, indicating that the emitting excited states formed by electrical and optical excitation are the same. The 41

M. G. Harrison, J. Gruner, and G. C. W. Spencer, Phys. Rev. B55,7831 (1997).

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PHYSICS OF ORGANIC ELECTRONIC DEVICES

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0 0

+

w

0.0

0

1

Photon Energy (eV) FIG.1.3. Electroluminescence (solid line) and optical absorption spectrum (dashed line) of MEH-PPV. The chemical structure of MEH-PPV is shown in the inset.

general features in the optical spectra, shown in Fig. 1.3, such as the strength and width of the absorption spectrum and the shift between the peak of the absorption and luminescence, are similar for most conjugated organic materials. The energy gap of these materials is typically between 1.5 eV and 3.5 eV and therefore the emission spectrum of organic light-emitting diodes can span the visible spectral region.' The intrinsic radiative lifetime of the luminescent, singlet excitons in these materials is between 1 ns and 20 The solid state, photoluminescence quantum efficiency varies from about 10% to near 100%.2*43-47 The optical index of refraction and static R. Priestley, A. D. Walser and R. Dorsinville, Optics Communications 158, 93 (1998). D. Z. Garbuzov, V. Bulovic, P. E. Burrows, and S. R. Forrest, Chem. Phys. Lett. 249, 433 (1996). 44 M. Remmers, D. Neher, J. Gruner, R. H. Friend, G. H. Gelinck, J. M. Warman, C. Quattrocchi, D. A. dos Santos, and J. L. Bredas, Macromolecules 29, 7432 (1996). " N. C. Greenham, I. D. W. Samuel, G. R. Hayes, R. T. Phillips, Y. Kessener, S. C. Moratti, A. B. Holmes, and R. H. Friend, Chem. Phys. Lett. 241, 89 (1995). 46 G. KoppingGrem, G. Leising, M. Schimetta, F. Stelzer, and A. Huber, Synthetic Metals 76, 53 (1996). 47 R. G. Sun, Y. Z. Wang, D. K. Wang, Q. B. Zheng, E. M. Kyllo, T. L. Gustafson, and A. J. Epstein, Appl. Phys. Lett. 76, 634 (2000). 42

43

8

I. H. CAMPBELL A N D D. L. SMITH

FIG. 1.4.Organic light-emitting diode device structure.

dielectric constant of these materials are typically about 1.7 and 3, respect i ~ e l y-. ~ ~

2. ORGANIC ELECTRONIC DEVICES The two principal organic electronic devices are light-emitting diodes and field-effect transistors. Organic light-emitting diodes are vertical structures that utilize electrical transport across a thin organic film. In contrast, organic field-effect transistors are lateral structures that utilize transport in the plane of an organic thin film. This distinction is significant because the distance through which current is driven in the two types of structures differs by orders of magnitude and the electrical transport properties of the organic thin films are often anisotropic. This is particularly true for polymer films, where the polymer chains lie predominantly in the plane of the film.52 Figure 1.4 shows the structure of a typical organic light-emitting diode. The diode consists of a transparent metal/organic film/metal structure. The transparent metal contact is usually indium tin oxide (ITO) or a thin metal (e.g., 10 nm of Au). The organic film, either small molecule or polymer, is typically about 100 nm thick. The top metal contact is a low work function metal or metal alloy such as Ca or Mg:Ag. The diode structure is typically 48 V. Bulovic, V. B. Khalfin, G. Gu, P. E. Burrows, D. Z. Garbuzov, and S. R. Forrest, Phys. Rev. B58,3730 (1998). 49 A. J. Campbell, D. D. C. Bradley, J. Laubender, and M. Sokolowski, J. Appl. Phys. 86, 5004 (1999). S o S.J. Martin, D. D. C. Bradley, P. A. Lane, H. Mellor, and P. L. Burn, Phys. Rev. B59, 15133 (1999). W.Holzer, M. Pichlmaier, E. Drotle5, A. Penzkofer, D. D. C. Bradley, and W. J. Blau, Optics Communications 163, 24 (1999). 5 2 D.McBranch, I. H. Campbell, D. L. Smith, and J. P. Ferraris, Appl. Phys. Left. 66, 1175 (1995).

''

PHYSICS O F ORGANIC ELECTRONIC DEVICES

9

fabricated as follows: A glass substrate is sputter coated with semi-transparent ITO, the organic film is then either evaporated or spin cast onto the semi-transparent contact, and finally the top contact is evaporated onto the organic film. The bottom and top contacts are patterned so that their spatial overlap defines the area of the LED. The bottom contact is generally patterned using optical lithography and the top contact is usually patterned using a shadow mask. The top contact is not patterned using photolithography because of the sensitivity of the organic thin-film to the processing procedures. Many variations of the basic organic light-emitting diode device structure have been investigated. Organic LEDs with two transparent contacts have been used to construct transparent displayss3- ” and microcavity designs have been used to adjust organic LED spectral properDoped polymer anode^^^^^^ and thin (1 nm) inorganic layers at the metal/organic interface6’V6 have been used to improve device lifetime. The basic organic LED structure can also be used to produce organic photodetectors and photovoltaic ~ e l l s . ~ To v ~date, ~ - ~these ~ devices have not been completely successful because it is difficult to separate the optically generated charge carriers and produce a p h o t o ~ u r r e n t . ~ ~ ~ ~ - ~ ’ Organic transistors are in a comparatively early stage of development and there is considerable variety in the details of their structure and fabrication. Figure 1.5 shows the structure of a prototype organic thin-film transistor fabricated on a crystalline Si substrate. The transistor consists of an organic thin-film on top of a gate insulator contacted by metal source and drain 5 3 G. Gu, V. Bulovic, P. E. Burrows, S. R. Forrest, and M. E. Thompson, Appl. Phys. Lett. 68, 2606 (1996). 54 G. Parthasarathy, P. E. Burrows, V. Khalfin, V. G. Kozlov, and S . R. Forrest, Appl. Phys. Lett. 72, 2138 (1998). ” G. Gu, G. Parthasarathy, and S . R. Forrest, Appl. Phys. Lett. 74, 305 (1999). 5 6 P. E. Burrows, V. Khalfin, G. Gu, and S . R. Forrest, Appl. Phys. Lett. 73, 435 (1998). 5 7 R. H. Jordan, L. J. Rothberg, A. Dodabalapur, and R.E. Slusher, Appl. Phys. Lett. 69, 1997 (1996). 5 8 S. A. Carter, M. Angelopoulos, S. Karg, P. J. Brock, and J. C. Scott, Appl. Phys. Lett. 70, 2067 (1997). 5 9 J. Gao, A. J. Heeger, J. Y. Lee, and C. Y. Kim, Synthetic Metals 82, 221 (1996). 6o L. S. Hung, C. W. Tang, and M. G. Mason, Appl. Phys. Lett. 70, 152 (1997). 6 1 F. Li, H. Tang, J. Anderegg, and J. Shinar, Appl. Phys. Lett. 70, 1233 (1997). 6 2 G. Yu and A. J. Heeger, J. Appl. Phys. 78,4510 (1995). 6 3 J. J. M. Halls, K. Pichler, R. H. Friend, S. C. Moratti, and A. B. Holmes, Appl. Phys. Lett. 68, 3120 (1996). 64 M. Granstrom, K. Petritsch, A. C. Arias, and R. H. Friend, Synthetic Metals 102,957 (1999). 6 5 S. Barth, H. Bassler, H. Rost, and H. H. Horhold, Phys. Rev. B56,3844 (1997). 66 S. Barth and H. Bassler, Phys. Rev. Lett. 79, 4445 (1997). 6 7 A. Kohler, D. A. dosSantos, D. Beljonne, 2. Shuai, J. L. Bredas, A. B. Holmes, A. Kraus, K. Mullen, and R. H. Friend, Nature 392, 903 (1998).

10

I. H. CAMPBELL AND D. L. SMITH

FIG.1.5. Organic field-effect transistor device structure.

electrodes. The gate insulator is often silicon dioxide about 100 nm thick grown on a doped silicon wafer that serves as the gate contact. The organic film is usually made from small molecules that typically have higher carrier mobility than polymers due to increased molecular ~ r d e r . ~ ,In ~ ~ ~ ~ operation, current is carried in a thin, approximately 5 nm, region of the organic thin-film next to the gate insulator.73The properties of the first few layers of the organic film are thus critical in determining the transistor performance. Most organic FETs are p-type devices that use Au for the source and drain contacts. The channel length, i.e. the spacing between the source and drain contacts, has been varied from about 0.1 pm to over 100pm.9972 A typical prototype transistor structure is fabricated as follows: A heavily doped silicon wafer is oxidized to produce the gate insulator, the source and drain contacts are vacuum evaporated and defined using optical lithography, and finally the organic film is thermally evaporated covering the gate insulator, and metal source and drain contacts. Organic transistors have also been fabricated on plastic substrates using both sputter deposited oxides and spin cast polymers as the gate insulator. In some cases, the fabrication sequence is changed so that the organic thin-film is deposited prior to the source and drain contacts. The source and drain contacts are then evaporated onto the organic film using shadow masks to define their structure.

6 8 L. Torsi, A. Dodabalapur, L. J. Rothberg, A. W. P. Fung, and H. E. Katz, Phys. Rev. B57, 2271 (1998). 69 G. Horowitz, F. Garnier, A. Yassar, R. Hajlaoui, and F. Kouki, Advanced Materials 8, 52 (1996). 70 R. Hajlaoui, D. Fichou, G. Horowitz, B. Nessakh, M. Constant, and F. Garnier, Advanced Materials 9, 557 (1997). 7 1 F. Garnier, G. Horowitz, D. Fichou, and A. Yassar, Supramolecular Science 4, 155 (1997). 7 2 J. A. Rogers, A. Dodabalapur, Z. N. Bao, and H. E. Katz, Appl. Phys. Lett. 75, 1010 (1999). 73 M. A. Alam, A. Dodabalapur, and M. R. Pinto, IEEE Transactions on Electron Devices 44, 1332 (1997).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

3.

SCOPE OF

11

THISARTICLE

This article discusses the basic device physics that governs the operation of organic light-emitting diodes and field-effect transistors. The field of organic electronic device physics is relatively new and is rapidly developing. Not all of the issues in the field are settled and some points remain controversial, but the essential device physics is becoming relatively clear. The operating principles of organic light-emitting diodes are fundamentally distinct from conventional inorganic semiconductor-based LEDs. The rectification and light-emitting properties of inorganic LEDs are due to the electrical junction between oppositely doped, p- and n-type regions of the inorganic semicond ~ c t o r . ’In ~ contrast, organic light-emitting diodes are formed using an undoped, insulating organic material. The rectification and light-emitting properties of the polymer LED are caused by the use of asymmetric metal contacts. One metal contact is only able to inject electrons efficiently and the other contact only injects holes efficiently. The injected electrons and holes recombine in the undoped organic film, emitting light. High work function metals inject holes more efficiently than electrons and, similarly, low work function metals inject electrons more efficiently than holes. Therefore, the high work function metal is the anode and the low work function metal is the cathode. This defines the forward and reverse bias directions of the diode and leads to current rectification, bipolar charge injection, and light emission in organic LEDs. The essential physical processes that need to be understood to describe organic LEDs are charge injection into the organic material from the metallic contacts, charge transport in the organic layer, and electron-hole recombination processes in the organic film. The operation of organic field-effect transistors is similar to that of inorganic thin-film transistors. In both devices, an undoped thin-film is contacted by metallic source and drain electrodes and separated from a gate electrode by a thin gate insulator. Applying a bias to the gate contact induces charge in the thin-film from the source and drain contacts. The conductivity of the thin-film is determined by the amount of induced charge, which is controlled by the gate bias. The main differences between the organic and inorganic structure are the type of metallic contacts used and the transport properties of the undoped thin film. Inorganic structures typically use doped semiconductor contacts made from the same material used for the insulating thin film.74Using the same material for the contacts and the insulating film ensures good electrical contact between the electrodes and the thin film. In contrast, organic FETs use metallic contacts 74

S. M.Sze, Physics of Semiconductor Devices, John Wiley & Sons, New York (1981).

12

I. H. CAMPBELL AND D. L. SMITH

instead of doped organic materials. This can result in significant contact resistance effects in the transistor current-voltage characteristics. The transport properties of thin organic films are dominated by carrier hopping from site to site in the disordered organic film. This leads to an effective carrier mobility that depends on the density of charge induced by the gate in the organic film. The essential physical processes that need to be understood to describe organic FETs are the behavior of the metallic contacts and the charge transport properties of the organic film. This article describes the physics of organic electronic devices by presenting the essential electronic structure and charge transport properties of organic materials separately and then combining them to describe the operating characteristics of devices. The article is organized as follows: Section I1 reviews the electronic structure of x-conjugated organic materials; Section I11 presents the essential features of metal/organic interface electronic structure; Section IV describes the electrical transport properties of thin organic films; Section V presents a device model of organic light. emitting diodes and discusses field-effect transistor operation; Section VI summarizes the conclusions.

II. Electronic Properties of n-Conjugated Organic Materials For the materials used to fabricate the organic electronic devices discussed in this article, the electronic states that play the essential roles in device operation are derived primarily from the p orbitals of carbon in an sp2-p hybrid configuration. The hybridized sp2 orbitals form u bonds essential to the molecular structure, but electronic excitations or charged states involving these orbitals are very high in energy. The low-energy neutral excitations and charged states, of interest for electronic devices, are formed from the p orbitals involved in n bonding.2 - The low-energy electronic excitations involving these p orbitals are typically in the 1.5-3.5 eV range for the materials used in devices. Both conjugated polymers and smaller conjugated molecules are used to make organic electronic devices. Although there are significant fabrication differences for these two classes of materials, their basic electrical behavior is similar. Devices fabricated from a polymer and a corresponding oligomer (an oligomer is a molecule consisting of a small number of the repeat units of a polymer, essentially a very short chain polymer) have similar electrical proper tie^.^^ This result is not surprising because structural and chemical 75

M. D. Joswick, I. H. Campbell, N. N. Barashkov, and J. P. Ferraris, J. Appl. Phys. 80,2883

(1996).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

13

defects limit the length over which conjugation is maintained in conjugated polymers. From an electronic structure point of view the polymer is like a group of oligomers bonded together, with weak interactions between the electronic states in the n-bonded manifold of the different conjugated segments. The electronic states of interest for devices are localized on the individual molecules or in the case of polymers on the separate segments over which conjugation is maintained. Organic electronic devices are formed from dense thin films of the conjugated molecules or polymers. The electronic properties of the dense films are not always the same as those of the isolated molecules or single polymer chains. The interactions between the molecules in the films can have important effects on the electronic structure. Current quantum chemical techniques can accurately describe many aspects of the electronic structure of isolated molecules. A theme of current research in this area is aimed at determining which features of the electronic structure of the isolated molecules are maintained and which are modified in the condensed phase. The section is organized as follows: First, an elementary picture of the electronic structure of n-conjugated hydrocarbons is presented; electron-ion and electron-electron interactions not included in the simple picture are then discussed; and, modifications of the molecular electronic structure in the solid state are described.

4. ELEMENTARY PICTURE OF THE ELECTRONIC STRUCTURE An elementary molecular orbital (MO) picture of the electronic structure of n-conjugated materials is first discussed. Because many of the conjugated materials used in organic electronic devices contain phenylene (benzene) rings, benzene and biphenyl, a molecule consisting of two benzene rings bonded together, are used as examples. Figure 11.1 is a schematic of the molecular structure of benzene. The solid lines represent the 0 bonds formed

- -

2P

P -P -2P

FIG. 11.1. Chemical structure of benzene and its n-electron molecular orbital energies.

14

I. H. CAMPBELL AND D. L. SMITH

primarily from the sp2 hybridized orbitals of carbon and the s orbitals of hydrogen. Neutral excitations and charged states involving these orbitals lie outside of the energy range of interest. The p, orbitals of carbon, which are orthogonal to the plane of the page, form the n bonds indicated by the circle. The low-energy excitations and charged states of device interest are formed from these orbitals. Six spatial states (twelve including spin) can be formed from the p, orbitals. In a simple molecular orbital picture with nearest neighbor interactions, the 6 MOs (not normalized) are (l,l,l,l,l,l), (O,l,l,O,- 1,- l), (2,1,- 1,-2,- l,l), (-2,1;1,-2,1,1), (0,l- l,O,l,- 1) and (1,- l J , - l J , - 1) with corresponding MO energies -28, -8, -8, 8, 8, and 28. Here the site ordering for the MOs is (1,2,3,4,3’,2’),where the site numbers are indicated in the figure and 8 is the magnitude (a positive number) of the hopping integral. The MO energy levels are schematically illustrated in Fig. 11.1. MOs 2 and 3, and 4 and 5 are degenerate. In the electronic ground state of the molecule, MOs 1, 2, and 3 are each occupied with two electrons. MOs number 2 and 5 have no amplitude on sites 1 and 4. The energy gap between occupied and empty MOs is 28 and the sum of the occupied MO energies is -88. Figure 11.2 shows a schematic of the molecular structure of biphenyl, which consists of two benzene molecules bound together. A simple pertur-

2P + C16

-

2P - C16

FIG. 11.2. Chemical structure of biphenyl and its n-electron molecular orbital energies.

15

PHYSICS OF ORGANIC ELECTRONIC DEVICES

bation theory description of the MO energies of biphenyl is shown next to the molecular structure schematic. The lowest energy biphenyl MOs are made up from the first benzene MO (L1 for the first MO on the left benzene and R1 for the first MO on the right benzene): (L1+ R1) and (L1 - R1) with energies - 28 - C/6 and - 28 C/6, respectively. Here C is the magnitude of the hopping integral between carbon-4 on the left benzene and carbon-1 on the right benzene. In the same way symmetric and antisymmetric biphenyl MOs can also be made from benzene MOs 3 , 4 and 6: (LN RN) C/3, - p - C/3 and (LN - RN), where N = 3, 4, or 6 with energies - p for N = 3; p - C/3, p C/3 for N = 4; and 2 p + C/6, 2p - C/6 for N = 6. Because benzene MOs 2 and 5 have no amplitude on carbons 1 and 4, they do not mix when forming biphenyl in the simple nearest neighbor interaction model. The inter-benzene hopping matrix element C depends on the angle between the planes of the two benzene molecules, C = C,lcos 81, where 8 is the angle between planes of the two molecules and C, is the magnitude of the matrix element when the benzene molecules are coplanar. The 6 lowest energy MOs are each occupied by 2 electrons in the neutral ground state. The energy gap between occupied and empty MOs is 2(p - C/3) and the sum of the occupied MO energies is -168 The energy gap of biphenyl is lowered compared with benzene by - 2C/3. This lowering of the energy gap is a consequence of the more spatially extended nature of the states in biphenyl. The extent of this gap lowering depends on the angle between the benzene molecules, but the sum of the MO energies does not depend on the angle at this level of approximation. The lowest energy state with an extra electron added to the biphenyl molecule has the extra electron in the (L4 R4) MO with energy p - C/3. The sum of the occupied MO energies for this negatively charged state is - 15p - C/3. The lowest energy state with an electron removed from the biphenyl molecule has a hole in the C/3. The sum of the occupied MO (L3 R3) MO with energy - p energies for this positively charged state is also - 15g - C/3. The sum of the occupied MO energies depends on the angle between the benzene molecules for both the positively and negatively charged states, but not for the neutral state. This leads to a coupling between the angular orientation and electronic charges. This molecular orbital picture is highly idealized but its qualitative results are preserved in more complete calculations. Figure 11.3 shows the result76 of a quantum chemical calculation (AM1 of the angular dependence of the energy of the biphenyl molecule (arbitrary energy zero) in the ground

+

+

+

+

+

+

76

+

Z. G . Yu, D. L. Smith, A. Saxena, R. L. Martin, and A. R. Bishop, Phys. Rev. Lett. 84, 721

(2ow. 77

H. J. S. Dewar, E. G. Zoebisch, and E. F. Healy, J. Am. Chem. SOC.107, (1985).

16

I. H. CAMPBELL AND D. L. SMITH 0.6 I

.

.

1

0.3

-0.3

-0.6 I 0

90

180

270

I

360

Angle (degree)

FIG. 11.3. AM1 results for the total energy as a function of torsion angle for biphenyl. Solid, dashed, and dot-dashed lines are for anion, neutral, and cation states, respectively (from Ref. 76).

electronic state for the neutral, positive, and negative ions. For the neutral molecule, there is a weak dependence of the energy on angle (in the simple description above the energy is independent of this angle). The lowest energy angle is approximately 40 degrees. The angular dependences of the energy for the two ions are very similar, they are much stronger than for the neutral molecule and have a minimum at 0 and a maximum at 90 degrees. The shape of the angular dependence curves is similar to lcos81 expected from the simple MO model. The minimum energy angle of the neutral molecule does not coincide with that for the ions so that a rotation will occur when an electron or hole is transferred to a neutral molecule to form an ion.

5. ELECTRON-ION AND ELECTRON-ELECTRON INTERACTIONS

The lowest energy geometry of a molecule depends on its electronic configuration. The minimum energy bond lengths and angles are different for the excited and ground electronic states of a neutral molecule and for the ground states of a neutral and a charged molecule. Structural relaxation of the molecule occurs when the electronic state of the molecule is changed. This relaxation affects, for example, the optical absorption and emission spectra near the fundamental absorption edge. As seen in Fig. 1.3, vibrational replicas are observed in both the absorption and emission spectra of MEH-PPV. These replicas occur because vibrational excitations can be induced in the final state of the transition. The theory describing optical

PHYSICS OF ORGANIC ELECTRONIC DEVICES

17

FIG.11.4. Benzenoid (top), quinoid (middle), and electron polaron (bottom) bond configurations in PPV.

transitions in which the initial and final electronic states have different lowest energy geometries is well established.' - 4 Because of the strong coupling between electronic states and molecular geometry that occurs in the n-conjugated materials used for electronic devices and also because of the molecular disorder usually present in dense films of these materials, the optical absorption and emission spectra are typically broad. Structural relaxation, compared to the neutral ground state, also occurs for charged states.'-4 The double bonds that occur in n-conjugated materials are delocalized and flexible in that it is possible to shift between different configurations comparatively easily. Figure 11.4 schematically shows two double bonding configurations, the benzenoid (top) and quinoid

18

I. H. CAMPBELL AND D. L. SMITH

(middle) configurations, for PPV. The benzenoid configuration has the lower energy for the neutral state. When an electron is added or removed from the molecule, the double bonding is disrupted, leading to an admixture of the two configurations for the charged states. The situation is schematically illustrated in the bottom of Fig. 11.4. An extra electron is put into a p, orbital to form a lone pair, which disrupts the double bonding scheme. The p, orbital that had been paired with the one containing the extra electron in the benzenoid configuration now pairs with an orbital in a benzene ring to form a quinoid Configuration. Because the quinoid configuration is higher energy than the benzenoid configuration, the system must eventually return to the benzenoid configuration. A singly occupied, unpaired p, orbital results when the configuration changes. (The figure is very schematic; the electrons aren't actually localized in a single orbital and the configuration doesn't abruptly change from one form to another.) The structurally relaxed negatively charged state is called an electron polaron. If an electron were removed from the neutral molecule, a hole polaron would be formed. The schematic for the hole polaron is similar to that shown for the electron polaron except that the doubly occupied p, orbital is replaced with an empty one. The relaxation energy depends on the length of the conjugated segment, decreasing as the length increases. Theoretical estimates for PPV give relaxation energies varying from about 0.15 eV for a three-ring oligomer to about 0.05 eV for a long chain.78 Structural relaxation occurs very rapidly when an electron is added to or removed from a n-conjugated organic molecule and from a device point of view the charged states that result can essentially always be considered as structurally relaxed. The singly charged states of device interest are electron and hole polarons. For simplicity these states are usually just called electrons and holes. The structural relaxation that occurs around charged states can lead to an effective attractive interaction between two polarons. If this attractive interaction is stronger than the repulsive Coulomb interaction for charges with the same sign, a doubly charged bound state, which has been called a bipolaron, can result. Bipolarons in conjugated polymers have been extensively discussed.'*79-85A schematic for an electron bipolaron is similar to 78 79

Z. Shuai, J. L. Bredas, and W. P. Su, Chem Phys. Lett. 228,301 (1994). S. A. Brazovskii and N. Kirova, Synthetic Metals 57,4385 (1993). S. Brazovskii, N. Kirova, Z. G. Yu, A. R. Bishop, and A. Saxena, Optical Materials 9,502

(1998). A. Saxena, S. Brazovskii, N. Kirova, Z. G. Yu,and A. R. Bishop, Synthetic Metals 101,325 (1999). 8 2 Z. G . Soos, S. Ramasesha, D. S. Galvao, and S . Etemad, Phys. Rev. B47, 1742 (1993). 83 Y. Shimoi and S . Abe, Synthetic Metals 78,219 (1996). 84 Y.Shimoi and S . Abe, Phys. Rev. B50, 14781 (1994). Y. Shimoi and S . Abe, Phy. Rev. B49, 14113 (1994).

PHYSICS O F ORGANIC ELECTRONIC DEVICES

19

that for the electron polaron in the bottom of Fig. 11.4 except that both of the p, orbitals explicitly shown are doubly occupied. For the bipolaron state there is only one region of the higher energy quinoid configuration, rather than two as would occur for two isolated polarons. The reduction of the high-energy quinoid configuration is the physical origin of the attractive interaction. A hole bipolaron schematic is similar to that for the electron bipolaron except that both of the p, orbitals shown are empty. If charged bipolarons were strongly bound, they could have a significant influence on the behavior of organic electronic d e v i ~ e s . ~Section ~ , ~ ~ I11 , ~ discusses ~ possible device implications of these kinds of states and approaches to detecting them. However, for the materials typically used for organic electronic devices, bipolarons are found not to be strongly bound and not to significantly influence device behavior.88 There is an attractive interaction between an electron polaron and a hole polaron due both to the Coulomb interaction and also to the structural relaxation around the charged states. A bound state results that is usually called an exciton. (This state is sometimes called a neutral bipolaron if the interaction due to structural relaxation is emphasized.) There is an extensive - lo5 literature on excitons in x-conjugated organic materials.78*82*83s85*89

N. Kirova and S. Brazovskii, Synthetic Metals 76, 229 (1996). P. S. Davis, A. Saxena and D. L. Smith, J. Appl. Phys. 78,4244 (1995). I. H. Campbell, T. W. Hagler, D. L. Smith, and J. P. Ferraris, Phys. Rev. Lett. 76, 1900 (1996). 8 9 M. Chandross, S. Mazumdar, S. Jeglinski, X. Wei, Z. V. Vardeny, E. W. Kwock, and T. M. Miller, Phys. Rev. B50, 14702 (1994). 9 0 M. Chandross, S. Mazumdar, M. Liess, P. A. Lane, 2. V. Vardeny, M. Hamaguchi, and K. Yoshino, Phys. Rev. B55, 1486 (1997). 9 1 M. Chandross and S. Mazumdar, Phys. Rev. B55, 1497 (1997). 92 P. G . DaCosta and E. M. Conwell, Phys. Rev. B48, 1993 (1993). 93 Y. N. Gartstein, M. J. Rice, and E. M. Conwell, Phys. Rev. B52, 1683 (1995). 94 E. M. Conwell, Synthetic Metals, 83, 101 (1996). 9 5 E. Moore, B. Gherman, and D. Yaron, J. Chem Phys. 106, 4216 (1997). 9 6 D. Yaron, E. E. Moore, Z. Shuai, and J. L. Bredas, J. Chem Phys. 108,7451 (1998). 9 7 E. E. Moore and D. Yaron, J. Chem Phys. 109, 6147 (1998). 98 E. Moore and D. Yaron, Synthetic Metals 85, 1023 (1997). 99 Z. Shuai, S. K. Pati, W. P. Su, J. L. Bredas, and S. Ramasesha, Phys. Rev. B55, 15368 (1997). l o o M. J. Rice and Y. N. Gartstein, Phys. Rev. Lett. 73, 2504 (1994). Y. N. Gartstein, M. J. Rice, and E. M. Conwell, Phys. Rev. B52, 1683 (1995). lo’ S. Brazovskii, N. Kirova, A. R. Bishop, V. Klimov, D. McBranch, N. N. Barashkov, and J. P. Ferraris, Optical Materials 9, 472 (1998). l o 3 J. L. Bredas, J. C o d , and A. J. Heeger, Advanced Materials 8,447 (1996). H. S. Woo, 0. Lhost, S. C. Graham, D. D. C. Bradley, R. H. Friend, C. Quattrocchi, J. L. Bredas, R. Schenk, and K. Mullin, Synthetic Materials 59, 13 (1996). lo’ D. Beljonne, 2. Shuai, R. H. Friend, and J. L. Bredas, J. Chem. Phys. 102,2041 (1995).

20

I. H. CAMPBELL AND D. L. SMITH

The spin-orbit interaction is very small in conjugated organic materials because light elements are involved and spin is a good quantum number. The spins of the electron and hole involved in an exciton can be either singlet or triplet paired. There is often an allowed optical transition between the electronic ground state and the lowest energy singlet exciton, but not between the electronic ground state and a triplet exciton. The triplet exciton is more strongly bound than the singlet exciton. The magnitude of the energy splitting between the singlet and triplet excitons can be an important parameter in organic LEDs because it helps determine the stability of the triplet exciton. The molecular orbital picture is based on a mean field treatment of electron-electron interactions and assumes that electron correlation effects do not dominate the electronic structure. Electron correlation can be significant in n-conjugated organic materials. If electron correlation does not change the qualitative nature of the ground state, low-energy neutral excited states, and low-energy charged states, its effects can usually be incorporated into a mean field description parametrically. However, there are cases in which electron correlation does qualitatively change the nature of these states. For example, in some polyenes (chains consisting of (CH,), units) electron correlation causes an optical dipole forbidden state to be the lowest energy excited singlet. Using symmetry labels appropriate for polyenes (C,,, point group), the ground state has A, symmetry and the lowest dipole allowed singlet excited state has B, symmetry. This B, state is expected to be the lowest energy singlet excited state on the basis of mean field electronic structure theory. When electron correlation is taken into account another singlet excited state, with A, symmetry, (not well described in mean field theory) can be pulled to lower energy than the B" ~ t a t e . ~ ~ . ' ~The ~ - 'A,' ~ excited state does not have an allowed dipole transition to the electronic ground state; there is an allowed two-photon transition between these states. The B, excited state is stabilized relative to the A, excited state by the phenylene rings that occur in many of the materials used for electronic device^.'^' For LEDs, it is important that the lowest singlet excited state have an allowed dipole transition to the ground l o 6 B. Lawrence, W. E. Torruellas, M. Cha, M. L. Sundheimer, G. I. Stegeman, J. Meth, S. Etemad, and G. Baker, Phys. Rev. Let?. 73, 597 (1994). lo' Z. G. Soos, S. Ramasesha, and D. S. Galvao, Phys. Rev. Let?. 71, 1609 (1993). l o * Z. G. Soos, D. S. Galvao, and S. Etemad, Advanced Materials 6, 280 (1994). l o g Z. G. Soos, S. Ramasesha, D. S. Galvao, R. G. Kepler, and S. Etemad, Synthetic Metals 54,35 (1993). 1 1 0 M. Chandross, Y. Shimoi, and S. Mazumdar, Chem. Phys. Lett. 280, 85 (1997). A. Chakrabarti and S. Mazumdar, Phys. Rev. B59,4839 (1999). D. Yaron and R. Silbey, Phys. Rev. B45, 11655 (1992).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

21

state so that materials in which the A, state is the lowest energy singlet exciton cannot be used for organic LEDs.

6. SOLIDSTATEPROPERTIES The electronic structure of individual molecules can be significantly modified in the dense solid state films used for devices. Quantum chemical techniques can accurately describe many aspects of the electronic structure of individual molecules. Infinite-chain polymers can present a problem for these methods, but the conjugation breaks that occur for real polymers limit the size of the systems that are of practical interest. Understanding which features of the electronic structure of the isolated molecules are maintained and which are significantly modified in the condensed phase and deducing the electronic properties of the condensed phase materials from those of the individual molecules or single-chain polymers is a current theoretical challenge. In many cases, the properties of localized neutral excited states, such as excitons, are not strongly modified going from the individual l 3 Single molecule calculations molecule to the condensed have been successful in describing near edge optical absorption spectra. By contrast the energies of charged states in dense films are strongly modified by screening. As a result, single molecule calculations of the ionization potential or electron affinity do not accurately describe the corresponding observations in dense films. Because the energies of charged states are strongly modified in the condensed phase, it is not straightforward to extract the energy gap of a dense film from single molecule calculations. Because exciton binding energies can be significant, extracting the energy gap from optical absorption data is also not straightforward. The energy gap is an important material parameter for device design. Polarons and perhaps bipolarons are the important charged states in organic materials. If bipolarons were strongly bound, so that a bipolaron had a significantly lower energy than two isolated polarons, they would have an important influence on the behavior of electrical devices. For example, they would play an important role in Schottky barrier formation.79986s87 Because of the possibility of strongly bound bipolarons, it is convenient to define two energy gaps, one corresponding to polaron formation and a second corresponding to bipolaron formation. The singleparticle energy gap, corresponding to polaron formation, is analogous to the energy gap in a conventional semiconductor. It is the energy difference between the electronic ground state and a state consisting of spatially 'I3

R. L. Martin, J. D. Kress, I. H. Campbell, and D. L. Smith, Phys. Rev. (accepted).

22

I. H. CAMPBELL AND D. L. SMITH

separated electron and hole polarons. The charge transfer energy gap, corresponding to bipolaron formation, is smaller than the single-particle energy gap by the bipolaron binding energy. As discussed in Section 111, it is the maximum equilibrium built-in potential that can be supported by the organic material. The materials used for organic electronic devices are, for the most part, highly disordered. The disorder is the result of different local environments of the molecules, different molecular geometries imposed by steric interactions with nearest neighbors, and chemical defects that break up conjugation. As a result of this disorder there is an ensemble of energies for both neutral excitations and charged states. Strong structural relaxation of the molecule occurs when a charge is added or removed and the electron transfer integrals between neighboring molecules are not large. As a result the low-energy charged electronic states of device interest are localized on individual molecules or on the individual conjugated segments of polymers. Band structure calculations have been performed for some of the conjugated materials of device interest.' 14- ' I 6 These calculations do not include disorder or structural relaxation for charged states. They indicate narrow bands corresponding to intermolecular transport. Such calculations are of interest for describing these materials in an ordered limit. There are examples of particularly well-ordered organic materials in which delocalized bandlike states are important.' l 7 But the low-energy states of device interest in the materials typically used for organic electronic devices are localized and electrical transport results from hopping between these localized states. In crystalline semiconductors, covalent chemical bonds are necessarily broken at nonepitaxial interfaces. The lattice matching conditions necessary for epitaxial interfaces are very restrictive. The electronic structure of nonepitaxial interfaces involving crystalline semiconductors are complex and often defect dominated. For example, interface or defect states in the energy gap of the semiconductor usually dominate the electronic properties of metal/semiconductor interfaces. In contrast, covalent chemical bonds are not necessarily broken at interfaces involving organic electronic materials and restrictive lattice matching conditions analogous to those for crystalline semiconductors do not arise. These less restrictive properties are critical to the success of organic electronic devices. They affect the type of substrate that can be used and the properties of both organic/organic and metal/ organic interfaces. Because lattice matching is not an issue, thin films of 'I4

'I5 'I6

P. Vogl and D. K. Campbell, Phys. Rev. B41, 12797 (1990). P. G. DaCosta, R. G. Dandrea, and E. M. Conwell, Phys. Rev. B47, 1800 (1993). M. Rohlfing and S. G . Louie, Phys. Rev. L e f f .82, 1959 (1999). J. H. Schon, S. Berg, Ch. Kloc, and B. Batlogg, Science 287, 1022 (2000).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

23

organic electronic materials may be prepared on almost any substrate. Plastic, low-cost, transparent, and/or flexible substrates have been investigated. Organic/organic interfaces are analogous to inorganic semiconductor heterojunction interfaces except that restrictive lattice matching conditions do not apply and, in many cases, the properties of the organic molecules at the interface are not strongly perturbed. It is thus straightforward to produce near ideal organic/organic heterojunctions simply by depositing (by spin casting or thermal evaporation) one organic material onto another. Analogous to semiconductor heterostructures, these junctions can be critical to efficient device design. At the organic/organic heterojunction, there can be discontinuities in material properties such as the conduction and valence energy levels and the carrier mobility. These discontinuities are used, for example, to confine charge carriers to light-emitting layers to enhance LED efficiency. Metal/organic interfaces are very important for organic devices because they determine the type and amount of charge injected into the undoped, insulating organic layer. Metal/organic interfaces are designed to produce good electrical contacts. It is essential that the metal/organic Schottky energy barriers be made small enough to make good electrical contacts. If the Schottky energy barriers were pinned, as usually happens in inorganic semiconductors, it would be impossible to produce small energy barriers for efficient injection of both carrier types. In some cases, there is effectively a weak interaction between the metal and the organic at the interface. For these interfaces, the electronic structure of the interface components is not strongly modified and it is possible to produce good electrical contacts for electrons by using low work function metals and for holes by using high work function metals. There is strong evidence of chemical reactions at some metal/organic interfaces."8 - I 2 ' However, when an interfacial chemical reaction occurs, it does not necessarily introduce states in the energy gap, formed from the x orbitals, that would pin the Schottky barrier. States associated with CJbonding are usually in a different energy region than the x orbitals and are unlikely to influence the energy gap region. A disruption of the x bonding could produce states in the energy gap, but more often leads to an alternate 7t bonding scheme without making a state in the energy gap. The electronic structure of organic interfaces is thus completely different from that of interfaces involving inorganic T. Kugler, A. Johansson, I. Dalsegg, U. Gelius, and W. R. Salanceck, Synthetic Metals 91, 143 (1997).

C. Fredriksson and J. L. Bredas, J. Phys. Chem. 98,4253 (1993). P. Dannetun, M. Fahlman, C. Fauquet, K. Kaerijama, Y. Sonoda, R. Lazzaroni, J. L. Bredas, and W. R. Salaneck, Synthetic Metals 67, 133 (1994). l Z 1 R. Lazzaroni, J. L. Bredas, P. Dannetun, C. Fredriksson, S. Stafstrom, and W. R. Salaneck, Electrochimica Acta 39, 235 (1994). l9

lZo

24

I. H. CAMPBELL A N D D. L. SMITH

semiconductors, and qualitatively different interface electrical properties are observed. 111. MetaVOrganic Interface Electronic Structure

Organic electronic devices consist of undoped, insulating thin films of conjugated organic materials into which charged carriers are injected from metallic electrodes. The operation of the devices is determined by the charge injection properties of the metal/organic interface and the electrical properties of electron and hole polarons in the organic film. One of the most basic questions concerning the electronic structure of a metal/organic interface is the energy required to inject electrons and holes from the metal contact into the organic material; that is, the difference between the Fermi energy of the metal contact and the energies of the electron and hole polaron states of the organic material. These energy differences are called the electron and hole Schottky energy barriers in analogy with the corresponding injection barriers at metal/semiconductor contacts. In inorganic semiconductors, such as Si and GaAs, Schottky energy barriers are weakly dependent on the type of metal contact for a given semiconductor; i.e., the Schottky energy barriers for various metals on Si are ~imilar.’~ This is not the case at metal/organic interfaces;88s’22- I z 4 there is a qualitative difference in the observed behavior of Schottky energy barriers in conjugated organic materials compared with inorganic semiconductors. Indeed, the essential operating principle of organic diodes is based on the asymmetry in the Schottky energy barriers of the two contacts making up the structure. Organic diode structures are fully depleted metal/organic/metal structures. At zero bias there is a built-in potential in these structures equal to the difference between the electron (or hole) Schottky energy barriers of the two metal contact^.*^^'^^-^^^ There is an electric field in the organic layer that is equal to this built-in potential divided by the thickness of the organic film. Measurements of built-in potentials, in combination with Schottky energy barrier measurements for a few specific metals, provide a method to determine the Schottky energy barriers for a wide range of metal contacts. Schottky energy barriers and built-in potentials play an important role in determining light-emitting diode and field-effect transistor characteristics, as discussed in Section V, and it is important to measure them directly to understand device performance.

”’ X. Wei, S. A. Jeglinski, and 2. V. Vardeny, Synthetic Metals 85, 1215 (1997). lz3

G. G. Malliaras, J. R. Salem, P. J. Brock, and J. C. Scott, J. Appl. Phys. 84, 1583 (1998). I. D. Parker, J. Appl. Phys. 75, 1656 (1994).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

25

The charged states of organic thin films, e.g. electron and hole polarons, have not been as widely studied as neutral excited states, and the energies of charged states in organic thin films are just becoming established. Optical measurements directly probe charge-neutral excited states, such as excitons, but charged states are only indirectly probed by optical spectroscopy and other experimental approaches must be used to determine their properties. For example, it is not straightforward to determine the single-particle energy gap of a conjugated organic material by optical absorption spectroscopy because the exciton binding energy can be significant and the excitons do not have a hydrogenic spectrum in which the threshold for absorption into the continuum states can be easily r e c ~ g n i z e d . ~ ~lZ7~ ~ However, *'~~Schottky energy barrier and built-in potential measurements in device structures can be used to determine the electronic energy structure of organic films. The single-particle energy gap, analogous to the energy bandgap in inorganic semiconductors, is the energy difference between the electronic ground state and a state consisting of an electron polaron and a hole polaron that are spatially separated. It can also be viewed as the sum of the energies required to (1) remove an electron from the material, leaving a hole polaron and putting the electron in a specific reference state, and (2) take an electron from the same reference state and put it into the organic material to form an electron polaron. If the reference state is the Fermi energy of a metal contact, these two energies are the electron and hole Schottky energy barriers and thus, the single-particle energy gap is the sum of the electron and hole Schottky barriers of a metal contact. The charge transfer energy gap, which can be different from the single-particle energy gap if bipolarons are strongly bound, can be determined from built-in potential measurements. In the absence of interface states that pin the Schottky energy barriers, the maximum built-in potential in an organic structure is a direct measure of the charge transfer energy gap. This section discusses built-in potentials and Schottky energy barriers in metal/organic/metal structures for three representative organic materials: MEH-PPV, pentacene, and Alq. Built-in potential and Schottky energy barrier measurements are presented and the use of these results to determine the electronic energy structure of the organic film is described. The electronic energy structure is then interpreted using molecular electronic structure calculations. Finally, based on these results, an approach to controllably manipulating Schottky barriers at metal/organic interfaces is demonstrated. 12' 126

12'

J. L. Bredas, J. Cornil, and A. J. Heeger, Advanced Materials 8, 447 (1996). J. Cornil, A. J. Heeger, and J. L. Bredas, Chem. Phys. Lett. 272, 463 (1997). E. A. Silinsh and V. Capek, Organic Molecular Crystals, AIP Press, New York (1994).

26

1. H. CAMPBELL AND D. L. SMITH

7. BUILT-INPOTENTIALS IN METAL/ORGANIC/METAL STRUCTURES As demonstrated by capacitance-voltage (C-V) measurements, undoped metal/organic/metal structures can be fabricated in which the organic layer is fully depleted at reverse, zero, and small forward bias.49~'28*'29 Th'1s means that the electric charge in the material is small enough that it does not significantly perturb the electric field in the structure and, therefore, the electric field is spatially uniform across the device. At zero bias, there is a built-in potential (Vbi) in these structures equal to the difference between the electron (or hole) Schottky energy barriers of the two metal contacts. There is an electric field in the organic layer that is equal to this buit-in potential divided by the thickness of the organic film. These fully depleted structures can be used to make measurements that determine the built-in potentials and Schottky energy barriers for different metals. Figure 111.1 (top) shows an energy level diagram of a metal/organic/metal structure with asymmetric metal contacts in thermal eq~ilibrium.'~'The vertical axis is energy and the horizontal axis is position. The slanting solid lines (slanting dashed lines) on the top and bottom represent the formation energies for negatively and positively charged polarons (bipolarons), respectively. The solid horizontal line represents the spatially constant electrochemical potential. The ovals on each side of the diagram represent the energy levels of possible, localized interface states. The formation energies are functions of position because of the electrostatic potential change across the structure. There is a built-in electrostatic potential (Vhi) in the device equal to the difference between the electron (4e) (or hole [4J) Schottky energy barriers of the two metal contacts, i.e. Vbi = 4e,l- $I, or, equivalently, &, - 4h,l.The single-particle energy gap is the sum of the electron and hole Schottky energy barriers of a single contact, e.g. 4e,l 4h,l. In thermal equilibrium, the electrochemical potential is constant across the structure. The electrochemical potential can be divided into the sum of two pieces, the electrostatic potential and the chemical potential. By measuring the built-in electrostatic potential change across the structure at equilibrium (Vhi), one can determine the change in chemical potential. The charge transfer energy gap between the bipolaron levels is the maximum change in chemical potential that can be supported by the organic material. If the chemical potential goes above the electron or below the hole bipolaron formation energy per particle, these intrinsic states will charge to

+

I. H. Campbell, D. L. Smith, and J. P. Ferraris, Appl. Phys. Lett. 66, 3030 (1995). Y. F. Li, J. Gao, G. Yu, Y. Cao, and A. J. Heeger, Chem. Phys. Lett. 287, 83 (1998). I3O I. H. Campbell, P. S. Davids, J. P. Ferraris, T. W. Hagler, C. M. Heller, A. Saxena, and D.L. Smith, Synthetic Metals 80, 105 (1996). 12'

12'

PHYSICS OF ORGANIC ELECTRONIC DEVICES

21

FIG. 111.1. Energy level diagrams of an asymmetric metal/organic-film/metalstructure at equilibrium. Energy is on the vertical and position is on the horizontal axis. The solid lines represent the formation energy of charged polarons and the dashed lines represent the formation energies per particle of charged bipolarons. The energy levels are shown including the built-in electrostatic potential (top) and with the built-in electrostatic potential subtracted (bottom). The Schottky energy barriers for electrons (4.) and holes (43 metal contacts are indicated (top). The empty (filled) ovals represent the energies of electron (hole) trap interface states. (Reprinted from Ref. 130, copyright 1996, with permission from Elsevier Science.)

a high density. Figure 111.1 (bottom) shows the same energy level diagram when the effect of the electrostatic potential has been subtracted. The slanting solid line is the spatially varying chemical potential. The net change in chemical potential across the structure is equal to the built-in electrostatic potential. The chemical potential cannot be pushed above the formation energy per particle of the lowest energy intrinsic negatively charged excitation or below the formation energy per particle of the highest energy intrinsic positively charged excitation. Neither can the chemical potential be pushed above a high density of interface states that can charge negatively or below a high density of interface states that can charge positively.

28

I. H. CAMPBELL AND D. L. SMITH

Work Function Difference (eV)

FIG. 111.2. Calculated built-in potential as a function of metal work function difference for an organic film with a single-particle energy gap of 2.4 eV; weakly bound bipolarons and no traps (solid line), bipolarons with a 0.5 eV binding energy and no traps (dashed line), and weakly bound bipolarons with 10’’ ~ 1 3 electron 1 ~ ~ traps in the middle of the single-particlegap (dotted line) (from Ref. 130).

If there is no state charging, the built-in potential across a metal/organic/ metal structure will be the difference between the work functions of the two metals. A smaller built-in potential implies that charging has occurred. A smaller built-in potential could be due to charging of either intrinsic states of the organic material or interface states. If small built-in potentials not related to intrinsic charged excitation energies are observed, it is a clear indication that interface state charging is the limiting process. For the organic materials used for electronic devices, the built-in potentials are often found to scale directly with the metal work functions over a wide range and built-in potentials nearly as large as the single-particle energy gap are observed. A model describing built-in potentials in metal/organic/metal structures, limited by charged intrinsic excitations and not by interface states, is presented in Ref. [87]. Polarons and bipolarons are the important charged excitations. The model includes possible extrinsic charged states such as those introduced into MEH-PPV by doping with C60.Fig. 111.2 shows the calculated built-in potential across a 50-nm organic film sandwiched between 2 metal contacts as a function of the difference in work function between the metals forming the contacts.*’ The single-particle energy gap is 2.4 eV. The work function of one of the contacts coincides with the negative polaron formation energy and the work function of the other contact is varied. The calculations are for room temperature. The solid line is for negligible bipolaron binding energy and no traps, the dashed line is for a

29

PHYSICS OF ORGANIC ELECTRONIC DEVICES

0.5-eV bipolaron binding energy and no traps, and the dotted line is for negligible bipolaron binding and 1 . lozo cm-3 electron accepting traps at an energy midway between the negatively and positively charged polaron levels. Because of thermal occupation of the polaron levels, essentially no built-in potential occurs until the work function drops to about 0.2 eV below the negative polaron formation energy. The built-in potential then increases approximately linearly with work function until it saturates at about 2.1 V. The saturation value of the built-in potential is smaller than the single-particle energy gap because of thermal occupancy of the polaron levels. When bipolarons with a significant binding energy are included, the maximum built-in potential decreases by slightly less than the bipolaron binding energy. The inclusion of a high density of trapping sites, such as occurs by C,, doping of MEH-PPV, also decreases the maximum built-in potential. These theoretical results provide a reference point for the experimental results presented following. 8. BUILT-INPOTENTIAL AND SCHOTTKY ENERGYBARRIER TECHNIQUES MEASUREMENT Capacitance-voltage measurements show that the charge density in the organic films can be low enough that the electric field across the bulk of the film is uniform. The size of the uniform electric field, and thus the built-in potential, can be measured by electroabsorption. The electroabsorption response of the organic film at a given photon energy is proportional to the imaginary part of the nonlinear susceptibility, Irn~(~)(hv), and the square of the electric field -AT A M ( ~ v ) -(hv) a I m ~ ( ~ ’ ( h v ) P T where M is the absorption coefficient, hv is the photon energy, T is the transmission, and E is the electric field.’ 3 1 In the experiment, the electric field consists of a DC component and an applied AC component E = Ed, -I-EacCOS(Rt)

(34

and the electroabsorption response is

-AT (hV) T

CC

+

Im~(~)(hv)(E”,(l + COS(2Rt))/2 -k 2EacEd,CO.S(~t) E:,)

(3.3)

where Ed, is the DC electric field, E,, is the amplitude, and SZ is the angular 13’

D. E. Aspnes and J. Rowe, Phys. Rev. B5,4022 (1972).

30

I. H. CAMPBELL AND D. L. SMITH

frequency of the applied AC electric field. In the presence of a DC electric field, the electroabsorption response is modulated at both the fundamental and the second harmonic frequency of the applied AC bias. The response at the fundamental frequency is -AT (hv; n)a 1 m p ( h v ) ( 2a~c ~ d c cos ) (nt) T

(3.4)

and the response at the second harmonic of the AC frequency is -AT (hv; 2Q) a Im~(~’(hv)(E2,~/2) cos (2nt) T

(3.5)

If there is no DC electric field then the response is modulated only at the second harmonic frequency of the applied AC bias. The DC field consists of the electric field from the equilibrium built-in potential and an applied DC bias. The size of the built-in potential can be determined by measuring the ratio of the electroabsorption response at the fundamental and at the second harmonic frequency for a known AC bias and no applied DC bias. The function describing the material optical properties I m ~ ‘ ~ ) ( hisv )the same at both the fundamental and the second harmonic frequencies and divides out when the ratio is taken. The size of the built-in potential can also be found by applying an external DC bias and monitoring the electroabsorption signal at the fundamental frequency of the AC bias to determine the applied DC bias required to cancel the built-in potential. In the second method, it is necessary that significant charge injection not occur at the bias voltage necessary to cancel the built-in potential. This is usually the case. The first approach is more versatile than the second and can be used to measure, for example, electric fields in multilayer devices under high carrier injection conditions. The second method is easier to apply and more accurate when it is appropriate. The two approaches give consistent results when they can both be used. To use electroabsorption to measure the built-in potential, the material must have a sufficiently large nonlinear response; i.e., I m ~ ( ~ ) ( hmust v ) be large enough to yield significant absorption changes. For some organic molecules used in devices, such as Alq, this condition is not satisfied. For these materials, a second approach, measuring the photocurrent as a function of bias, can be used to determine the built-in potential. In this approach the photocurrent resulting from above gap optical absorption is measured as a function of bias on the structure. The photocurrent signal changes sign when the applied bias reverses the sign of the electric field in the material and vanishes when the bias cancels the equilibrium built-in field. Because there are significant injection currents under bias, modulation

PHYSICS OF ORGANIC ELECTRONIC DEVICES

31

techniques must be used to distinguish the photocurrent. When both electroabsorption and photocurrent vs bias techniques can be used, they give consistent results. Internal photoemission directly measures individual Schottky energy barriers'32 rather than just the difference between Schottky barriers as determined from built-in potential measurements. However, internal photoemission in organic materials works for a limited range of Schottky barrier values. In internal photoemission, optically generated hot electrons (or holes) in a metal contact yield a photocurrent as they traverse a metal/ insulator interface. The photocurrent yield is Yield cc (hv - 4J2

(3.6)

where 4s is the Schottky barrier and hv is the photon energy.'32 The Schottky barrier is determined by extrapolating the photocurrent yield to zero as a function of photon energy. In fully depleted organic materials internal photoemission can be used to determine both electron and hole Schottky energy barriers in the same structure by changing the bias direction. This is in contrast to inorganic semiconductors, where an n-type sample must be used to find the electron Schottky barrier and a p-type sample must be used to find the hole Schottky barrier. In one bias direction, the photocurrent is due to electrons excited over the electron Schottky barrier whereas in the other bias direction, the photocurren; is due to holes excited over the hole Schottky barrier. The single-particle energy gap is the sum of the electron and hole Schottky barriers. Because these two barriers can be measured in the same device, the single-particle energy gap can be determined from measurements on a single device structure. This minimizes problems with irreproducible device fabrication. There is a limit to the range of the Schottky barriers that can be measured using internal photoemission. If the Schottky barrier is too large, it is necessary to use such high-energy photons that photocurrent from absorption in the low-energy tail of the organic material absorption spectrum becomes a problem. If the Schottky barrier is too small, injection currents become a problem. Thus only Schottky barriers near the center of the single-particle energy gap can be determined using internal photoemission. The photocurrent thresholds determined in the presence of an electric field are smaller than the zero electric field Schottky barriers because of the image charge potential created when an electron or hole leaves the metal. The electric field lowers photocurrent thresholds by

A4s = e(eE/&)'I2

(3.7)

13' R. Williams, in Injection Phenomena (R. K. Willardson and A. C. Beer, eds.), Academic Press, New York, (1970), Chapter 2.

32

I. H. CAMPBELL AND D. L. SMITH

where E is the electric field in the organic material and E is its static dielectric ~ 0 n s t a n t . It l ~is~therefore important to know the electric field in the device structure, which is directly determined by built-in potential measurements. The electric field dependence of the photoemission threshold can be investigated by making measurements as a function of bias voltage. These measured results are found to agree with Eq. 111.7 after accounting for the built-in electric field and using a value for the dielectric constant independently determined from capacitance measurements. 9. BUILT-INPOTENTIAL AND SCHOTTKY ENERGYBARRIER MEASUREMENT RESULTS This section presents the results of internal photoemission and built-in potential measurements of metal/organic Schottky energy barriers in three representative organic electronic materials: MEH-PPV, pentacene, and Alq. In addition to determining the Schottky energy barriers to charge injection, which are critical for device applications, these measurements are also used to determine the energies of the fundamental charged states in these organic materials. The MEH-PPV results are presented in detail to serve as a model for the analysis of the other materials. The results for pentacene and Alq molecules are then summarized in light of the discussion of the MEH-PPV results. Electroabsorption measurements of C,, doped MEH-PPV are presented to illustrate the effects of extrinsic electronic states within the energy gap of an organic material. a. M E H - P P V Figure 111.3 shows the electroabsorption spectrum near the absorption edge of an Al/MEH-PPV/Al structure at the fundamental (upper panel) and second harmonic (lower panel) of the applied AC bias.'30*'33 The absorption spectrum is shown for comparison. The magnitude of the electroabsorption signal at the fundamental frequency depends on the size of the DC bias, as shown in the upper panel of Fig. 111.3, whereas the magnitude of the electroabsorption signal at the second harmonic frequency does not depend on the size of the DC bias. The spectral shape is the same at both the fundamental and second harmonic frequencies. Because there is no built-in potential in this symmetric structure, the electroabsorption signal at the fundamental frequency vanishes when no DC bias is applied.

1 3 3 I. H. Campbell, J. P. Ferraris, T. W. Hagler, M. D. Joswick, I. D. Parker, and D.L. Smith, Polymers for Advanced Technologies 8, 417 (1997).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

33

Energy (eV)

FIG. 111.3. Electroabsorption signal as a function of photon energy for an AI/MEH-PPV/AI structure measured at the fundamental (upper panel) and at the second harmonic frequency (lower panel) of the applied AC bias. The three curves in the upper panel are for different values of an applied DC bias. The electroabsorption signal at the second harmonic frequency does not depend on DC bias. The absorption spectrum is shown in both panels for comparison (from Ref. 130).

Figure 111.4 (upper panel) is a plot of the magnitude of the electroabsorption signal as a function of DC bias for a series of metal/MEH-PPV/Al structure^.^^*'^^^' 3 3 The electroabsorption signal is nulled at a voltage corresponding to the bias necessary to cancel the internal electric field produced by the different metal contacts. The signal is nulled when the applied bias equals the built-in potential, Vbi. The DC bias is referenced to the A1 contact, i.e. the Au/Al structure is nulled when the A1 is biased - 1V relative to the Au contact. The bias required to null the electroabsorption signal corresponds closely to the difference between the metal work functions except for the saturation that occurs in the Sm-A1 structure. (The metal work functions are listed in Fig. 111.7.) Figure 111.4 (lower panel) is an analogous plot of the magnitude of the electroabsorption signal as a function of DC bias for a series of metal/MEH-PPV/Ca structure^.^^^'^^^'^^ The bias is referenced to the metal contact, i.e. the Al-Ca structure is nulled when the A1 is biased 1.3 V relative to the Ca contact. The bias required to null the electroabsorption signal corresponds closely to the difference

34

I. H. CAMPBELL A N D D. L. SMITH

FIG. 111.4. Magnitude of the electroabsorption response at a photon energy of 2.1 eV as a function of bias for metal/MEH-PPV/Al structures (upper panel) and metal/MEH-PPV/Ca structures (lower panel) (from Ref. 88).

between the metal work functions except for the saturation that occurs in the Pt-Ca structure. The Pt-Ca structure requires 2.1 V to cancel the built-in electrostatic potential. Thus, MEH-PPV can support a change in chemical potential at least as large as 2.1 eV at room temperature so that the separation between the formation energy per particle of the lowest energy intrinsic negatively charged excitation and of the highest energy intrinsic positively charged excitation must be larger than 2.1 eV. For an organic material with a single-particle energy gap of 2.4 eV (typical of MEH-PPV as discussed following) and no charged excitations with lower energy than the polarons, one expects saturation of the built-in potential at about 2.1 eV (see Fig. 111.2). These results demonstrate that bipolarons are not strongly bound in MEH-PPV." Figure 111.5 shows the result of internal photoemission measurements in which the square root of the photocurrent yield is plotted as a function of photon energy for an Al/MEH-PPV/Ca structure in reverse bias." In

PHYSICS OF ORGANIC ELECTRONIC DEVICES

35

Photon Energy (eV)

FIG. 111.5. Internal photoemission response as a function of photon energy for an Al/MEHPPV/Ca structure biased to collect electrons. The inset shows the electric field dependence of the photoresponse threshold (from Ref. 88).

reverse bias, the photocurrent is due to electrons excited over the electron Schottky energy barrier depicted in the inset in Fig. 111.5. The solid line is a least squares fit to the photocurrent that extrapolates to 1.0 eV. Figure 111.6 is a plot of the square root of the photocurrent as a function of photon energy for the same Al/MEH-PPV/Ca structure as in Fig. 111.5, and a Cu/MEH-PPV/Ca structure, both in forward bias." In forward bias, the photocurrent is due to holes excited over the hole Schottky energy barrier as depicted in the inset in Fig. 111.6. The solid lines are least squares fits to the photocurrents that extrapolate to 1.1 eV and 0.80 eV for A1 and Cu, respectively. The electric field dependence, taking account of the built-in potential in the structures, of the photoemission thresholds shows the expected behavior from Eq. 111.7, as shown in the inset of Fig. 111.5. The

FIG. 111.6. Internal photoemission response as a function of photon energy for an Al/MEHPPV/Ca structure and a Cu/MEH-PPV/Ca structure biased to collect holes (from Ref. 88).

36

I. H. CAMPBELL AND D. L. SMITH Work Function Sm 2.1 Ca 2.9 -

MEH-PPV 2.90

- Sm, Ca 3.0 A l A'? ._. .._ - Ag 4.3 -

A1 4.3 Ag 4.3 =

Cu 4.6 AU 5.1 Pt 5.6 -

Fermi Energy -.

c_ u_ 4.6

5.30

= Au 5.1 Pt 5.2

FIG.111.7. Energy level diagram for MEH-PPV and a series of metal contacts deduced from the electroabsorption and internal photoemission measurements. The line at 2.9 eV (5.3 eV) corresponds to electron (hole) polarons in MEH-PPV. The measured Fermi energies of the metals in contact with MEH-PPV are shown on the right. The work functions of the metals are shown on the left (from Ref. 88).

zero field Schottky barriers are A1 electrons 1.2 eV, A1 holes 1.2 eV, and Cu holes 0.9 eV. The single-particle energy gap of MEH-PPV, determined by the measured electron and hole Schottky barriers on the A1 structure, is 2.4 eV.88 The single-particle energy gap is about 0.2 eV larger than the absorption threshold of 2.2 eV. The energy difference of 0.2 eV between the single-particle energy gap and the absorption threshold is the exciton binding energy.88*'34 The electroabsorption and photoemission results provide a consistent picture of the electronic structure of MEH-PPV. Figure 111.7 is an electronic energy diagram of MEH-PPV derived from these measurements. The uncertainty in energy values is about kO.1 eV. The metal contacts to MEH-PPV are accurately described by the ideal Schottky picture, in which the electron (hole) Schottky barrier is determined by the energy difference between the work function of the metal and the electron (hole) polaron level of the material. The metal work functions listed in Fig. 111.7 were measured in situ using Kelvin probe techniques. The Kelvin probe measurements were relative to (111) Au that was taken to be 5.3 eV;'35 the Au films used for contacts were polycrystalline and had a slightly smaller work function. The Kelvin probe results were close to the standard literature values.'35 The electron (hole) polaron level is at 2.9 eV (5.3 eV). The charged bipolaron binding energies are less than 0.1 eV.

134 S. F. Alvarado, P. F. Seidler, D. G. Lidzey, and D. D. C. Bradley, Phys. Rev. Lett. 81, 1082 (1998). 13' H. B. Michaelson, in CRC Handbook of Chemistry and Physics (R. C. Weast and M. J. Astle, eds.) CRC Press, Boca Raton, Florida (1982), p. E-79.

PHYSICS OF ORGANIC ELECTRONIC DEVICES

3 5 -1

37

c

Au

c

a

0

-1

1

Work Function Difference (eV)

FIG. 111.8. Calculated (solid line) and measured (points) potential difference across metal/ MEH-PPV/A1 structures as a function of the work function difference of the contacts (from Ref. 88).

Figure 111.8 shows the calculated and the measured built-in potential as a function of the work function difference with respect to Al. The calculations used the energy levels of Fig. 111.7 and the model of Ref. [87]. Figure 111.9 is a similar plot with the work function difference referenced to Ca. The calculated results are not sensitive to bipolarons for bipolaron binding energies less than 0.1 eV. There is good agreement between the measured and calculated results, indicating that the energy level scheme provides a quantitative description of both the internal photoemission and electroabsorption results. C,,-doped MEH-PPV structures were investigated to show the effect of an extrinsic trapping level. C,, introduces an electron trap below the electron polaron level in MEH-PPV, but does not add charged carriers.

:

xp Au

-2

-1

0

Work Function Difference (eV)

FIG. 111.9. Calculated (solid line) and measured (points) potential difference across metal/ MEH-PPV/Ca structures as a function of the work function difference of the contacts (from Ref. 88).

38

I. H. CAMPBELL AND D. L. SMITH

IEQ1ACm MEH-PPV

MEH-PPV

FIG. 111.10. Schematic representation of the built-in potentials in undoped (a) and C,,-doped (b) MEH-PPV. The upper panels show the relative alignment of the electron and hole polaron energy levels of MEH-PPV, the electron acceptor level of C,,, and the work functions of Pt and Ca metals before the metals and the polymer are in contact. The lower panels show the built-in potential for the structures after contact (from Ref. 136).

Figure 111.10 is a series of schematic energy level diagrams indicating the effect of C,, doping on the built-in potential in MEH-PPV.63*'36-'38The upper panel in Fig. 111.10 shows the relative alignment of the electron and hole polaron energy levels of MEH-PPV and the work functions of Pt and Ca before the metals and the polymer are in contact. The Pt/MEH-PPV/Ca structure after contact is shown in the lower panel of Fig. 111.10. After contact, there is a built-in potential in the polymer film, Vbi, only slightly smaller than the energy gap of MEH-PPV. Figure 111.10 is analogous to Fig. 111.10 except for C,,-dOped MEH-PPV. Before contact, the relative alignment of the energy levels is the same with the addition of the C,, electron acceptor energy level that lies within the energy gap of MEH-PPV. The energy separation between the hole polaron level and the C,, electron accepting level is labeled AC60. After contact, there is a built-in potential in the structure slightly smaller than AC60. In this case, the chemical potential at the Ca contact is pinned near the electron acceptor level of C,, and the Pt contact is pinned near the hole polaron level of MEH-PPV as before. C. M. Heller, I. H. Campbell, D. L. Smith, N. N. Barashkov, and J. P. Ferraris, J. Appl. Phys. B81, 3227 (1997). 13' N. S. Sariciftci and A. J. Heeger, Synthetic Metals 70, 1349 (1995). E. Maniloff, D. Vacar, D. McBranch, H. L. Wang, B. Mattes, and A. J. Heeger, Synthetic Metals 84, 547 (1997).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

39

Bias (V) FIG. 111.11. Magnitude of the electroabsorption signal at a photon energy of 2.1 eV as a function of external DC bias for metal/MEH-PPV/Ca structures (upper panel) and for metal/C,,-doped MEH-PPV/Ca structures (lower panel) (from Ref. 136).

By measuring the built-in potential in structures employing metal contacts that are not pinned at the hole polaron level of MEH-PPV it is possible to determine the energy difference between the C,, electron acceptor level and the work function of the metals. Figure 111.11 shows the magnitude of the electroabsorption signal as a function of DC bias for both metal/MEH-PPV/Ca and metal/C,,-doped MEH-PPV/Ca structures.'36 For all of the C,,-doped structures, the built-in potential is reduced by about 0.6 eV compared to the results for undoped MEH-PPV. It is clear that the chemical potential is pinned at the low work function Ca contact because changing the higher work function metal changes the built-in potential demonstrating that the high work function contact is not pinned. Figure 111.12 shows the calculated built-in potential across both an undoped and a C,,-doped MEH-PPV film sandwiched between two metal contacts as a function of the difference in work function between the high work function metal and the fixed Ca contact.' 3 6 The experimental built-in

40

I. H. CAMPBELL AND D. L. SMITH I

I

I

I

Work Function Difference (W,-W,J

I

(ev)

FIG.111.12. Calculated and measured built-in potential as a function of metal work function difference for undoped and C,,-doped MEH-PPV. The calculated built-in potential for undoped (upper solid line) and C,,-doped (lower dashed line) MEH-PPV are in good agreement with the measured built-in fields for undoped (squares) and for C,,-doped (diamonds) metal/polymer/Ca structures (from Ref. 136).

potentials are also shown in Fig. 111.12. The electron and hole polaron energy levels of MEH-PPV and the work functions of the metals used in the calculation are those shown in Fig. 111.7. The C,, molecular density was 4 . 1019cm-3.The model describes the data well for a C,, acceptor energy level 1.7 eV above the MEH-PPV hole polaron level. For MEH-PPV, there are no intrinsic charged excitations in an energy range almost as large as the single-particle energy gap; that is, between the formation energy of negatively and positively charged polarons. For the metals investigated, the size of the built-in potential in MEH-PPV films is not limited by charged interface states, That is, interface electron traps, which could charge negatively, are not introduced significantly below the metal work function and interface hole traps, which could charge positively, are not introduced significantly above the metal work function when the metal work function is in the single-particle gap. This observation does not imply that there is no chemical interaction between the MEH-PPV and the metals, and there is evidence that chemical reactions do occur," 39*140 but 8v1

13'

T. Kugler, W. R. Salaneck, H. Rost, and A. B. Holmes, Chem. Phys. Lett. 310, 391 (1999).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

41

Bias (V)

FIG. 111.13. Magnitude of the electroabsorption signal at a photon energy of 1.8 eV as a function of external D C bias for metal/pentacene/Ca structures.

rather that interface states which limit the built-in potential are not generated by the reactions. C,,-doping introduces an electron accepting state in the single-particle energy gap of MEH-PPV. This state limits the built-in potential in C,,-doped MEH-PPV films and therefore it can be identified by built-in potential measurements. The work function of some metals, such as Pt, lie outside of the single-particle energy gap of MEHPPV. The built-in potential of a device structure consisting of an MEH-PPV film with one of the electrodes being Pt will therefore saturate and be less than the difference in work functions of the two contacts. b. Pentacene Figure 111.13 shows the electroabsorption signal as a function of DC bias voltage for three metal/pentacene/metal structures with increasing built-in potentials. The built-in potential is about 1.3 V for Al/Ca and about 1.8 V for both Au/Ca and Pt/Ca structures. Figure 111.14 shows a plot of the square root of the measured photocurrent yield for the electron Schottky barrier of Ca (Ca top contact) and the hole barrier of A1 (A1 top contact) for pentacene. The measurements were performed with a bias applied to the test structures producing an electric field of about 2 x lo5 V/cm in the pentacene layer. The zero electric field Schottky barriers are about 0.6 eV for electrons from Ca and about 0.6 eV for holes from Al. The energy level diagram for metal/pentacene contacts deduced from the internal photoemission and electroabsorption measurements is shown in Fig. 111.15. The work functions of the metals are shown on the left and the 140

I. G. Hill, A. Rajagopal, A. Kahn, and Y. Hu, Appl. Phys. Lett. 73, 662 (1998).

42

I. H. CAMPBELL AND D. L. SMITH

FIG. 111.14. Internal photoemission response as a function of photon energy for a Ca/ pentacene/Ca structure biased to collect electrons (upper panel) and an Al/pentacene/Al structure biased to collect holes (lower panel).

Work Function

Ca 2.9

-

Pentacene 2.4

A1 4.3 AU 5.1 -

Pt 5.6 -

Fermi Energy

-Ca 3.0 - A14.3 - Au, Pt4.8

4.9

FIG. 111.15. Energy level diagram for pentacene and a series of metal contacts deduced from the electroabsorption and internal photoemission measurements. The line at 2.4 eV (4.9 eV) corresponds to electron (hole) polarons in pentacene. The measured Fermi energies of the metals in contact with pentacene are shown on the right. The work functions of the metals are shown on the left.

PHYSICS OF ORGANIC ELECTRONIC DEVICES

-0.5

0.0

0.5

1.0 1.5 Bias (V)

2.0

43

2.5

FIG. 111.16. Magnitude of the photocurrent signal as a function of bias voltage at a photon energy of 2.7 eV for metal/Alq/metal structures. The photocurrent signal goes to zero when the applied bias cancels the built-in potential (from Ref. 141).

measured energy positions of the metals in contact with pentacene are shown on the right. The ionization potential (hole polaron level) of pentacene is about 4.9 eV, as determined from UPS measurement^.'^^ From the internal photoemission and photocurrent vs bias measurements, the electron polaron level is about 2.4 eV and the hole polaron level is about 4.9 eV; the energy gap is about 2.5 eV. The absorption threshold is 1.7 eV and therefore the exciton binding energy is about 0.8 eV. No evidence of bipolarons is observed, as expected for small molecules. c. Alq The electroabsorption signal in Alq is small and photocurrent as a function of bias voltage was used to determine the built-in potential for a series of metal/Alq/metal structures. Figure 111.16 shows the AC photocurrent signal as a function of DC bias voltage for five structures with increasing built-in potential^.'^' The built-in potential in the structure is the bias at which the photocurrent is a minimum. The built-in potential is near zero for Ca/Ca and Sm/Ca structures, about 0.4 V for Al/Ca, about 1.6 V for Au/Mg, and about 2 V in Pt/Ca structures. The built-in potential of 2 V in Pt/Ca structures shows that the difference between the electron Schottky barriers of Pt and Ca is 2 eV. Figure 111.17 is a plot of the square root of the measured photocurrent yield for electron Schottky barriers of Ca, Mg, and A1 (using Ca top contacts) and a similar plot for the hole Schottky barriers of Cu and Au 14'

I. H. Campbell and D. L. Smith, Appl. Phys Lett. 74, 561 (1999).

44

I. H. CAMPBELL A N D D. L. SMITH

Electron Barrier

+

Hole Barrier

0.5

1 .o 1.5 Photon Energy (eV)

2.0

FIG.111.17. Internal photoemission response as a function of photon energy for metal/Alq/ Ca structures biased to collect electrons (upper panel) and metal/Alq/Al structure biased to collect holes (lower panel) (from Ref. 141).

(using A1 top contacts).141 The measurements were performed with a bias applied to the test structures producing an electric field of about 2 x lo5 V/cm in the Alq layer. The solid line is a least-squares fit to the data that extrapolates to electron injection thresholds of about 0.5 eV for Ca and Mg, and 0.9 eV for Al, and hole injection thresholds of 0.7 eV and 1.3 eV for Au and Cu, respectively. The image charge potential lowers the extrapolated threshold in the films by about 0.1 eV for each case. The energy level diagram for metal/Alq contacts deduced from the internal photoemission and photocurrent vs bias measurements is shown in Fig. 111.18. The work functions of the metals are shown on the left and the measured energy positions of the metals in contact with Alq are shown on the right. The ionization potential (hole polaron level) of Alq is about 6.0 eV as determined from ultraviolet photoemission m e a ~ u r e m e n t s . ' ~ ~ , ~ ~ ~ From the internal photoemission and photocurrent vs bias measurements, the electron polaron level is about 3.0 eV and the hole polaron level is about 6.0 eV; the energy gap is about 3.0 eV. The optical absorption threshold of Alq is about 2.7 eV43 and therefore the exciton binding energy is roughly 142 143

A. Schmidt, M. L. Anderson, and N. R. Armstrong, J. Appl. Phys. 78, 5619 (1995). A. Rajagopal, C. I. Wu, and A. Kahn, J. Appl. Phys. 83, 2649 (1998).

PHYSICS O F ORGANIC ELECTRONIC DEVICES

Work Function

Sm 2.7 eVCa 2.9 eV Mg 3.6 eVA14.3 eV Cu 4.6 eV -

Fermi Energy Ec 3.0 eV

- Ca, Sm, Mg 3.6 eV - Al4.0eV '443

Au 5.1 eVPt 5.6eV-

45

Ev 6.0 eV

- Cu4.6 eV - Au5.2eV - Pt 5.6eV

FIG. 111.18. Energy level diagram for Alq and a series of metal contacts deduced from the photocurrent and internal photoemission measurements. The line at 3.0 eV (6.0 eV) corresponds to electron (hole) polarons in Alq. The measured Fermi energies of the metals in contact with Alq are shown on the right. The work functions of the metals are shown on the left (from Ref. 141).

0.3 eV. The three low work function metals investigated, Sm, Ca, and Mg, all had an electron Schottky barrier of about 0.6 eV. In contrast to MEH-PPV, the electron Schottky barrier in Alq is pinned for these low work function metals. For metals with work functions larger than Mg the ideal Schottky model provides a generally accurate description of the energy barrier. The smallest barrier to electron injection was about 0.6 eV for Ca, Mg, and Sm, and the smallest barrier to hole injection was about 0.4 eV for Pt. No evidence of bipolarons is observed, as expected for small molecules. 10. SOLIDSTATEAND MOLECULAR PROPERTIES The built-in potential and internal photoemission measurements give important information about the energies of the excited and charged states in the organic materials. Specifically the exciton binding energies of MEH-PPV, pentacene, and Alq were found to be 0.2, 0.8, and 0.3 eV, respectively; and charged bipolarons in MEH-PPV are weakly bound, if at all. The exciton binding energy of pentacene was previously known from optical measurements of charge transfer state^,^ but this approach could not be applied to MEH-PPV or Alq because the corresponding charge transfer states do not appear in their optical spectra. The device results agree with the earlier optical result for pentacene. Previous estimates of the exciton binding energy of MEH-PPV varied over a wide range, from essentially zero to well over 1 eV. The disagreement was partially the result of different definitions. In some

46

I. H. CAMPBELL AND D. L. SMITH

theoretical discussions the exciton binding energy was taken as a measure of the importance of correlation effects in the molecule. To estimate exciton binding energies theoretically, it is necessary to correctly describe both the exciton, a neutral excited state, and a state consisting of electron and hole polarons spatially separated, essentially two charged states. Single molecule calculations can describe the exciton state reasonably accurately without large corrections from solid state effects. However, there are large polarization effects to the energies of charged states that must be added to molecular calculations to describe the charged states. When these polarization effects are taken into account, the measured charged state energies can be described and the measured exciton binding energy explained. This point has been illustrated in a series of calculations on Alq. In Ref. [113], a hybrid density-functional-theory approach was used to calculate the ground state electronic properties and a time-dependent density-functional-theory approach was used to investigate the excited state electronic properties of the Alq molecule. The calculated molecular results were compared with measurements on dense solid state films of Alq. The molecular calculations describe the optical absorption spectrum near the fundamental absorption threshold without significant corrections from solid state effects, but large dielectric corrections must be included for the molecular calculations to describe the measured ionization potential and single-particle energy gap. When these dielectric corrections are made, using the calculated molecular polarizability, both the measured ionization potential and single-particle energy gap are well described. Vertical excitation energies and oscillator strengths were calculated for the first ten excited singlet states of Alq. The optical absorption spectrum of a thin solid film of Alq is shown in Fig. 111.19.”’ For comparison, graphical representations of the calculated singlet excitation energies and oscillator strengths are also shown in Fig. 111.19. Overall, the agreement between calculation and experiment is very good for the low-energy transitions. The calculated onset of absorption is at 2.77 eV, in close agreement with the observed onset. The most intense transitions are underestimated by a few tenths of an eV. Above about 3.5 eV the agreement with experiment is not as satisfactory, probably due to a basis set artifact. Overall, the fundamental absorption edge in the dense solid state film is well described by the molecular calculations. The calculated molecular ionization potential (IP,) is 6.60 eV and the calculated molecular electron affinity (EA,) is 0.83 eV for vertical transitions in which the molecular geometry of the ground state is also used for the ions. The calculated structural relaxation energies are 0.09 eV for the positive ion and 0.11 eV for the negative ion. The static polarizability was found to be basically isotropic with c1 = 327 a.u. The experimental solid state ionization potential of Alq is between 5.6 and

PHYSICS OF ORGANIC ELECTRONIC DEVICES

47

Photon Energy (eV) FIG. 111.19. The near gap optical absorption spectrum of Alq. The vertical lines represent the energies and oscillator strengths calculated using time-dependent DFT (from Ref. 113).

6.0 eV,'42-'45 nearly 1 eV smaller than the calculated molecular ionization potential. The measured solid state energy gap, 3.0 eV, is more than 2 eV smaller than the computed difference between the molecular ionization potential and electron affinity, IP, - EA, = 5.8 eV. These differences are due to the additional stabilization associated with the charged states in the solid state film resulting from polarization of the neighboring molecules and from structural relaxation. Polarization stabilizes both the positive ion, decreasing the ionization potential of the solid, and the negative ion, increasing the electron affinity. Such polarization corrections are much less important for the neutral excited states that appear in optical property calculations. The solid state ionization potential and single-particle energy gap of the Alq film can be calculated using the theoretical electron affinity and ionization potential of the molecule and correcting for solid state polarization effects and structural relaxation. The permanent dipole moments of the various Alq molecules are oriented so that their interaction energy with the localized charge cancels and on average makes no net contribution to the stabilization energy. This interaction of the charge with the permanent dipole moments leads to an inhomogeneous broadening, but not a net shift, of the energy distribution. The induced dipoles are oriented by the localized charge and are all directed toward (or away from) the charge and have the 144 Y. Hamada, T. Sano, M. Fujita, T. Fujii, Y. Nishio, and K. Shibata, Jpn. J. Appl. Phys. 32, L514 (1993). 145 M. Probst and R. Haight, Appl. Phys. Lett. 71,202 (1997).

48

1. H. CAMPBELL A N D D. L. SMITH

same sign energy contribution. The stabilization energy due to interaction with the induced dipoles is (3.8)

ai

is the induced dipole moment at molecule i and ri is the where intermolecular distance. To estimate this induced polarization correction we enclose the charged Alq molecule of interest in a spherical cavity also containing the ten nearest Alq molecules in the x-ray crystal structure. The radius of this cavity is ro = 1.1 nm. The polarization energy from the ten nearest neighbor molecules is calculated directly. The additional stabilization associated with the remaining molecules is treated using a continuum approximation and the Clausius-Mossotti relation. The stabilization energy is then

where the sum is over the ten neighbor molecules and the second term is the contribution from the more distant molecules. The solid state ionization potential and single-particle energy gap are then Ips

=Ipm

Eg

= (Ip,

-

(3.10)

Epolar - Estruct(h)

- EAm) - 2Epolar

- Estruct(e)

- Estruct(h)

(3.11)

where Estruct(e,h) is the structural relaxation energy for the electron (hole) polaron. The sum above, to the ten nearest molecules, gives a contribution to the stabilization energy of 0.70 eV, and the continuum contribution is 0.44 eV. The total polarization correction for each ion is 1.14 eV. The calculated structural relaxation for the hole polaron is 0.09 eV. The solid state ionization potential is then 6.60 eV - 1.14 eV - 0.09 eV = 5.37 eV, in reasonable agreement with experimental values ranging from 5.6 to 6.0 eV. The sum of the calculated structural relaxation energies is 0.20 eV. The energy gap is then 6.60 eV - 0.83 eV - (2 x 1.14 eV)-0.2 eV = 3.29 eV. This is in reasonable agreement with the experimental result of 3.0 eV f0.2 eV. These results show that single molecule calculations can describe localized neutral excitations in solid state films reasonably well, but that large polarization corrections are necessary to describe charged states in solid state films using single molecule calculations. When these corrections are included, the measured ionization potential, energy gap, and exciton binding energy are reasonably described.

PHYSICS OF ORGANIC ELECTRONIC DEVICES

49

11. MANIPULATING SCHOTTKY ENERGY BARRIERS USING DIPOLELAYERS

The Schottky energy barriers between a metal and a conjugated organic material are important parameters for device operation. Small Schottky barriers are required for efficient electrical injection. For some materials, such as MEH-PPV, small barriers can be reached using common metals for electrodes, although reactive low work function metals such as Ca are needed to achieve small electron barriers. For other materials such as Alq, small energy barriers cannot be reached using common metals for the electrodes. It would be useful to controllably manipulate Schottky barriers so that small barriers can be reached on all materials and the use of reactive metals can be avoided. 146- 149 Self-assembled monolayer (SAM) techniques can be used to attach a monolayer of polar molecules to the surface of a metal. Because of the ordering inherent in SAM structures, the molecular dipoles are oriented relative to the metal surface. The metal work function can therefore be controlled using the oriented SAM dipole layer. Because the Schottky model holds for many organic materials, Schottky barriers can also be controlled using the SAM dipole layer. The schematic energy level diagrams shown in Fig. 111.20 illustrate the basic idea.'46,'47 Figure III.20a represents the untreated metal/organic interface (i.e., there is no SAM dipole layer on the metal). Figures. III.20b and 111.20~show the effect of inserting an oriented dipole layer between the metal and the organic film. In Fig. III.20a, the dipole layer is oriented so that the electron Schottky energy barrier is decreased, and in Fig. III.20b the dipole layer is oriented so that the electron Schottky energy barrier is increased. Fig. III.20d is a magnified view of the interface showing a SAM with an electric field across it representing the effect of the dipole layer. Schottky barrier control was demonstrated using three alkane-thiol adsorbates to form the self-assembled monolayers: CH,(CH,),SH [CH, SAM], NH,(CH,),,SH [NH, SAM], and CF,(CF,),(CH,),SH [CF, SAM]. The chemical structure of the CH, SAM is shown at the top of Fig. 111.21; the chemical structure of the other molecules is similar except the end groups are changed to give different dipole moments.146 When the thiol adsorbate forms a monolayer film on the surface of a column Ib metal, the hydrogen attached to the sulfur in the molecule is removed and the sulfur bonds to the metal. These adsorbates were chosen because their self14' I. H. Campbell, S. Rubin, T. A. Zawodzinski, J. D. Kress, R. L. Martin, D. L. Smith, N. N. Barashkov, and J. P. Ferraris, Phys. Rev. B54, R14321 (1996). 14' I. H. Campbell, J. D. Kress, R. L. Martin, D. L. Smith, N. N. Barashkov, and J. P. Ferraris, Appl. Phys. Lett. 71, 3528 (1997). 14' F. Nuesch, Y. Li, and L. J. Rothberg, Appl. Phys. Lett. 75, 1799 (1999). 149 F. Nuesch, F. Rotzinger, L. SiAhmed and L. Zuppiroli, Chern. Phys. Lett. 288, 861 (1998).

50

I. H. CAMPBELL AND D. L. SMITH

FIG. 111.20. Schematic energy level diagrams of metal/organic interfaces: panel a, untreated interface; panel b (panel c), dipole layer that decreases (increases) the electron Schottky energy barrier; and panel d, magnified view of the interface (from Ref. 146).

assembly properties have been extensively studied and they form dense, well-ordered monolayers, and because the set includes molecules with dipole moments of both signs. Silver was chosen as an electrode because its work function is near the center of the MEH-PPV energy gap so the electron Schottky energy barrier can be either increased or decreased. The change in work function of the chemically treated electrodes was measured with respect to pristine Ag using a Kelvin probe. In the Kelvin probe technique, the substrate metal surface and a metallic, vibrating probe tip constitute the two plates of a capacitor. The vibration of the probe tip induces an AC current. This current is nulled when the voltage applied to the tip is equal to the difference in surface potential between the tip and the substrate. Figure 111.21 shows the Kelvin probe current as a function of relative substrate bias (V,,, - VAg) for a pristine Ag electrode and for Ag electrodes modified by the three S A M S . ' The ~ ~ difference V,,, - V,, repre-

PHYSICS OF ORGANIC ELECTRONIC DEVICES

51

H-S(CHZ)~CH~

+

*~

2.24 D

FIG. 111.21. Kelvin probe current as a function of substrate bias for a pristine Ag electrode and for Ag electrodes modified by the three SAMs. The applied substrate bias has been shifted by the difference in the surface potentials of the Kelvin probe tip and the pristine Ag substrate. The molecular structure of the CH, SAM and its calculated dipole moment is shown above the panel (from Ref. 146).

sents the change in surface potential with respect to the pristine Ag electrode. The CH,, NH,, and CF, SAMs change the surface potential with respect to pristine Ag by -0.70 V, -0.45 V and 0.85 V, respectively. The CH, and NH, SAMs decrease the effective work function and the CF, SAM increases the effective work function of the Ag electrode. The expected surface potential shift due to a molecular dipole layer has the form (3.12)

where N is the areal density of molecules, pmo1 is the dipole moment of the molecule, pAgiSis the screened dipole moment of the Ag'S- bond, and E is a static dielectric constant. For these SAMs, N takes on values between about 3 and 5 . 1014 cm-2 and E is between 2 and 3. The value of pAg+sis difficult to determine precisely, but is expected to be nearly the same for the three SAMs. The dipole moments of the SAMs were calculated using quantum chemistry techniques. The calculated dipole moments were 2.24D, 1.77D, and - 1.69D for the CH,, NH,, and CF, SAMs, respectively. These calculated dipole moments give the observed trends and approximate magnitudes of the surface potential shifts.

52

I. H. CAMPBELL AND D. L. SMITH

Diode Bias (VApp-VAg)

FIG. 111.22. Electroabsorption signal as a function of diode bias for Ag/MEH-PPV/Ca structures with a pristine Ag electrode and for Ag electrodes modified by the three SAMs. The applied substrate bias has been shifted by the built-in potential in the pristine Ag/polymer/Ca structure (from Ref. 146).

To determine the effect of the SAM layers on Schottky barriers, the chemically treated electrodes were incorporated in diode structures with a MEH-PPV layer roughly 50 nm thick and a top Ca contact. Figure 111.22 shows the measured electroabsorption signal as a function of diode bias (VApp- VAg)for a pristine Ag electrode and for Ag electrodes modified by the SAMs.146 Because the calcium/polymer Schottky energy barrier is constant, the difference, VApp- VAg, represents the change in electron Schottky energy barrier with respect to the pristine Ag electrode. The CH,, NH,, and CF, SAMs change the electron Schottky energy barrier by -0.60 V, -0.45 V, and 0.50 V, respectively. These results demonstrate tuning of the Schottky energy barrier of Ag on MEH-PPV in an organic diode structure over a range of more than 1 eV. Figures 111.21 and 111.22 show that changing the effective work function of the Ag substrate produces a corresponding change in the Ag/polymer electron Schottky energy barrier. The shift of the surface potentials seen in Fig. 111.21 is essentially the same as that of the Schottky energy barriers seen in Fig. 111.22, except for the CF, SAM. The effective work function of the Ag/CF, SAM film is greater than the ionization potential of MEH-PPV so the electron Schottky energy barrier saturates at a shift of 0.50 V. These alkane-thiol molecules have large energy gaps that block charge injection. As a result, the diodes with alkane-thiol modified contacts did not have favorable injection properties even though, in some cases, they had small Schottky barriers. Conjugated-thiol SAMs, with modest energy gaps, can improve charge injection. The conjugated-thiol-based SAMs both modify the metal/organic Schottky energy barrier and are sufficiently transparent to electrons to allow improved charge injection. This point is

PHYSICS OF ORGANIC ELECTRONIC DEVICES HS

53

F

15

Diode Bias (V)

FIG. 111.23.Current-voltage characteristics of Cu/MEH-PPV/Ca structures for a pristine Cu electrode and for Cu electrodes modified by the two conjugated SAMs. The fluorinated SAM improves charge injection and the non-fluorinated SAM degrades charge injection. The inset shows electroabsorption measurements of the device built-in potentials; the F SAM increases the built-in potential and the H SAM decreases the built-in potential consistent with the Kelvin probe measurements. The molecular structure of the fluorinated SAM is shown above the panel (from Ref. 147).

emphasized in Fig. III.20d, which shows the electron and hole energy levels of two different SAMs represented by the pairs of solid and dashed lines. The large energy gap alkane-thiol blocks charge injection, while the smaller energy gap conjugated-thiol does not significantly impede electron transfer. Two thiol-adsorbates were used to form conjugated self-assembled mono[F SAM] and HS(C,H,C,),C,H,-H [H layers: HS(C6H,C,),C6H,-F SAM]. The chemical structure of the F SAM adsorbate is shown at the top of Fig. 111.23; the H SAM chemical structure is equivalent except the fluorine atom is replaced by hydrogen. 14’ These adsorbates were chosen because they are known to form dense, well-ordered monolayers and the two molecules shift the Schottky energy barrier in opposite directions. The Cu electrode has a work function within the energy gap of MEH-PPV, which allows the hole Schottky energy barrier to be either increased or decreased, to either improve or degrade charge injection. The inset in Fig. 111.23 shows the measured electroabsorption signal as a function of diode bias for a pristine Cu electrode and for Cu electrodes modified by the two SAMs.’,’ The built-in potential of the pristine Cu/polymer/Ca structure is about 1.5 V; i.e., the electroabsorption signal is zero at 1.5 V. The built-in potential increases to about 1.7 V for the F SAM structures and decreases to about 1.2 V for the H SAM structures. The difference in the built-in

54

I. H. CAMPBELL AND D. L. SMITH

potentials represents the change in hole Schottky energy barrier. Figure 111.23 shows the current density as a function of diode bias for a pristine Cu/polymer/Ca diode and for analogous diodes employing Cu electrodes modified by the two SAMs.14’ The structure with the F SAM has substantially higher current than the pristine Cu structure for a given voltage and, similarly, the current is considerably reduced in the H SAM structure. Because the calcium electrode and the polymer thickness is the same for each diode, these current-voltage characteristics are representative of the change in hole injection from the Cu electrodes. They demonstrate significant improvement in hole injection for the fluorinated SAM and, similarly, a substantial decrease in hole injection for the non-fluorinated SAM. These results demonstrate control and improvement of charge injection in organic electronic devices by utilizing self-assembled monolayers to manipulate the Schottky energy barrier between a metal electrode and the organic electronic material.

IV. Electrical Transport Properties

The charge injection and carrier transport properties of organic materials govern the electrical characteristics of organic electronic devices. Without charge injection, conjugated organic materials have negligible intrinsic carrier concentrations and very high room temperature resistivity. In these insulating, disordered organic films, the carrier mobility is dominated by hopping transport between localized molecular site^.^.^ Although organic materials can be electronically doped, the dopant ions significantly modify the intrinsic properties of the organic material, making carrier mobilities determined in doped materials largely irrelevant to the undoped films used in devices.38 Therefore, measurements of the mobility must be performed on undoped, insulating films. Conventional Hall effect measurements have not been useful for determining carrier mobilities in these insulating organic materials and photo-Hall measurements are complicated by short carrier lifetimes and relatively large exciton binding energies. Two general approaches are used to measure the carrier mobility: (1) measuring the transit time of optically injected carriers across thin film^^*''^-'^^ I. H. Campbell, D. L. Smith, C. J. Neef, and J. P. Ferraris, Appl. Phys. Lett, 74,2809 (1999). M. Redecker, D. D. C. Bradley, M. Inbasekaran, and E. P. Woo, Appl. Phys. Lett. 73,1565 (1998). 15’ M. Redecker, D. D. C. Bradley, M. Inbasekaran, and E. P. Woo, Appl. Phys. Lett. 74, 1400 (1999). 151

PHYSICS O F ORGANIC ELECTRONIC DEVICES

55

and (2) fitting the current-voltage characteristics of d e v i ~ e s . ~ , ~ ~ , ~ ~ , ' ~ ~ The electronic structure of conjugated organic thin films consists of a distribution of localized electronic states with different energies. The site energy distributions are believed to be approximately Gaussian with standard deviations typically between 0.1 eV and 0.2 eV.4 The localized sites are either individual molecules or isolated conjugation segments of a polymer chain. Electrical transport results from carrier hopping between neighboring sites. At room temperature, equilibration between neighboring sites of Merent energy is generally fast enough that carrier transport can be described using a mobility p i ~ t u r e However, .~ at lower temperatures or for systems with very large disorder, the mobility description may not be valid.4 Hopping transport in a disordered system leads to a mobility that can depend strongly on both electric field and carrier d e n ~ i t y The . ~ mobility increases with increasing electric field and with higher carrier densities. As the electric field increases, more states become available for energetically favorable hopping transitions, thus increasing the mobility. At high carrier densities, the mobility is increased because charge transport occurs predominantly in a region with a higher density of states and therefore increased number of energetically favorable hopping sites. Organic diodes and organic field-effect transistors depend on transport normal to and parallel to the plane of the film, respectively. Because the molecular packing in organic thin films is often asymmetrical, particularly for polycrystalline and polymer films, they are likely to have different mobilities normal to and parallel to the plane of the film.162Organic diodes and FETs also operate in distinct electric field and charge density regimes. Organic diodes typically operate at electric fields of several times lo5 V/cm and at carrier densities of up to a few loi7cm-3.163 In contrast, field-effect transistors operate at lateral electric fields of a few lo4 V/cm and at carrier G. G. Malliaras, J. R. Salem, P. J. Brock, and C. Scott, Phys. Rev. B58,R13411 (1998). P. E. Burrows, Z. Shen, V. Bulovic, D. M. McCarthy, S. R. Forrest, J. A. Cronin, and M. E. Thompson, J. Appl. Phys. 79, 7991 (1996). lSS A. J. Campbell, D. D. C. Bradley, and D. G. Lidzey, J. Appl. Phys. 82, 6326 (1997). lS6 A. J. Campbell, M. S. Weaver, D. G. Lidzey, and D. D. C. Bradley, J. Appl. Phys. 84,6737 (1998). 15' P. W. M. Blom, M. J. M. DeJong, and J. M. Vleggaar, Appl. Phys. Lett. 68, 3308 (1996). lS8 P. W. M. Blom, M. J. M. DeJong, and M. G. Van Munster, Phys. Rev. B55, R656 (1997). lS9 H. C. F. Martens, H. B. Brom, and P. W. M. Blom, Phys. Rev. B60,R8489 (1999). 160 L. Bozano, S. A. Carter, J. C. Scott, G. G. Malliaras, and P. J. Brock, Appl. Phys. Len. 74, 1132 (1999). 16' M. A. Lampert and P. Mark, Current Injection in Solids, Academic, New York (1970). 16' C. Y. Yang, F. Hide, M. A. DiazGarcia, A. J. Heeger, and Y. Cao, Polymer 39,2299 (1998). 163 B. K. Crone, I. H. Campbell, P. S. Davids, D. L. Smith, C. J. Neef, and J. P. Ferraris, J. Appl. Phys. 86, 5767 (1999). lS3 lS4

56

I. H. CAMPBELL A N D D. L. SMITH

densities above 10" cm-3.74 For carrier densities below about 1017 cm-3, typical of diode operation, the mobility does not depend strongly on the carrier density.76 In this low density limit, interactions between carriers and state filling effects are not significant. In contrast, for carrier densities above about 10" cm-3, typical of FET operation, occupation of the lower energy hopping sites begins to enhance the carrier mobility.76 Because of the anisotropy in molecular structure and differences in the operating regimes, the mobilities determined from LED and FET measurements are not necessarily equivalent. Three techniques have been used to determine carrier mobilities in organic electronic materials: time-of-flight (TOF) current transient measurem e n t ~ fitting ,~ of single-carrier space charge limited (SCL) diode currentvoltage (I-V) characteristic^,'*^^*^^^^ 5 3 - 161 and analysis of field-effect transistor current-voltage characteristics.' These three mobility measurement techniques sample different electric field and charge density regimes. The time-of-flight technique can measure the carrier mobility for electric fields from about 5 x lo4 V/cm to 1 x lo6 V/cm but is restricted to very low volume averaged carrier densities, about 1013 cmP3 or less. Because the TOF technique is restricted to very low carrier densities, it is difficult to measure the carrier mobility in the presence of extrinsic trap states. Fitting single-carrier space charge limited diode current-voltage characteristics can probe the mobility for electric fields from about lo5 to lo6 V/cm and for carrier densities from about 1OI6 to 10l8 ~ m - These ~ . measurements are much less sensitive to trapping effects than TOF measurements because the comparatively large density of injected carriers can fill moderate densities of traps without significantly perturbing the mobility measurement. In SCL diode measurements, the electric field and carrier density are functions of position within the device structure. Therefore, fitting the measured I-V characteristics requires assumptions about the carrier density and electric field dependences of the mobility. If both TOF, which requires low trap densities, and SCL diode mobility measurements can be performed, they usually yield consistent results. In FET mobility measurements, the lateral electric fields that are responsible for current flow are typically in the lo4 V/cm range and the charge is confined to a thin region of the organic film adjacent to the gate insulator. This leads to very high charge carrier densities of about lo'' ~ m - These ~ . carrier densities are high enough to significantly modify the mobility due to changes in the occupation of hopping sites. In addition, the results are sensitive to the local molecular structure near the interface that may differ significantly from that typical of bulk films.164-16' The FET mobilities are often significantly larger than 164 D. D. C. Bradley, M. Grell, A. Grice, A. R. Tajbakhsh, D. F. OBrien, and A. Bleyer, Optical Materials 9, 1 (1998).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

57

TOF and SCL diode results on the same material. The higher mobility inferred from FET measurements may be due to the different carrier density regimes in which the measurements are made or to variations in the local molecular structure in the two kinds of devices. FET mobility results will be discussed in the transistor part of Section V. The carrier mobilities of different organic materials, as measured by TOF and SCL diode techniques, vary over a large range, from lo-* cm2/vs to ~ m ’ / v s .They ~ are orders of magnitude smaller than those in typical inorganic semiconductor^.^^ The low mobility of the organic materials plays a critical role in determining the characteristics of organic devices. Measured carrier mobilities for films made from a given organic material can differ both because of different processing conditions for the films and because of different synthesis conditions used to make the organic mat e 1 i a 1 . l ~It ~ is difficult to compare detailed mobility results between different research groups who may use different processing conditions and materials made under different synthetic conditions. It is possible to standardize processing conditions so that consistent mobility results can be achieved for films fabricated from material made in a given synthetic run. However, especially for conjugated polymers, films fabricated from materials made in different synthetic runs, even when using the same equipment and nominally the same approach for the synthesis, often give somewhat different mobility results. The electrical properties of the polymer film are sensitive to the film morphology, which can be affected by small changes in the polymer molecular weight distribution that varies slightly from batch to batch. This section focuses on mobility measurements of three representative conjugated organic materials: MEH-PPV, Alq, and pentacene. Carrier mobilities determined by TOF and SCL techniques are compared. The electric field dependence of the mobilities is discussed and theoretical models used to interpret the measurements are described. 2. TW-OF-FLIGHTMOBILITY MEASUREMENTS

Time-of-flight is an established technique to measure carrier mobilities in insulating materials. In this technique, a semitransparent blocking contact/ insulating film/blocking contact structure is used. An optical pulse incident S. Guha, W. Graupner, R. Resel, M. Chandrasekhar,H. R. Chandrasekhar, R. Glaser, and G. Leising, Phys. Rev. Lett. 82, 3625 (1999). 166 L. Athouel, R. Resel, N. Koch, F. Meghdadi, G. Froyer, and G. Leising, Synthetic Metals 101,627 (1999). 16’ P. A. Lane, M. Liess, X. Weis, J. Partee, J. Shinar, A. J. Frank, and Z. V. Vardeny, Chemical PhrJics 227, 57 (1998). Ice H. J. Schon, C. Kloc, R. A. Laudise, and B. Batlogg, Appl. Phys. Lett. 73, 3574 (1998).

58

1. H. CAMPBELL A N D D. L. SMITH

on the material through the semitransparent contact creates a thin sheet of electron-hole pairs next to that contact and, depending on the sign of the applied bias, electrons or holes are driven across the sample. The absorption depth of the optical excitation must be small compared to the film thickness and the optical pulse duration must be short compared to the transit time of the charged carriers across the sample. Low-intensity optical pulses are used so that the photogenerated charge carrier density does not significantly perturb the spatially uniform electric field in the structure. The carrier mobility, p, is determined from the measured carrier transient time, Z, by p=-

d2 ZV

where d is the film thickness and V is the applied voltage. The structures used for TOF measurements consisted of a thin, semitransparent A1 contact on a glass substrate, an organic film a few pm thick, and a top, thick A1 contact. A nitrogen laser pumped dye laser producing 500 ps pulses was tuned to the peak of the absorption coefficient for the organic material so that the absorption depth of the optical pulse was much smaller than the film thickness. The bandwidth of the current preamplifier was two orders of magnitude greater than the reciprocal of the transit time. The product of the structure capacitance and amplifier input impedance was at least two orders of magnitude smaller than the transit time. The total charge injected into the film was about 0.01 CV, where C is the capacitance of the structure and V the applied voltage. Figure IV.l is a log-log plot of the T O F hole current density in MEH-PPV as a function of time after optical excitation for applied biases of 10 V, 40 V, and 100 V at room temperat~re.”~The transit time was determined by the intersection of the asymptotes to the plateau and declining slope of the current transient. The MEH-PPV film was 1.8 pm thick. Figure IV.2 shows the hole mobility as a function of electric field determined from the TOF measurements (markers) and a least-squares fit (solid line) to the Poole-Frenkel form

where E is the electric field and po and E , are parameters describing the mobility. The Poole-Frenkel form for the electric field dependent mobility is frequently observed in organic molecular solids and polymers4. Figure IV.2 shows that the Poole-Frenkel form describes the measured TOF results reasonably well. The fit to the TOF data yielded the parameters p, = 2.1 x lo-’ cm2/vs and E , = 8.7 x lo4 V/cm. The TOF current transi-

PHYSICS OF ORGANIC ELECTRONIC DEVICES

1

-

- . , - - . - I

59

- . - . - - - I

lo0V -

,

,

I

I

, 1 1 1 1 1

I

, ,

o-~

1

Time (s)

FIG. IV.1. Time-of-flight hole current transients for MEH-PPV at three applied voltages. The structure was semitransparent A1 (10 nm)/MEH-PPV (1.8 (m)/Al(lOO nm) (from Ref. 150).

ents are close to the dispersive regime at the largest electric fields4. In linear-linear current transient plots it becomes difficult to distinguish the current plateau at electric fields above about 4 x lo5 V/cm. The error bars on the mobility shown in Fig. IV.2 are estimated from T O F measurements on several different devices. It was not possible to measure hole T O F transients at significantly lower temperatures. At 250K the T O F plateau and falling edge could no longer be clearly distinguished over a significant range of electric fields. Neither was it possible to measure electron T O F transients. Space charge limited diode I-V measurements, discussed following, show that the electron mobility of MEH-PPV is much smaller than the hole mobility, and trapping effects may be significant for electrons.

60

I. H. CAMPBELL AND D. L. SMITH

0.0'

0

1

2

3

4

5

I

6

Electric Field (lo5 Vkm) FIG. IV.2. The hole mobility of MEH-PPV as a function of electric field derived from TOF measurements. The markers are TOF results and the solid line is a least-squares fit of Eq. 4.2 to the TOF results (from Ref. 150).

Figure IV.3 is a log-log plot of the T O F electron current in Alq as a function of time after optical excitation for applied biases of 20 V, 50 V, and 100 V at room temperat~re."~The transit time was determined by the intersection of the asymptotes to the plateau and declining slope of the current transient. Figure IV.4 shows the electron mobility as a function of electric field determined from the T O F measurements (markers) and a least-squares fit (solid line) to the mobility assuming the Poole-Frenkel form. Figure IV.4 shows that the Poole-Frenkel form describes the measured T O F results reasonably well. The fit to the TOF data yielded p, = 1.5 x 10-'cm2/Vs and E , = 1.5 x lo4 V/cm. It was not possible to measure the hole mobility in Alq using the T O F technique. Significant hole trapping that obscures the current transients is widely observed in Alq.'69 Figure IV.5 shows the measured TOF hole mobility of pentacene at room temperature as a function of electric field (markers). The mobility is essentially independent of field over the range of electric fields considered. cmZ/Vs. It was not The mean value of the TOF data yielded p = 1.2 x possible to measure the electron mobility in pentacene using the T O F technique in these films. 169 R. G. Kepler, P. M. Beeson, S. J. Jacobs, R. A. Anderson, M. B. Sinclair, V. S. Valencia, and P. A. Cahill, Appl. Phys. Lett. 66,3618 (1995).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

I

........ 2

........

I

2

4 6 8

61

.

I

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4 6 8

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-

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. 4 '6'8"

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. 4 '6'8"

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. .

. .

.4

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. . .

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4

Time (s) FIG. IV.3. Time-of-flight electron current transients for Alq at three applied voltages. The structure was semitransparent Al (10 nm)/Alq (2 (m)/Al (100 nm) (from Ref. 113).

These T O F mobility results are typical of organic electronic materials with a low density of charge carriers in the film. Strong field dependence and low mobility, typical of hopping conductivity in a broad density of states, is observed for both holes in MEH-PPV and electrons in Alq. For more ordered systems, the mobility is higher and less strongly field dependent. For example, the pentacene hole mobility is significantly larger and has much weaker field dependence than the MEH-PPV hole and the Alq electron mobility.

62

I. H. CAMPBELL AND D. L. SMITH

Electric Field (105V/cm) FIG. IV.4. The electron mobility of Alq as a function of electric field derived from TOF measurements. The markers are TOF results and the solid line is a least-squares fit of Eq. 4.2 to the TOF results (from Ref. 113).

Electric Field (10'V/cm) FIG. IV.5. The hole mobility of pentacene as a function of electric field derived from TOF measurements.

PHYSICS OF ORGANIC ELECTRONIC DEVICES

63

FROM SINGLE-CARRIER SCL DIODE I-V CHARACTERISTICS 13. MOBILITY

The sensitivity of the time-of-flight technique to small densities of extrinsic traps frequently prevents its use to measure carrier mobilities. This problem can be overcome by using single-carrier diode current-voltage characteristics in the space charge limited current flow regime. To demonstrate the validity of this technique, we compare measured and calculated currentvoltage characteristics of MEH-PPV and pentacene hole only devices using the T O F hole mobility measurements presented preceding. The independently measured hole mobilities were used, without adjustable parameters, to calculate the current-voltage characteristics of device structures with space charge limited hole current. The I-V characteristics were described using the device model of Ref. [170]. For the SCL contacts, the model reduces to a numerical evaluation of space charge limited current with a field dependent mobility that includes carrier drift and diffusion (the diffusion component is small). Figure IV.6 shows measured (solid) and calculated (dashed) currentvoltage characteristics for a Pt/MEH-PPV/Al structure.' 5 0 The T O F mobility was used and there were no adjustable parameters in the calculation. ''O

P. S. Davids, I. H. Campbell, and D. L. Smith, J. Appl. Phys. 82, 6319 (1997)

Bias (V) FIG. IV.6. Measured (solid) and calculated (dashed) current-voltage characteristics for a Pt (10 nm)/MEH-PPV (2 pm)/Al (100 nm) structure. Positive bias corresponds to space charge limited hole injection from Pt. The calculation used the fit to the TOF mobility shown in Fig. IV.2 without adjustable parameters (from Ref. 150).

64

I. H. CAMPBELL AND D. L. SMITH

Bias (V) FIG. IV.7. Measured (solid) and calculated (dashed) current-voltage characteristics for two Pt/pentacene/Ca structures. Positive bias corresponds to space charge limited hole injection from Pt. The calculations used the TOF mobility shown in Fig. IV.5 without adjustable parameters.

Positive bias corresponds to space charge limited hole injection from the Pt contact and negligible electron injection from the A1 contact that has a large electron Schottky barrier. There is good agreement between the measured and calculated I-V characteristics over 5 orders of magnitude in current. The current-voltage measurements were made using devices with 2-pm thick MEH-PPV films prepared identically to those used for the TOF measurements to ensure that the microstructures of the organic films were equivalent.I6’ The average charge density in the TOF measurement is about 2 orders of magnitude smaller than that in the space charge limited I-V characteristic. At an applied bias of 50 V the average calculated hole density ~ , the average hole density in the for the diode was about 10’’ ~ m - whereas TOF measurement was about l O I 3 crnp3.Because the mobility derived from the TOF measurements at low density accurately describes the I-V characteristics at much higher density, trapping effects are not important for holes in this material. The carrier density in these 2-pm thick devices under space charge limited current flow is about 2 orders of magnitude smaller than in typical polymer LEDs that have organic layers about 100 nm thick. Figure IV.7 shows measured (solid) and calculated (dashed) currentvoltage characteristics for 150-nm and 300-nm thick Pt/pentacene/Ca structures. The TOF mobility was used and there were no adjustable parameters in the calculation. Positive bias corresponds to space charge limited hole injection from the Pt contact and negligible electron injection from the Ca

PHYSICS OF ORGANIC ELECTRONIC DEVICES

65

contact that has a large electron Schottky barrier. There is good agreement between the measured and calculated I-V characteristics. The highest charge density in the TOF measurement is about 4 orders of magnitude smaller than that in the space charge limited I-V characteristic. Because the mobility derived from the TOF measurements at low density accurately describes the I-V characteristics at much higher density, trapping effects are not important for holes in this material. The results for hole dominated diodes made from MEH-PPV and pentacene show that if a mobility is known from TOF measurements it can be used to accurately describe the current-voltage characteristics of singlecarrier SCL diodes. Single-carrier SCL diode current-voltage characteristics are next used to determine the mobility for cases in which trapping interfered with the TOF measurements. To determine the electron mobility of MEH-PPV, for which it was not possible to measure the electron mobility using the time-of-flight technique, a series of electron only structures with space charge limited current flow was measured and fit using an electric field dependent and charge density independent electron mobility. Current-voltage characteristics were measured for a series of Ca/MEHPPV/Ca electron only devices in which the polymer thickness was varied. The current is space charge limited because the energy barrier to injection of electrons from Ca into MEH-PPV is small. Figure IV.8 shows current density versus bias voltage for a thickness series of Ca/Ca electron only device^.'^^.'^^ The experimental results are shown as solid lines and the model results as dashed lines. The same electron mobility parameters, po = 5 x lo-’’ cm’/Vs, and E , = 1.0 x lo4 V/cm, were used for all the structures. The model describes current-voltage characteristics for Ca/Ca devices over a range of thicknesses, and over several orders of magnitude of current density. The thickness scaling is not V 2 / L 3as expected from the analytic result for space charge limited current that does not include the field dependence of the mobility.’ 61 Figure IV.9 shows the calculated electron density and electric field profiles for the 100 nm Ca/Ca device at a bias such that the current density is 5 x lo-’ A/cm2. Electrons are injected from the left contact and collected at the right contact. These profiles, for a single bias point, demonstrate the range over which the electric field and charge density vary in SCL conditions. For single-carrier SCL diodes, the carrier mobility is the main physical quantity that determines the current-voltage characteristics. Of course, the device geometry and the dielectric constant must also be known, but they can be independently determined. Therefore, the carrier mobility can be

’”

B. K. Crone, P. S. Davids, I. H. Campbell, and D. L. Smith, Appl. Phys. Lett. 84, 833 (1998).

66

I. H. CAMPBELL AND D. L. SMITH

0

10 Bias (V)

20

lo-' h

"s 3

10-2

L.

'3

I

1o4

10

1

Bias (V)

FIG. IV.8. Measured (solid line) and calculated (dashed line) current-voltage characteristics for 25-, 60-, and 100-nm thick Ca/MEH-PPV/Ca electron only devices on linear (upper panel) and log-log (lower panel) scales (from Ref. 163).

determined from the I-V curves of these devices. If SCL contacts cannot be made, as is the case for both electrons and holes in Alq, it is also necessary to know the injection properties of the contact. In the diode part of Section V, it is shown that if the Schottky barrier is known, single-carrier diode I-V curves can be used to find the carrier mobility for non-space charge limited cases. 14.

MOBILITY

MODELS

The Poole-Frenkel form for the field dependence of mobility was first observed in TOF measurements on molecularly doped polymer^.^^^^' 729173 17'

173

D. M. Pai, J. Chem. Phys. 52, 2285 (1970). W. D. Gill, J. App. Phys. 43, 5033 (1972).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

'I

CdCa

)

67

50

100

Position (nm)

FIG.IV.9. Calculated electron density (upper panel) and electric field (lower panel) profiles for a 110-nm Ca/MEH-PPV/Ca device biased to provide 5 x lo-' A/cm2 device current density. The electron injecting contact is on the left (from Ref. 163).

These materials consist of isolated molecular dopants in an inert polymer host. The dopant molecules do not introduce charged carriers, i.e they do not dope the material in the usual semiconductor sense of the term, but they provide low-energy sites that the injected carriers occupy. In the molecularly doped polymers, transport results from carrier hopping between the dopant molecules, the host polymer provides a matrix for the active dopant sites. Because similar field dependence is observed in molecularly doped polymers and the solid state films used for electronic devices, it is plausible that the physical mechanisms controlling the electrical transport in these two classes of materials are similar. Bassler and coworkers extensively studied mobility in these types of materials using Monte Carlo simulations of the Gaussian disorder model (GDM).' 7 4 * 1 In the GDM, electrical transport results from carrier hopping between localized sites with the site energies randomly distributed according to a Gaussian distribution. The GDM describes some features of 174

17'

H. Bassler, Phys. Star. Sol. (b) 175, 15 (1993). D. Hertel, H. Bassler, U. Scherf, and H. H. Horhold, J. Chem. Phys. 110, 9214 (1999).

68

I. H. CAMPBELL A N D D. L. SMITH

the observed mobility. However, the magnitude of the electric fields over which strong field dependence was found in the simulations did not correspond well with experiment. The magnitude of the fields at which the GDM showed strong field dependence, qualitatively similar to the experimental observations, was significantly larger than the field regimes at which that behavior was observed experimentally. Gartstein and Conwell' 7 6 showed that a spatially correlated site energy distribution for the carriers can explain the observed field dependence. Physically, strong field dependence occurs when the potential drop (eE8) across a relevant length scale ([) for the system becomes comparable to kT. With uncorrelated site energies, the only relevant length scale is the distance between hopping sites. This distance is relatively small and strong field dependence only occurs at high fields. Spatial correlations introduce a new longer length scale, the length over which the site energies are correlated. Thus spatial correlations can lead to strong field dependence of the mobility at lower fieldsthan would occur for uncorrelated site energies. Many of the dopant molecules used in molecularly doped polymers have large permanent electric dipole moments. Dunlap and coworkers177- I a 1 proposed a model for the mobility of molecularly doped polymers based on the long-range nature of the interaction between charged carriers and the dipole moments of the molecular dopants. In this model the energetic site disorder is the result of different electrostatic potentials at the various sites due to the random distribution in orientation of the dipole moments of the nearby dopant molecules. Because the charge-dipole interaction is long range, sites that are spatially close also have nearly the same energy so that there is a correlation between site position and energy. This model has been successful in describing many aspects of the mobility of molecularly doped polymers. Some of the materials used for organic electronic devices, such as Alq, also have large permanent dipole moments and the charge-dipole interaction model probably applies to them as well as to the molecularly doped polymers for which it was originally designed.I13 However, other materials used for organic devices, such as PPV, do not have permanent dipole moments and therefore this model does not appear to apply to them in detail.

176

17' 17*

Y. N. Gartstein and E. M. Conwell, Chem. Phys. Lett. 217, 41 (1994). D. H. Dunlap, P. E. Parris, and V. M. Kenkre, Phys. Rev. Lett. 77, 542 (1996). S. V. Novikov, D. H. Dunlap, V. M. Kenkre, P. E. Parris, and A. V. Vannikov, Phys. Rev. .

Lett. 81, 4472 (1998). P. E. Parris, M. Kus, D. H. Dunlap, and V. M. Kenkre, Phys. Rev. E56, 5295 (1997). P. E. Parris, D. H. Dunlap, and V. M. Kenkre, J. Polymer Science B35,2803 (1997). I * ' V. M. Kenkre, M. Kus, D. H. Dunlap, and P. E. Parris, Phys. Rev. E58, 99 (1998).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

69

A second physical mechanism, fluctuations in molecular geometry such as the phenylene ring-torsion in PPV, also leads to spatially correlated site energies and applies to conjugated materials without permanent dipoles. The spatial energy correlation is the result of strong intermolecular restoring forces for ring-torsion fluctuations in dense films of closely packed molecules. By contrast, the intramolecular restoring force for a ring-torsion is small as seen, for example, in the AM1 calculations for biphenyl shown in Fig. 11.3. For neutral biphenyl, the energy is almost independent of the torsion angle. Because the restoring force is primarily intermolecular, ring-torsions on neighboring molecules tend to move together. If an extra electron or hole is added to the system, the energy of the charged state depends strongly on the torsion angle, as also seen in the AM1 calculations of biphenyl in Fig. 11.3. Thus, there is a strong coupling between the energy of a carrier at a site and the ring orientation at that site. Different sites will have different ring orientations and therefore this coupling leads to disorder in the site energies. Because the rings on near neighbors move together, there is a spatial correlation in the site energies. Details of the site energy density of states and spatial correlation functions for this model are worked out in Ref. [76]. The results are found to be similar to those for the charge-dipole interaction model. Both models give a site energy density of states that is nearly Gaussian and a two-site energy correlation function that falls off as the reciprocal of distance between sites. As a result the field dependence of mobility predicted by the two models is very similar, although the physical origin of the energy disorder is different. There is a difference in expected temperature dependence in the two models because in the charge-dipole interaction model the disorder is independent of temperature whereas in the molecular geometry fluctuation model, the disorder increases with increasing temperature. A 1D Master equation with nearest neighbor hopping that can be used to describe field-dependent mobility has been exactly solved by Derrida.’*’ In the continuum limit, the mobility can be calculated using this solution to the Master e q ~ a t i 0 n . l ~The ’ result is

where y = peE, E is the electric field, e is the magnitude of the electron ) the site energy at position y , p,, is the mobility if all site charge, ~ ( y is energies were the same, and p = l / k T Using a Gaussian approximation to B. Derrida, J. Statistical Phys. 31, 433 (1982).

70

I. H. CAMPBELL A N D D. L. SMITH

calculate the correlation function and the (l/y) dependence for the correlation function (c(y)c(O)) given by both the charge-dipole interaction and the molecular geometry fluctuation models gives

(4.4) where D is essentially the standard deviation of the density of site energies, (l/a) is a momentum cutoff, and K , ( z ) is the first-order modified Bessel function of the third kind. (Slightly different prescriptions were used for cutting off momentum integrals when going to the continuum limit in Refs. [177] and [76], which leads to a small difference in numerical factors for the forms stated in those papers.) In the charge-dipole disorder model the standard deviation of the site energy distribution is 0 = ,/e2P2no/12nc2a where P is the dipole moment and no is the densit of dipoles. In the molecular geometry fluctuation model 0 = ,/ where * v is the linear coupling constant between site energies and the ring-torsion and K is the angular restoring force constant for the ring-torsion. In the chargedipole interaction model 0 is temperature independent, whereas in the molecular geometry fluctuation model 0 is proportional to JkT. Using an asymptotic expansion for K , ( z ) , the high field mobility becomes' 77 p

&-8202 e ( 2 ~ 0 eEa) JF

(4.5)

In this 1D solution both models give the same result for the field-dependthat is, the Poole-Frenkel form for the field ence for the mobility, In p dependence. Because 0 has different temperature dependence in the two models, the mobilities have different temperature dependencies. The 1D results are not generally valid for dense three-dimensional (3D) systems. In 3D, the carriers can take optimal paths to avoid high-energy barriers, whereas in the 1D model there is only one path the carriers can take. The steady state Master equation describing the carrier transport for this system is N

0=

3,

1 IOijPj(l - Pi) - WjiPi(l - Pj)l

(4.6)

j

Here Pi is the probability for the polaron to be on site i and oijis the polaron hopping rate from site j to site i. Double occupation at a site is excluded. After finding the solution for Pi, the average carrier velocity is found from CijOijPj(l - Pi)Rij V = (4.7) x j Pi

-

where

-

aijis the position difference between sites j and i, and the mobility is

PHYSICS OF ORGANIC ELECTRONIC DEVICES

71

10" h

P

NE

1o - ~

0

W

3.

10"

FIG. IV.10. Calculated logarithm of mobility as a function of El'' with different polaron-torsion couplings. Solid, dashed, and dot-dashed lines correspond to v = 0.1, 0.2, and 0.3 eV, respectively (from Ref. 76).

found from ?j = p z . The 3D Master equation is in general too complex for analytic solution and two numerical approaches are commonly used, Monte Carlo simulation and direct solution of the Master equation using sparse matrix techniques. Compared with Monte Carlo simulations, the sparse matrix approach has some advantages: it guarantees the steady state solution; it is more convenient for considering density-dependent effects; and it is often numerically more efficient. For cases in which both approaches can be used they give the same result. The field-dependent mobility in the dilute limit can be found by linearizing the Master equation. Figure IV.10 shows a calculation of mobility as a function of electric field for the molecular geometry fluctuation model with three values for the coupling parameter, v, between the ring-torsion and the site energy. The ring-torsion restoring force, K , was chosen to describe the measured hole mobility of MEH-PPV using v = 0.3 eV. The values for these parameters are consistent with quantum chemical estimates using model systems.76The curves are reasonably close to linear, showing that the model gives approximately the Poole-Frenkel form. Figure IV.11 shows the density of states for the site energies with the same values of the coupling parameters as in Fig. IV.10. For a system with a stronger coupling and therefore a broader density of site energies, the mobility is low and has strong field dependence, whereas for a system with weak coupling and therefore a narrower density of site energies, the mobility is higher and has weaker field dependence.

12

I. H. CAMPBELL A N D D. L. SMITH

12.0 -

h

4.0 -

-0.4

-0.2

0

r (W

0.2

0.4

FIG. IV.11. Calculated site density of states as a function of energy for different polaron-torsion couplings. Solid, dashed, and dot-dashed lines correspond to v = 0.1, 0.2, and 0.3 eV, respectively (from Ref. 76).

The results of Fig. IV.10 suggest that there is correlation between the magnitude of the mobility and the strength of the field dependence because stronger energetic disorder leads to both lower magnitude of the mobility and stronger field dependence. Molecular geometry fluctuations, such as ring-torsions, are a source of energetic disorder. If the molecular structure is constrained in such a way that the restoring force for ring-torsions is increased or their coupling to the site energies is reduced, energetic disorder is reduced. As a result, the magnitude of the mobility is enhanced and its field dependence is weakened. Comparing TOF hole mobility measurements in MEH-PPV and poly(9,9-dioctylfluorene) (PFO) shows this correlation. The phenylene rings in an isolated neutral MEH-PPV chain can rotate easily. In PFO, two rings are fixed together by bridging bonds so they can only rotate together. As a result, both the intermolecular restoring force to ring-torsion is increased, because two rings rather than one collide with a neighboring molecule, and the coupling of the ring-torsion to the site energy is reduced, because the charge can more easily delocalize on two rings than on a single ring. Figure IV.12 compares mobility measurements of MEHPPVlSo and P F 0 1 5 1with calculations based on the molecular geometry fluctuation model. The observed hole mobility in P F O is about two orders higher than that for MEH-PPV and the field dependence is much weaker. This is the expected qualitative behavior. In the calculations, the hole mobility data of MEH-PPV was fit by adjusting the parameter K describing the ring-torsion intermolecular restoring force and the parameter v descri-

PHYSICS OF ORGANIC ELECTRONIC DEVICES

13

bing the strength of the coupling between the ring-torsion and the site energy around the values estimated from AM 1-level quantum chemistry calculations of model systems. The value of K was then increased to fit the PFO data. In principle, both an increase in K and a decrease in v should occur when going from MEH-PPV to PFO. But such changes in either of the two parameters give very similar results. Good fits to the mobilities of both materials are achieved for physically reasonable values of the parameters. The mobility can depend on carrier density, because when some carriers fill deep potential sites from which hopping is difficult, the other carriers become more mobile. The density dependence of mobility can be studied by solving the nonlinear Master equation. Figure IV. 13 illustrates the carrier density effects on the mobility in the molecular geometry fluctuation model with parameters appropriate for holes in MEH-PPV. The mobility is enhanced by almost one order of magnitude with increase of the carrier density to n = 6.9 x 10'' cm-3 at E 4 x lo4 V/cm. In the low-field regime, where the field-assisted carrier hopping is less efficient than in the high-field regime, the carrier density effect on mobility is more pronounced. Diode measurements show that at electric fields of a few times lo6 V/cm there is not a strong carrier density dependence of the mobility in MEH-PPV for densities up to about 10" ~ m -By ~ contrast, . field effect transistor measurements have suggested that the mobility increases strongly with increasing

-

1o - ~

: h

"E

U

I

L

200400600800

E ' (V'ncrn-'n) ~

200400600600

E'" (V'ncrn-'n)

FIG. IV.12. Logarithm of hole mobility as a function of The left panel shows experimental (dots) and calculated (solid line) results for MEH-PPV. The right panel shows experimental (dots) and calculated (solid line) results for PFO (from Ref. 76).

74

I. H. CAMPBELL AND D. L. SMITH

FIG. IV.13. Calculated logarithm of mobility as a function of El'' with different carrier densities. Dotted, short-dashed, long-dashed, and dot-dashed lines correspond to carrier ~, The solid line shows the results of densities n = 0.08, 0.5, 2, and 6.9 x 10l8~ r n - respectively. solving the linearized Master equation (from Ref. 76).

carrier density at low fields for carrier densities above about 10" ~ m -The ~ . calculated results in Fig. IV.13 are consistent with these device measurements and explain why this behavior is expected when the different field/ density regimes are sampled. Figure IV.14 shows the effect of deep traps on the mobility. Traps are randomly distributed with a concentration 2 x 10'' cm-3 and a trap energy level 0.5 eV below the center of the Gaussian site energy density of states. The molecular geometry fluctuation model with parameters appropriate for holes in MEH-PPV is used. Because of the traps, the mobility is small in the low-field regime for small carrier densities. When the carrier density is sufficiently large to saturate the traps, the mobility is enhanced dramatically. These results show how a small density of traps can have a very large effect on T O F measurements, in which the carrier densities are very small, but do not significantly affect single-carrier SCL diode measurements, in which the carrier densities are much larger. In 3D systems, a carrier can optimize its path to avoid high-energy barriers and achieve a higher mobility. Figure IV.15 illustrates current patterns in the low-field and the high-field regions. The molecular geometry fluctuation model with parameters appropriate for holes in MEH-PPV is used. The figure shows a projection of the 3D lattice onto the x-y plane (the field is in the x direction) by summing over the currents in the z direction.

PHYSICS OF ORGANIC ELECTRONIC DEVICES

1o

75

-~

h

10-6 0

W

5

lo-’ i

FIG. IV.14. Calculated logarithm of mobility as a function of El’’ with different carrier densities for a system with randomly distributed traps. The trap concentration is 2 x 10’’cm-j and the trap level is -0.5 eV. Short-dashed, long-dashed, dot-dashed, and dotted lines correspond to carrier densities n = 4.7, 2.4, 1.2, 0.3 x 1017 cm-j, respectively. The solid line shows the results of solving the linearized Master equation without traps (from Ref. 76).

The width of each line in the figure is proportional to the current across the bond. Darker lines indicate that the current is opposite to the standard directions (from left to right and from down to up). In the low-field regime, the carriers take complex paths involving many chains. When such irregular paths occur, a 1D model, where the path is always along the field, is not appropriate. In the high-field regime, where the field is strong enough to overcome the energy barriers, the carrier paths are essentially one-dimensional. Because the energetic disorder is electrostatic in the charge-dipole interaction model, it should be the same, except for a sign reversal, for electrons and holes. Thus the charge-dipole interaction model predicts that the mobility for electrons and holes should have similar electric field dependence. This model should be applicable if the energetic disorder is dominated by the random orientation of the molecular dipoles. For Alq, the molecular dipole moment is large, 5.3 Debye,’I3 and the intermolecular spacing is about 1 nm, which leads to disorder with a standard deviation of about 0.1 eV, comparable to the total energetic disorder. Thus, it is likely that dipolar disorder is the major source of energetic disorder in Alq films. Although it is not possible to measure both carrier mobilities of Alq using TOF, the device measurements presented in Section V give similar electron

76

I. H. CAMPBELL AND D. L. SMITH

FIG. IV.15.Current patterns for different applied fields. The width of a line is proportional to the current across the bond. Upper and lower panels are for E = 0.5 x 10’ and 2 x lo6 V/cm, respectively (from Ref. 76).

and hole mobilities with similar electric field dependence as expected if dipolar disorder is dominant. In MEH-PPV, holes have a higher mobility and a weaker electric field dependence than electrons, suggesting that the energetic disorder is greater for electrons than for holes. Dipolar disorder is not expected to be a major source of the energetic disorder in MEH-PPV films. MEH-PPV has a flexible molecular structure and molecular geometry fluctuations such as ring-torsion should be a significant source of energetic disorder. For the

77

PHYSICS OF ORGANIC ELECTRONIC DEVICES

simple example of the biphenyl molecule discussed in Section 11, there is an electron-hole symmetry and the coupling of electron and hole states to ring-torsion is the same. This electron-hole symmetry is broken in more complex molecules such as MEH-PPV and it is not expected that electrons and holes have the same field dependence. The hole mobility in pentacene is comparatively high and essentially independent of electric field. (The electron mobility in pentacene is not well studied.) This is the behavior expected from a material that does not have spatially correlated energetic disorder. Pentacene does not have a permanent dipole moment and it is structurally rigid. As a result, neither charge-dipole interactions nor molecular geometry fluctuations are expected to make major contributions to energetic disorder. Therefore the energetic disorder in pentacene is expected to be comparatively weak and not spatially correlated. V. Organic Diodes and Field-Effect Transistors

A number of organic electronic devices have been proposed, but organic diodes, in particular LEDs, and FETs, have been the most extensively explored. Both of these classes of devices use thin films of undoped insulating materials. Organic diodes are vertical transport devices consisting of an organic film sandwiched between metal contacts. The carriers are transported vertically from one contact to the other across the thickness of the film. Organic FETs are lateral devices consisting of a conducting gate contact, a gate insulator, source and drain electrodes electrically isolated from the gate contact by the gate insulator, and an organic film in contact with the source and drain electrodes. Carriers are transported laterally along the organic film across the gap between source and drain electrodes. For both classes of devices, carriers in the organic film originate from the metallic contacts. Organic diodes are quite different from conventional semiconductor p-n junction diodes. Organic FETs are more nearly similar to inorganic thin-film FETs. Detailed device models have been demon70*17 1 9 183*184 but device models strated for organic diodes' '3 - ' '7*1 5 -~ ~ ~ ~ ' specifically for organic FETs are less extensively d e ~ e l o p e d .8 ~

'','

'"

P. W. M. Blom, M. J. M. DeJong, and S . Breedijk, Appl. Phys. Lett. 71,930 (1997). B. K. Crone, P. S. Davids, I. H. Campbell, and D. L. Smith, J. Appl. Phys. 87, 1974 (2000). L. Torsi, A. Dodabalapur, and H. E. Katz, J. Appl. Phys. 78, 1088 (1995). G. Horowitz, R. Hajlaoui, H. Bouchriha, R. Bourguiga, and M. Hajlaoui, Advanced Materials 10, 923 (1998). R. Tecklenburg. G. Pasch, and S . Scheinert, Advanced Materials for Optics and Electronics 8, 285 (1998).

78

I. H. CAMPBELL A N D D. L. SMITH

The materials used for organic electronic devices are disordered. As a result there is an ensemble of energies for the charged states. The low-energy charged states of device interest, polarons, are localized on individual sites and electrical transport results from polaron hopping between these localized states. At room temperature, equilibration between neighboring sites of different energy is generally fast enough that carrier transport can be described using a mobility picture. Hopping transport in a disordered system leads to a mobility that can depend strongly on both electric field and carrier density. Organic diodes and organic FETs operate in distinct electric field and charge density regimes. Organic diodes operate in a high-field, low-carrier density region whereas the FETs operate in a lowfield, high-carrier density region. For the operating conditions relevant to organic diodes, the carrier density dependence of the mobility can be neglected, but the field dependence must be included. The carrier density dependence of the mobility must be included for the operating conditions relevant to organic FETs. The field dependence of the mobility at the low carrier densities relevant to organic diodes has been determined, for some materials, by a combination of TOF and single-carrier SCL diode currentvoltage measurements. In organic devices, carriers are injected into an undoped film from metal contacts. The Schottky energy barriers between the contact metal Fermi energy and the polaron levels in the organic material have been determined for a variety of cases. To determine interfacial current densities, it is in general also necessary to know transition rates for electrons and holes to cross the metal/organic interface. Reliably determining such transition rates, either by direct measurement or by microscopic theory, is a challenging Fortunately, for most cases of interest the current densities at which the devices are operated are not large enough to drive carrier densities at the metal/organic interface far out of local equilibrium. As a result, carrier injection enters device models as a boundary condition for interface carrier densities with these interface densities determined by local thermodynamic equilibrium. ''O An understanding of the microscopic details of carrier transport at the interface is thus not required for the device model. This is a major simplification that has allowed successful device modeling. In organic LEDs, electrons are injected from one side of the organic film and holes are injected from the other. The electrons and holes form excitons

I9O 19'

'93

V. I. Arkhipov, E. V Emelianova, Y. H. Tak, and H. Bassler, J. Appl. Phys. 84, 848 (1998). U. Wolf, V. I. Arkhipov, and H. Bassler, Phys. Rev. B59, 7507 (1999). V. I. Arkhipov, U. Wolf, and H. Bassler, Phys. Rev. B59,7514 (1999). U. Wolf, S. Barth, and H. Bassler, Appl. Phys. Lett. 75, 2035 (1999). E. M. Conwell and M. W. Wu, FPhys. Rev. Lett. 70, 1867 (1997). Y. N. Gartstein and E. M. Conwell, Chern. Phys. Lett. 255, 93 (1996).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

79

that recombine, perhaps radiatively. For an organic LED device model, it is necessary to include a description of carrier recombination. The recombination process is bimolecular. A Langevin form for the bimolecular recombination coefficient is commonly used.'94 The section first describes a device model for organic diodes. The model is then compared with experimental results on single-layer, single-carrier structures; with single-layer bipolar structures; and with multilayer structures. Experimental results on organic FETs are then discussed. 15. ORGANIC DIODES

a. Device Model The device model'70 for organic diodes includes charge injection, transport, recombination, and space charge effects in the organic material. It can describe contact limited current, space charge limited current, and cases in between. Because of the geometry of organic diodes, a one-dimensional device model is appropriate. The transport of electrons and holes in the organic film are described by time dependent continuity equations coupled to Poisson's equation

an at

1 aJ,

e

ax aE

=G-R

47ce (p - n)

-= -

ax

(5.3)

with drift-diffusion forms used for the current densities,

and

(

4:)

J , = ep, p E - - 'eT

Here, n (p) is the electron (hole) density, J , (J,) is the electron (hole) current density, G ( R ) is the carrier generation (recombination) rate, p, (p,) is the electron (hole) mobility, e is the magnitude of the electron charge, E is the lQ4 V. N. Abakumov, V. I. Perel, and I. N . Yassievich, in Nonradiative Recombination in Semiconductors, North-Holland, Amsterdam (1991),p. 108.

80

I. H. CAMPBELL A N D D. L. SMITH

electric field, E is the static dielectric constant, k is Boltzmann’s constant, T is temperature, x is the position coordinate along the film growth axis, and the diffusivities have been written in terms of the mobilities using the Einstein relation. The carrier mobilities are field dependent in the organic materials used to form organic LEDs and are reasonably well described by the Poole-Frenkel form P = Po exp

(&)

The Einstein relation between diffusivity and mobility does not necessarily apply in regions where the mobility has strong field dependence. This question was studied for the specific case of the charge-dipole interaction model for the mobility, discussed in Section IV, in Ref. [lSO]. The Einstein relation was found to apply at high fields, in slightly modified form, for this model. The same arguments and conclusions apply to the molecular geometry fluctuation model. In practice, the diffusion terms make a very small contribution to the calculated device current.’95 Electron-hole recombination is bimolecular, R = y (np). A Langevin form is used for the recombination coefficient

y=- 471e~rn &

(5.7)

where ,urnis the larger of p, or p,. The generation rate is determined from the recombination rate using detailed balance, G = y(nepe), where (n,p,) is the product of equilibrium electron and hole densities. For the materials used in organic diodes, typically with energy gaps near 2 eV, carrier generation is very small. The equations are spatially discretized using the Scharfetter-Gummel approach. The resulting first-order differential equations are integrated forward in time using, for example, Gear’s method. To find the steady state solution at a given applied voltage bias, a time-dependent potential ramp that stops at the desired voltage is applied to the right contact and the equations are integrated forward in time starting from thermal equilibrium until steady state is reached. The position independence of the total particle current J = J , J , is used to verify that steady state has been reached. Poisson’s equation is integrated, with the calculated steady state values for electron and hole densities, to verify that the correct value of the potential drop across the device for the applied voltage bias is achieved. To calculate

+

‘51’

J. M. Lupton and I. D. W. Samuel, J. of Phys. D. - Appl. Phys. 32, 2973 (1999).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

81

a current-voltage (I-V) curve a series of voltage ramps is applied. In performing the time integrations it is useful to have an explicit expression for the time dependence of the electric field aE(x) at

~

L

a'(L) at

47ce (J(x)

-

E

JoL

J(x)dx)

(5.8)

where L is the length of the device and 4 is the electrostatic potential whose value at the right contact, ' ( L ) , is set by the voltage ramp and whose value at the left contact, '(O), is fixed at zero. Equation 5.8 follows from the time derivative of Poisson's equation. It is used to find E(0) to start the spatial integration of Poisson's equation at each new time step. The time-dependent formulation is used to avoid the difficulties that arise with two-point boundary conditions in a steady state formulation. For the equilibrium description, the material is viewed as made up of a series of localized electronic sites. Nondegenerate cases are considered so that the occupation probability of a site by electrons or holes is significantly less than unity. Let no be the density of localized sites times the number of ways that a site can be occupied by an electron or hole (i.e., its degeneracy). If E, (E,) is the energy of the electrons (holes) (so that, ( E , - E,) is the energy gap of the polymer) the equilibrium densities are e

=

O

e - ( E c - e9(x) - Wkr

(5.9)

and pe

=

e(E,. - e&.x) - IL)/kT

(5.10)

where t+b is the Fermi energy. To find the equilibrium solution, Poisson's equation is integrated using these expressions for the carrier densities. In organic materials, polaron states are disordered with an approximately Gaussian density of states. For simplicity, a single polaron energy is used in the device model. The effect of including a Gaussian density of states in the device model was investigated in Ref. [163]. It was found that the effects of the Gaussian state density could be incorporated by a small, temperaturedependent, shift of the effective barrier height. The size of the shift was smaller than the experimental uncertainties in the measured barrier heights. The contacts enter the model as spatial boundary conditions specifying the particle currents at the boundaries x = 0 and x = L. There are three particle current components at each metal/organic material interface: a thermionic emission current component from the metal into the organic material, a backflowing interface recombination current component that is the time-reversed process of thermionic emission, and possibly an athermal current component such as a tunneling current. Specifically consider the

82

I. H. CAMPBELL AND D. L. SMITH

electron current at x

= 0,

where J t h is the thermionic emission current density, Ji, is the interface recombination current density, and J,, is the tunneling current density. The thermionic emission current density has the general form Jth

= -e

1 P,f, A

Qi,/ -

(1 - Pl),

(5.12)

i,P

where i labels electronic states in the metal incident on the interface, Pi is the probability that this state is occupied, f, is the electron flux carried by the state, e labels electron polaron states in the organic material, oi,{is the capture cross section for the state i incident on the interface to be captured into the polaron state e, A is the interface area, and PPis the probability the polaron state !t is occupied. The backflowing interface recombination current density has the general form

(5.13) where ( l / ~ / ,is~ the ) transition rate for a polaron in state t to go into the state i in the metal. ( l / ~ ( , and ~ ) oi3(are related by detailed balance fiai./ =

(k)

(5.14)

where i* is the time-reversed state of i; that is, if an electron in state i is going toward the interface, an electron in state i* is going away from the interface. Thus (5.15) The electron distribution in the metal is not driven out of equilibrium by the current flowing in the device, so that the first term in Eq. 5.15 balances a quasi-equilibrium distribution in the organic material near the interface (5.16) where PyEqis the quasi-equilibrium distribution in the organic material. If intersite hopping is fast enough to keep the carrier distribution in the organic material in local quasi-equilibrium (a necessary condition for the mobility description in the first place), Eq. 5.16 takes the form J,h

- Ji, =

-K(N(OpEq - N(O))

(5.17)

PHYSICS OF ORGANIC ELECTRONIC DEVICES

83

where the kinetic coefficient, K , has the form

(5.18)

N(0)QEq is the electron polaron density at the interface determined by quasi-equilibrium with the metal and N(0) is the actual electron polaron density at the interface. The boundary condition for electron current at x=Ois J,(O) = -K(LV(O)'~~- N(0))

+ J,,

(5.19)

For most cases of interest the kinetic coefficient, K , and the interface densities are very large and the boundary condition reduces to the requirement of quasi-equilibrium at the interface

N(0)QQ = N(0)

(5.20)

In the device model calculations, specific forms for the three current components are used.'70 However, in almost all cases of interest the results reduce to the quasi-equilibrium result. This is a major simplification because the results do not then depend on the detailed microscopic forms for the current components, which are somewhat idealized. The other three interfacial particle currents, which constitute the remaining three boundary conditions, have forms analogous to the x = 0 electron boundary condition. Because of the image force, the interfacial energy barrier depends on the electric field at the interface, (5.21)

where @ is the Schottky energy barrier at zero field and E is the static dielectric constant of the organic material. (The image force barrier lowering term is only included when the electric field has the correct sign for barrier lowering.) As seen in Fig. 111.5, this barrier lowering shows up directly in the internal photoemission measurements of barrier height. It can have significant consequences on the calculated device behavior and it is important that it be included in the device model. At steady state, the continuity equations can be integrated spatially to obtain a recombination current J ,

J, =

jI

eR.dx

= J,(L)

-

JJO) = JJO) - J,(L)

(5.22)

Electrons (holes) injected at x = 0 (x = L) to give J,(O) (J,(L)) either recombine in the device and contribute to the recombination current, J , , or

84

I. H. CAMPBELL AND D. L. SMITH

completely traverse the device and contribute to J,(L) (Jp(0)).Both electron current at the hole injecting contact J,(L) and hole current at the electron injecting contact J p ( 0 ) result in a parasitic loss which lowers the quantum efficiency. In the ideal case of unity quantum efficiency J , = IJI, and J,(L) = J,(O) = 0. Two important figures of merit for organic LEDs are the recombination efficiency q, (the ratio of the recombination current J , to the total device current IJI) and the recombination power efficiency qrp (the ratio of power output from recombination to electric power input). These have the form q ='

J

IJI

(5.23) (5.24)

The total recombination current J , will be higher than the radiative recombination current because some excitons recombine nonradiatively. Spatial dependence of the radiative and nonradiative recombination rates is possible. For example, dipolar quenching of radiative recombination can occur near the metallic interfaces and a spatial variation in the density of nonradiative recombination centers can occur during deposition.' 96,197 Carrier density dependence of the recombination processes is also possible, for example by saturation of recombination centers. However, the spatial and density dependence of radiative and nonradiative rates are not known so the ratio of the radiative recombination to the total recombination (Q) is taken as constant. The quantities q, and qrP are related to the quantum efficiency and power efficiency, respectively, by multiplying qr and qrp by the ratio of radiative to total recombination. qq = Qq, and q p = Qqrp,where qq is the quantum efficiency and q, is the power efficiency. In a simple case, where all the singlet excitons decay radiatively, all the triplet excitons decay nonradiatively, and 1/4 of the excitons formed are singlets and 3/4 are triplets, the ratio of radiative to total recombination will be Q = 1/4. b. Single-Carrier Devices Device model results for representative single-carrier organic diodes, using material parameters similar to those of MEH-PPV, are first presented to illustrate the importance of the Schottky energy barrier on device behavior. Then detailed comparisons between measured and calculated current-volt196 V. Choong, Y. Park, Y. Gao, T. Wehrmeister, K. Mullen, B. R. Hsieh, and C. W. Tang, Appl. Phys. Lett. 69, 1492 (1996). 19' H. Becker, S. E. Burns, and R. H. Friend, Phys. Rev. B56, 1893 (1997).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

85

FIG. V.l. Linear-linear (upper panel) and log-linear (lower panel) plots of calculated current density as a function of bias voltage for 120-nm MEH-PPV devices with a 1.4 eV barrier to electron injection and 0.1,0.2,0.3,0.4,0.5,and 0.6 eV barriers to hole injection (from Ref. 170).

age characteristics are presented. Calculated results for a 120-nm thick device with a barrier for electron injection fixed at 1.4 eV approximately corresponding to an A1 contact (at this large value for the barrier few electrons are injected) and the barrier for hole injection varied from 0.1 to 0.6 eV in 0.1 eV steps is shown in Fig. V.l.I7O The upper panel shows a linear-linear plot and the lower panel a log-linear plot of the calculated I-V curves. The calculated results for 0.1, 0.2, and 0.3 eV barriers are nearly the same. For these cases, the current flow is essentially space charge limited. As the energy barrier is further increased the current is decreased, indicating that the current flow becomes injection limited. These results show that for typical organic diode device parameters, it is important that injection barriers be kept less than about 0.4 eV. Figure V.2 shows the calculated hole densities for four bias voltages as a function of position for a device with a 0.6 eV barrier to hole injection and a 1.4 eV barrier to electron injection, approximately corresponding to a Cu/ MEH-PPV/A1 structure (upper panel), and a device with a 0.1 eV barrier to hole injection and a 2.3 eV barrier to electron injection, approximately corresponding to a Au/MEH-PPV/Au structure (lower panel).I7O The bias voltages in the upper panel are 20 V (solid line), 15 V (dotted line), 10 V

86

I. H. CAMPBELL AND D. L. SMITH

Position (Angstroms)

FIG. V.2. Calculated hole density as a function of position for a Cu/MEH-PPV/AI device (upper panel) and the Au/MEH-PPV/Au device (lower panel). In the upper panel the bias voltages are 20 V (solid line), 15 V (dotted line), 10 V (dashed line), and 5 V (dot-dash line). In the lower panel the bias voltages are 8 V (solid line), 6 V (dotted line), 4 V (dashed line) and 2 V (dot-dash line). The hole injecting contact is at the right (from Ref. 170).

(dashed line), and 5 V (dot-dash line). In the lower panel the bias voltages are 8 V (solid line), 6 V (dotted line) 4 V (dashed line), and 2 V (dot-dash line). In both panels, the hole injecting contact is at the right. The hole density for the device in the upper panel is essentially constant spatially. The hole density increases rapidly with increasing bias but for all the bias values shown is rather small and does not significantly influence the electric field in the device. For the bias range shown the hole density in the device is the quasi-equilibrium value at the hole injecting contact, including barrier height lowering by the image force. The increase in hole density with bias is due to the image force barrier lowering. The behavior shown in the upper panel is characteristic of contacted limited diodes. The hole density for the

PHYSICS OF ORGANIC ELECTRONIC DEVICES

4;)O

2;)O

87

6bO 860 Id00 1 i O O

Position (Angstroms) Au/MEH-PPV/AU

101

0

I

400 600 800 1000 Position (Angstroms)

200

FIG. V.3. Calculated electric field as a function of position for a Cu/MEH-PPV/AI device (upper panel) and the Au/MEH-PPV/Au device (lower panel). The values of voltage bias are the same as in Fig. V.2. The hole injecting contact is at the right (from Ref. 170).

device in the lower panel varies strongly with position. The hole density at the hole injecting contact is equal to the true equilibrium value. (Because of the sign of the electric field in this case, there is no image force lowering of the injection barrier and no tunneling injection). The hole density changes significantly with bias and it is large enough to strongly influence the electric field in the device. The behavior shown in the lower panel is characteristic of space charge limited diodes. Figure V.3 shows the calculated electric fields as a function of position for the Cu/MEH-PPV/Al structure in the upper panel and the Au/MEH-PPV/ Au structure in the lower panel at the same bias voltages as in Fig. V.2.l7' For the device in the upper panel, the electric field is an essentially constant function of position, whereas for the device in the lower panel the electric field is a strongly varying function of position. For the device in the upper panel, the electric field at the hole injecting contact has the correct sign to lead to image force lowering of the injection barrier. Because of the high hole density near the hole injecting contact in the device in the lower panel, the sign of the

88

I. H. CAMPBELL AND D. L. SMITH

0.10

Bias (V) FIG. V.4. Measured (solid lines) and calculated (dashed lines) current density as a function of voltage bias for MEH-PPV devices about 110 nm thick with Au as the electron injecting contact and Pt, Au, Cu, and A1 as the hole injecting contact (from Ref. 198).

electric field is reversed and it does not have the correct sign to lead to image force lowering of the injection barrier. These results are characteristic of contact limited and space charge limited diodes, respectively. The dependence of the device current on organic film thickness can be used to distinguish contact limited from space charge limited current flow. The device current in the contact limited regime scales as (I/applied - Vbi)/L because the electric field is constant across these devices, and the mechanisms that determine device current -carrier mobility, injection barrier lowering, and carrier density- all scale with electric field. In the space charge limited regime, the electric field is not constant across the device, and the current scaling is complex and depends more strongly on device thickness. It does not follow that ( I/applied - Vbi)'/L3, as predicted by constant mobility space charge limited current calculations, because of the strong electric field dependence of the carrier mobilities.16' It is therefore straightforward to experimentally distinguish contact limited from space charge limited current flow by the length scaling of device currents. Now, detailed comparisons between measured and calculated I-V characteristics for single-carrier diodes are presented. First the effect of changing barrier heights by using different contact metals is considered. Figure V.4 compares measured and calculated I-V curves for MEH-PPV devices about 110 nm thick in which the cathode contact is fixed to be Au and the anode contact is varied, including Pt, Au, Cu, and Al.19' The mobility parameters p, and E , were determined by fitting to these data. The same mobility parameters were used for the calculation of all four devices. The Schottky

"* I. H. Campbell, P. S. Davids, D. L. Smith, N. N. Barashkov, and J. P. Ferraris, Appl. Phys. Lett. 72, 1863 (1998).

PHYSICS OF ORGANIC ELECTRONIC DEVICES

lo6 03

menlonlc

89

Interface Recombination

Bias (V)

a €

z

7

Bias (V)

1

1010-6 20

25

30 35 Bias (V)

40

FIG. V.5. Calculated values of the injection current components and the total device current as a function of bias for the Pt (upper panel), Cu (central panel), and A1 (lower panel) hole injecting contact devices (from Ref. 198).

energy barriers for hole injection into MEH-PPV were essentially zero for Pt, 0.2 eV for Au, 0.6 eV for Cu, and 1.1 eV for Al. The I-V curves for the devices with hole injection from Pt and from Au are essentially the same. This is expected because the energy barriers for hole injection are small enough in both cases that the current is space charge limited. When the Schottky barrier is increased in the device with a Cu contact, the current at a given bias voltage is much reduced, showing that the contact limits the current flow in this case. The current is further reduced when an A1 contact with a still larger barrier is used. The magnitude of the three components of the calculated injection current, that is the thermionic, interface recombination, and tunneling current components, together with the net device current for the devices with Pt, Cu and A1 contacts, is shown in Fig. V.5.19* The upper panel shows

90

I. H. CAMPBELL A N D D. L. SMITH

the case of a Pt injecting contact, where the energy barrier to injection is very small (0.05 eV was used in the calculation) and the current flow is space charge limited. In this case the magnitude of the thermionic emission and interface recombination currents are both very large and they almost exactly cancel each other. The net device current, which is the difference in magnitude between the thermionic emission and interface recombination currents, is much smaller than either of these two components separately. There is no tunneling current because the electric field at the contact has the wrong sign for tunneling. The thermionic emission current and interface recombination currents are independent of bias because the applied electric field is completely screened by a large carrier density at the interface. The calculated behavior of the injection current components for the case of the Au injecting contact (not shown), which has a 0.2 eV Schottky barrier, is qualitatively similar to that for the Pt contact. This behavior of the injection current components is characteristic of space charge limited diodes. The middle panel of Fig. V.5 is for the device with a Cu contact where the energy barrier is 0.6 eV and the current flow is contact limited. The magnitude of the thermionic emission and interface recombination components are still much larger than the net device current. The electric field near the contact is no longer screened by a high density of carriers near the interface as for the device with a Pt contact, and there is image force lowering of the barrier. Therefore the thermionic emission current is bias dependent and in addition tunneling is possible. Over most of the bias range shown, thermionic emission is much larger than tunneling, but tunneling is a stronger function of bias than is thermionic emission, and at the highest bias shown these two current components become comparable. In the range where thermionic emission dominates tunneling, a combination of thermionic emission and interface recombination (the time-reversed process of thermionic emission) keeps the interface hole density at its quasi-equilibrium value. In this regime the microscopic details of the current components do not influence the device model results. The behavior of the injection current components for the Cu/MEH-PPV/Au device is characteristic of contact limited diodes with moderate injection barriers. The lower panel of Fig. V.5 is for a device with an A1 contact where the hole barrier is 1.1 eV. All current components are strongly reduced by the large Schottky barrier and it is necessary to go to much larger bias to get a reasonably large current. Because tunneling is a more rapidly increasing function of bias than thermionic emission, tunneling is larger than thermionic emission at the bias required for significant current flow in this device with a large energy barrier. Interface recombination nearly cancels the tunneling injection current. The net device current is approximately proportional to the tunneling component but is much smaller in absolute

91

0

5

10

15

Bias (V) FIG. V.6. Measured (solid line) and calculated (dashed line) current density versus bias voltage for a Ca/Alq/Ca electron only device with a 100-nm thick Alq layer. The inset shows the same results on a log-log plot. Also shown is a calculated I-V characteristic assuming a SCL contact with a 0.1 eV energy barrier (dot-dash) (from Ref. 113).

magnitude. In this regime, the hole density at the injecting contact is not determined by quasi-equilibrium and the microscopic details of the current components are important for the device model results. The behavior of the injection current components for the Al/MEH-PPV/Au device is characteristic of contact limited diodes with large injection barriers. For practical diodes, it is very desirable to use small injection barriers so the large injection energy barrier regime is not of great practical interest. For Alq, neither electron nor hole contacts are SCL for common metal electrodes. The current-voltage characteristics of electron only Alq diodes were measured and compared to model calculations using the independently determined energy barrier (from internal photoemission, Section 111) and mobility (from TOF, Section IV). Figure V.6 shows measured (solid) and calculated (dashed) current-voltage characteristics for a Ca/Alq/Ca structure with a 100-nm thick Alq layer.'13 There is good agreement between the measured and calculated I-V characteristics. For the Ca contact, an energy barrier of 0.62 eV was used to describe the measured I-V characteristic. This energy barrier is in good agreement with the energy barrier of 0.6 eV f 0.1 eV determined from internal photoemission and built-in potential measurements. Also shown in Fig. V.6 is the calculated I-V characteristic assuming a SCL contact with a 0.1 eV energy barrier (dot-dash). The space charge

92

I. H. CAMPBELL AND D. L. SMITH

-10

0 10 Bias (V)

20

FIG. V.7. Measured (solid) and calculated (dashed) current density versus bias voltage for a Pt/Alq/Ca hole only device with 50-nm, 100-nm, and 150-nm thick Alq layers. The inset shows that the current scales with electric field including the built-in potential (from Ref. 141).

limited contact would give a substantially higher current density for a given voltage. As discussed in Section IV, the charge-dipole disorder model implies that the electric field dependence of the electron and hole mobilities of Alq should be comparable. It was not possible to measure the hole mobility of Alq using time-of-flight techniques and there are no space charge limited hole contacts to Alq. To estimate the hole mobility, the I-V characteristics of a thickness series of Pt/Alq/Ca devices were measured and fit using the hole mobility as the only adjustable parameter (the other parameters were determined in Sections I11 and IV). Figure V.7 shows the measured and calculated I-V characteristics for 50-nm, 100-nm and 150-nm thick Alq 1 a ~ e r s . IThe ~ ' inset shows that the current scales with electric field including the built-in potential. The fit to the hole mobility accurately describes the measured characteristics as a function of applied bias and film thickness. The hole mobility determined from this procedure is similar to the electron mobility but the exact mobility parameters are sensitive to the precise value of the hole Schottky barrier. The measured hole Schottky barrier is 0.4eV k 0.1 eV and this uncertainty in the barrier leads to relatively large uncertainties in the fit mobility. Nevertheless, within the experimental error, the electric field dependence of the electron and hole mobilities are similar as expected by the charge-dipole interaction model. The single-carrier single-layer device model results have a simple form when the current is contact limited. As an example, for an electron only device the electron current density is independent of position and equal to the net device current density, J,. When the current is contact limited both the electric field and the electron density are nearly constant across the device and the device current density is well approximated by the drift

PHYSICS OF ORGANIC ELECTRONIC DEVICES

93

component of the electron current density, J , = epN(O)E, where p depends on the electric field. From Eq. 5.19 the electron density is N ( 0 ) = N(0)QQ

+ J,, K

- Jd

~

(5.25)

The device current density is negligible compared to the sum of the thermionic and tunneling current densities and can be dropped in the preceding equation. There are two limiting regimes: one in which the thermionic emission current density is large compared to the tunneling current density and one in which the tunneling current density is large compared to the thermionic emission current density. In the first limit, the carrier density at the contact is given by its quasi-equilibrium value. This is a particularly simple case because all of the complexities of the injection process have canceled out and the interface carrier density is determined by statistics alone. In this case both N(O), because of the image force barrier lowering, and p have e f i field dependence. The second limit is more complex because the interface carrier density depends on the details of the injection processes. Summarizing, for typical organic LED device parameters, current is space charge limited if the Schottky energy barrier to injection is less than about 0.4 eV and contact limited if it is greater than that. Injection currents have a component due to thermionic emission and a component due to tunneling. Thermionic emission is more important for smaller injection barrier structures. If thermionic emission dominates injection, a combination of thermionic emission and interface recombination establishes a quasi-equilibrium carrier density at the interface. This is the case of interest for most practical devices. If tunneling dominates, a combination of tunneling and interface recombination establishes the carrier density at the interface. The net device current is small compared to the largest injection component. Interface recombination almost exactly cancels the injection current; that is, most carriers that are injected into the polymer fall back into the metal contact. As a result of the low mobilities of conjugated polymers only a very small fraction of the injected carriers are extracted from the contact region. c. Bipolar Devices SCL electron only Ca/MEH-PPV/Ca diodes were discussed in Section IV, and were used to extract the electron mobility parameters. Having determined both electron and hole mobilities from single-carrier diodes, these mobilities are used in the device model to describe bipolar, light-emitting structures. Figure V.8 shows current density versus bias voltage for a series of Pt/Ca bipolar devices with MEH-PPV layer thickness from 40 nm to 110

94

I. H. CAMPBELL AND D. L. SMITH 0.1 WCa

1

I

;*

3.-2. * A

0.05

CI

8

E

U

nI 0

4

8

Bias (V) lo-'

lo4 1

10

Bias (V) FIG. V.8. Measured (solid line) and calculated (dashed line) current density versus bias voltage for 40-, SO-,loo-, and 110-nm thick Pt/MEH-PPV/Ca bipolar devices on linear (upper panel) and log-log (lower panel) scales (from Ref. 163).

nm.163 The Pt and Ca contacts provide low-energy barriers for hole and electron injection, respectively. The data is described using the device model with the carrier mobility parameters determined from the single-carrier devices with no additional fitting parameters. The current is dominated by holes in these devices because the hole mobility is much larger than the electron mobility. The model describes the data over a range of device thicknesses and over several orders of magnitude of device current. Figure V.9 shows current density versus bias voltage for Pt/Ca and Cu/Ca bipolar devices, as well as Pt/Al and Cu/Al hole only devices.'63 The Pt/Ca and Cu/Al devices are 100 nm thick, the Pt/Al is 90 nm thick, and the Cu/Ca is 80 nm thick. The Pt/Ca device has space charge limited contacts for both electrons and holes, whereas the Cu/Ca device has space charge limited contacts for electrons, but holes are contact limited due to the larger energy

PHYSICS OF ORGANIC ELECTRONIC DEVICES

0

5

10

15

95

20

Bias (V) lo-'

lo-*

1o

.~

1o4

1

10

Bias (V) FIG. V.9. Measured (solid line) and calculated (dashed line) current density versus bias voltage for MEH-PPV devices with various contacts: Pt/AI and Cu/AI hole only devices and Pt/Ca and Cu/Ca bipolar devices on linear (upper panel) and log-log (lower panel) scales. Holes are injected from the Pt, Au, and Cu electrodes. Electrons are injected from the Ca electrode. Devices are about 100 nm thick (from Ref. 163).

barrier to injection of holes from Cu into MEH-PPV. The model describes the data well over several orders of current density using the mobility parameters determined from single-carrier devices. The Pt/Ca and Pt/Al devices have similar currents, the Pt/Al current is somewhat higher because it is thinner and has a smaller built-in potential. The Cu/Ca device has a substantially larger current than the Cu/Al devices. This is due in part to the thickness difference, but primarily because the Cu/Ca device current has contributions from both electrons and holes, whereas the Cu/Al device current is hole only. Figure V.10 shows calculated carrier density and electric field profiles for Pt/Ca and Cu/Ca bipolar devices, for biases that give a current density of 6x A / c ~ ' . ' In ~ ~both cases electrons are injected from the left at x = 0 and holes from the right at x = L. The electron and hole densities are given by the quasi-thermal equilibrium values at the injecting contacts. For the

96

I. H. CAMPBELL A N D D. L. SMITH

Position (nm)

Position (nm)

FIG. V.10. Calculated hole (solid line) and electron (dashed line) carrier density (upper panel) and electric field (lower panel) profiles for Pt/MEH-PPV/Ca and Cu/MEH-PPV/Ca devices. The electron injecting contact is at the left and the hole injecting electrode is at the right (from Ref. 163).

Pt/Ca device the electrons and holes are space charge limited and have high carrier densities at the injecting contacts that suppress the electric field. The electron density drops rapidly across the device. The slope of this drop is determined by a combination of the electron mobility and carrier recombination. The holes dominate the current density across virtually the entire device. For the Cu/Ca device the electron contact is space charge limited, whereas the holes are contact limited. The electric field and carrier densities near the hole injecting contact at x = L are relatively constant. The electron density is high near the electron injecting contact at x = 0, and screens the electric field. At this current density, the electron density is about 3 orders of magnitude larger than the hole density, however the electron mobility is about a factor of 300 lower than the hole mobility. The electrons and holes both contribute significantly to the device current. Figure V.11 shows calculated current density profiles for electrons and holes for the Cu/Ca device at applied biases of 6V, 9SV, and 13V, with current densities of 7x A/cmZ,3 x l o v 3A/cm2, and 6 x lo-' A/cmZ,re~pectively.'~~ The bipolar devices emit light and the measured luminance can be

97

PHYSICS OF ORGANIC ELECTRONIC DEVICES

I

0

40

80

Position (nm) FIG.V . l l . Calculated electron (dashed line) and hole (solid line) current density profiles for a Cu/MEH-PPV/Ca device. The current density profiles are shown for bias voltages of 6, 9, and 13V, corresponding to current densities of 7 x and 6 x 10-2A/cm2 respec3x tively (from Ref. 163).

compared with the expectations of the device model. Figure V.12 shows the measured and calculated external luminance as a function of device current for 40-nm and 110-nm thick Pt/Ca devices.’63 The symbols are the measured device luminance using a silicon photodiode placed flush against the LED substrate. The lines are calculated external luminance obtained by multiplying the calculated recombination current, J,, by the optical energy gap and by a factor 5 that is the fraction of recombination events leading to externally measurable light emission. 5 includes nonradiative recombination, total internal reflection inside the organic layer, and absorption as the emitted light goes through the semitransparent contact. It is taken to be 5 = 1/110 to fit the measured luminance for the 110-nm device. It is difficult to determine 5 quantitatively, but this value is a reasonable estimate. The model reproduces the linear behavior of luminance as a function of current density and the decrease in luminance with decreasing device thickness. However, the model underestimates the magnitude of the drop in luminescence with decreasing thickness. This difficulty with the model may be due to the assumption that radiative recombination efficiency is uniform across a device and the same for different device thickness.

98

I. H. CAMPBELL A N D D. L. SMITH 0.1

WCa

-

h

"&

$- 0.05-

.....

0

llOnm 4onm

li 1

0

0

0.025

0.05

current Density (A/cm2)

FIG. V. 12. Measured (symbols) and calculated (lines) external luminance versus device current density for 40-nm and 110-nm thick Pt/MEH-PPV/Ca devices (from Ref. 163).

d. Multilayer Devices The calculated carrier profiles shown in Fig. V.10 illustrate a potential problem for organic LEDs. Because the electron mobility is much smaller than the hole mobility for MEH-PPV, most of the device current is carried by holes and the recombination is strongly peaked near the electron injecting contact. These properties are undesirable for two reasons: Many of the injected holes traverse the device without recombining (i.e., J,(L) is comparable to JJO) in Eq. V.22); and recombination takes place near a contact where dipole quenching and nonradiative losses can reduce luminescence efficiency. Similar problems occur if it is only possible to make space charge limited contacts for one of the carrier types. In general, asymmetric injection or transport properties between electrons and holes causes one of the carriers to dominate the current flow. A dominant carrier type usually results in parasitic currents that do not produce recombination. Multilayer structures can be used to overcome these difficulties.33~349184~199-202 For example, a bilayer structure can be used to present an energy barrier to the dominant carrier and prevent it from traversing the device without recombining. Multilayer structures can also prevent the region of high recombination from occurring near an electrode; they are typically used to confine the carrier recombination to a thin region (10 nm) near an internal organic/ Ig9 2oo

L. S. Hung and C. W. Tang, Appl. Phys. Letf. 74, 3209 (1999). M. Strukelj, F. Papadimitrakopoulos, T. M. Miller, and L. J. Rothberg, Science 267, 1969

(1995).

C. Giebeler, H. Antoniadis, D. D. C. Bradley, and Y. Shirota, J. Appl. Phys. 85, 129 (1999). E. Bellmann, S. E. Shaheen, R. N.Grubbs, S. R. Marder, B. Kippelen, and N. Peyghambarian, Chemistry of Muferiuls 11, 399 (1999). 201 202

PHYSICS OF ORGANIC ELECTRONIC DEVICES

3.0eV

99

3.oev

organic A Iorganic B

I I

5.1eV-

-5.3eV 5.4eV

j

5.4-6.OeV

0.1

I

"a

0

'

I

I I I

10

I

'

I

20

Bias (V) FIG. V.13. Energy level diagram for single-carrier bilayer devices with a variable energy barrier to hole transport (above the panel). The holes are injected into organic B from a metal with an energy level at 5.3 eV that provides space charge limited current. Calculated current-voltage characteristics for the hole only bilayer devices as a function of the hole energy level discontinuity at the organic heterojunction (from Ref. 184).

organic interface. The thin region where the recombination occurs is often doped with a small luminescent organic molecule that serves as the dominant radiative recombination path. This is a convenient method of varying the optical emission energy in organic LEDs. To illustrate functions that can be achieved with multilayer structures, consider a bilayer hole only device with the energy level diagram shown in Fig. V.13. The conduction energy level is 3.0 eV for both organic materials. The valence energy level is 5.4 eV for material B, and is varied from 5.4 to 6.0 eV in 0.2 eV increments for material A. The two organic layers are 50 nm thick and have the same hole mobility parameters. The left (right) metal contact has an energy level of 5.1 eV (5.3 eV). The I-V characteristics are

100

I. H. CAMPBELL AND D. L. SMITH

4-

%-

g p

.s

3

4

-

- - - ,0.6eV I

32

_ _ _ _ - - - - - _J 0- ._w I

......................................... 00

0.2eV

-

0 eV

50

100

Position (nm)

FIG. V.14. Calculated hole density (upper panel) and electric field profiles (lower panel) for hole only bilayer devices at a current density of 0.1 A/cm2 as a function of the hole energy level discontinuity at the interface. The holes are injected from the right at x = 100 nm and the organic heterojunction is at x = 50 nm (from Ref. 184).

calculated for forward bias where the holes are injected on the right and there is an energy barrier for holes to traverse the structure going from material B into material A. (Details of how current flow at the heterostructure interface is included in the device model are discussed in Ref. [184].) Figure V.13 shows the calculated current-voltage characteristics for the hole only two-layer devices as a function of the energy barrier between the two organic materials.' 84 Successively higher voltages are needed to achieve a given current density as the energy barrier is increased. Even the modest 0.2 eV heterojunction barrier causes a significant increase in the voltage required to obtain a given current. Figure V.14 shows the calculated hole density and electric field profiles at a current density of 0.1 A/cm2 for the hole only bilayer devices as a function of the energy barrier between the two organic r n a t e r i a l ~ . 'The ~ ~ holes are injected into material B at the right contact. The hole density profile in material B remains unchanged as the energy barrier is increased, except near the heterojunction interface where the hole density increases as the barrier increases. The hole density in material A is relatively uniform across the

101

PHYSICS OF ORGANIC ELECTRONIC DEVICES

20

I

I

I

I

I

h

E

< >

15

m

-z 'p

10

Q)

i i

Luminescent Layer

0

2

4

6

8

10

Bias (V) FIG. V.15. Calculated electric field as a function of bias in the center of the blocking layer and in the center of the luminescent layer of a 0.5 eV single-carrier barrier structure (from Ref. 204).

layer and its magnitude decreases as the barrier increases. For a 0.4 eV barrier the hole density is over 2 orders of magnitude lower than for the case with no barrier. The field across material B does not change significantly as the energy barrier is increased. However, the field in organic material A increases and is nearly constant spatially across the layer. The increased voltage required to maintain a given current density is dropped across material A. A large field is needed to maintain a constant current density when the carrier density in material A decreases because of the energy barrier. For devices with a hole barrier there is a large accumulation of holes at the interface. The spike in the hole density at the interface causes a rapid change in the electric field at the interface. The field in the hole barrier layer (material A) is significantly larger than in the hole injection layer (material B). The electric field in the middle of both the blocking layer and the hole injection layer is plotted as a function of bias for a 0.5 eV barrier device in Fig. V.15. As the bias is increased, the electric field in the blocking layer increases rapidly. In contrast, the electric field in the hole injection layer remains relatively small and increases slowly with bias. The addition of a blocking layer causes an accumulation of charge at the blocking interface and, because of this charge accumulation, changes the electric field distribution in the structure. The effect of charge accumulation at a blocking layer in a multilayer organic device structure has been directly observed using electroabsorption

102

I. H. CAMPBELL AND D. L. SMITH

Bias (V)

FIG. V.16. Measured electric field as a function of bias in the blocking layer and in the luminescent layer of a 200-nm single-carrier barrier structure (from Ref. 204).

techniques.'03 To illustrate the approach, a structure was investigated that consisted of a large energy gap electron blocking layer, poly(p-phenylene diamine), and a smaller energy gap luminescent layer, MEH-PPV, each 100-nm thick sandwiched between Ca contacts. The internal electron energy barrier is about 1 eV. The Ca/luminescent layer contact has a negligible Schottky energy barrier and the Ca/blocking layer contact has large electron and hole Schottky energy barriers. Figure V.16 shows the measured electric field in each of the layers as a function of forward bias, i.e. electrons are injected into the luminescent layer. This is an analogous situation to the calculations shown in Fig. V.15 except that, in this case, electrons are blocked rather than holes. These experimental results demonstrate the effects of blocking layers by directly measuring the redistribution of electric field in the structure due to the accumulation of charge at the blocking layer/luminescent layer interface. Inserting an organic layer near a contact can also enhance current flow by serving as a transport layer. The transport layer can have increased carrier mobility, a reduced Schottky barrier, or both. The upper panel of Fig. V.17 shows the effect of increasing electron mobility in a layer near the electron contact of a two-layer electron only device.'04 (The energy barrier to electron injection at the metallic contact is 0.5 eV and there is no energy barrier at the heterojunction interface.) The solid line is the calculated I-V characteristic when the electron mobilities of the two layers are the same. I. H. Campbell, M. D. Joswick, and I. D. Parker, Appl. Phys. Lett. 67, 3171 (1995). I. H. Campbell and D. L. Smith, in Semiconducting Polymers: Chemistry, Physics and Engineering (G.Hadziioannou and P. Van Hutten, eds.), John Wiley & Sons, New York (2000). '03 '04

PHYSICS OF ORGANIC ELECTRONIC DEVICES

a E

9

103

,

Enhanced mobllity

0.06 0.04

0.02 0

o

2

4

6

8 1 0 1 2

Bias (V)

Position (nm) FIG. V.17. Calculated current density as a function of bias (upper panel) and electron density as a function of position at 12 V bias (lower panel) for a two-layer electron only 0.5 eV barrier structure. The mobility in the left layer is increased by a factor of 10 in the enhanced mobility structure (dotted line) (from Ref. 204).

The dotted line is the calculated I-V characteristic when the electron mobility in the layer near the electron injecting contact is increased by a factor of ten. The lower panel of Fig. V.17 shows the calculated electron density as a function of position at a 12 V bias.z04The electron current is constant across the structure. Because the electron mobility changes abruptly at the interface between layers, the electron density must also change abruptly at this interface. The electron density is larger in the lower mobility material at the right of the device. In a two-carrier structure, the holes will also be concentrated in the right layer. Thus the enhanced electron mobility helps to concentrate the electrons in the region where the hole density is high. To illustrate the effects of multilayer structures on device efficiency, consider a two-carrier device made from a two-layer structure using an electron contact on the left with a 0.5 eV injection barrier and a hole contact on the right with a 0.1 eV injection barrier. If there is no heterostructure barrier for holes, the hole current is space charge limited because of the

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0

2

4

6

8

1

0

Bias (V) FIG. V.18. Calculated current (solid line) and recombination current (dashed line) density as a function of voltage bias for a single-layer structure, a two-layer structure with a hole blocking layer, and a two-layer structure in which the hole blocking layer also serves as an electron transport layer (from Ref. 204).

small energy barrier to hole injection. The electron current, however, is contact limited because of the comparatively large energy barrier to electron injection. If the electron and hole mobilities are the same, the device current will be dominated by holes. Such a structure will not be an efficient LED because most of the injected holes will traverse the device without recombining with the comparatively small number of injected electrons. If a heterojunction energy barrier for holes is included, the holes are confined in the layer on the right by this barrier and the number of holes traversing the device without recombining is reduced. If in addition the layer on the left has increased electron mobility, electron current will increase and the voltage necessary to reach a given current will decrease. Figure V.18 shows the calculated current density (solid lines) and recombination current density (dashed lines) as a function of bias for a single layer structure, a structure with a 0.3 eV hole barrier, and a structure with a 0.3 eV hole barrier and also with a factor of 10 enhanced electron mobility in the left material layer.204 For the single-layer device the recombination current density is only about a fifth of the total device current; the current is dominated by holes and only about a fifth of them recombine. The other four-fifths of the holes traverse the device without recombining and are lost at the electron injecting contact. When the 0.3 eV hole blocking layer is included, the holes cannot easily cross the energy barrier. The recombination current density is essentially equal to the total current density, meaning that essentially all the carriers injected into the device recombine. In this case the device quantum efficiency is limited by the efficiency of radiative recombination. However, the bias necessary to reach a given current density has been increased by the hole blocking layer. When the electron mobility

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of the organic layer on the left is also increased, so that it serves as both a hole blocking layer and also as an electron transport layer, both the total current and the recombination current are increased. The enhanced electron mobility has increased the electron current. In this structure the current in the left layer is carried only by electrons; the holes are effectively blocked from this layer. The electrons recombine when they enter the layer on the right, which has a large density of holes. It is not necessary to use an electron blocking layer in this structure because the hole injection is high so that there are a large number of holes in the structure with which the electrons can recombine. In the model calculations previously presented, the energy offset between the electron and hole energy levels at organic heterojunctions was taken as an adjustable parameter. In organic materials, the heterojunction energy offsets are generally given by the difference between the energy levels of the constituent materials with respect to v a c ~ u m . ' Although ~ ~ * ~ ~there ~ can be small energy changes due to polarization effects at the i n t e r f a ~ e , this '~~~~~~ energy alignment is expected for the weak van der Waals interactions between the organic molecules. However, organic heterojunction interfaces can have excited state intermolecular interactions. For example, excited states called exciplexes can be formed consisting of an electron in one organic layer and a hole in the other materia1.206*207 The exciplexes can have optical properties very different from the constituent materials and they can be the dominant recombination state in multilayer devices where most of the electrons and holes recombine at a heterojunction interface. e. Transient Response and High Current Density Operation In many applications, such as video displays, the frequency response of the diode is important. The response time of an organic LED is determined by the carrier transit and recombination times.208-210The carrier transit time is approximately (z = L 2 / p v which is about s for typical organic device parameters. Although the carrier mobility in disordered organic films is very low, the length that the carriers traverse is short and they can therefore be modulated at relatively high frequencies. The turnoff time for I. G. Hill and A. Kahn, J. Appl. Phys. 84, 5583 (1998). D. D. Gebler, Y. Z. Wang, J. W. Blatchford, S. W. Jessen, D. K. Fu, T. M. Swager, A. G . MacDiannid, and A. J. Epstein, Appl. Phys. Lett. 70, 1644 (1997). '07 D. D. Gebler, Y. Z . Wang, D. K. Fu, M. Swager, and A. J. Epstein, J. Chem. Phys. 108, 7842 (1998). ' 0 8 D. J. Pinner, R. H. Friend, and N. Tessler, J. Appl. Phys. 86, 5116 (1999). '09 V. R. Nikienko, V. I. Arkhipov, Y. H. Tak, J. Pommerehne, H. Bassler, and H. H. Horhold, J. Appl. Phys. 81,7514 (1997). 'lo V. I. Nikitenko, Y. H. Tak, and H. Bassler, J. Appl. Phys. 84, 2334 (1998). '05 '06

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light emission is considerably faster because the carrier recombination times are between 1 ns and 20 ns. The devices previously described were operated at steady state with current densities of the order 0.1 A/cm2 or below. Steady state operation is of interest for many display applications. However, there are also applications where high-intensity pulsed operation is desired. In this mode of operation considerably higher electric fields and carrier densities occur in the diodes. The extent to which material parameters, such as mobility, and device behavior extracted from measurements performed at much lower current densities can be extended to these high bias conditions is addressed using pulsed high-current measurements. The MEH-PPV structures used for the pulse bias measurements were designed to minimize series resistance. The devices consist of thin MEH-PPV films ( < l o 0 nm) sandwiched between thick Pt and Ca electrodes. Test structures made with thinner Ca films had similar I-V characteristics at low current density, but were dominated by series resistance effects at high current density. The electron mobility of MEH-PPV is much smaller than the hole mobility and because of this large mobility difference, holes dominate the current flow in these structures. The electrical properties of the devices are essentially space charge limited hole only devices. The injected electrons recombine close to the cathode and do not significantly alter space charge in the device. The measured current-voltage characteristics were acquired using both steady state and pulsed techniques at room temperature. The pulse duration and the rise time of the electrical excitation were 250 ns and 15 ns, respectively, and the repetition rate was 10 Hz. The transient current was measured using an inductively coupled current probe. The devices were designed so that Joule heating was not significant. Figure V.19 is a log-linear plot of the measured (solid and crosses) and calculated (dashed) current density as a function of applied bias for structures with 40-nm and 60-nm thick MEH-PPV layers.2" The steady state I-V measurements are shown as solid lines and the pulsed measurements as crosses. There is good agreement between the measured and calculated I-V characteristics over six orders of magnitude in current. The fit to the I-V measurements yielded mobility parameters p0 = 7.8 x lo-' cm2/vs and E , = 4 x lo4 V/cm. Figure V.20 shows the calculated hole density and electric field as a function of position for the 40-nm device at biases of 3 V (0.1 A/cm2) and 15V (1 x lo3 A/cm2).211The hole density is shown on the left vertical axis and the electric field on the right. The hole injecting contact (Pt) is at the origin and the electron injecting contact (Ca) is at the other end of the device (40 nm). In both cases, the electric field and hole density are nonuniform due to the I. H. Campbell, D. L. Smith, C. J. Neef, and J. P. Ferraris, Appl. Phys Lett. 75,841 (1999).

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Bias (V) FIG. V.19. Measured (solid, CW, and crosses, pulsed) and calculated (dashed) currentvoltage characteristics for Pt/MEH-PPV/Ca structures with polymer thickness of 40 nm and 60 nm. Positive bias corresponds to hole injection from Pt (from Ref. 211).

space charge of the injected holes and the hole density is below 10'' cm-3 in the bulk of the material. The electron density (not shown) is only significant near the electron injecting contact due to the low mobility of electrons in this material. These calculated carrier density profiles do not depend strongly on the assumed charge carrier mobility. For a given device

--

_ - - - I

I lo160

10

20 30 Position (nm)

-1

40

FIG. V.20. Calculated carrier density (solid) and electric field (dashed) profiles for the 40-nm thick device of Fig. V.19 at applied biases of 3V (0.1 A/cm2) and 15V (lo3 A/cm2), respectively. The Pt contact is at left and the Ca contact is at the right (from Ref. 211).

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geometry and applied voltage the magnitude of the current depends on the value of the mobility but the injected charge density does not. The maximum electric field and carrier density is about 4 x lo6 V/cm and 1 x 10" ~ m - respectively. ~ , These results demonstrate that an electric field dependent mobility, without carrier density dependence, provides an accurate description of hole transport in this polymer over this range of field and carrier density. 16. FIELD-EFFECT TRANSISTORS

Organic field-effect transistors have been investigated since the late 1980s. Recently, they have achieved performance comparable to inorganic thin-film transistors. The structure of an organic FET is shown in Fig. 1.5. The organic FET operates in a manner analogous to inorganic thin-film transistors employing undoped semiconductor layers and doped semiconductor contacts. Although a detailed, quantitative model for organic FETs has not yet been demonstrated, much of the essential device physics is clear. Organic FETs operate in a charge injection mode where the charge is injected into the organic material from the metallic source and drain contacts. The charge injection mode is distinct from both charge accumulation and charge inversion modes of operation. Conventional doped, inorganic FETs operate in a charge inversion regime. For example, a p-channel transistor uses p-type semiconductor contacts and an n-type doped semiconductor layer in which the channel is formed.74 When sufficient gate bias is applied to the structure, an inversion layer is formed consisting of a thin p-type region in the n-type doped semiconductor adjacent to the gate insulator (the channel). The p-type doped semiconductor contacts make good electrical contact only to this inversion layer. In the on state, the current flows through the inversion layer. In the off state, the leakage current is very low because the structure consists of a reverse biased diode in series with a forward biased diode. This high off state resistance is one of the principle advantages of doped inorganic transistors. In contrast, organic FETs operate in a charge injection regime. Because the organic material in which the channel is formed is undoped, all of the charge in the channel is injected from the contacts. Space charge limited contacts, as described in Section 111, are used. At zero gate bias, the organic material contains no free charges except for a thin region (a few nm) near the contacts that contains charge thermally excited from the metals into the organic material. When a gate bias is applied to the structure, charges are injected into the organic material from both source and drain contacts, forming a thin sheet of charge adjacent to the gate insulator (the channel). In the on state, the current flows through this thin sheet of charge. In the

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off state, the leakage current is low because of the high intrinsic resistivity of the undoped organic film. The charge injection regime is not equivalent to a charge accumulation regime. Transistors operating in an accumulation regime utilize doped organic or semiconductor layers but, instead of forming an inversion layer, the channel is formed by biasing the gate to accumulate charge of the same type as the layer doping. For example, a p-type transistor has the gate biased so that additional holes accumulate adjacent to the gate insulator. Devices operating in the accumulation regime have large leakage currents and their total current is sensitive to the organic layer thickness. The poor leakage current and sensitivity to film thickness occur because the contacts are not rectifying, as they are in the inversion regime, so current can flow through the doped regions of the organic material independent of the gate bias. It is therefore not desirable to operate in an accumulation regime. A device model of an organic FET needs to include the charge transport properties of the organic material and the electrical properties of the metallic contacts. The most important properties that must be included are the charge density and electric field dependence of the carrier mobility and the contact interface electronic structure. Because the charge density and electric field in the organic material vary by many orders of magnitude, including an accurate description of the mobility is important for calculating the charge density and potential profiles within the device. At present, the charge transport properties of organic materials, particularly at the high carrier densities typical of FETs, are not very well understood. A detailed numerical device model is likely to help explain discrepancies between conventional FET models and organic device measurements. To date, organic FETs have been analyzed using inorganic thin-film transistor models that use electric field and charge density independent mobilities and ignore the details of the contacts.’ The drain-source current in such a model is

I,

W L

= - pCi[VGs -

V,] V,,

W I,, = - pCi[VGs- V,]’ 2L

Linear Region

(5.26)

Saturation Region

(5.27)

where W is the device width, L is the channel length, Ci is the capacitance per unit area of the insulating layer, VGsis the gate-source voltage, V,, is the drain-source voltage and V, is the offset voltage. The linear region applies for V,, << (VGs - V,) and the saturation regime for V,, > V,, where I,, is constant, independent of V,,. The offset voltage accounts for several factors, including the effect of work function differences between the gate- and

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I. H. CAMPBELL AND D. L. SMITH

0

-1

-2

-3

-4

-5

v,, (V) FIG.V.21. Transistor current-voltage curves for a pentacene FET fabricated on a Si substrate that serves as a gate electrode, with a YSZ gate insulator and Pt source and drain electrodes.

drain-source contacts and trapped charges in the gate insulator. It is typically determined by fitting to measured FET I-V characteristics. The charge carrier mobility is usually determined in the saturation region from the slope of vs V,, and the known structure parameters. Using this approach, the best organic FET mobilities at room temperature are about 1 cm2/vs, comparable to amorphous silicon. To show an example of organic FET properties, devices were fabricated on p-type Si substrates using yttria stabilized zirconia (dielectric constant about 25) as the gate insulator and pentacene as the organic material. The structures used Pt source and drain contacts, a 150-nm thick gate insulator, and a gate length of 10 pm with W / L = 350. The drain-source current as a function of gate bias is shown in Fig. V.21. The offset voltage is close to zero as expected, because the ionization potential of pentacene and the valence band of silicon are both about 5 eV and the insulator is relatively free of trapped charge. The mobility determined from these I-V characteristics using Equations (5.26) and (5.27) is about lo-' cm2/Vs in both the linear and saturation regimes of the transistor. This mobility is almost two orders of magnitude higher than that determined from the time-of-flight and space charge limited current-voltage measurements shown in Section IV. The origin of this higher mobility may be due either to an increase in the mobility with carrier density or to an anisotropic mobility characteristic of

PHYSICS OF ORGANIC ELECTRONIC DEVICES

111

-5

-4

-3

-1

0 0

-1

-2 “D

-3

-4

(v)

FIG. V.22. Transistor current-voltage curves for a pentacene FET fabricated on a transparent polycarbonate substrate. (Reprinted with permission from Ref. 32. Copyright 1999 American Association for the Advancement of Science. Readers may view, browse, and/or download this material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or in part, without prior written permission from AAAS.)

the film structure. Reliably assessing the film structure, e.g. using x-ray diffraction, is difficult because the conduction occurs in a layer only a few nm thick adjacent to the insulator. Organic FETs can be made flexible, low cost, lightweight, and transparent. High-performance pentacene FETs have been fabricated on transparent, plastic polycarbonate substrates. The current-voltage characteristics of these transistors are shown in Fig. V.ZL3’ The mobility of these FETs was 0.2 cm’/vs as determined in the saturation regime. Mechanically flexible, organic transistors with good performance have also been fabri~ated.’~ These transistors used polymers for the substrate, the gate, drain, and source contacts, the gate insulator, and the active material in which the channel is formed. The most important properties of field-effect transistors are their frequency response and transconductance. The response time of organic FETs is determined by the carrier transit time. The transit time is z = L’/pV’,, which is about lop6s for typical FET parameters. The response time of the FET is therefore similar to that of LEDs. The higher mobility observed in

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FETs compensates for the greater distance between the electrodes. The transconductance, i.e. dI,/a V, at constant drain voltage, of the organic FETs (in the linear region), again using a constant mobility model, is g = WpCiVD/L. For the high dielectric constant FETs operating at low voltage previously discussed the transconductance is about mhos. Organic transistors are being explored for chemical sensor applications. Because the current in an organic transistor is carried in a thin organic layer about 3 nm thick, it is straightforward to produce FETs with very thin active organic layers that can be directly exposed to the ambient. The current-voltage characteristics of the transistor are sensitive to vapor molecules. The vapor molecules can change both the mobility and density of the charge carriers in the organic film. Each molecule produces distinct changes in the transistor response that can be used to identify the vapor molecule. The chemical specificity and strength of the sensing interaction can be tuned by adding chemical functional groups to the active organic molecules or by incorporating other functional organic materials into or on top of the active organic layer. An example of organic transistor response is shown in Fig. V.23. The I-V characteristics are shown for the following exposure conditions: ambient, and 1 part-per-thousand each of methanol, ethanol, p-xylene, o-xylene, and tetrahydrofuran (THF). In all cases, exposure to the vapor decreases the drain-source current. The changes in the drain-source current are a function of the drain-source voltage, i.e. the shapes of the I-V curves are different. The change in shape of the I-V characteristic shows that it is possible to distinguish molecules from their distinct responses as a function of voltage. To illustrate this, consider the differential response, R, of the transistor. The differential response is the derivative of the drain-source current with respect to the drain-source voltage of the reference ambient response divided by the derivative of the drain-source current with respect to the drain-source voltage of the sensor gas response at constant gate voltage or

R=-

(5.28) VG

where IR(n[ vR(,)] is the drain-source current [drain-source voltage] of the reference ambient (sensor gas) response. The lower panel in Fig. V.23 shows the differential response of the transistor upon exposure to different vapors. The xylene, THF, and alcohol molecules have substantially different differential response as a function of voltage. The 2 xylene molecules have peaks at about 2-3 V and the 2 alcohol molecules have peaks at about 5-6

PHYSICS OF ORGANIC ELECTRONIC DEVICES

0

-2

-6

-4 'DS

-8

113

-10

1('

FIG.V.23. Effect on the current-voltage curves of exposing a pentacene FET to various organic solvents (upper panel), dserential response (Eq. 5.27) of the transistor current-voltage curves (lower panel).

V. The THF molecule has a sharper peak at about 3 V. The large amount of information available from a single transistor in a chemical sensor array will add significantly to its ability to discriminate and identify molecules. The response time of these organic thin-film transistors is a few seconds at concentrations of about 1 part-per-thousand.

VI. Summary and Future Directions

This article discussed the essential device physics governing the operation of organic electronic devices focusing on aspects relevant to light-emitting

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diodes and field-effect transistors. In about a decade of research, most of the basic physical mechanisms governing organic LED performance have been identified and many important problems have been resolved. Organic electronic devices use undoped, insulating organic materials as light-emitting and charge-transporting layers. Carriers are injected into the insulating organic materials from metallic contacts. The organic materials are conjugated small molecules or polymers that are either vacuum evaporated or cast from solution to produce disordered thin films. There are no lattice matching issues for substrates or heterostructures such as occur for inorganic semiconductors. The ease and flexibility of fabrication are major advantages of organic electrical devices. The basic device physics of organic electronic devices is distinct from doped inorganic crystalline semiconductor based devices. The rectification of inorganic diodes is due to the electrical junction between oppositely doped, p- and n-type regions of the inorganic semiconductor. In contrast, the rectification of organic diodes is caused by the use of asymmetric metal contacts. One metal contact is only able to inject electrons efficiently and the other contact only injects holes efficiently. Therefore, the properties of metal/organic contacts, rather than doping profiles, determine the behavior of the device. Transistor action in conventional, inorganic semiconductor FETs is due to the formation of an inversion layer in the doped semiconductor under the gate insulator. In undoped organic FETs, transistor action is due to injection of charge into the insulating organic film. The injected carriers form a thin sheet of charge adjacent to the gate insulator that is the conducting channel. Charge transport in the disordered organic thin films occurs by hopping from site to site rather than by band transport. This hopping mechanism leads to mobilities in organic films that are orders of magnitude lower than inorganic semiconductor mobilities and that depend strongly on electric field and carrier density. For the conditions typical of organic LEDs, the carrier mobilities vary from about lo-’ cmZ/Vsto lo-’ cm’/Vs. These low mobilities require LED device designs that utilize transport normal to thin films (about 100 nm thick) rather than parallel to the film. Organic LEDs have been demonstrated with display level brightness at bias voltages below 3 V, internal quantum efficiencies above 25%, and continuous operating lifetimes in excess of 9 years. They are now used in a number of display products and their development for commercial applications is accelerating. Organic field-effect transistors have not yet been commercially developed. To be technologically useful, FETs require carrier mobilities of about 1 cm2/vs or higher. To achieve these relatively high mobilities in organic thin films it is necessary to use either high electric fields, well-ordered thin organic films, or high carrier densities. Organic FETs have recently been

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produced with carrier mobilities over 1 cm’/Vs and operating voltages below 5 V, comparable to widely used amorphous Si thin-film transistors. These devices are becoming promising for large-area, low-cost electronics applications. The scientific and technological interest in these materials and devices is accelerating and new products and applications continue to emerge. Examples include optically pumped solid state organic lasers and perhaps solid state light-emitting electrically pumped organic diode lasers,”’ -’’I electrochemical cells,222-’x and broad spectrum white light-emitting diode^.'"^^'^ The basic device physics of organic LEDs is becoming relatively clear and the focus of much device research is shifting to understanding the unique aspects of organic FETs.”~ An organic transistor device model incorporating the relevant charge injection and transport properties of organic materials is needed. The success of organic electronic devices has led to wide interest in designing new organic materials with properties suited for specific applications. The objective is to be able to design an organic molecule or polymer F. Hide, M. A. DiazGarcia, B. J. Schwartz, M. R. Anderson, Q. B. Pei, and A. J. Heeger, Science 273, 1833 (1996). ’13 F. Hide, B. J. Schwartz, M. A. DiazGarcia, and A. J. Heeger, Chern. Phys. Lett. 256,424 (1996). 2 1 4 M. D. McGehee, M. A. DiazGarcia, F. Hide, R. Gupta, E. K. Miller, D. Moses, and A. J. Heeger, Appl. Phys. Lett. 72, 1536 (1998). ’15 N. Tessler, G. J. Denton, and R. H. Friend, Nature 382, 695 (1996). 2 1 6 N. Tessler, N. T. Harrison, and R. H. Friend, Advanced Materials 10,64 (1998). N. Tessler, D. J. Pinner, V. Cleave, D. S. Thomas, G. Yahioglu, P. LeBarny, and R. H. Friend, Appl. Phys. Lett. 74,2764 (1999). 2 1 * V. G . Kozlov, P. E. Burrows, G. Parthasarathy, and S. R. Forrest, Appl. Phys. Lett. 74, 1057 (1999). 219 C. Zenz, W. Graupner, S. Tasch, G. Leising, K. Mullen, and U. Scherf, Appl. Phys. Lett. 71,2566 (1997). ’O S. V. Frolov, M. Shkunov, Z. V. Vardeny, and K. Yoshino, Phys. Rev. B56,R4363 (1997). 2 2 1 S. V. Frolov, 2. V. Vardeny, and K. Yoshino, Phys. Rev. B57,9141 (1998). 2 2 2 Q. B. Pei, G. Yu, C. Zhang, Y.Yang, and A. J. Heeger, Science 269, 1086 (1995). 223 J. Gao, G. Yu,and A. J. Heeger, Appl. Phys. Lett. 71, 1293 (1997). 2 2 4 S. Tasch, L. Holzer, F. P. Wenzl, J. Gao, B. Winkler, L. Dai, A. W. H. Mau, R. Sotgiu, M. Sampietro, U. Scherf, K. Mullen, A. J. Heeger, and G . Leising, Synthetic Metals 102, 1046 (1999). L. Holzer, B. Winkler, F. P. Wenzl, S. Tasch, L. Dai, A. W. H. Mau, and G. Leising, Synthetic Metals 100, 71 (1999). 2 2 6 Y. Yang and Q. B. Pei, Appl. Phys. Lett. 70, 1926 (1997). 227 2. L. Shen, P. E. Burrows, V. Bulovic, S. R. Forrest, and M. E. Thompson, Science 276, 2009 (1997). 2 2 8 R. S. Deshpande, V. Bulovic, and S. R. Forrest, Appl. Phys. Lett. 75,888 (1999). 229 B. Crone, A. Dodabalapur, Y.-Y. Lin, R. W. Filas, Z. Bao, A. LaDuca, R. Sarpeshkar, H. E. Katz, and W. Li, Nature 403,521 (2000). ‘12

’”

’”

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0

2

4

6

8 1 Bias (V)

0

1

2

Input Power w/cmz)

FIG. VI. 1. Calculated device current density (solid lines) and recombination current density (crosses) as a function of bias (upper panel), and output power as a function of input power (lower panel) for single-layer LEDs with different mobilities. The LEDs are 100 nm thick, have equal electron and hole mobilities with E , = 5 x lo4 V/cm for all cases, and with p,, values of and lo-’ cm’/vs.

so that thin films made from it have specific electrical and optical properties. This goal requires a thorough understanding of the properties of the molecule or polymer and how those properties determine the thin-film behavior. An important objective is to increase the carrier mobility of the organic thin films. For example, increasing the carrier mobility would dramatically improve the power efficiency of the LEDs. To illustrate this point, model calculations of the device current density, recombination current density, and input and output power for single-layer LEDs with different mobilities are shown in Fig. VI.l. The LEDs are 100 nm thick and have equal electron and hole mobilities; the carrier mobility has the Poole-Frenkel form with Eo = 5 x lo4 V/cm for all cases and with po values of lop6, and lo-’ cm’/vs. The diode with a mobility prefactor of

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cm’/Vs is typical of current LEDs. The upper panel of Fig. VI.l shows the calculated device current density (lines) and recombination current density (crosses) as a function of bias for the three diodes. In all cases, the recombination current is nearly equal to the device current but the bias required to reach a given current decreases rapidly as the mobility prefactor is increased from to lo-’ cm’,Vs. This decrease in bias significantly improves the device power efficiency. The lower panel of Fig. VI.l shows the output power as a function of input power for the 3 diodes (solid lines); the dashed line represents a device with 100% efficiency, i.e. equal input and output power. At an input power of 10 W/cm’, the power efficiencies are 20%, 50%, and 95% (increasing with higher mobility). Increasing the mobility to the lo-’ cmZ/Vs range will allow devices with near ideal power efficiency for input powers up to about 20W/cm2. This could enable the development of high efficiency, low cost, lighting and other high brightness applications. The development of organic electronic materials and devices is at an exciting time, in some ways analogous to the early development of inorganic semiconductor devices, in which the interaction between physical understanding, improved materials, and new device measurements led to rapid progress in both scientific understanding and technological application. Organic electronic materials and devices have many unique properties, such as large area processing, mechanical flexibility, tunable light emission, chemical sensing interactions, and biocompatibility, which make them attractive for a wide range of applications that are largely inaccessible to conventional inorganic semiconductor devices. The next decade is likely to see continuing rapid progress and exciting new developments in this rich area of science and technology. ACKNOWLEDGEMENTS The authors are grateful to many collaborators who have made major contributions to this work, including Alan Bishop, David Brown, Brian Crone, Paul Davids, Thomas Hagler, Christian Heller, Michael Joswick, Joel Kress, Richard Martin, Duncan McBranch, Avadh Saxena, Zhi Gang Yu, and Thomas Zawodzinski at Los Alamos, and Nikolai Barashkov, John Ferraris, and Charles Neef at the University of Texas at Dallas. This research was supported by the Los Alamos Directed Research and Development Program, the Defense Advanced Research Projects Agency, and the Department of Energy.

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SOLID STATE PHYSICS. VOL. 55

Charge Density Wave Formation in Nanocrystals PHILIP

KIM'. JIAN ZHANG'.

AND CHARLES

M . LIEBER3

'Department of Physics. University of California. Berkeley. CA 94720 'Optikos Corporation Cambridge. M A 02141 3Division of Engineering and Applied Sciences and Department of Chemistry and Chemical Biology. Harvard University. MA 02138

.

Contents

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Charge Density Waves in Macroscopic and Mesoscopic Scale . . . . . . 3. Fabrication of Nanostructures . . . . . . . . . . . . . . . . . . . . . I1. Charge Density Waves in Bulk Transition Metal Dichalcogenides . . . . . . 4. Instability of Electron Gas in Low Dimensions and CDW Formation . . 5. Crystal Structure of Transition Metal Dichalcogenides . . . . . . . . . 6. CDW Formation in Transition Metal Dichalcogenides . . . . . . . . . 111. Creation of CDW Nanocrystals by STM Tip . . . . . . . . . . . . . . . . 7. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . 8. Nanocrystal Formation and Characterizations . . . . . . . . . . . . . 9. Structural Model for the Solid-Solid Phase Transition . . . . . . . . . 10. Driving Force for the Solid-Solid Phase Transition . . . . . . . . . . . IV . Finite Size Effect and Fermi Surface Roughening . . . . . . . . . . . . . . 11. Size-dependent Effect in Nanocrystals . . . . . . . . . . . . . . . . . 12. Fermi Surface Nesting in Nanocrystals . . . . . . . . . . . . . . . . . V . Nanocrystal Fabrication in Other TMD Systems . . . . . . . . . . . . . . 13. Nanocrystal Formation in H-layer of 4Hb-TaSe2 . . . . . . . . . . . . 14. Nanocrystal Formation in 2H-TaS2 . . . . . . . . . . . . . . . . . . 15. Local Transformation of 1T-TaS2 and 1T-TaSe, . . . . . . . . . . . . 16. Nanocrystal Formation in Other TMD Materials . . . . . . . . . . . . VI . Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120 120 122 124 126 126 128 129 132 133 133 137 141 142 142 143 148 148 151 152 154 156 157

119 ISBN 0-12-607755-X ISSN 0081-1947/01 $35.00

Copyright It' 2001 by Academic Press All rights of reproduction in any form reserved

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1. Introduction

1. PREFACE Physics in nanometer scale structures has attracted much interest during the last few decades because of enhanced quantum effects in these small systems. The study of nanostructures is, in part, motivated by the trend toward smaller devices in the semiconductor industry. As the elements in devices achieve nanometer dimensions, it is necessary to investigate the role of quantum confinement of electrons to understand new phenomena that manifest themselves in these nanoscale structures.' The artificial nanometer structures used in this study range from semiconductor or metallic nanostructures fabricated by modern lithographic technologies to chemically synthesized nanocrystals. Various mesoscopic phenomena such as single electron tunneling, energy level quantization,' phase coherence, and quantum transport3 have been intensively studied in these confined nanostructures. Furthermore, the ability to control the precise size and shape of the nanostructure has also provided many opportunities for scientific discovery, such as strong size-dependent optical and electrical properties of chemically synthesized quantum dot.4 In addition to the investigation of enhanced quantum effects in confined structures, these systems enable us to understand the change of physical properties evolved from isolated atoms or small molecules to an infinite bulk phase. For example, well-defined discrete quantum levels of an atom/ molecule are related to the electronic band structure of the bulk. The ionization energy of individual molecule corresponds to the work function of the bulk. Physics of atoms and molecules has been studied over decades, and we have a relatively good understanding now. However, it is the mesoscopic scale that bridges this microscopic scale to the macroscopic scales in the bulk. Localization-delocalization of electrons, many-body electron interaction, and phase coherency in mesoscopic systems are some of key elements to understanding such transitions from systems of a few electrons in an atom or molecule to systems of loz3electrons in the bulk. To trace the evolution of electronic structures from the bulk phase to the mesoscopic scale, it is useful to describe the system in Bloch reciprocal space. Continuum of wavevector in Bloch reciprocal space, which is a good

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For a general review, see, for example, M. Reed, Sci. Am. 268, 118 (1993). For a review, see M A. Kastner, Rev. M o d . Phys. 64, 849 (1992); R. C. Ashoori, Nature 319, 413 (1996). C. W. Beenakker and H. van Houten, Solid State Physics 44, eds. H. Ehrenreich and D. Turnbull, Academic Press, Boston (1991). A. P. Alivisatos, Science 271, 9333 (1996).

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quantum number in the bulk metal, is no longer a good quantum number in a nanostructure. Due to the confinement of electrons, only states that satisfy the boundary condition are allowed, and this discrete nature of allowed states in Bloch reciprocal space becomes important even at relatively high temperatures. A smooth Fermi surface of bulk metal should be modified to accommodate this effect, as the system size approaches atomic scales. Therefore, the macroscopic quantum states, such as superconductivity and charge density wave (CDW), which is due to electron-electron and electron-hole pairing across the Fermi surface of bulk material respectively, might have a different nature in mesoscopic systems. As an example, superconductivity in a confined mesoscopic system has In these been extensively studied recently by several research studies, it was shown that the superconducting state depends strongly on the boundary condition imposed by the sample size and shape in the mesoscopic regime, where the superconducting coherent length, 5, is comparable to the size of system L,.’ As the system size decreases (L, << 0, the discrete energy-level spectrum should change superconducting properties significantly.6Experimentally, superconductivity in the mesoscopic scale can be identified by the presence of an energy gap, AE, in the quasiparticle tunneling spectrum, which is significantly larger than the energy spacing between electronic eigenstates, SI7Black et al. reported a superconducting gap, R, in 5 nm A1 nanoparticles, which is larger than the bulk gap value and can be driven to zero by applying a sufficiently large magnetic field.* This experimental observation demonstrates, the effect of confinement and quantization on a collective quantum state in the mesoscopic system. CDWs at mesoscopic scales have been studied by transport measurements in the mesoscopic regime where the phase coherence length of CDW is comparable to the system size.9 These studies showed that there are some

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V. V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X.Qiu, C. Van Haesendonck, and Y. Bruynseraede, Nature 373,319(1995);C. Strunk, V. Bruyndoncx, V. V. Moschalkov, C. Van Haesendonck, Y. Bruynseraede, and R. Jonckheere, Phys. Rev. B. 54, R12701 (1996); V. Buyndoncx, L. Van Look, M. Verschuere, and V. V. Moshchalkov, Phys. Rev. B. 60, 10468 (1999). M. T. Touminen, J. M. Hergenrother, T. S. Tighe, and M. Tinkham, Phys. Rev. Lett. 69,1007 (1992);also Phys. Rev. B. 47, 11599 (1993); M. Tinkham, J. M. Hergenrother, and J. G. Lu, Phys. Rev. B. 51, 12649 (1995). Eventually, the superconductivity will be extinguished in molecular scale structure, where AE-R. See, for example, J. A. A. J. Perenboom, P. Wyder, and F. Meier, Phys. Rep. 78, 174 (1981). * C.T. Black, D. C. Ralph, and M. Tinkham, Phys. Rev. Lett. 76, 688 (1996). A. J. Steinfort, H. S. J. van der Zant, A. B. Smits, 0. C. Mantel, P. M. L. 0. Scholte, and C. Deker, Phys. Rev. B. 57, 12530 (1998); 0.C. Mantel, C. A. W. Bal, C. Langezaal, C. Dekker, and H. S . J. van der Zant, Phys. Rev. B. 60, 5287 (1999).

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similarities between the transport phenomena in superconductors and the collective sliding motion of CDWs." In addition to these similarities, the local variation of CDWs in mesoscopic systems, which can be probed by scanning probes, can provide direct information regarding to the electronic structure of the system. As we will discuss in the next section, the formation of density wave is directly related to the Fermi surface shape through Fermi wavevector nesting" Therefore, the charge density wave formation in nanoscale systems can be used to probe the sensitive change of eigenstates near the Fermi level, as the system size varies. In this review, we will explore the evolution of electronic structures in CDW nanocrystals.

2. CHARGEDENSITY WAVESIN MACROSCOPIC AND MESOSCOPIC SCALE Before we discuss CDW formation at mesoscopic scales, it is useful to review briefly CDWs in bulk materials. CDWs in the bulk are characterized by a periodic modulation of the electronic density and ion positions in quasi-onedimensional (1D) or two-dimensional (2D) conductors. They incorporate a periodic modulation of the valence charge of the crystal and are usually accompanied by a small periodic lattice distortion." The mechanism of the spontaneous symmetry breaking associated with CDW formation is closely related to the instability of a low-dimensional electron gas, which will be discussed in detail in the next section. This instability originates from the topological nature of the Fermi surface of low-dimensional electron gases. In the case of a 1D metallic system, where the Fermi surface consists of only two points, the conduction electrons are collectively coupled to the underlying periodic lattice. In simple terms, this symmetry-breaking charge modulation lowers the energy of the occupied electron states and raises that of the unoccupied states, so that the CDW state becomes stable below a certain critical temperature T,. Peierls has shown" that the ground state of this coupled electron-phonon system is characterized by a gap in the single-particle excitation spectrum and by a collective mode formed by electron-hole pairs at the Fermi points, which results in CDWs. Experimentally, 1D CDWs have been observed in many organic or inorganic metallic l o These similarities can be found when the role of current in CDW replaces the role of voltage in superconductors. For example, the collective CDW current and frequency relation are similar to the AC Josephson relation between voltage and frequency in superconductors. I ' G. Gruner, Density Waues in Solids, Addison-Wesley Publishing Company, New York (1994). l 2 R. E. Peierls, Quantum Theory ofSolids, Oxford University Press, Cambridge (1955).

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chain material^.'^ In 2D metals, however, the CDW formation is less favorable due to the reduced singularity at the Fermi surface, which allows only an incomplete gap to open at the 2D Fermi surface. Only materials with nontrivial Fermi surface structure, which enable a large number of electron-hole pairings across the Fermi surface (Fermi surface nesting), are expected to form 2D CDW phases at low temperatures. In the 1970s, a series of 2D layered transition metal-dichalcogenide (TMD) compounds were synthesized by DiSalvo and coworker^.'^ Extensive x-ray diffraction and electron diffraction by Wilson and coworkers have unambiguously identified 2D CDW states in these materials,lS followed by numerous scanning probe experiments.'6-'9 Recently, other new classes of 2D CDW states have been observed in several metallic monolayers absorbed on the surface of semiconductors.'' The CDW formation in nanoscale systems may bring interesting physics. When the size of a crystal approaches nanometer dimensions, the quantum confinement effects of electrons in the crystal cannot be neglected, even at relatively high temperatures. For semiconductor nanostructures, the Fermi level lies within the energy gap, and the relevant physics is governed by localized state at the band edges. The size-dependent quantum effects of these semiconducting nanocrystals are readily observable through optical4 or electrical transport measurements." On the other hand, quantum confinement effects in a metallic system are more difficult to observe due to the l 3 For a general review of 1D CDW systems, see, for example, Crystal Chemistry and Properties of Materials with Quasi-One-Dimensional Structures, ed. J. Rouxel, D. Reidel Publ. Co., Boston (1987). l4 F. J. DiSalvo and T. M. Rice, Physics Today (April 1979), p. 32. l 5 J. A. Wilson, F. J. DiSalvo, S. Mhajan, Adv. Phys. 24, 117 (1975); R. L. Withers and J. A. Wilson, J . Phys. C.: Solid State Phys. 24, 17 (1975). l 6 R. V. Coleman, B. Giambattista, P. K. Hansam, A. Johnson, W. W. McNairy, and C. C. Sloug, Advances in Phys. 37(6), 559 (1988); R. V. Coleman, W. W. McNairy, C. G. Slough, P. K. Hansma, B. Drake, Surt Sci 181, 112 (1987). l 7 R. E. Thomsn, U. Walter, E. Ganz, J. Clarke, A. Zettl, P. Rauch, and F. J. DiSalvo, Phys. Rev B. 38, 10734 (1988); R. E. Thomson, U. Walter, E. Ganz, P. Rauch, A. Zettl, and J. Clarke, J . Microsc 152, 771 (1988). l 8 G. Gammie, S. Skala, J. S. Hubacek, R. Brockenbrough, W. G. Lyons, J. R. Turker, and J. W. Lyding, J . Microsc. 152,497 (1988). X . L. Wu, P. Zhou, and C M. Lieber, Phys. Rev. Lett. 61, 2604 (1988); X. L. Wu, P. Zhou, and C. M. Lieber, Nature 335, 55 (1988). 2 o J. M. Carpinelli, H. H. Weitering, E. W. Plummer, and R. Stumpf, Nature, 381, 398 (1996); J. M. Carpinelli, H H. Weitering, M. Bartkowiak, R. Stumpf, and E. W. Plummer, Phys. Rev. Lett. 79, 2859 (1997). 2 1 D. L. Klein, P L. McEuen, J. E. B. Katari, R. Roth, and P. Alivisatos, Appl. Phys. Lett. 68, 2574 (1996); D. L. Klein, R. Roth, A. K. L. Lim, A. P. Alivisatos, and P. L. McEuen, Nature 389, 699 (1997).

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small energy level spacing near the Fermi level. Previously, these effects have been observed only in metallic dots with sizes 5 1nm at very low temperatures.22For a CDW quantum dot, however, size-dependent quantum effects can be studied even for a relatively large system, because CDW wavevectors depend sensitively on the shape of the Fermi surface rather than the energy level spacing itself.23 Taking advantage of this, investigation of CDW formation in TMD nanocrystals provides a good opportunity to study the evolution of electronic structures of quantum systems from the macroscopic bulk to mesoscopic/molecular length scales.

3. FABRICATION OF NANOSTRUCTURES The development of new methods for the preparation of nanostructures is important in nanoscale research because it is often the creation of these structures that limits studies of potentially interesting physical phenomena. Researchers have been creating submicron structures by thin film and lithographic technologies, which have been developed over several decades in the semiconductor industry. This approach facilitates the creation of devices with carefully tailored geometries and electron densities. Various semiconducting and metallic quantum dots have been created using a 2D electron gas in semiconductor heterostr~cture,~ small metallic grains,” and ion-beam d e p o ~ i t i o n Transport .~~ measurements of these structures have provided a wealth of information about the effects of Coulomb interactions and spatial quantization on electron transport at the mesoscopic scale. Another approach is to grow nanostuctures chemically. A variety of semiconducting or metallic nanocrystals have been fabricated with dimensions that can be controlled at the atomic level.4 In the combination of lithographically fabricated electrodes, individual or small collections of such materials have been studied showing well-defined quantum levels and Coulomb interaction between the crystals and electrodes.21s26 An alternate approach to electrical measurements has been to use scanning probe microscopes.27 Scanning probe microscopes, such as scann2 2 D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. Lett. 74, 3241 (1995); D. C. Ralph, C. T. Black, and M. Tinkham, Phys. Rev. Lett. 78, 4087 (1997). 23 J. Zhang, J. Liu, J. L. Huang, P. Kim, and C. M. Lieber Science 274, 757 (1996). 24 W. Chen, H. Ahmed, and K. Nakazoto, Appl. Phys. Lett. 66, 3388 (1995). 2 5 T. C. Shen, C. Wang, G . C. Abeln, J. R. Tucker, J W. Lyding, Ph. Avouris, and R. E. Walkup, Science 268, 1590 (1995). 26 H. Park, A. K. L. Lim, P. Alivisatos, J. Park, and P. L. McEuen, A p p l . Phys. Lett. 75, 301 (1999). 2 7 J. K. Gimzewski and C. Joachim, Science 283, 1683 (1999).

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ing tunneling microscopes (STMs) and atomic force microscopes (AFMs), can in principle both create and probe the properties of very small nanoscale structures. For example, the STM has been used to manipulate individual atoms and molecules into ~ t r u c t u r e s , ~to~ -probe ~ ~ quantum behavior in several nanostructures 3z*33,and to lithographically pattern s ~ r f a c e s tAFMs . ~ ~ have also been used to create nanostructures, including electronic3 and mechanicaP devices, by selective oxidation and nanomachining. Recently, it has been demonstrated that these SPM approaches can fabricate nanometer-scale structures by inducing a collective motion of atoms at the surfacez3 and by locally breaking chemical bonds of the sample” using tip-sample interaction at higher bias voltage. The abilities of SPM, which can create and probe nanoscale structure locally with an atomic precision, make this method valuable in the investigation of mesoscopic systems. In this review, we will mostly focus on STM experiments on CDWs in TMD nanocrystals. First, in Section 11, we will review the basic principles of CDW formation in 2D layered materials, specifically metal-dichalcogenide materials. In Section 111, we will discuss the creation of 1T-TaSe, nanocrystals at the surface of 2H-TaSe, by a STM tip-induced solid-solid phase transformation. The details of the fabrication process and the characterization of these nanocrystals in bulk surfaces will be presented. Section IV is the main part of this paper. In this section, the sizedependent effects of this 2D electronic system will be discussed through the irregularity of the CDW wavelength. A model of Fermi-surface roughening due to the confinement of 2D electrons in this quantum dot will be presented and compared to experimental results. Finally, in Section V, we will discuss a possibility of extending nanocrystal fabrication to other TMD systems.

28 29 30 31

D. M. Eigler and E. K. Schweizer, Nature 344, 524 (1990). L. J. Whitman, J. A. Stroscio, R. A. Dragoset, and R. J. Celotta, Science 251, 1206 (1991). I.-W. Lyo and Ph. Avouris, Science 253, 173 (1991). T. A. Jung, R. R. Schlittler, J. K. Gimzewski, H. Tang, and C. Joachim, Science 271, 181

(1996). 32 33 34

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M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature 363, 524 (1991). Ph. Avouris and I.-W. Lyo, Science 264, 942 (1994). A D. Kent, T. M. Shaw, S. von Molnr, and D. D. Awschalom, Science 262, 1249 (1993). E. S. Snow and P. M. Campbell, Science 270, 1639 (1995). P. E. Sheehan and C. M. Lieber, Science 272, 1158 (1996).

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II. Charge Density Waves in Bulk Transition Metal Dichalcogenides 4. INSTABILITY OF ELECTRON GASIN LOW DIMENSIONS A N D CDW

FORMATION

CDW formation in low-dimensional metallic systems has an origin in the instability of a low-dimensional electron gas, which exhibits collective charge and spin mode in its ground state. In low dimensions, a particular topological structure of the Fermi surface can lead to an instability of the electron gas, which is called “Fermi surface nesting”. This instability can be explained by the singularity in the wavevector-dependent electron response function x(q). From Lindhard linear response theory,37x(q), which describes rearrangement of the charge density in the external perturbation, is given by

where Ek is the electronic energy at wavevector k, d is the dimensionality of the system, and 1 = exp((E - EF)/kBT) + 1 is the Fermi function at temperature ?: For the free electron gas, using the energy dispersion relation Ek = h2k2/2m, calculations have been explicitly carried out, and the resulting ~ ( qis) depicted in Figure 11.1. As shown in the figure, x(q) is singular when q = 2k, where the denominator Ek - Ek-ZkF vanishes at the Fermi surface. Similar to the van Hove singularity (VHS) of the density of states (DOS), the singularity of ~(q)is more pronounced in lower 10 dimensions. In lD, ~ ( q diverges, ) implying that an arbitrarily small external perturbation may lead to a charge redistribution, and thus the electron gas is unstable with respect to the formation of the charge density waves (CDW) or the electron spin density waves (SDW). The divergence of 1D x(q) at q = 2k, is due to the particular topology of the Fermi surface in this dimension (two points at fk,), which enables perfect Fermi surface nesting. In fact, it was shown in a rigorous manner’’ that this 1 D electron gas instability leads to a broken symmetry state at T = 0 (for example, superconducting, CDW, or SDW state). The detailed nature of this lowtemperature broken symmetry state depends on the relative interaction strength of small ( q = 0) and large (q = 2k,) momentum transfer between the states at EF.38 N. W. Ashcroft and N. D.Mermin, Solid State Physics, Holt, Rinehart and Winston, New York (1976). 37

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FIG. 11.1. Wavevector dependent Lindhard response function for 3D (solid line), 2D (broken line), and 1D (dotted line) electron gas.

Figure 11.1 shows x(q) of electron gas in various dimensions. Specifically, due to the particular topology of the 1D Fermi surface, x(q) diverges at q = 2k, in the 1D electron gas, and hence perfect Fermi surface nesting makes it possible for holes at the right Fermi point (k = k,) to pair with electrons at the left Fermi point (k= -kF), and vice versa. Peierls proposedI2 that this instability of the 1D electron gas, combined with the spontaneous lattice distortion with 2k, periodicity, could result in a CDW ground state. In this CDW state, a gap opens up at E , in the single-particle excitation spectrum, turning the 1D metal into an insulator (Peierls metalinsulator transition). This spontaneous symmetry breaking CDW state has been observed in many quasi 1D metals including transition metal chalcogenides, transition metal bronzes, and some organic conducting chain compounds.' In higher dimensions, however, the number of electronic states that satisfy the Fermi surface nesting condition with a single Fermi wavevector k, is significantly reduced. Thus, the singularity of x(q) at q = 2k, becomes weak in 2D (discontinuity in the first derivative) and even weaker in 3D (discontinuity in the second derivative). Incomplete Fermi surface nestings in 2D and 3D metals implies that only a small fraction of electrons in the 38 In the actual formation of the CDW, SDW, or superconducting state, the phonons must interact with the electrons. In most cases the phase transition to these broken symmetry states is stabilized by weak inter-chain coupling, which makes the system weakly 3D in nature.

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system will gain electronic energy by opening a gap at the Fermi level, while the cost of elastic energy, by distorting the atomic lattice, should remain roughly the same. Therefore, in 2D materials, a specific topology of the Fermi surface is required to form a stable CDW state, while a CDW state rarely exists in 3D due to the inherently small Fermi surface nesting.

5. CRYSTAL STRUCTURE OF TRANSITION METALDICHALCOGENDES 2D CDWs have been experimentally observed in TMD with layered structures. The chemical composition of these compounds can be written as MX,, where M stands for group Vb metal elements (V, Nb, Ta) or group IVb metal (Ti), and X stands for chalcogenides (S, Se or Te). The schematic structure of TMDs is illustrated in Figure 11.2(a). The atoms are covalently bonded to each other within three-atom-thick layers, forming either octahedral coordination or trigonal prismatic coordination (Figure 11.2(b,c)). The layers are weakly bonded to each other mainly through chalcogenchalcogen van der Waals interactions. Due to this layered structure, TMDs have a strong anisotropy perpendicular to the layers, and thus, a strongly anisotropic band structure that confines electrons to the layers. For this

FIG. 11.2. (a) Schematic structure of transition metal dichalcodegenide. X and M. represent chalcogenides and a transition metal atom, respectively. (b) Octahedral and (c) trigonal prismatic coordination for T and H. polymorph. Dark and light spheres represent chalcogenides and a transition metal atom, respectively.

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reason, electrons in TMDs can be considered as electrons in a 2D system, weakly interacting with neighboring planes. Another major structural characteristic in the TMD materials is the existence of 12 polymorphisms (polytypes) of the compounds. Two major polytypes of TMD compounds are T-Phase compounds, in which the metal atoms occupy the octahedral holes defined by a two chalcogenides atom sheet (Figure 11.2(b)), and H-phase compound, in which the metal atoms occupy the trigonal prismatic holes instead (Figure 11.2(c)). Other polytypes of the compounds are also possible because of many variations in the stacking order of the T-phase and H-phase sandwich layers. The next simplest TMD structure is 4Hb-compounds whose unit cell is expanded along the direction normal to basal plane direction. Within the unit cell, 4Hb-compounds have alternative T-phase and H-phase layers stacked together, and then the whole 4Hb-crystal is built by repeatedly stacking this two-layered unit. The structural parameters of T-phase and H-phase are similar. For example, 2H-TaSe2 has the crystalographic structural parameters, in-plane lattice constant a = 3.43 A and distance between layers c/2 = 6.35 A, while 1T-TaSe, has u = 3.48 A and c = 6.26 A. Despite this structural similarity, the electronic properties of these two phases, which are defined primarily by trigonal prismatic or octahedral bonding in a single layer, are quite distinct: 2H-TaSe2 exhibits a weak CDW that forms at 122 K and becomes commensurate with the atomic lattice (3a x 3a) at 90 K, whereas 1T-TaSe, exhibits a very strong CDW that forms initial1 around 600 K and becomes at 473 K. This commensurate with the atomic lattice ( & u x f i u ) difference of CDW states is due to the subtle change of the Fermi surface structure as we will discuss in the next section. 6. CDW FORMATION IN TRANSITION METALDICHALCOGENIDES Electronic band calculations on these materials39940 showed that the Fermi surface structure is nearly k, independent due to the strong 2D nature of the materials, where k, is the wavevector perpendicular to the layers. Figure 11.3 shows a schematic diagram of a Fermi surface cross section of TMDs in the basal plane. Large portions of the Fermi surface can be translated to other parts of the Fermi surface by a single nesting vector qo, which satisfies the condition for the 2D CDW formation. Furthermore, from symmetry considerations, two more such nesting vectors, rotated by 120" and 240" with 39 40

L. F. Mattheiss, Phys. Rev. B. 8, 3719 (1973). H. W. Myron and A. J. Freeman, Phys. Reo. B . 15, 885 (1977).

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FIG.11.3. Schematic diagram of Fermi surface cross section of the TMD in the basal plane showing the first Brillouin zone and one segment of the second Brillouin zone. qo depicts the dominant Fermi surface nesting vector.

respect to qo, are also possible. Therefore, these simple theoretical considerations predict that the CDWs form a triangular superlattice generally incommensurate with the underlying atomic lattice due to the nontrivial relation between the nesting vector and reciprocal lattice vectors. The interaction of a CDW with the underlying atomic lattice provides rich physics in TMD materials. At high temperatures, the CDWs are incommensurate with the atomic lattice due to the large thermal motions of CDWs, which average out the interaction to the atomic lattice. As the temperature decreases, however, the incommensurate CDW state turns into a commensurate state where CDWs are locked onto the atomic lattice. The details of this commensurate-incommensurate phase transition process are quite subtle, and differ from material to material. For example, the most intensively studied TMDs, such as 1T-TaS, and 1T-TaSe,, which are structurally similar materials, exhibit quite different phase diagram^?^ Both materials show a CDW transition at a temperature of about 600 K. While 1T-TaSe, has a single incommensurate-commensurate transition at 473 K, 1T-TaS, exhibits several intermediate phases41 before its CDW becomes 41

X. L. Wu and C. M. Lieber, Science 243, 1703 (1989)

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commensurate at 150 K. On the other hand, 2H-TaSe2 exhibits a weak CDW state that forms at 122 K and becomes commensurate with respect to the atomic lattice at 90 K, whereas 2H-TaS2 shows an onset temperature of 75.3 K for the formation of incommensurate CDW state which exists down to the low temperatures. 4Hb-phase TMD exhibits the combination of the properties of both 1T- and 2H-compounds due to its structure characteristics, which stacks alternating T- and H-phase layers. The interplay between point defects in the underlying lattice and various CDW phases provides even richer physics in addition to these complicated phases. It was shown that point defects in the atomic lattice can effectively pin the CDWs, and even generate defects in a CDW lattice itself.41 This CDW pinning to lattice defects is quite similar to the pinning of an elastic manifold by random impurities. This local structural distortion of a CDW phase has been effectively studied by STM, because this technique not only provides the real space image of the charge modulation of the CDW and underlying atomic lattice, but also probes local electronic structures such as the CDW gap.43 Furthermore, these layer materials provide an ideal system for an STM experiment because a simple mechanical cleavage can provide a fresh, clean and atomically flat surface. Simultaneously probing the CDW lattice and the underlying atomic lattice makes it possible to analyze directly the structural relation of the CDW lattice and the atomic lattice. Figure II.4(a) displays an STM image of CDW lattice commensurate with the underlying atomic lattice of 2HTaSe, at 4 K. This image shows a 3a x 3a commensurate superlattice of CDW with respect to the atomic lattice and the maxima of the CDW located at the centers of the triangles formed by three adjacent Se atoms, which implies that the CDW maxima are pinned in the Ta atoms at the second atomic sheet in the sandwiched layer (Figure 11.4(c)). Figure II.4(b) shows a CDW and atomic lattice of 1T-TaSe, at 4 K. The CDW forms a commensurate state with a f i a x f i a superlattice at a 13.9" angle relative to the underlying atomic lattice. The locations of the CDW maxima also suggest that the T-phase CDW maxima are pinned on the positions of the Ta atoms in the middle atomic sheet in the sandwiched layer (Figure 11.4(d)). STM images of the CDW state and underlying atomic lattice also provide qualitative information about the intensity of the CDW state. The comparison of the images in Figure, 11.4(a) and (b) suggests that the CDW in the 1T-TaSe,, which has 2.0 A apparent charge modulation height, has much stronger intensity than that of the CDW in the 2H-TaSe2, which has H. Schafer, Angew. Chem. Intl Ed. 10, 43 (1971). H.-J. Guntherodt and R. Wisesendanger, Scanning flcnneling Microscopy I. Springer-Verlag. Berlin (1991).

42 43

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(a)

FIG.11.4. STM image of commensurate CDW states in (a) H-TaSe, and (b) T-TaSe, bulk crystal. (c) and (d) show the corresponding structural models of commensurate CDW states in H-TaSe, and T-TaSe, materials, respectively.

0.4A apparent charge modulation height, at 4 K; That is, there is much greater gapping of the Fermi surface in the 1 T vs 2H materials. As we have discussed thus far, the CDWs in the bulk TMDs have been used for studying 2D electronic systems. Recently we have reported a novel fabrication process of T M D nanocrystals using STM.23In the next section, we will discuss details of this nanocrystal fabrication process, CDW formation in a finite-size T M D crystal, and the size-dependent confinement effects of this 2D electronic system embedded on the surface of bulk T M D materials. 111. Creation of CDW Nanocrystals by STM TIP

In this section, the detailed process of creating nanostructures will be discussed. In this process, the STM tip is used to drive a solid-solid phase transformation between two distinct 1T- and 2H-phase of TaSe,, and the STM is subsequently used to probe the atomic structure and electronic

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properties of the created nanocrystals. This nanocrystal fabrication utilizes the collective motion of the top layer of atoms by an intense electric field from a sharp STM tip apex to fabricated 2D nanocrystals in a controlled way as discussed next.

7. EXPERIMENTAL METHODS All the single crystals studied in our experiment were grown by the chemical vapor transport method with iodine as a transporting agent.42 Characterization of the bulk single crystals was conducted using x-ray diffraction, magnetic susceptibility, and transport measurements to ensure the highquality of the single crystals for STM experiments. The experiments were carried out using a home-built ultrahigh vacuum (UHV) STM system at 4 K. Single crystals were cleaved in UHV chamber within a room temperature region, then were transferred in situ to the STM stage in UHV, which was maintained at 4 K. STM images were typically obtained with a tunneling current of 50 pA and a bias voltage of less than 500 mV to minimize the tip-sample interaction during imaging. The bias voltage was applied to the STM tip. The tunneling current versus tip-sample separation was also measured through the experiments and the resulting barrier heights indicated a vacuum tunneling junction. Two different methods were used to modify the 2H-TaSe, surfaces and to create T-phase nanocrystals. The first method uses a loop oscillation of STM feedback control. The bias voltage was set to a preset value (> 1.2 V), and then the STM feedback loop gain was increased to cause the feedback loop oscillations for a short period of time ( - 100 ms). The reduction in the tip-sample separation during oscillation results in an enhanced electric field under the tip apex. After this modification step, the STM was switched back to the imaging mode (reduced bias voltage and feedback gain) to investigate the modified surface. The second method uses a bias pulse to the STM tip. In this method, the feedback loop was open, the tip was advanced a preset distance to the surface, and then bias voltage pulses were applied to the tip to increase the local electric field under the tip momentarily. The duration and magnitude of pulses typically range from 24 to 200 ms and from 1.4 V to 3.0 V, respectively. Both methods worked in our experiments to increase the electric field locally, and successfully drive a local solid-solid phase transition, and to create nanocrystals.

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8. NANOCRYSTAL FORMATION AND CHARACTERIZATIONS

A typical large-scale image of the cleaved 2H-TaSe, crystal at 4 K (Figure III.l(a)) shows both the hexagonal atomic lattice of the surface Se atom

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FIG. 111.1. (a) STM image of 2H-TaSe,. Both atomic lattice and 3a x 3a CDW superlattice are visible. (b) Image of the surface region in (a) after performing the feedback loop oscillation method with a bias volage of - 1300mV. Both images were recorded with tunneling current of 50 pA and bias volage of 500mV applied to the tip. The small open circles in the images highlight the orientations of CDW lattices in H- and T-phase regions. (Adapted from Reference [23] Copyright 1996 American Association for the Advancement of Science. Readers may view, browse, and/or download this material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or in part, without prior written permission from AAAS.)

sheet and the 3a x 3a superlattice corresponding to the commensurate CDW state expected for 2H-TaSe2 material. "*' After recording this image, we positioned the STM tip at the central area and made a modification on the sample surface by the feedback loop oscillation described preceding. A subsequent image of the same area shows a dramatic change at the central modified area. In particular, the central modified area has roughly hexagonal shape and exhibits a large corrugation amplitude (a peak to peak amplitude of 2.4A) and a triangular lattice with lattice constant of 1.25 nm (Figure III.l(b)). This image also shows the distinct domain structures and line defects at the central modified area. The orientation of each of the different high-amplitude CDW regions changes f 14" relative to the weak 3a x 3a CDW superlattice of the surrounding 2H-TaSe2 crystal. These characteristics of the high corrugation lattice in the central modified area are consistent with the well-studied CDW state that exists in 1T-TaSe,,'5.'6 and thus it is plausible that the transformed region corresponds to T-phase material. Atomic resolution STM images provide further evidence for this H-phase to T-phase transformation. We notice that the CDW state in the transformed region exhibits domain structures in which the CDW orientation shows distinct angle difference relative to CDWs in adjacent domain. The STM images show that the angles of the CDW orientations in the different

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FIG. 111.2. (a) Atomic resolution image of a CDW twin domain boundary in a T-TaSe, nanocrystal. (b) Corresponding structural model to illustrate the nature of CDW twin domain boundary. (c) Atomic resolution image of a CDW discommensuration line. (d) Corresponding model for understanding the structure of a CDW discommensuration line.

domains are roughly 28". Furthermore, the atomic resolution image at the domain boundary (Figure III.2(a)) illustrates that the underlying atomic lattice is defect free and that the CDW states are commensurate on both sides of the domain boundary. These observations suggest that the CDW domains are twin domains. As illustrated in the structural model in Figure 111.2(b), the commensurate f i a x ,/%a CDW superlattice in T-TaSe, can occur with either a clockwise (a-phase) or counter clockwise @phase) 13'54' degree rotation relative to the underlying atomic lattice to form two different orientation CDW domains,' which agrees well with the experimental observation (Figure 111.2(a)). Therefore, the CDW orientational angle change across the twin boundary should be 27.8", which is close to the observed angle k 14". In addition to the CDW twin domain boundaries, we also observed CDW line defects in the T-phase nanocrystals that separate two CDW domains

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with the same orientation and a certain phase shift. An atomically resolved image of a line defect shown in Figure III.2(c) indicates that the CDW lattices on both sides are commensurate with the same orientation and that the CDW phases shift, which corresponds to one atomic lattice vector across the line defect (depicted as broken solid line in Figure 111.2(d)). The atomic scale model of this line defect is consistent with a CDW discommensuration line, which was proposed to form during the incommensurate to commensurate CDW phase transition in 2H-TaSe,.44s45The observations of CDW twin domains and CDW discommensuration lines suggest that the CDW domains may nucleate at different sites and then grow together to form either twin domain boundaries or CDW discommensurate lines depending on the relative CDW orientation and phase relation at the nucleation sites. Because the CDW in T-TaSe, crystal is strongly pinned to the atomic lattice at temperatures far below the CDW incommensurate to commensurate transition temperature (- 473 K), the CDW twin domain boundaries and the CDW discommensuration lines are trapped in the nanocrystals and are relatively stable during STM imaging in our experiment. The structure of the interface between a T-phase nanocrystal and its H-phase TaSe, surrounding provide further insight into nanocrystal structure and the transformation mechanism. Figure III.3(a) shows that the T/H boundary of the transformed region is an extremely sharp interface and that the CDWs in both T- and H-phase do not have strong distortions at the interface. This high-resolution image also shows that the atomic lattices in both regions are hexagonal with the same lattice constant, a 3.4 A, and the same lattice orientation. Furthermore, the atoms at the interface form roughly square cells in contrast to the hexagonal cells formed in the other areas. The atomic rows in the transformed region do, however, exhibit shifts relative to the surrounding H-phase lattice. These results are consistent with the known lattice constants of the T- and H-phases of TaSe,I5 and imply that the local region of H-phase TaSe, was converted to T-phase through a coherent shift of Se atom sheets in the top Se-Ta-Se layer, as we will discuss in the next section. Lastly, cross-sectional profiles (e.g. Figure III.3(b)) taken through the T-phase nanocrystal and surrounding H-phase region demonstrate that there is no step observed at the boundary between the two regions, and therefore, both phases are in the same layer. Taken together, these results suggest that the central T-phase TaSe, nanocrystal was created by a local solid-solid transformation of the 2H-TaSe, crystal driven by the STM tip.

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44 45

W. L. McMillan, Phys. Rev. B. 12, 1187 (1975). R. N. Bhatt and W. L. McMillan, Phys. Reo. B. 12, 2042 (1975).

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FIG. 111.3. (a) Atomically-resolved topographic image of the interface between H-phase and T-phase regions in TaSe,. H-phase in the upper left region shows a weak 3a x 3a CDW superlattice while T-phase in lower right region shows a strong &a x &a CDW state. The longer black line running cross the image from upper left to lower right follows the direction of the T-phase CDW lattice, which rotated 13" relative to the atomic lattice in the H-phase CDW lattice. (b) Corrugation profile corresponds to the longer black line in (a). (Adapted from Reference [23] Copyright 1996 American Association for the Advancement of Science. Readers may view, browse, and/or download this material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or in part, without prior written permission from AAAS.)

9. STRUCTURAL MODELFOR

THE

SOLID-SOLID PHASE TRANSITION

Several important questions arise from our proposed solid-solid phase transition mechanism: (i) how does the H-phase TaSe, structure convert to the T-phase, and (ii) how does the STM tip interact with the sample surface

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FIG. 111.4. (a) An atomic model highlighting the initial (white) and final (black) positions of Se atoms at the surface-vacuum interface in the H- to T-phase transformation. The displacement vectors for ech of these Se atoms are denoted by black arrows. The Ta atoms are depicted as dark-gray spheres, and the blck spheres with a white dot depict Se adatoms. (b) Ball and stick model that illustrates the collective motion of the Se atoms in the conversion of the local Ta atom coordination from trigonal prismatice (H-phase) to octahedral (T-phase). (c) A portion of the model and an experimental image focusing on the interface between T-phase (left) and H-phase (right) TaSe, regions.

to drive this transformation? Correlated motion of Se atoms at the solidvacuum interface in the topmost layer of H-TaSe, could be driven by the electric field of the STM tip. Although the direction of the force arising from the electric field is radially symmetric about the tip, there are three energetically favorable directions for Se atoms to displace due to the lattice symmetry of the TaSe, basal plane. These directions are [l,O,l,O], [l,l,O,O], and [O,l,l,O] as indicated in Figure III.4(a). The minimum displacement to

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convert H- to T-Phase involves collective motion of Se atoms by u/$ along one of the three directions. This displacement converts the trigonal prismatic coordination of H-phase TaSe, to the octahedral coordination of T-phase TaSe, (Figure III.4(a) and (b)). This displacement changes the Ta coordination in the T M D sandwich layer, and subsequently produces a large change in electronic structure that is manifested by the distinct CDW In states from the H-phase ( 3 ~ x 3 ~to) the T-phase ( f i u x f i u ) . addition, our model of this local structural transformation predicts a single-layer T-phase TaSe, nanocrystal with a small number of Se vacancies or adatoms. Figure III.4(a) illustrates the motion of Se atoms in response to a repulsive force that is expected when a negative bias voltage is applied to the STM tip. The Se atom vacancies predicted in this model (with a negative bias) have not yet been observed, although they are expected to be difficult to detect for the following reasons. First, the percentage of vacancies is small; we estimated < 1% vacancies in a single 40 nm T-phase crystal. Second, it is difficult to observe all of the atomic sites in the T-phase nanocrystals region due to the strong CDW modulation. In addition, this same model predicts that Se adatoms would be produced when a positive bias voltage was used to drive the transformation (because the Se atom motion would be toward the tip). The small number of adatoms generated in this reverse direction will not be readily observed because they would be expected to exhibit significant mobility on the surface. This model predicts that the Se atoms at the interface will form a rectangular rather than hexagonal structure and that the surface Se rows from the T- and H-phase regions will have a $ 4 6 (1.08) mismatch at the interface (Figure 111.4(c)). This model also requires no shifting of the Ta sublattice. Analysis of the experimental data (Figure III.4(c)) shows that the Se atoms at the sharp boundary between T- and H-phase regions define a rectangular cell and that the mismatch of atomic rows in both regions (highlighted by solid lines) is 1.1 & 0.1 8. The experimental data (Figure III.4(a)) show that the Se atoms at the interface form a rectangular cell that is only slightly distorted from a square, and thus these results contrast with the rectangular model cell suggested by Figure III.4(c). The difference between the experimental results and the model is believed to be due to relaxation of the Se atom positions at the interface between the T- and H-phases. Further experimental evidence that supports the proposed model were found at the junction between the two different T/H boundaries of modified region. According to our proposed structural transform model, the angle between two T/H interfaces must be 60" or an integer multiple of 60" due to the crystal symmetry of the TaSe, lattice. Hence, the boundaries of these phases and the angle between the boundaries are expected to be correlated. For example, 60" angle boundaries should have the same interface structure

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FIG. 111.5. (a) and (b) are portions of the structural model to illustrate the existence of the two different angles and the correlation of the angle and the types of the T- and H-phase interfaces. (c) and (d) are high-resolution STM images of portions of nanocrystals that demonstrate the predicted interface structures from the model.

as shown in the structure model in Figure IIIS(a), while 120" angle boundaries should have structurally different interfaces: One boundary has higher Se atom density than the other Figure IIIS(b)). It is expected that the distortion of the underlying atomic lattice should be more prominent for this 120" interface with higher density Se atoms in the STM images. Indeed, it was found in the atomic resolution STM images that the 60" angle boundaries consist of the same interface structure (Figure IIIS(c)), whereas those of 120" consist of two different types of interface structure (Figure III.5(d)).46 The relatively good agreement between the experimental data and our model indicates that this model should provide a useful framework for understanding this solid-solid phase transition. 46 Higher Se density boundaries should have prominent distortion, which manifests itself as a protrusion of atomic rows parallel to the interface.

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10. DRIVING FORCE FOR

THE

141

SOLID-SOLID PHASETRANSITION

The general structural features of the H- to T-phase local solid-solid transformation identified in the previous section do not address the nature of the interaction between the STM tip and the sample surface, which is responsible for driving this transformation. We believe that this transformation is driven in large part by the large electric field at the apex of the STM tip. This proposition is supported by the following experimental observations. (1) The size of the T-phase nanocrystals increases with increases in the bias voltage used to induce the transformation. (2) No transformation from H- to T-phase was observed when the STM tip was brought into contact with the surface or pushed into the surface with low applied voltages. Hence, it is apparent that purely mechanical deformations of the sample cannot drive this structural transformation. Figure 111.6 shows the modification voltage dependence of the nanocrystal formation. As indicated by an arrow, the graph shows that there is a modification-voltage threshold ( 1.2 V) to initiate the solid-solid phase transition, and thereafter the size of the transformed T-phase crystals increases with increasing modification-voltage. The presence of the relation between the created nanocrystal size and applied bias voltage suggests that the electric field is important at least in the initiating process of the transformation, because the electric field around the STM tip apex is determined by the applied bias voltage and the detail shape of the tip. To find a possible driving force for this phase transition, we have estimated the temperature increase of the sample surface during the trans-

-

FIG. 111.6. Dependence of the created nanocrystal size on the modification voltage of the STM tip The arrow represents the modification-voltage threshold initiating the solid-solid phase transformation of 2H-TaSe,.

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form. In our modification process, the maximum current at peak value was 1pA at a bias voltage of 1.3 V, which corresponds to a power, P = 1.3pW transferred to the local area underneath of the STM tip. Assuming that local equilibrium is maintained, the temperature rise during the transformation process is estimated to be 2 to 10 K.47 This temperature increase is much smaller than that of H- to T-phase transition of Bulk TaSe, crystals (21000 K), and thus it is very unlikely that the rise of local equilibrium temperature during the surface modification process contributes to the transformation. However, the peak power tunneling current 1pA during the fabrication process corresponds to l O I 3 electrons per second in a localized (10 nm) region under the tip position.48 This highly focused tunneling current density can interact with the dipole moment of Se-Ta bond, leading to an inelastic tunneling process. Because the ratio of the inelastic tunneling electrons to the elastic tunneling ones is -10-3,49*50 the number of electrons participating in the inelastic tunneling can be 10" per second. These inelastic tunneling electrons induce multiple vibration exitations in Se-Ta bonds to cause inequilibrium heating.5l S 5 ' We believe that the strong local electric field underneath the STM tip and inelastic electron tunneling mediated bond rearrangement are likely the main driving forces for the observed T-phase nanocrystal creation. Finally, it is also important to recognize that the intense electric field in STM experiments is localized (10 nm) under the tip position>8 while the T-phase nanocrystals created in our studies can be considerably larger in size: 7 to > 100 nm. This suggests that more detailed studies will be required to elucidate how this highly localized phenomena can create rather extended structures.

-

-

-

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IV. Finite Size Effect and Fermi Surface Roughening 11. SIZE-DEPENDENT EFFECTIN NANOCRYSTALS

Nanostructures, such as semiconductor and metallic quantum dots, have been studied intensively in recent years. For intrinsic semiconductor nanos47 This estimation was done by solving heat diffusion equations with the bulk thermal conductivity of TaSe, obtained from B. N. J. Persson and J. E. Demuth, Solid State Communication 51, 769 (1986). 48 J. A. Stoscio and D. M. Eigler. Science 254, 1319 (1991). 49 B. N. J. Persson and A. Baratoff, Phys. Rev Lett. 59, 339 (1987). R. E. Walkkup, D. M. News, and Ph. Avouris, Phys. Rev. B. 48, 1858 (1993). M. Ueta, H. Kanazaki, K. Kobayashi, Y. Toyozawa, and E. Hanarnura, Excitonic Processes in Solids, Springer, Berlin (1987). s2 T.-C. Shen, C. Wang, G. C. Abeln, J. R. Tucker, J. W. Lyding, Ph Avouris, and R. E. Walkup, Science 268, 1590 (1995).

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tructures, the Fermi level lies in the energy gap, and the relevant physics is governed by the band edges where the confinement of wave functions dominates the low-energy optical and electronic behavior. For example, the band gap in CdS semiconductor nanocrystal can be chanfed between 2.5 to 4.5 eV as the size is varied from the bulk crystal to molecular regime.s3 In contrast, the Fermi level of a metallic cluster lies in the middle of the band, where the energy level spacing is small owing to the large density of states. Therefore, detecting quantum confinement effects in metallic nanocrystals or metallic clusters is more difficult. Although it is possible to study a cluster that consists of more than 20,000 metallic atoms,54 most experimental studies of metallic nanostructures have been focused on metal clusters consisting of several to hundreds of atoms” where the size-dependent electronic structure can be readily observed because of the relatively large level spacing. In theoretical studies, it has been shown that the jellium model, which ignores the detailed ionic structure, explains the electronic properties of these metallic clusters remarkably well as a function of the cluster size.56 However, in these previous experimental and theoretical studies, there has been little work addressing size-dependent Fermi surface evolution due to the absence of information about ionic structure. In metallic bulk crystals, the size and shape of the Fermi surface are uniquely determined by the crystal structure. As the size of the crystal decreases, however, the discreteness of allowed states in the Bloch reciprocal space becomes important. For small crystals of molecular size, the concept of a Fermi surface is inadequate due to the ill-defined Bloch wavevectors in such small systems. Therefore, there might be changes in Fermi surface relative to bIulk in nanocrystals that contain 5 lo4 electrons. 12. FERMI SURFACE NESTING M NANOCRYSTALS In our STM experiments, the capability to create 2D nanocrystals of varying size, and to probe the electronic structure simultaneously, provides a great opportunity to elucidate these basic issues. High-resolution STM images with well-defined atomic and CDW structures in nanocrystals also allow us to study the CDW formation in various sizes of nanocrystals. As we discussed in Section 11, the CDW is a broken symmetry state of low-dimensional metals, coupling electron-electron interactions with elecs 3 L. E. Brus, Appl. Phys. A . 53, 465 (1991); T. Vossmeyer, L. Katisikas, M. Giersig, I. G. Popovis, H. Weller, J . Phys. Chem. 98, 7665 (1994). 5 4 T. P. Martin. T. Bergmann, H. Grohlich, and T. Lauge, Phys. Rev. Lett. 65 ‘748 (1990). 5 s W. A. de Heer, Rev. Mod. Phys. 65, 677 (1993). s6 M. L. Cohen, M. Y. Chou, M. D. Knight, and W. A. de Heer, J . Phys. Chem. 91 3141 (1987).

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tron-phonon interactions. The instability of the Fermi surface, due to Fermi surface nesting, is believed to play a central role in CDW formation. For an infinite bulk system, the Fermi surface nesting vector Q is related to the CDW wavelength I , . 2n

Q=-

4

(3)

Thus, the CDW length is sensitive to the topology of the Fermi surface. However, for a sufficiently small crystal, the concept of a Fermi surface, which is valid for an infinite bulk system, is ill-defined. Nevertheless, by considering the discrete level spacing due to the quantum confinement within a nanocrystal, a dispersion in nesting vectors is expected, as we will discuss following. In the experiment, it is possible to control the size of T-phase TaSe, nanocrystals by adjusting the applied voltage during the tip-induced modification as discussed in Section 111. Shown in Figure IV. 1 are typical T-phase nanocrystals that have dimensions from 70 to 7 nm. Qualitatively, the largest T-phase TaSe, nanocrystal (Figure IV. l(a)) exhibits a relatively uniform commensurate CDW state. In the smallest T-phase TaSe, nanocrystal (Figure IV.l(d)), both the intensity and wavelength of the CDW are obviously distorted relative to the uniform state observed in bulk single crystals of T-phase TaSe,. The CDW amplitude is larger at the center of the triangular nanocrystal, and the wavelength appears to decrease from 1.2nm at the center to 1.0 nm at the edge. For a quantitative analysis, the STM images were digitized to locate the CDW maxima. Figure IV.2 shows an example of this process. Once the positions of CDW maxima were located, the bond length between CDW maxima was computed. To avoid considering unusually stretched bonds, only the CDW maxima that have six nearest neighbors were considered in calculating the dispersion of CDW wavelengths. From the distribution of this CDW maxima bond length, the average values of Ac and dispersion of CDW wavelength AA, were obtained. As shown in the Table IV.l, A& increases by reducing the size of the nanocrystal. The four T-phase nanocrystals in Figure IV.1 have descending characteristic dimensions L of 70, 17, and 7 nm, and an increasing measured AIc of 0.6, 1.5, and 2.7& respectively. This peculiar behavior can be explained by considering the evolution of the Fermi surface with the size of the nanocrystal, as we will discuss following. T-phase nanocrystals have two major distinctions compared to metallic clusters (or crystals): (1) T-phase nanocrystals are embedded in the H-phase. Only small elastic distortions were observed at the interface between the T- and H-phase (Figure IV.l(c)). Therefore, the wave functions near the edge of the nanocrystals are bulklike. This is a sharp contrast to the case of metal clusters, whose energy

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FIG.IV.1. STM images of IT-TaSe, nanocrystals of different size embedded in 2H-TaSe2 single crystal. The nanocrystals were made with modification voltages of (a) -1.4OV. (b) - 1.30V, and (c) - 1.20V. Scale bars correspond to (a) lOnm, (b) 2nm, and (d) 1 nm. Adapted from Reference [23] Copyright 1996 American Association for the Advancement of Science. Readers may view, browse, and/or download this material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or in part, without prior written permission from AAAS.)

levels are dominated by the surface states.57( 2 ) In most cases, the T-phase nanocrystals have reasonably good symmetrical shape, which follows the underlying crystallographic symmetry. Thus, it is possible to apply periodic boundary conditions to consider electronic structure inside the nanocrystals. From (1) and (2), it is expected that the wavevector k is still a good quantum number in the nanocrystals, and thus the eigenstate of an electron in the nanocrystal can be described by k in the Brillouin zone (BZ). However, due to the finite size of the nanocrystals, the eigenstates in the BZ are not continuous and have the spacing -2nlL between the states, where L is the characteristic dimensions of the nanocrystal. With the same analogy of an

’’

A. Rubio, J. A. Alonso, X. Blase, L. C. Balbas, and S. G. Louie, Phys. Rev. Lett. 77 247 (1996).

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FIG. IV.2. Processed STM image of nanocrystals showing nearest bonding between the CDW maxima. Only the bonds form the CDW maxima that has six nearest neighbors were considered for calculation. Graph shows the histogram of CDW wavelength variation calculated from the image.

infinite (bulk) system, we can define pseudo-Fermi surface which mimics the bulk Fermi surface in the limit that L + co (Figure IV.3). Therefore, the, analogy of Fermi surface nesting can be applied to the CDW formation in the nanocrystals. Due to the discreteness of the eigenstates, there is an inevitable dispersion AQ in the nesting vector Q of the pseudo-Fermi surface of the nanocrystal,

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TABLE I. CDW WAVELENGTH VARIATIONINSIDEDIFFERENT NANOCRYSTALS L (nm)

A& from Experiment

(A)

A& from Equation 5 (A)

~

7 17

2.7

2.2

1.5

70

0.6

0.9 0.1

In real space, combining (3) and (4) results in the CDW wavelength dispersion A&

2 AAc = L

(5)

Note that the above equations eventually break down when L- Ac, and when L -+ co,where a bulk property is resolved (i.e. AAc 0). Based on this pseudo-Fermi surface model, we can estimate the CDW A%cin various sizes of nanocrystals in Figure IV.l, whose L range from 7 nm to 70 nm [Table IV.11. As shown in Table IV.l, (5) gives the correct order of magnitude estimate for the measured AAc, To understand the details of CDW behavior in the nanocrystal, such as amplitude variations and distortions, more studies are required. For example, the electronic and atomic structure at the interface between H- and T-phase TaSe,, the exact shape of the nanocrystal, and defects inside the nanocrystal should have a strong effect on the electronic

-

r

.......... .

0 0 0

.kJxy.-.-.-.-

0.

0.0

0 0 0 0 0

FIG. IV.3. (a) Schematic diagram for Fermi surface and Fermi surface nesting in the bulk TaSe, crystal. (b) Schematic diagram for the pseudo-Fermi surface (solid line) and Fermi surface nesting (solid arrows) for the nanocrystal. Filled and empty circles correspond to occupied and unoccupied eigenstate respectively. The dashed line depicts the bulk Fermi surface.

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wavefunction, and subsequently, change the charge density in the nanocrystals. Furthermore, for small enough nanocrystals ( L I , E 1nm), Fermi surface nesting is inadequate, annd Jahn-Teller distortions, which explain spontaneous symmetry breaking in a molecule,58 might be the better description of CDW phenomena in such small n a n o c r y ~ t a l sAlthough .~~ we could successfully analyze our results semi-quantitatively, these important issues are not fully understood and need more experimental and theoretical studies to elucidate this interesting issue further. V. Nanocrystal Fabrication in Other TMD Systems

In the previous section, we described investigations of the creation of T-phase TaSe, nanocrystals imbedded in the H-phase layer using biased STM tip. In these studies, a coherent model for the structural transformation was proposed, which is based on the unique structural properties of TaSe,. Naturally, we can extend this method to the creation of 2D nanocrystals in other TMD materials with similar crystal structures. These materials include 4Hb-TaSe2, 2H-TaS,, 1T-TaSe,, 1T-TaS,, NbSe,, aMoTe,, and b-MoTe,. As we will discuss in this section, it has been shown that similar STM tip-induced solid-solid phase transitions were discovered in 4Hb-TaSe2, 2H-TaS,,60 and 1T-TaS,.61 For 1T-TaSe, and 1T-TaS,, the nanocrystal fabrication method can generate a locally perturbed CDW lattice with extended discommensurate lines, which were described in Section I11 at low temperatures,60-62while the solid-solid phase transitions to nanocrystals took place at room temperature.61 In this section, we will briefly discuss the CDW nanocrystal formation in these materials by extending and applying the method used in previous sections. 13. NANOCRYSTAL FORMATION IN H-LAYER OF 4Hb-TaSe2 The structure of 4Hb-TaSe2 can be described as a composition of stacks of alternating H-phase and T-phase single TaSe, layers as discussed in Section 11. In previous STM studies 1 6 9 6 3 it was found that either H-phase and S. D Kevan, Nature 381 376 (1996). The smallest T-phase TaSe, nanocrystal we made has the size -7 nm [Figure IV.l(d)]. As we already discussed, down to this size the idea of rough Fermi surface is still valid. 6 o J. Zhang, Ph. D. Thesis, Harvard University (1998). 6 1 J. Kim, C. Park, W. Yamaguchi, 0. Shiino, K. Kitazawa, and T. Hasegawa Phys. Rev. B. 56, R15573 (1997). 6 2 J. Liu, Ph. D. thesis, Harvard University (1997). 58

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FIG. V.l. (a) T-TaSe, domain created by an oscillating STM ip with a bias voltage of 1.3V on the H-phase layer of 4Hb-TaSe2 crystal at 4K. (b) High-resolution image of the T/H interface of created nanocrystal. (c) Line profile across the black lines in (b). This profile indicates that the T-phase domain is embedded in the H-layer of TaSe,.

T-phase single plane of 4Hb-TaSe2 surface can be randomly exposed to the surface upon a sample cleavage. Prior to performing the surface modification at 4 K, we chose an area that showed 3a x 3a CDW superlattice on the freshly cleaved 4Hb-TaSe2 to locate an H-phase layer. Figure V.l(a) shows the T-phase nanocrystal domains in the H-phase layer of 4Hb-TaSe2 created by oscillating the STM tip with a bias voltage of 1.3 V as described in Section 111. In a -60 nm region, hexagonal ,/%a x ,/%a CDW lattices with strong corrugation amplitude (an apparent height of 2.0& are seen indicating that this region is now T-phase TaSe,.64 High-resolution images of the interface between the modified area and H-phase surroundings exhibit a clear and sharp boundary (Figure V.l(b)). Furthermore, as depicted in Figure V.l(c), the line profile across the interface shows that the created T-phase TaSe, is in the same layer of H-phase layer. Therefore, this modified region is structurally transformed T-phase embedded in the H-phase top layer of 4Hb-TaSe2 crystal, and not an exposure region of the underlying T-phase layer. Further characterization of these T-phase regions including the threshold modification voltage, the relationship between the nanocrystal size and biased voltage, and the interface structure, confirm that the process involves

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I. Ekvall, J. Kim, and H. Olin, Phys. Reo. B. 55, 6758 (1997). In addition, the central area of the modified surface exhibits a hole with lOnm diameter, which is possibly a region damaged by a strong tip-sample mechanical interaction. 63

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FIG. V.2. Bias-dependent images of the T/H interface on the 4Hb-TaSe2 crystal surface. The image (a) was taken at a bias voltage of 300 mV. (b) was taken at a bias - 300 mV, which could image the sub-layer of T-phase CDW lattice through the top H-phase layer.

a local structural transform of the H-phase layer to a T-phase nanocrystal, similar to that of 2H-TaSe2 discussed in the previous section. These results imply that the properties of a single H-layer alone in 2H- or 4Hb-TaSe2 determine the local solid-solid structural transformation, and the interlayer coupling between H- and T-layer or H- and H-layer is not essential to the transformation process. Imaging the T/H interface with different bias voltages exhibits an interesting phenomena. In particular Figure V.2 displays changes in the STM image of the T/H interface of 4Hb-TaSe2 nanocrystal upon changing the bias voltage. At a bias voltage of 300 mV, the atomic lattice and 3a x 3a CDW lattice can be clearly seen in the H-phase region (right side in (a)), which corresponds to the H-layer of 4Hb-TaSe2 crystal surface (unmodified). When the bias voltage is reversed to - 300 mV, this interface image shows drastic change in the H-phase region (right side in (b)), while the modified T-phase region and interface itself do not exhibit a preciable change. Instead of the 3u x 3a CDW lattice structure, a weak $u x*a T-phase CDW lattice is observed in this unmodified region. The presence of a weak f l u X f i a T-phase CDW lattice at this bias was reported in a previous study on this material,65 and is believed to be an image of the T-phase layer underneath the top H-phase layer. Therefore, this observation further confirms that the distinctively strong T-phase CDW image in the modified region (left-hand side of (a)) is indeed an image of transformed H-phase, rather than an image of the underlying T-phase layer (right side of (b)). 65

W. Han, E. R. Hunt, 0. Pankratov, and R. F. Frindt, Phys. Rev. B. 50, 14746 (1994).

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14. NANOCRYSTAL FORMATION IN 2H-TaS2 The similarity of the atomic and chemical structure and electronic properties of TaSe, and TaS, makes 2H-TaS2 a good candidate for extending our STM tip-induced nanocrystal fabrication process. Indeed, after we perform the same surface modification on 2H-TaS2, T-phase TaS, nanocrystals embedded in the H-phase TaS, layer, which are similar to the T-phase TaSe, in 2H-TaSe2, are observed. Figure V.3 shows the existence of two different types of boundaries as in the TaSe, nanocrystal systems. These two T/H boundaries form 120" angles, and both have sharp interfaces without significant distortion. However, as in TaSe, nanocrystals, one (upper boundary) has a pronounced distortion on the atomic lattice of the interface, which makes observed S atomic rows in parallel with the interface protrude from the surface (bright in STM image), while the other (lower boundary) has a rather moderate distortion. Further analysis of the atomic lattices in both boundaries shows that the relative shift of S atomic rows across the interface and the S atoms at the boundaries form a rectangular array rather than the hexagonal lattice as suggested for TaSe, in the previous structural model for the nanocrystal creation in Figure 111.4. Therefore, we believe that the structural transformation model proposed to illustrate the STM tipinduced local solid-solid phase transform in a TaSe, system can be applied to a TaS, system as well. A systematic study of the modification voltage dependence of the T-TaS, nanocrystal formation suggests that the T-TaS, nanocrystals can be created

FIG. V.3. (a) STM image of boundary of T-phase TaS, nanocrystal embedded in 2H-TaS2. This nanocrystal was created by the STM tip at a bias voltage - 1.OV. (b) Boundary of created T-phase TaS, region with a bias votage larger than in (a) (- 1.5V). Large area (-400nm) of 2H-TaS2 surface was transformed to a region that consists of smaller T-phase single domain separated by discommensurate lines.

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more easily than T-TaSe, nanocrystals. The threshold bias voltage for creating TaS, is 1 V, which is significantly lower than that of TaSe, (1.2 V, see Figure 111.6, for example). The size of the TaS, nanocrystals is also much larger than that of the TaSe, at the same modification bias voltage. We could routinely obtain very large (- 600 nm) transformed T-phase region at moderate modification voltages (- 1.2 V). The largest transformed T-phase TaS, region has dimension of 800 nm at the modification voltage of 1.6 V, which is significantly larger than that of TaSe, nanocrystal(200 nm). This result suggests that the energy barrier to create a T-phase nanocrystal from H-phase is significantly lower for the TaS, system than the TaSe, system. Considering the higher electron affinity of S atoms compared to that of Se atoms, the dipole moment of the Ta-S bond should be greater than that of the Ta-Se bond. Therefore, the strong electric field under the biased STM tip could exert a larger force on the top S atoms (versus Se atoms). Finally, it is worth mentioning that the CDW state in a large transformed T-phase TaS, region often shows a CDW domain structure with domain boundaries consisting of CDW twin boundaries (different direction of CDW lattice line across the boundary) and discommensuration line (phase shift of the CDW lattice across the boundary) networks (Figure V.3(b)). This observation sharply contrasts the observation in the TaSe, system, where only isolated discommensurate lines and CDW twin domain boundaries exist. The size of this domain is 10 nm, and the CDWs at this boundary are either strongly suppressed or distorted, suggesting that the underlying atomic layer also has an extended defect such as twin boundary.

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15. LOCALTRANSFORMATION OF 1T-TaS,

AND

1T-TaSe,

We have applied the same surface modification method described in the previous section on both 1T-TaSe, and 1T-TaS, crystal surface. In this experiment, we found that it is more difficult to transform T-phase TMD to H-phase TMD. Figure V.4(a) shows the modified surface of 1T-TaSe, after the surface modification method with a bias voltage of 1.3 V. Clearly, the surface region underneath the biased STM tip was damaged, and CDW point defects and discommensuration lines radially emitted from the central damaged region were created rather than phase transformation to a 2HTaSe, nanocrystal. We have performed the modification process up to 5 V at 77K, but have not obtained 2H-phase nanocrystals. A detailed atomic resolution image of the end of the discommensuration line suggests that these CDW discommensurate lines have a direct relation with the underlying atomic lattice (Figure V.4(b)). As shown in Figure V.~(C), the formation of CDW discommensuration lines can be mediated by an atomic lattice

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FIG. V.4. (a) STM image of modified region of 1T-TaSe, surface. The applied bias for the oscillating STM tip was 1.3V. The central region of crystal surface was damaged by intense electric field and CDW point defects, and discommensurate line radially emitted from central region is visible. (b) Close-up image of the end of the discommensurate line in (a) with atomic resolution. (c) An atomic model illustrating the formation of CDW discommensurate line mediated by an atomic lattice dislocation. Solid circles represent Se atoms at the surface and open circles indicate the CDW maxima. The dashed line shows the CDW discommensurate line and the solid line indicates the phase shift of CDW domains across the discommensurate line.

dislocation. Because the T-phase CDW lattice is commensurate with the atomic lattice (-a x @a), around the dislocation in the atomic lattice, the CDW lattice carries a phase shift, which results in the formation of a CDW discommensuration line. Further application of the surface modification method to 1T-TaS, crystals shows drastic changes in the modified region. Unlike the isolated CDW discommensuration line formation in 1T-TaSe,, after the modification process 1T-TaS, exhibits CDW domain structure separated by a CDW discommensuration line network (Figure V.5). On these discommensuration lines, the strong T-phase CDW maxima are significantly suppressed. These domain structures with discommensurate line network are similar to the CDW nearly commensurate structure in bulk 1T-TaS,, except for the fact that the nearly commensurate CDWs in bulk IT-TaS, have a domain size of 10 nm66 at room temperature and are commensurate at low temperatures, while the characteristic length scale of the domain structures created by STM tip at low temperatures is 55nm.67

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X. L. Wu and C. M. Lieber, Phys. Rev. B. 41, 1239 (1990).

It was also found that the pinning strength of these domain structures to the underlying atomic lattice is rather moderate. We could restructure domain network by performing the second surface modification process near the first one. This subsequent surface modification process on the previously created CDW domain area changed the whole domain structure, implying CDW domains can easily slide, and therefore the pinning of CDW lattice to the underlying lattice should not be strong. "

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FIG.V.5. (a) STM image of modified regon of 1T-TaS, surfae. Instead of forming isolated discommensurate line as in 1T-TaSe,, CDW domain network was formed separated by CDW discommensurate line. The applied bias for the oscillating STM tip was 1.3 V. (b) Close-up view of T-phase CDW domains and discommensurate lines in (a).

In a recent experiment done by Kim et ~ l . , ~it' was suggested that a 1T to 2H local solid-solid phase transformation of TaS, is also possible with a highly biased STM tip at room temperature. In this experiment, they found the threshold voltage that can initiate the transformation is 8 V, which is significantly larger than that of 2H to 1T conversion (1.2 V as we discussed in Section 111). The size of the 2H-TaS, nanocrystal made in their structures varies from 10 to 50 nm at a bias voltage of 10 V. Similar to the 1T-TaS, nanocrystals in 2H- TaS,, the created H-phase TaS, nanocrystals have sharp triangular phase boundaries along the crystallographic directions, and high-resolution STM image near the phase boundary suggests that the H-phase nanocrystal formation results from a coherent shift of the surface S atomic sheet.

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16. NANOCRYSTAL FORMATION IN OTHER TMD MATERIALS There are other families of TMD materials that would produce interesting systems if locally modified regions could be created using the STM tipinduced phase transformation. These materials include NbSe, and MoTe,. Single-crystal 2H-NbSe, is an interesting system that exhibits incommensurate CDW state below 33 K and also a superconducting state below 7.2 K. This coexistence of CDW and superconducting states at low temperatures has drawn much research interest in this material, especially in STM s t ~ d i e s d . ' ~The ,~~ structure , ~ ~ of 2H-NbSe, is similar to that of 2H-TaSe,.

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FIG. V.6. (a) Atomic resolution STM image of cc-MoTe, crystal surface showing a hexagonal atomic lattice. (b) Atomic resolution STM image of b-MoTe, crystal surface showing zigzag atomic chain due to the distortion of Mo atoms from octahedral coordinate.

However, unlike TaSe,, no T-phase NbSe, has been found in nature, which might imply that the T-NbSe, phase is thermodynamically unstable. The motivation of applying the surface modification method described in the previous section is to investigate the possibility of creating and trapping a local T-phase quasi-stable region embedded in stable bulk H-phase NbSe, single-crystal surface. Another interesting TMD material is MoTe,. MoTe, exhibits two different phases, a and b, similar to H and T phases in TaSe,, with quite distinct electronic properties (Figure V.6).70,71b-MoTe,, which is a metastable low-temperature phase, has a distorted coordination of Mo atoms in the same layer forming one-dimensional metallic bonding, and leads to zigzag chain-like stripes (Figure V.6(b)). Electrical transport measurements show that a-MoTe, is semiconducting while b-MoTe, exhibits metallic behavi~r.~O Extensive studies of STM tip-induced structural transition in NbSe, and MoTe, materials, however, have not exhibited any evidence for locally transformed region or creation of nanocrystals embedded in the host crystal.60 Modification of NbSe, always produces severe damage to the H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles Jr., and J. V. Waszczak, Phys. Rev. Lett. 63, 214 (1989). 6 9 H. F. Hess, R. B. Robinson, nd J. V. Waszczak, Phys. Rev. Lett. 64, 2711 (1990). 70 T. J. Wieting and M. Schluter (eds.), Electrons and Phonons in Layered Crystal Strucures, Reidel Publishing, Boston (1977). 7 1 R. M. A. Lieth (ed.), Preparation and Crystal Growth of Materials with Layered Strucures, Reidel Publishing, Boston (1977).

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crystal surface, possibly due to the sublimation induced by an electron b~mbardment.~,These results imply that thermodynamic stability is required for both initial (2H-phase) and final (1T-phase) bulk material to initiate STM tip-induced solid-solid phase transition. Applying the nanocrystal fabrication method on both ct and fi phase MoTe, has not been successful either.23 A tentative explanation is that the structure of the p-MoTe, has a significant distortion from the octahedral coordination of Mo atoms, which might produce a large potential barrier to transforming between tl and fi phases by collective motion of the top layer of Te atoms. In this section, we reviewed solid-solid transformation induced by a bias voltage applied to a STM tip in various TMD materials. The transformation was accomplished through collective motion of many chalcogen atoms at once in the strong local electric field of the biased STM tip. This nanocrystal formation is closely related to the layered structure of TMD material, although the detailed nature of the transformation might depend on the specific crystal structure as in the case of NbSe, and MoTe,. Together with local variation of CDW lattice by intercalated foreign atoms in host TMD material^,^^,^^ this novel method of nanocrystal formation will elucidate CDW physics in nanometer scales in future study. VI. Summary and Future Work

We have reviewed CDW formation in 2D nanostructures. Size-dependent effects of CDW lattice in this 2D nanometer scale electronic system were investigated by studying the CDW wavelength dispersion. It was found that the quantization of allowed electronic states in Bloch reciprocal space of these nanostructures introduces Fermi surface roughening. This model provides an explanation to the experimentally observed CDW wavelength irregularity in the TMD nanocrystals. 2D CDW nanocrystals were created on the surface of host TMD material by a local solid-solid phase transition. This nanometer scale transformation was accomplished through the collective motion of many chalcogen atoms under the strong local electric field of the biased STM tip. STM tip-induced local phase transformations in various TMD materials have been discussed. The size of fabricated nanocrystals ranges from 7 to more than 600 nanometers within the surface layer of host TMD crystals. Atomic resolution images elucidate the structural changes between two different regions and were used to develop an atomic model 72 73 74

S. Kondo, S. Heike, M. Lutwyche, and Y. Wada, J . Appl. Phys. 78, 15 (1995). F. W. Boswell, G . A. Scholz, and J. C. Bennett, Phys Rev. B. 56, 1175 (1997). I. Ekvall, H. E. Brauer, E. Wahlstrom, and H. O h , Phys. Rev. B. 59, 7751 (1999).

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that describes a pathway for the production of nanocrystals from the host crystal precursor through a solid-solid phase transition. Future work may extend to the quantum transport measurements in these systems using a combination of SPM and modern lithographic technique. Recently, there has been a progress in the transport measurement of mesoscopic 1D CDW materials.75 In these studies, CDW sliding was investigated to elucidate a collective CDW transport, which leads to strongly nonlinear conduction. Considering rich physics in the 2D CDW nanocrystals, mesoscopic transport measurements in nanostructured 2D TMD materials should provide an opportunity to significantly advance understanding of the dynamics of 2D CDW in this length scale. A new approach to fabricating such structures will include efforts to isolate a single layer of crystals from host materials either ~ h e m i c a l l yor~ p~ h y ~ i c a l l y . ~ ~ Acknowledgments

We acknowledge J.-L. Huang, and J. Liu for their extensive help and strong support in this work. This work was supported by the National Science Foundation (C.M.L.).

0. C. Mantel, F. Chalin, C. Dekker, H. S. J. van der Zant, Y. I. Latyshev, B. Pannetier, and P. Monceau, Phys. Rev. Lett 84, 538 (2000); N Markovic, M. A. H. Dohmen, and H. S. J. van der Zant, Phys. Reo. Lett. 84, 534 (2000). 7 6 D. W. Murphy and G . W. Hull Jr., J. Chem. Pyx 62,973 (1975); P. Joensen, R. F. Fidt, and S. R. Morrison, Muter. Res Bull. 21, 457 (1986). 7 7 X. Lu, H. Huang, N . Nemchuk, and R. S. Ruoff, Appl. Phys. Lett. 75, 193 (1999). 75

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SOLID STATE PHYSICS, VOL. 55

Vibrational Properties of Shape-Memory Alloys ANTONIPLANES AND LLU~S MA~OSA Departament d’Estructura i Constituents de la Materia. Facultat de Fisica. Diagonal, 645. 08028 Barcelona. Catalonia (Spain)

I. Introduction 11. Martensitic Transitions and Shape-Memory Materials 1. Preliminaries 2. Phase Diagram 3. Thermodynamics and Kinetics 111. Precursor Effects IV. Lattice Dynamics 4. Elastic Behavior a. Symmetry Adapted Strains 5. Vibrational Anharmonicity: Gruneisen Parameters V. Experimental Results 6. Second-order Elastic Constants a. Room-temperature values b. Temperature dependence 7. Phonon Dispersion 8. Third-order Elastic Constants and Vibrational Anharmonicity VI. Phase Stability 9. Total Entropy Change 10. Electronic and Vibrational Contribution 1 1 . Energy Change VII. Modelling 12. Strain-phonon Coupling Model 13. Localized Soft-mode Models a. Nucleation theories b. Defect-assisted nucleation c. Remarks on the transition mechanism VIII. Magnetic Coupling 14. Properties of Ni,MnGa 15. Phonons and Elastic Constants 16. Models for the Intermediate Transition IX. Conclusions

160 162 162 164 172 175 181 181 184 186 187 188 188 192 199 210 217 217 22 1 228 233 235 245 245 247 25 1 252 254 256 260 265

159 ISBN 0-12-607755-X ISSN 0081-1947iOl $35.00

Copyright I~ 2001 by Academic Press All rights of reproduction in any form reserved.

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1. Introduction

The crystal structure of a given material depends on its energy and also its entropy, which plays an increasingly important role as temperature is raised.’ In the case of nonmagnetic metallic systems, the entropy has two major contributions: the electronic contribution arising from electronic states near the Fermi level, and the vibrational contribution, related to the vibrations of the atoms around their equilibrium positions in the crystal lattice.’ A significant amount of work has been carried out aimed at finding the relevance of each term in the determination of the stable phase of a specific system under imposed external conditions (temperature, pressure, e t ~ . ) .The ~ . ~present review is concerned with a class of bcc-based metallic alloys, commonly called shape-memory alloys. These materials have the unique property of being able to recover from large deformations by slightly increasing their temperature. The physical mechanism behind this effect is a diffusionless, first-order structural transition, usually referred to as the martensitic transition. The problem of martensitic transitions in shape-memory alloys has been considered in many excellent review papers dealing with several aspects of this The point of view adopted here for the study of this class of systems connects the martensitic transition with the loss of stability of the high-temperature bcc phase. More than 50 years ago Zener’ was the first to point out that, in general, there is more vibrational entropy in the bcc phases, which have lower Debye temperatures, than in their close-packed counterparts. Within this framework, the important point to be stressed is that the phonons corresponding to the atomic motions associated with the structural change have rather low energy. On the one hand, these low-lying phonons confer a large vibrational entropy to the bcc phase and thus are the origin of its thermodynamic stability at high temperature. On the other hand, as a result of the weak restoring forces for these specific atomic motions, these phonons facilitate the mechanical instability that brings the

M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press. London, (1954). D. C. Wallace, Thermodynamics of Crystals, John Wiley & Sons, New York (1972) G. Grimvall, Thermophysical Properties of Materials, North-Holland, Amsterdam (1986) 0. Eriksson. J. M. Willis, and D. C. Wallace, Phys. Rev. B, 46, 5221 (1992) A. L. Roitburd, Solid State Physics 33, 317 (1978). L. Delaey, in Materials Science and Technology, Vol. 5 Phase Transformations in Materials, ed. P. Haasen, VCH, Weinheim (1991), p. 339. C. Zener, Elasticity and Anelasticity of Metals, University of Chicago Press, Chicago (1948), p. 32.

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bcc structure toward a close-packed phase (martensite). In this review we discuss the problem based mainly on the reported behavior of phonon dispersion curves, elastic constants, and entropy measurements for different families of shape-memory materials. Among these families, the favorite choice here will be Cu-based alloys. In the present context, they are especially appealing because the electronic contribution to their entropy is negligibly small.' They have been studied in depth during recent years and, at present, appear as paradigmatic examples of shape-memory materials. For the sake of generality, these results will be compared, when possible, with those corresponding to other shape-memory systems. The experimental results will be analyzed within the framework of phenomenological models and theories developed to account for the specific characteristics of martensitic transitions. More recently, ferromagnetic shape-memory materials have received a great deal of interest. The lattice dynamics in these systems are influenced by the coupling between magnetic and structural degrees of freedom. Because the intent is to analyze this problem by comparing these materials with non-magnetic materials, only systems with a bcc-based high temperature structure will be considered. At present, this requirement restricts the discussion to the case of Ni-Mn-Ga Heusler alloys. We will consider these systems separately in the last section of this review. This review is intended to provide researchers with an update of recent research on fundamental aspects of shape-memory alloys, relevant for a deeper understanding of the mechanisms of the martensitic transition. Because this subject has been developed in a rather independent way by solid state physicists and material scientists, it is our aim that this chapter will be of interest to both communities. After an introduction to the general problem of the martensitic transition (11), followed by a detailed discussion of precursor effects in Section 111, we provide a brief survey of the key ingredients of lattice dynamics in Section IV, useful for the comprehension of the following sections. A thorough compilation of significant experimental data is presented in Section V. Emphasis will be made in determining and showing both common and particular features of different alloys. In Section VI we discuss a number of thermodynamic results in relation to the phase stability problem, mainly with reference to the excess of entropy of the high-temperature phase. In Section VII, recent models accounting for the martensitic transition are described. Finally, in Section VIII, ferromagnetic shape-memory alloys are considered. We outline the main conclusions in Section IX.

* T. B. Massalski, U. Mizutani, Prog. Mnter. Sci. 22, 151 (1978).

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II. Martensitic Transitions and Shape-Memory Materials 1. PRELIMINARIES

Martensites form when a high-temperature crystalline phase undergoes a first-order phase transition to a lower symmetry crystalline phase. This structural phase transition usually proceeds by means of the nucleation of the new phase within the initial one, and may be induced either by changing the temperature or by applying a mechanical stress. There are two important properties distinguishing martensitic transitions from other symmetrybreaking crystal-crystal transitions; the transition does not involve diffusion of atoms, nor substantial volume ~ h a n g e . From ~ ~ ' ~this point of view such a class of structural transitions must be classified as displacive. Ferroelastic transitions also meet these requirements. The main difference from the generally accepted martensitic, is that the lattice distortion in ferroelastic materials is much smaller.* At this point it is important to distinguish them from the complementary subclass of replacive transitions that involve the local and/or long-range thermally activated diffusional rearrangement of atoms among lattice sites. In this latter case, ordering and/or compositional change occur and provide the necessary driving force for the transition. In replacive transitions, the natural thermodynamic order parameters are apparent (i.e. composition, etc.) and it is not surprising that the application of chemical thermodynamics has been fully developed and has been highly successful in providing a framework for describing why and how these transitions take place." For displacive transitions their diffusionless nature leads to the important consequence that the constituent atoms do not interchange their relative positions. This is, obviously, very relevant in the case of compounds with more than one constituent because it ensures that chemistry is not involved in determining the free energy that drives these phase transitions. In comparison with replacive transitions, displacive transitions have been much less theoretically developed, in part because of the problem of adequately identifying the relevant order parameter. In the particular case of martensitic transitions, the fact that the volume change at the transition is usually negligible implies that the change of crystallographic structure can be achieved by means of a dominant shear mechani~m.'~.'~ J. W. Christian, The Theory of Transformations in Metals and alloys, 2nd ed. Pergamon Press, Oxford (1975). l o J. W. Christian, G. B. Olson, and M. Cohen, J. Phys. IV (Paris), 5, C8-3 (1995). * Ferroelastic transitions also meet these requirements. The main difference from the generally accepted martensitic is that the lattice distortion in ferroelastic materials is much smaller. D. DeFontaine, Solid State Physics 34, 74 (1979). l 2 W. G. Burgers, Physica 1, 561 (1934).

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163

The transition symmetry change produces a lattice misfit along interfacial boundaries, accommodated by elastic strain. This transition-induced strain is the decisive factor that determines the main kinetic and morphological characteristics of the transition. In this respect an important feature to be pointed out is the fact that the symmetry properties of the high-temperature phase result in quite a large degeneracy g of the low-temperature phase. This means that the structural transition can give rise to g phases. They differ in their crystallographic orientation with respect to the initial phase, but are energetically equivalent in the absence of an externally applied field (which breaks the degeneracy). In general, under these circumstances, the lowtemperature phase is not a single crystal, but rather the different symmetryrelated phases (so-called domains or variants) form a complex heterophase. The corresponding domain arrangement (self-accommodating) eliminates the long-range character of the strain field (volume-dependent energy) that would result from a coherent single-variant martensitic particle embedded in the parent Many characteristic features of martensites, including shape-memory properties, are dominated by the interplay between these different domains.' 5 * 1 6 Martensitic transitions have been observed in a wide variety of crystalline solids ranging from organic systems to simple metals and alloys. We will define the category of shape-memory systems to include those bcc-based metals and alloys that undergo a martensitic transition with an evident first-order character and such that there is a reversible lattice correspondence between the bcc and the product structure. The martensitic transition in these alloys takes place, in a certain composition range, from the open bcc phase to a close-packed structure. In general, the bcc phase is only thermodynamically stable at temperatures well above the martensitic transition, but it can be retained at lower temperatures by means of a suitable cooling treatment (quenching). Consequently, the martensitic transition in these systems takes place in an initially metastable phase. This fact is especially important in relation to the technological applications of these materials, in particular for applications above room temperature. At low enough temperatures the metastability of the bcc phase is unimportant, and the bcc phase can be treated as a thermodynamic equilibrium phase. In shape-memory materials, the martensitic transformation occurs with little thermal hysteresis, and spreads over a certain temperature range. The C. M. Wayman, Introduction io the Crystallography of Mariensiiic Transformations, McMillan Series in Materials Science New York, 1964. l4 B. Horovitz, G. R. Barsch, and J. A. Krumhansl, Phys. R e v . , 43, 1021 (1991). l 5 H. Tas, L. Delaey, and A. Deruyttere, Metall. Trans. 4, 2833 (1973). l 6 T. Saburi, and C. M. Wayman, Acta metall. 27, 979 (1979).

164

ANTONI PLANES AND L L U ~ SMAROSA

temperatures at the beginning and end of the transformation on cooling (M, and M,) and those corresponding to heating (A, and A,) are quantities commonly used to characterize the transition. These features are not always relevant for the discussion of the results presented in this paper and, in these cases, we will assume that the transition takes place at a generic temperature T,. If not specified, this temperature will be taken to coincide with M,. As previously mentioned, the remarkable thermomechanical properties, known generically as shape-memory properties, owe their origin to the martensitic transition. These properties make this class of materials very attractive from a technological point of view because they function as sensors as well as actuators and are promising candidates for smart materials." The shape-memory effect is related to the fact that when these materials are deformed in the low-temperature phase, they are able to recover their original shape by the reverse transition upon heating. In the high-temperature phase, the same systems display another unique property called superelasticity. This refers to the possibility of recovering, upon loading and unloading, a large strain (in many cases > 10%)associated with the stress-induced transformation. Several articles dealing with this interesting feature of martensitic transitions' '*19 have been published. They include a detailed description of the shape-memory effect, superelasticity, and other interesting related properties such as the two-way shape-memory effect or the high damping capacity, and will not be considered here. 2. PHASE DIAGRAM Many shape-memory alloys belong to the Hume-Rothery class of materials.8*20*21 They are based on noble metals (Cu, Ag and Au), usually alloyed with sp-valent elements. The equilibrium phase diagrams of all these materials are remarkably similar. This is a consequence of the fact that their phase stability is largely dominated by the average number of conduction electrons per atom, ela, but only to a lesser degree do they depend on the particular elements in the binary or ternary alloy system. As an example, in Figure 11.1 we present a section of the phase diagram (equilibrium and P. A. Besselink, J. Phys. IV 7 , C5-581 (1997). L. Delaey, R. V. Krishnan, H. Tas, and H. Warlimont, J. muter. Sci. 9, 1521, 1536, 1545 (1974). l9 Shape Memory Materials, ed. K. Otsuka and C. M. Wayman, Cambridge University Press, Cambridge (1998). H. Warlimont and L. Delaey, Prog. Muter. Sci. 18, 1 (1974). 2 1 D. Pettifor, Bonding and Structure of Molecules and Solids, Oxford U.P., Oxford (1995). Chap. 6. l7

l8

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

1500

165

L+a

1300

- 1100 Y

Y

t! 900

' 3

c

8 E

&?

700 500

0

10

20

30

40

at 'lo Zn FIG.11.1. Section of the phase diagram of Cu-Zn-A1 for a fixed atomic fraction (17 at%) of Al. Order-disorder and martensitic transition lines (solid lines) determine the metastable phase diagram. The hatched area corresponds to the equilibrium B-phase region, which shows the characteristic V-shape. A2, 8 2 , and L2, denote the metastable disordered and ordered /?-structures, and M denotes the martensitic phase. Greek symbols stand for equilibrium phases and L for the liquid phase. The equilibrium phase diagram is from Shape-Memory Eflects, Superelasticity and Damping in Cu-Zn-Al Alloys, Katholieke Universiteit Leuven, Department Metaalkunde, Report 78R1, 1978. The order-disorder lines are obtained from a mean field model proposed by R. C. Singh, Y. Murakami, and L. Delaey, Scripta. metall. 12, 435 (1978), and the martensitic transition line is determined from the expression proposed by A. Planes, R. Romero, and M. Ahlers, Acta metall. mater. 38, 757 (1990).

metastable) of the Cu-Zn-A1 shape-memory alloy, for a fixed atomic fraction of Al. The bcc structure is denoted as the P-phase and is the hightemperature stable phase within a broad composition range, centred at an electron-to-atom ratio e/a 1: 1.5. The composition width of the /?-phase stability region becomes narrower as temperature decreases and, in many cases, a temperature is reached at which the boundaries of this region meet each other (eutectoid point). This causes the /?-phase field to have a characteristic I/ shape. On slow cooling, the P-phase transforms eutectoidally into a mixture of the a (fcc structure) and y (Cu,Al, type of structure) phases. For kinetic reasons, however, the /?-phase can be retained as a metastable phase below its stability region by means of a fast enough

166

ANTONI PLANES AND L L U ~ SMAGOSA

FIG.11.2. (a) bcc structure with four sublattices denoted by u = I, 11, 111, and IV. p j is the occupation probability of the j atom in the (r sublattice. The A2, 8 2 , L2,, and DO, structures are defined as follows: A2 (Im3m): p j = p? = p j r r = py ( = x j ) V j ; B2 (Pm3m): p j = p? = py = pj' ( = x j ) V j L2, (Fm3m): py = pj' = p j = p? # py V j ; and DO, (Fm3m): pj' # p: = pj' = p y r Vj. (a), (b), and (c) respectively define the unit cells of the L2,, B2, and A2 structures (a* = 4 2 ) .

cooling. In this process, the bcc structure becomes configurationally ordered." Usually nearest-neighbor order develops at high temperature and the B2 superstructure is formed. In many cases, next nearest-neighbor order is established at a lower temperature and the L2, (or DO,) superstructure (see Figure 11.2) is commonly observed.* A feature to be pointed out is the fact that the unit cell dimensions of the L2, (and DO,) superstructure double those of the disordered and B2 structures.** It is important to mention that when the fl-phase becomes ordered, a possible reduction of diffusivity and/or a decrease of available free energy, drastically reduces the tendency of the equilibrium phases to precipitate.',

M. Ahlers, Prog. Ma.Sci. 30, 135 (1986). This is the usual situation in most Cu-based ternary systems with shape-memory properties. Important exceptions are Cu-Zn, in which only the B2 superstructure is formed, and Cu-AI-X (with a small fraction of X) close to the stoichiometric Cu,Al, where a DO, superstructure is directly formed on cooling from high temperature. ** The ordered structure is no longer bcc (the space groups are Pm3m and Fm3m for 8 2 and L2, respectively) but is generically called ordered bcc or simply bcc-based. 23 S. D. Kulkarni, Acta metall. 21, 1461 (1973). 22

*

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

167

1900 1500

1100

700

,,I 0

,

,

20

I

,

il

60 .at% Ni

40

-Tm

,

,

80

,

100

FIG. 11.3. Phase diagram of Ni-A1 alloy. From M. F. Singleton, J. L. Murray, and P. Nash, Binary Alloy Phase Diagrams, ed. T. B. Massalski, Metals Park, OH, American Society for Metals (1986). The martensitic transition (T,) line is determined using data from Y.K. Au and C. M. Wayman, Scripta metall. 6, 1209 (1972) and S. Rubini, C. Dimitropoulos, S. Aldrovandi, and F. Borsa, Phys. Rev. B. 46, 10563 (1992).

The situation is not very different in Ni-based alloys such as Ni-A124 or Ti-Ni.25 Within the composition range where these alloys display a martensitic transition (from 60 at% to 69 at% Ni in Ni-Al, and close to the equiatomic composition in Ti-Ni) , the /?-phase is only stable at high temperatures as well. In Figure 11.3 we show the corresponding phase diagram of Ni-Al. At a lower temperature T,, the martensitic transition takes place. In shape-memory alloys, this temperature is known to be strongly dependent on composition. Figures 11.1 and 11.3 include the martensitic transition line, which illustrates this fact. Very few studies have been devoted to understanding this problem from a microscopic point of view. For Hume-Rothery 24 B. Predel, Landolt-Bornstein New Series IV/Sa, edited by 0. Madelung, Springer-Verlag, Berlin, (1991) p. 212. 2 5 T. Saburi, in Shape Memory Materials, ed. K . Otsuka and C. M. Wayman, Cambridge University Press. Cambridge (1998), p. 97.

168

ANTONI PLANES AND L L U ~ SM A ~ O S A

alloys, the effect of composition on T, can be accounted for by the e/a dependence of the relative stability between equilibrium phases taking into account the effect of long-range order.26 An interesting different approach was proposed by P.A.- LindgHrd,” who related such a strong composition dependence of T, to the effect of disorder scattering of phonons caused by the mass difference between the elements of the compound. Because this transition is diffusionless, martensite inherits the ordered arrangement of the high-temperature /?-phase. Martensite is a close-packed phase that can be described by the application of a combination of shear A convenient choice2’ mechanisms to the high-temperature bcc considers the following combination; (1lo)[ 1TO] and (1r2)[T11] homogeneous shears together with a superimposed static modulation (or shuffle), which corresponds to a phonon mode on the transverse T A , branch, with a specific wavenumber q. This leads to the typical stacking sequences of close-packed planes with a basal plane derived from the { 1lo} planes of the /?-phase. For instance, the 9R (or 18R in the case of an ordered DO, or L2, /?-phase) and the 2H structures (in Ramsdell notation), are usually observed in noble metal-based martensites.,’ They correspond to the following sequence of close-packed planes: ABCBCACAB for the 9R and AB for the 2H, and are obtained for q = *qzs and q = qze, where qzB is the zone boundary wavenumber in the [ l l O ] direction of the /?-phase (N-point in the reciprocal space). In Figure 11.4 we schematically show the steps in order to accomplish the transformation from the bcc to the 9R phase. It is worth noting that this mechanism needs as a first step a volume-preserving Bain distortionzgresulting in a significant increase of energy, which indicates that the actual transition path may be different. Ahlers,’ proposed a model also giving the correct crystallography, which overcomes this problem. It consists of a low-energy long wavelength displacement on the (1 10) plane in the [lTO] direction, followed by a shuffle displacement on another { 1lo} plane. Both displacements are coupled because the decrease in energy is more favorable than if the q = 0 shear occurs first to an intermediate state. In Cu-based alloys, 9R (or 18R) and 2H martensites develop for compowhere (e/a)eu is the sitions with e/a <(e/a),, and e/a 2 (ela),, re~pectively,~’ electron-to-atom ratio value corresponding to the composition of the

M. Ahlers, Proc. Int. Conference on Solid-Solid Phase Transformations. Edited by M. Koiura, K. Otsuka and T Miyazaki, The Japan Institute of Metals, Sendai, Japan 1999, p. 831. ” P.-A. LindgHrd, J. Phys. IV 5, C2-29 (1995). S. Kajiwara, Muter. Trans. JIM 17, 435 (1986). 29 M. J. Kelly, J. Phys. I ? MetalPhys. 9, 1921 (1979). 30 M. Ahlers, 2. Metallk. 65, 636 (1974). 3 1 E. Obrado, LI. Mafiosa, and A. Planes, Phys. Rev. B, 56, 20 (1997). 26

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

-

[lTolp

HOMOGENEOUS

t

SHEAR

-

SHUFFLE

169

[001]9R

t

IllOlp

FIG. 11.4. Lattice distortion of the bcc (B2) toward the 9R close-packed martensitic structure. The positions of the atoms along the stacking sequence in the bcc and martensitic structures are shown. Notice that 8, = 109.47" changes to B,, = 120", characteristic of close-packed structures.

eutectoid point in the equilibrium phase diagram. This seems to be reminiscent of the e/a structure dependence in the equilibrium phase diagram typical of Hume-Rothery alloys. Table 11.1 summarizes the main crystallographic characteristics of a number of shape-memory alloys. As mentioned preceding, martensitic transitions can be induced either by changing the specimen temperature or by applying an external stress 0. Application of a uniaxial stress results in the formation of the martensitic variant best suited to accommodate the imposed strain. The relative stability between the high-temperature and the different martensitic structures not only depends on temperature and composition, but also on the level of

170

ANTON1 PLANES AND LLUiS MA6OSA

CHARACTERISTICS OF PARENTAND MARTENSITIC PHASES IN A TABLE11.1. CRYSTALLOGRAPHIC NUMBEROF SHAPE-MEMORY ALLOYS,AND PHONON-MODE ASSOCMTED WITH THE STACKING THENOTATION ADOPTEDFOR THE MARTENSITE IS THE MOSTWIDELY USEDIN THE SEQUENCE. LITERATURE Parent Phase (space group)

Martensite (space group)

Crystal system*

Phonon mode

L21lD0, (Fm3m) L2IlD0, (Fm3m)

2H (Pnmm) 18R (Wm)

Hexagonal

c1101

Orthorhombic

3[llO]

Cu,Zn, --x x 0.6

B2 (Pm3m)

9R (P2/m)

Orthorhombic

f[llO]

Ni,Al, - x 0.6 C x 6 0.69

B2 (Pm3m)

7R (P2/m)

Orthorhombic

f[llO]

N i , Ti, - x * * (near equiatomic composition)

82 (Pm3m)

B19 (P2/m)

Monoclinic

m101

AU - Cd (near equiatomic composition

82 (Pm3m)

5’

Monoclinic

f [ 1101

(P2/m)

B2 (Pm3m)

Hexagonal

:c1101

(Pnmm)

L21 (Fm3m)

18R (I2Im)

Orthorhomhic

3[llO]

Material Ternary Cu-based ela

> (ela),,

ela

< (ela),,

-

AuCuZn,

Y;

*Here we have listed the most commonly used lattice symmetry. For instance, although 9R and 18R martensites are indeed monoclinic, the large unit cells (containing 9 and 18 close-packed planes respectively) are almost orthorhombic. **Fully annealed alloys transform from the B2 to the E l 9 martensite, while thermally cycled or thermo-mechanically treated specimens transform in two steps, i.e. B2 --t R -+ B’19. The space group of the R-phase is P3. The intermediate transition is rnartensitic. The two-step behavior also occurs in aged Ni-rich alloys.

VIBRATIONAL PROPERTIES O F SHAPE-MEMORY ALLOYS

600 c

0

a

171

5 I

I

61R

L

"I

-

Temperature

5

6R+18R

--

18 R

1BR

2H+18R

200

240

280

Temperature

320

(K)

FIG. 11.5. Stress-temperature (u - 7') phase diagram for a (211-27.8at% A1-3. 8 at% Ni. The transformation lines have been determined from calorimetric measurements on thermally induced transitions under applied constant stress. This procedure gives the lines for forward and reverse transitions. Data from J. Ortin, L1. Maiiosa, C. M. Friend, A. Planes, and M. Yoshikawa, Philos. Mag. A 65,461 (1992). The inset shows the schematic u - T phase diagram of Cu-Al-Ni alloy system proposed by K. Otsuka, H. Sakamoto , K. Shimitzu, Acta Metall. 27, 585 (1979). Used by permission of Taylor & Francis, http://www.tandf.co.uk/journals/phm.htm.

applied stress.j2- 35 For a given composition and crystallographic orientation, a number of B-to-martensite and martensite-to-martensite transitions can be observed at different temperatures and levels of applied stress (tensile and compressive). Stress-temperature phase diagrams have been obtained for different alloy system^.^^*^^,^' The family of Cu-based systems has been studied in great detail, and in all cases the 2H structure is the stable martensitic phase at low stress level (for e/a 2 (e/a)eu). Figure 11.5 depicts this phase diagram for the Cu-Al-Ni alloy system. At different temperatures, the typical sequence of phases obtained by applying a tensile stress is presented. 32 K. Otsuka, C. M. Wayman, K. Nakai, H. Sakamoto, and K. Shimitzu, Actu muter. 24, 207 (1976). 33 W. Arneodo and M. Ahlers, Scripta metall. 27, 1287 (1979) 34 H. Kato, J. Dutkiewicz, and S. Miura, Actu metull. muter., 42, 1359 (1994). 35 V. Novak, J. Malimanek, and N. Zarubova, J. Phys. ZV 5, C8-997 (1995). 36 S. Miura, T. Mori, N. Nakanishi, Y. Murakami, and S . Kachi, Philos. Mug. A. 34,337 (1976). 37 S. Miura, F. Hori, and N. Nakanishi, Philos. Mug. A . 40,611 (1979).

172

ANTONI PLANES AND L L U ~ SMAROSA

3. THERMODYNAMICS AND KINETICS

The martensitic transition in shape-memory alloys has a thermoelastic character, which means that at each temperature and level of applied stress within the transformation range, thermoelastic equilibrium is a~hieved.~’ This equilibrium condition is defined by a local balance at the transforming interfaces between competing forces. In the forward (parent to martensite) transition, the driving force promoting martensite is balanced by the increase in elastic (strain and interfacial) energy. In the reverse transition the elastic energy previously stored promotes, together with the driving force, the reversal into the parent phase. An ideal thermoelastic transition would proceed without hysteresis. In shape-memory alloys the hysteresis is small and is a consequence of different dissipative effects operating during the transition. When the transition is thermally induced and a polyvariant martensite develops, the dominant mechanism of hysteresis has its origin on the relaxation of elastic strain energy arising from the accommodation of the transformational shape change (self-accommodating process) and from the interaction between martensitic variants. The relaxation of elastic strain energy has been found to depend on the specific crystallographic structure of the growing marten~ite.~’Hysteresis is, for instance, larger for the P+2H than for the /3%18R transition. Actually, this is a consequence of the different martensite growth mechanisms accounting for the accommodation of the transformation strain in each case: twinning and gliding mechanisms.6 A different scenario occurs when the transition is stress-induced; the intrinsic hysteresis arising from a single interface transformation has its origin in the interaction of the interface with dislocation^.^^ Actually, such an intrinsic hysteresis is very small. From a kinetic point of view, these transitions are athermal, which means that, contrary to thermally activated systems, thermal fluctuations do not play any relevant role.’ When cooling from the high-temperature phase, the transition starts at a given temperature. However, the system needs to be continuously cooled down to make the transformed fraction of the new phase increase. At any temperature in the two-phase region, the transition appears to be instantaneous in practical time scales, and hence the transformed fraction virtually does not depend on time, i.e., it increases each time the temperature is lowered. The transition is not completed until the temperature is lowered below a certain value. Hence, in these transitions the temperature plays the role of an external controlling parameter that deter-

’* A. Planes and J. Ortin, J. Appl. Phys. 71, 950 (1992). ’’J. Ortin and A. Planes, J. Phys. ZV (Paris), 1, C4-13 (1991). 40

F. C. Lovey, A. Amengual, V. Torra, and M. Ahlers, Philos. Mug. A 61, 159 (1990).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

173

mines the free-energy difference between the high- and low-temperature phases. This free-energy difference provides the driving force for the transition. Actually, the path followed by the system is an optimal path in which accomodation of the transformation elastic strain is almost accomplished. Accurate observation^^^^^^ have revealed that these transitions proceed through a series of discrete jumps (or avalanches) connecting metastable equilibrium states separated by very high energy barriers (>> k,T, where k, is the Boltzmann constant). In these states the thermoelastic equilibrium condition is satisfied. At each step, elastic energy is stored in the system and, at the same time, energy is released (latent heat irreversible energy). For this reason, at each new metastable situation, the temperature of the system must be lowered again to restart transformation. For low enough temperature rates, however, avalanches take place in time intervals much smaller than the time of appreciable variation of the driving force. Thus, the system spends the overwhelming majority of its time in a situation of thermoelastic equilibrium. During the avalanches the irreversible energy is (at least partially) released in the form of transient elastic waves. They are emitted in the ultrasonic range and are known as acoustic e m i ~ s i o n . These ~ ~ . ~acoustic ~ waves contain dynamic information of the mechanism associated with the structural change, or source mechanism, which has generated them. Several models have been developed aimed at describing the source mechanism during martensitic transition^.^^ - 4 7 Experiments corroborate that the acoustic radiation pattern mainly corresponds to { 1lo}( 170) martensitic shear mechanism, but that it also contains a small effect due to a volume change mechanism.48 A different point of view consists of statistically analyzing the acoustic emission signals generated during the transition. Within this approach it has been reported that the statistical distributions of signal amplitudes and durations exhibit power law distribution^.^^ Indeed, this result is indicative of the fact that the evolution of martensitic systems in the two-phase region

+

A. Planes, J. L. Macqueron, M. Morin, and G . Gutnin, Phys. Stat. Sol. ( A ) 66,717 (1981). A. Planes, J. L. Macqueron, M. Morin, Guenin, and L. Delaey, J. Phys. (Paris), 43, C4-615 (1982). 4 3 R. Pascual, M. Ahlers, R. Rapacioli, and W. Arneodo, Scripla. metall. , 9, 79 (1975). 44 J. Baram and M. Rosen, Philos. Mag. A . 44, 895 (1981). 45 H. N . G . Wadley and J . A. Simmons, J. Res. Nat. Bur. Stand. 98, 55 (1984). 46 Z. Yu and P. C. Clapp, J. Appl. Phys. 62,2212 (1987). 4 7 L1. Maiiosa, A. Planes, D. Rouby, and J. L. Macqueron, Acta metall. 38, 1635 (1990). 4 8 L1. Maiiosa, A. Planes, D . Rouby, M. Morin, P. Fleischmann, and J. L. Macqueron, Appl. Phys. Lett. 54, 2574 (1989). 49 E. Vives, J. Ortin, L1. Mafiosa, I. Rafols, R. Perez-Magrane, and A. Planes, Phys. Rev. Lett. 72, 1694 (1994). 41 42

174

ANTONI PLANES AND L L U ~ SMAFJOSA

takes place without characteristic length and time scales, which is a typical feature of criticality. In addition, for Cu-based shape-memory alloys, there is some evidence that the distribution exponents show some universal character, depending only on the symmetry of the low-temperature phase.” In a set of recent papers Sethna and collaborator^^^-^^ have suggested that the intrinsic disorder in the system is responsible for power-law distributions. The idea was substantiated by studying a simplified model exhibiting a fluctuationless first-order phase transition consisting of a Random Field Ising model at zero temperature with an applied field.51 By changing the external field the system follows deterministic dynamics corresponding to local energy relaxation. The transition is found to take place through avalanches, whose size distribution depends on the amount of disorder. For a small degree of disorder, the system exhibits a large (percolating) avalanche, while for a large degree, the avalanches are tiny and the transition extends over a broad range of the external applied field. The important point is that for a critical value of disorder the distribution of avalanches is a power law. Actually, similar behavior has been obtained in other models such as the Random Bond Ising the Blume-Emery-Griffith model with disorder,” and the diluted Ising The fact that the critical behavior is only obtained after fine tuning of the degree of disorder suggests the following question: Why do real martensitic systems appear to stay at the critical amount of disorder? Two explanations have been proposed to answer this question. The first assumes that internal relaxation during the growth process spontaneously modifies the disorder until the critical state is reached after a certain number of cycles.54 In spite of the fact that the density of defects in martensitic systems is known to be modified by cycling through the tran~ition,~at present, more experimental support is needed to confirm this interpretation. A second explanation is based on the fact that, in models, the critical region has been shown to be very large and, therefore, power-law behavior in a limited number of decades, as found in experiments, cannot be taken as a definite proof of ~riticality.’~ Furthermore, due to experimental limitations, it is not easy to discern slight deviations from the critical state, either sub- or supercritical. L1. Carrillo, L1. Maiiosa, J. Ortin, A. Planes and E. Vives, Phys. Rev. Lett. 81, 1889 (1998). “J. P. Sethna, K. A. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts, and J. D. Shore, Phys. Rev. Lett. 70, 3347 (1993). ”0.Perkovic, K. A. Dahmen, and J. P. Sethna, Phys. Rev. Lett. 75,4528 (1995). 53K.A. Dahmen and J. P. Sethna, Phys. Rev. B. 53, 14872 (1996). 54 E. Vives and A. Planes, Phys. Rev. B. 50, 3839 (1994). 5 5 E. Vives, J. Goicoechea, J. Ortin, and A. Planes, Phys. Rev. B. 52, R5 (1995). 5 6 B. Tadic, Phys. Rev. Lett. 77, 3843 (1996). 5 7 D. Rios-Jara and G. Guinin, Acta metall. 35, 109 (1987).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

175

111. Precursor Effects

From a general point of view, precursor or pretransitional effects are phenomena announcing that a system is preparing for a phase transition before it actually occurs.” They are related to possible anticipatory visits of the system into the approaching phase and are characteristic of systems that undergo second-order (or at least nearly second-order) transitions in which, due to critical fluctuations, length scales start to diverge before the transition. Because typical first-order transitions occur abruptly, preceding incursions into the new structure are, in principle, not expected. However, among the wide class of materials undergoing martensitic transitions, pretransitional phenomena are observed in several cases. This fact has suggested distinguishing, in a broad sense, different groups of martensitic material^:'^ (i) those, such as steels, in which the transition is strongly first order, (ii) materials with moderately first-order transitions, which include shapememory alloys and (iii) systems, such as the A-15 compounds and perovskite ferroelectric materials, that show a close to second-order transition. While the first class of materials does not display significant precursor effects, they have been observed in the latter two classes. Actually, this is rather surprising in the case of shape-memory alloys because the transition has an evident first-order character, as revealed by a significant latent heat (discontinuity of the entropy). Indeed, these systems show simultaneously distinctive attributes of first- and second-order transitions. Within this context it can be considered that understanding precursor phenomena in shape-memory alloys should provide a key to enable the comprehension of the unique features of the martensitic transition in these materials.60 In different systems, precursors take many different forms and are commonly observed as anomalous effects in x-ray, electron, and neutron scattering measurements. Examples are the partial softening of various distortive modes (including q = 0 homogeneous deformations and q # 0 phonon modes), the “central peak,” or the appearance of tweed patterns, among others. In this section we give a summary of the situation by referring to a number of significant effects observed using different experimental techniques in various alloy systems. The quasi-elastic “central peak” observed in neutron scattering refers to the appearance of elastic streaks and satellites, whose intensity increases as the transition is approached. It represents either static or rather long-lived A. D. Bruce and R. A. Cowley, Structural Phase Transitions, Taylor and Francis, London 1981. 5 9 L. E. Tanner and M. Wuttig, Mat. Sci. Engng. A127, 137 (1990). 6o J. A. Krumhansl, J. Phys. ZV (Paris), 5, C2-3 (1995). 58

176

ANTONI PLANES AND L L U ~ SMAROSA

4

0

0.1 0.2 Reduced wave number

0.3

FIG.111.1. T A , phonon dispersion curve (upper) and elastic scattering intensity (lower) along the same direction at different temperatures for a Ni6,,,AI3,,, alloy. The elastic peak appears at the same position in the q-space as the dip in the phonon branch. On approaching the martensic transition, the frequency of the phonon decreases and the elastic intensity increases. Data from S. M. Shapiro, B. X.Yang, Y. Noda, L. E. Tanner and D. Schryvers, Phys. Rev. B. 44,9301 (1991).

fluctuations in position and is associated with the existence of dips at specific wave numbers in the phonon dispersion curves. The intriguing feature is the coexistence of a central peak and a partially softening phonon. This phenomenon has been observed in many second-order (or nearly) displacive phase transitions5* such as Nb,Sn and others. In most of these cases the energy of the soft phonon is so low that it is difficult to distinguish between the phonon and the central peak. In shape-memory alloys the central peak has been clearly detected in Ni-A1.61 It is observed along the [550] direction close to the reduced wave number 5 = 0.16 (see Figure 111.1). At the same position, the T A , phonon branch displays a Kohn-like anomaly (dip) which, from first-principle calculations,62has been attributed 6 1 S. M. Shapiro, B. X. Yang. Y. Noda, L. E. Tanner, and D. Schryvers, Phys. Rev. B. 44, 9301 (1991).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

177

to strong electron-phonon coupling and specific nesting properties of the multiply connected Fermi surface. As we will see, these anomalies announce the selected structure of the martensitic phase. In this case, because the phonon softening is incomplete at the transition, the central and phonon peaks are clearly separated. In shape-memory systems the central peak is considered to be an extrinsic property attributed to crystal imperfections. Within this context, the theory proposed by Halperin and Varma63 appears to be relevant. It is based on the existence of defects that induce a force field F(r) on the undistorted lattice. The response to this field gives rise to a displacement field in the neighborhood of the defects. In linear response theory, the displacement amplitude can be expressed as

where wq is the soft mode frequency. The diffuse scattering intensity arising from this field is related to the square of the displacement amplitude. That is,

It is worth mentioning that in the limit q + 0, the scattering is referred to as Huang diffuse scattering. Actually, the behavior predicted by the above equation between the intensity of the “central peak” and the frequency of the soft mode, has been experimentally corroborated in Ni-A1 from neutron scattering measurements.61 This is shown in Figure 111.2. In x-ray diffraction, weak superstructure spots characterizing the martensitic phase have been detected in the diffraction pattern at temperatures well above the martensitic transition temperature in some systems. An interesting example is the case of the TiNi and TiNi(Fe) shape-memory alloys. The striking feature is that these spots are shifted slightly from the commensurate reciprocal points of the bcc l a t t i ~ e . The ~ ~ incommensuration -~~ changes from one Brillouin zone to another and does not possess inversion symmetry around the Brillouin zone center. This is understood if satellites are at the commensurate positions of the rhombohedra1 martensitic lattice. Therefore, the whole diffraction pattern has been interpreted as being originated G. L. Zhao and B. N. Harmon, Phys. Rev. B. 45,2818 (1992). B. I. Halperin and C. M. Varma, Phys. Rev. B. 14,4030 (1976). 64 M. B. Salamon, M. E. Meichle, C. M. Wayman, C. M. Huang, and S. M. Shapiro, in Proc. Znt. Con$ on Modulated Structures, AIP Conference, Proc. N.5, AIP, New York, (1979). 6 5 M. B. Salamon, M. E. Meichle, and C. M. Wayman, Phys. Rev. B. 31, 7306 (1985). 66 S. M. Shapiro, Y. Noda, Y. Fuji, and Y. Yamada, Phys. Rev. B, 30,4314 (1984). 62

63

178

ANTONI PLANES AND L L U ~ SM A ~ O S A

FIG.111.2. Inverse of the intensity of the elastic (central) peak measured at the position of the dip of the T A , branch for a Ni6z.5A137,5 alloy, as a function of ( f i ~ ) w ~ ; is the frequency corresponding to the anomalous phonon. From S. M. Shapiro, B. X. Yang, Y. Noda, L. E. Tanner, and D. Schryvers, Phys. Rev. B. 44, 9301 (1991).

by a two-phase mixture. Yamada et al.67968determined the corresponding atomic structure in real space and deduced that above the transition, the system contains a random (spatial and orientational) distribution of martensitic regions in the mesoscale range, coherently embedded in the bcc matrix. This precursor effect is therefore associated with the excitation of embryonic fluctuations of the low-temperature phase, which could originate about lattice defects. Tweed patterns are a typical precursor effect observed by TEM. Tweed is not a specific feature of martensitic materials. Actually, it was first detected by Tanner69 in a Al-Cu alloy undergoing a phase separation process. Tweed patterns have also been reported in various ceramic^,^' including the high-T, YBaCu071 among other materials. In a number of martensitic materials such as Ni-Ti,72 Fe-Pd,73*74and C U - A Ualloys, ~ ~ this pattern develops well 6 7 Y. Yamada. Proc. I n t . Conf: on Murtensitic 7kmsforrnutions, Japan Institute of Metals, Nara, Japan (1986), p. 89. Y. Yamada, Muter. Trans. JIM 33, 191 (1992). 6 9 L. E . Tanner, Philos. Mug. 14, 111 (1966). 7 0 A. H. Heuer and M. Rhiile, Actu metall. 12, 2101 (1985). 7 1 W. W. Schmahl, A. Putnis, E. Salje, P. Freeman, A. Graeme-Barber, R. Jones, K. K. Singh, J. Blunt, P. P. Edwards, J. Loram, and K. Mirza, Philos. Mug. Lett. 60, 241 (1989). 7 2 I. M. Robertson and C . M. Wayman, Philos. Mug. A . 48, 421, 443, 629 (1983). 7 3 S. Muto, S. Takeda, R. Oshima, and F. Fujita, J. Phys.; Condens. Mutter 1, 9971 (1989). 74 S. Muto, R. Oshima, and F. Fujita, Actu metall. muter. 4, 685 (1990). 7 5 K. Yosada and Y. Kanawa, Trans. JIM 18, 46 (1972).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

179

above the actual transition and becomes increasingly more pronounced as it is approached. Tweed is observed as a cross-hatched modulation formed by narrow diffuse striations parallel to the { l l O } planes, which reflect some pseudo-periodic deformation. The pattern is characterized by two apparent length scales: the longitudinal extent of the long diagonal striations and the transverse width. These distances are on the scale of tens to one hundred lattice constants. High-resolution TEM confirms the presence of microdomains whose internal non-uniform displacements mimic the structure of the product martensitic phase in the alloy. Shapiro and collaborator^^^ have shown that in Ni-A1 the tweed pattern leads to an intense diffuse scattering along the [TlO] direction about the (1,1,0) Bragg peak, which increases as the martensitic transition is approached. The fact that the tweed pattern reproduces itself with thermal cycling indicates that it must be determined by some material inhomogeneity in the parent phase. From the work of Kartha and collaborators it is acknowledged that compositional disorder plays a fundamental role in the generation of t ~ e e d . ~ This ~ - ~is' corroborated by experiments and by molecular dynamic simulations.80 For instance, tweed has been found experimentally in zirconium doped with a small amount of oxygen, but not in pure Zr.81 Kartha et a1 noticed that (i) statistical compositional disorder is unavoidable in alloys. It is then reasonable to consider that the composition of the alloy will vary around some average value due to the disorder frozen in the system. (ii) The transition temperature in shape-memory alloys is highly sensitive to alloy composition. Consequently, it is proposed that the statistical quenched-in disorder will determine the spatial variation of a local transition temperature, and this will lead to the deformation modulation that is formed above the bulk transition temperature at which long-range martensitic order is actually established. An interesting feature to be mentioned concerns the fact that the distortion of any local region produces long-range elastic fields. This effect may modify the transition conditions of surrounding regions. Hence the explanation for the formation of tweed involves a problem of cooperative behavior in a disordered, locally interacting but frustrated system. The situation is similar to that corresponding to a disordered magnet with competing interactions, where a spin glass phase 76 S. M. Shapiro, J. Z . Lareze, Y. Noda, S. C. Moss, and L. E. Tanner, Phys. Rev. Lett. 57, 3199 (1986). 7 7 S. Kartha, T. Castan, J. A. Krumhansl, and J. P. Sethna, Phys. Rev. Lett. 67, 3630 (1993). 7 8 J. P. Sethna, S. Kartha, T. Castan, and J. A. Krumhansl, Phys. Scr. T42, 214 (1992). 7 9 S. Kartha, J. A. Krumhansl, J. P. Sethna, and L. K. Wickmann, Phys. Rev. B. 52,803 (1995). C. S. Becquart, P. C. Clapp, and J. A. Rikin, Phys. Rev. B. 48, 7 (1993). A. Heiming, W. Petry, J. Trampeneau, M. Alba, C. Herzing, H. R. Schober, and G. Vogl, Phys. Rev. B. 43, 10933 (1991).

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ANTONI PLANES AND L L U ~ SM A I ~ O S A

can occur. Kartha and collaborators have then suggested the interesting idea that the tweed structure can be understood as a spin glass p h a ~ e . ~ ~ , ~ ' From this point of view the appearance of tweed has been argued to take place through a phase transition at which some nonlinear elastic susceptibility should increase a n o m a l o ~ s l y This . ~ ~ behavior may, for instance, be observable through measurements of higher-order elastic constants. At present, some experimental evidence exists that there is such a transition to an elastic glassy phase from measurements of anisotropic thermal expansion carried out by T. Finlayson (see Ref. [82] for details). In order to give an interpretation of the pretransitional effects observed in different systems, it appears that in all cases the role of defects is of crucial importance. Nevertheless, defects (or more generally quenched-in disorder) is not a sufficient ingredient in order to justify the existence of precursors. The problem can be understood when it is noticed that all systems showing premonitory phenomena have a common characteristic: restoring forces in specific lattice directions are weak. This shows up through ultrasonic measurements and inelastic diffraction measurements from which it is learned that one phonon branch and the corresponding elastic constant have anomalously low value^.^^*^^ Therefore, from the point of view of vibrational properties, these systems are highly anisotropic. Results that illustrate these features will be presented in Section V. They show that these systems are incipiently unstable with respect to specific long- and shortwavelength acoustic modes. Interestingly, the lattice distortion resulting from the martensitic transition is precisely related to such anomalous acoustic modesE5 While the weak restoring force condition is the universal feature enabling the existence of precursors, the wide variety of pretransitional effects that are observed in different systems are strongly determined by the specific characteristics of the transformation path and also by active defects existing in the system. Here we mean by active defects those that have the appropriate characteristics to modify (locally) the response of the system in the instability directions. These defects are expected to have a pronounced impact on the phase transition by locally inducing an extra softening in the system. Therefore, we consider that precursors are related to intrinsic features of shape-memory alloys, but the way they show up is influenced by active defects. J. A. Krumhansl, Proc. International Symposium and Exhibition on Shape Memory Materials, Kanazawa, Japan (May 1999). Mar. Sci. Forum, 327-328, (2000). 83 W. Petry, J. Phys. ZV (Paris), 5, C2-15 (1995). 84 S. M. Shapiro, B. X. Yang, G. Shirane, Y. Noda, and L. E. Tanner, Phys. Rev.Left. 62, 1298 (1989). J. A. Krumhansl, and Y. Yamada, Mar. Sci. Engng. A127, 167 (1990). 82

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

181

IV. Lattice Dynamics

In this section we provide a brief survey of lattice dynamics in solids, with the aim of making the present paper easy reading. A more detailed description can be found, for instance, in References [86,87]. The atoms in a solid oscillate around their equilibrium positions. It is assumed that the amplitudes of the vibrations will, in general, be small compared to the interatomic distances. Under this approximation, any nuclear motion is the superposition of a number of monochromatic waves, the normal modes. For crystalline solids, normal modes are propagating vibrational waves with wave vectors q determined by the crystal geometry. For each value q, the frequencies are determined by diagonalizing the force-constant matrix, which is closely related to the interatomic potential. The relation w = wj(q) is called the dispersion relation. The index j, which differentiates the various frequencies corresponding to the same propagation vector, characterizes the various branches of the dispersion relation. A certain branch of the dispersion relation is called longitudinal and transverse if the polarization vectors 4 (which determine the pattern of motion of the nuclei) are parallel and perpendicular to q, respectively. Thus, the vibrational modes are written as

In the long wavelength limit (q -,0), the solid behaves as a continuum, the low-frequency vibrations are simply sound waves, with a linear dispersion relation w = ujq, where v j is the appropriate velocity of sound. Classical elasticity theory will then be suitable to describe the vibrations of the solid in this limit. 4. ELASTIC BEHAVIOR

Elastic solids obey the generalized Hooke's law, which states that each stress component (oij)is a linear function of the strain component (cij)* [88].

C. Stassis, in Methods of Experimental Physics, Academic Press, New York (1986), p. 369. H. R. Schober and W. Petry, in Materials Science and Technology, Vol. 1, ed. R. W. Cahn, P. Hassen, and E. J. Kramer, VCH, Weinheim (1993). * Einstein notation for repeated indexes will be used from now on. J. F. Nye, Physical Properties of Crystals, Oxford University Press, Oxford (1987).

86

182

ANTONI PLANES AND L L U ~ SMAROSA

TABLEIV.l. NUMBER OF INDEPENDENT SOEC FOR Crystal system Tric1ini c Monoclinic Orthorhombic Tetragonal and trigonal Hexagonal Cubic Isotropic

THE DIFFERENTSYMMETRY GROUPS

Number of independent SOEC 21 13 9 6 or 7 (depending on the space group)* 5 3 2

*See page 348 of D. C. Wallace, Solid State Physics 25, 302 (1970).

The Cijk,are the effective elastic stiffnesses, and are the components of a fourth-rank tensor. The term second-order elastic constant (hereafter referred as SOEC) is also widely used. The elastic stiffness tensor has 81 components, but due to symmetry, the number of independent terms is reduced to 21. Symmetry also enables the use of the Voigt notation with only two indices for each elastic constant.88 Finally, crystal symmetry of each specific solid further reduces the number of independent SOEC. In Table IV.l we have listed the number of independent SOEC for the different crystal systems. SOEC determine the velocity of propagation of ultrasonic waves. For an anisotropic elastic body, Newton's law is expressed as

where um are the components of the displacement vector and p is the mass density. For a plane harmonic travelling wave of the form ui = Aiexp[i(qjxj - w t ) ] ,

(6)

with phase velocity u = 4 4 , the Christoffel equation is obtained,

which has non-zero solutions only if the determinant of the coefficients is zero. This constraint leads to a cubic equation in o2 (or alternatively in terms of u z ) for each propagation vector q. A plot of the solutions (velocities) obtained for all propagation directions yields the velocity surfaces that characterize the elastic behavior of the solid.

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

183

In general, for a given direction of propagation, the displacement vectors u are neither parallel (purely longitudinal waves) nor perpendicular (purely transverse waves) to the vector q. Nevertheless, purely longitudinal and purely transverse waves can propagate along some specific crystallographic directions. For instance, in the case of a cubic crystal, a purely longitudinal and two purely transverse waves can propagate along the [ l l O ] direction. The velocities of these plane waves, in terms of the SOEC are for the longitudinal wave

uL = u‘ =

u44

=

&

for the transverse wave with [lTO] polarization for the transverse wave with [Wl] polarization

(8)

where

c, = C l l + Cl,2 + 2c44

C ‘ = Cll

and

- Cl,

2

.

(9)

From these expressions it is clear that, from an experimental point of view, the measurement of the velocities of propagation of these three waves will suffice for the determination of the three independent SOEC of a cubic crystal. So far, the elastic constants have been defined through a phenomenological expression (Hooke’s law). It is also possible, and desirable, to define them in terms of appropriate derivatives of thermodynamic functions. These functions can be expanded in powers of the strain. By taking, for instance, the internal energy, the adiabatic second-order elastic constants are defined asa9

.$,=(””) . aEijaEkl

S

Higher-order elastic constants are defined in a similar way. For thirdorder elastic constants (TOEC)

89

D. C. Wallace, Solid State Phys. 25, 301 (1970).

ANTONI PLANES AND L L U ~ SMAROSA

184

Taking into account symmetry considerations, for cubic crystals six independent TOEC-Clll, C14,, Cl12,C,,,, C123, and C,,,-are obtained. Similar definitions hold for the isothermal elastic constants, with the free energy F instead of U. The difference between adiabatic, isothermal, and effective elastic constants is small and in most cases (as will be done in the present review) no distinction is made among them. a. Symmetry adapted strains The elastic contribution to the free energy (per unit volume) of a given crystal can be expanded as a Taylor series with respect to the strain tensor components. This expression reads

(12) It is convenient to express the elastic free energy in terms of the irreducible strain tensor components. For the different crystallographic point groups, these components have been determined by Liakos and Saunders.” For a cubic symmetry they are qo =

+ E~~ + E~~

‘1’ = (

2 - ~E~~ ~ c l~I ) / d which corresponds to a tetragonal distortion.

q2 = c l l

- E~~

‘13

I

which corresponds to a volume change.

which corresponds to a (110) [IT01 shear.

(13)

= &23

1 ‘, = E~~

‘15 = E l 2

which correspond to any shear on (100) planes.

The irreducible strain tensor components form a basis that diagonalizes the SOEC matrix. The eigenvalues are the elastic response of the solid for these symmetry-adapted distortions. For cubic crystals these responses are: the bulk modulus B = $ ( C l l 2C12) is the response to q o ; C‘ = g C l l - C12)is the response to q l and q2; and C,,, to q3, 1 ‘, and q5. In order for a crystal to be mechanically stable, these elastic moduli must be positive.’ By using the irreducible strain tensor components it is possible to obtain a symmetry-adapted expression for the free energy, which is a suitable starting point for the study of purely elastic phase transitions (proper

+

J. K. Liakos, and G. A. Saunders, Philos. Mug. A 46,217 (1982).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

185

ferroelastic system) in the spirit of the Landau In this context the strain tensor components play the role of the order parameter. As an example, we consider here the case of the purely elastic cubic to tetragonal In this particular case the distortive order parameter involves two components of the irreducible strain tensor for cubic symmetry, namely q1 and qz. Taking only leading terms in the nonlinear contribution to the free energy (actually, there is coupling to other strain components), the adequate Landau expansion up to fourth order takes the form

where C , and C , are third-order and fourth-order elastic constant combinations given by

The relevance of higher-order elastic constants will be discussed in a subsequent section of this paper. An interesting feature of the above free energy concerns the presence of the cubic invariant, which is indicative of a possible first-order character of the transition for the change of symmetry considered. In fact, for a second-order transition to occur, the second- (C’) and third-order ( C , ) coefficients in the expansion must vanish simultaneously, which is very i m p r ~ b a b l e Therefore, .~~ for a first-order transition, the elastic constant C‘ is the only parameter assumed to show a relevant temperature dependence. Usually this dependence is taken to be linear: C’ = a(T - T,). Minimization of 9 with respect to ql and qZ leads to three sets of solutions that correspond to the three tetragonal variants with tetragonal axes alon [1003, [OlO], and [001]96*97characterized by deformations (ql = (&3)&, qz = 4, (vl = ( 3 / 3 ) & , qZ = -4, and (ql = (2$/3)~, R. A. Cowley, Phys. Rev. B 13, 4877 (1976). R. Kragler, Physica B. 93, 314 (1978). J. D. Axe and Y. Yamada, Phys. Rev. B. 48, 159 (1981). 94 M. P. Brassington and G . A. Saunders, Phys. Rev. Lett. 48, 159 (1982). 9 5 P. W. Anderson and E. I. Blout, Phys. Rev. Left. 14, 217 (1965). 96 G. R. Barsch and J. A. Krumhansl, Phys. Rev. Lett. 53, 1069 (1984). ” G. R. Barsch and J. A. Krumhansl, Proceedings of the Infernational Conference on Marfensitic Transformations, ed. J. Perkins, Monterey, CA (1993), p. 53.

91 92

’’

186

ANTONI PLANES AND L L U ~ SMAROSA

q2 = 0). E determines the tetragonal distortion and is given by: E = 1 - c/a, where a and c are the lattice parameters in the tetragonal phase. The equilibrium transition temperature is obtained by equating the free energies of the high- and low-temperature phases, and is given by

2c3 To=T,+g,c14. 5. VIBRATIONAL ANHAFMONICITY: GR~JNEISEN PARAMETERS

In real crystals the phonon frequencies depend on the strain field of the solid. Such a dependence can be expressed in terms of the Gruneisen parameters &Y(P, q), which are defined as*

where q is the wavevector and p stands for the phonon polarization. The Gruneisen parameters quantify the vibrational anharmonicity of the crystal lattice, and macroscopic manifestations of such anharmonicity (as for example thermal expansion) can be expressed in terms of these parameters. (For a detailed treatment, refer to Ref. [2].) In the long-wavelength limit, the Gruneisen parameters can be experimentally determined from the changes in the ultrasonic velocities (up(N)) for strained solids, as

where p o is the mass density in the unstrained state and wp

(N) = CmumN m Nn u u uu

with Cmunu the adiabatic SOEC for the mode propagating along N with polarization along U.

* When dealing with uniform changes of the unit cell (produced for example by the application of hydrostatic pressure), the Griineisen parameters take the following expression, which is the most commonly used:

The relationship between this expression and equation (17) is given in Ref. [98].

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

187

There is a close relationship between the Griineisen parameters and highorder elastic constants because all quantify the vibrational anharmonicity in the elastic limit. In particular, abyp(N)can be computed from the complete set of SOEC and TOEC using the following e x p r e ~ s i o n : ~ ~

The expressions that enable computation of the TOEC and abyp(N)from experimental data of ultrasonic velocity changes produced by uniaxial stresses along specific crystallographic directions (M) and by hydrostatic pressure were derived by Brugger and Fritz98 for the different crystal symmetries. For a cubic crystal

with

and Sijklthe elastic compliance tensor components (the inverse of the SOEC tensor).

V. Experimental Results

In this section we will compile some of the most relevant experimental results dealing with the vibrational properties of shape-memory alloys. The lattice dynamics for wavevectors spanning along the whole Brillouin zone can be investigated by means of inelastic neutron experiments. This technique, however, provides poor results at the origin of the zone. In this region the solid behaves elastically, and ultrasonic methods are the most suitable experimental technique for obtaining the elastic constants. 98 99

K. Brugger and T. C. Fritz, Phys. Rev. 157, 524 (1967). K. Brugger, Phys. Rev. 137, 1826 (1965).

188

ANTONI PLANES AND L L U ~ SM A ~ ~ O S A

6. SECOND-ORDER ELASTICCONSTANTS a. Room-temperature values The room temperature values for B and C,, do not particularly depend on the alloy composition in the range where shape memory alloys display a martensitic transition* Moreover, these elastic constants are quite similar for each alloy family.'OO In contrast, the room temperature values reported for C' are more sensitive to composition, as illustrated in Figure V.l. In this figure, the room temperature values for B, C,,, and C' are plotted as a function of Be content in the Cu-Al-Be alloy system, and as a function of Ni content in the Ni-A1 system. An important result to be stressed is that the values of C' are anomalously low when compared to the other elastic moduli ( B and C,,). Such a low value is an indication that the bcc structure

*

For Au-Cu-Zn alloys these moduli may depend upon composition. A. Planes, L1. Mafiosa, and E. Vives, Phys. Rev. B 53, 3039 (1996).

loo

-

- 100 t

u"

::

90 85

0

115 -

-

5L 4

'2.5

3.0

3.5 4.0 at% Be

4.5

5.0

62.0

62.5

63.0

at% Ni

FIG. V.l. Elastic moduli for a family of composition related (a): Cu-Al-Be and (b) Ni-A1 alloys. Notice the low value of C', which shows stronger relative change than the other moduli. For Cu-Al-Be, the figure has been elaborated using data from A. Planes, L1. Maiiosa, J. Ortin, and D. Rios-Jara, Phys. Rev. B. 45,7633 (1992), and for Ni-AI, using data from T. Davenport, L. Zhou, J. Trivisonno, Phys. Rev. B. 59, 3421 (1999).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

189

shows weak restoring forces for shears on the { 1lo} planes along the (110) directions. Based on a thorough compilation of data for room temperature values of C' for different Hume-Rothery alloys, Verlinden and Delaey'o'*'02 proposed the following empirical formula to calculate C': C' =

25.5(GF - 0.27) exp(l1.5 - e/a1'/2)'

G F is a geometrical factor, accounting for the ion repulsion energy. The correlation between the experimental data and the values predicted by eq. (21) is shown in Figure V.2. It is apparent that the formula provides satisfactory results for alloys with B2 atomic order, but for alloys with DO, or L2, order, the calculated values are too large. Indeed, the origin of such a discrepancy lies in the fact that these structures result from next nearestneighbour ordering, and C' is mainly dependent on the interaction between these neighbors (actually, in the central force approximation, it is completely determined by this interaction). Verlinden and Delaey have suggested that the type of atomic order should be incorporated in this empirical relationship, but at present there are not enough experimental data to achieve such a task. In the bcc structure, the long-wavelength acoustic modes associated with C' are not the only modes with low values in their elastic moduli. To show this, it is instructive to compute the sound velocity surfaces obtained by solving the Christoffel equations (eq. 7) for an anisotropic elastic medium. These surfaces display two minima; one for the mode (lOl)[lOi] (associated with C'), and a second for a direction close to the pure mode (121)[1T1] (see Figure V.3(a) and (b)). This mode has been termed "special mode" by Nagasawa et al.lo3 and in the Burgers' picture" it corresponds to the second homogeneous shear necessary to bring the bcc structure to the martensitic phase. The corresponding elastic constant for this mode can be expressed as

1 c, = C' + -IC(C4, 2 with

K = 2(C,,

-

C'),

(22)

+ c,2)/(3c1, + 5c12 + 2c44).*

B. Verlinden and L. Delaey, Acra metall. 36, 1771 (1988). B. Verlinden and L. Delaey, Proc. Znt. Conf on Martensitic Transformation, Ed. The Japan Institute of Metals, Nara, Japan 1986, p. 768. * K can be expressed as sin2Os,where Os is the angle between the propagation direction of the special mode and the [Ool] axis. A. Nagasawa, N. N Nakanishi, and K. Enami, Philos. Mag. A 43, 1345 (1981). lo' lo'

190

ANTONI PLANES AND L L U ~ SM A ~ ~ O S A

12

-ae 0 8 i,

-

4

FIG. V.2. Correlation between the experimentally measured value of C' at room temperature and the value calculated using equation 21 for (0)Cu-Zn; (W) Ag-Zn, ( 0 )Au-Zn, (V)Ag-Cd; (A)Au-Cd, (0) Cu-Zn-Al, (4) AgMg, b) CuSn, (*)Cu-Ni-Zn, (*) Cu-Au-Zn, (e)Au-AgCd. The dashed line indicates the predicted result. The points showing larger discrepancy from this dashed line (* and 0) correspond to alloys displaying next nearest-neighbor ordering ( L2, or DO,). Results reprinted from B. Verlinden and L. Delaey, Acta metall. 36, 1771 (1988) Copyright 1988, with permission from Elsevier Science.

It has been argued1O4that the stability of the bcc structure is controlled by the ratio of the elastic constants that give the restoring forces to the two independent shears: A = C,,/C'. This ratio determines the elastic anisotropy. The low value of C' results in high values of the elastic anisotropy for shape-memory alloys, with the exception of Ti-Ni alloys close to the stoichiometric composition, which will be discussed later in this review. Furthermore, the dependence of the room temperature C' on composition leads to a composition-dependent elastic anisotropy. As mentioned preceding, the transition temperature in shape-memory alloys is extremely sensitive to composition. As a result of such a strong dependence, T, is a good parameter to characterize each specific sample. From this point of view, it lo4

C. Zener, Phys. Rev. 71, 846 (1947).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

191

FIG.V.3. Velocity surface sections of the (a) (lOT), and (b) (001), planes of the B-phase; (c)

(NO), and (d) (001), planes of the martensite corresponding to the (lOT), plane; (e) (lor), plane corresponding to the (001), plane. Solid lines correspond to T = T, and dashed lines to T = 200K (martensitic phase) and T = 400 K @-phase). The velocity dip derived from the C, mode in the 8-phase is indicated by the arrow at C1. The arrow C2 corresponds to the [111],[1~1], mode and appears due to the C’ mode in the B-phase. The outer curve in each case corresponds to the longitudinal mode, which has the highest velocity. From A. GonzalezComas, L1. Maiiosa, A. Planes, F. C. Lovey, J. L. Pelegrina, and G. Guenin, Phys. Reo. B56, 5200 (1997).

is interesting to plot room temperature data of A for different Cu-based alloys as a function of the corresponding transition temperature. It is seen that all data points collapse on a single curve,’’’ as illustrated in Figure V.4a. Such a fact does not occur for C’ , for which data for different alloy families lie on different curves. The elastic anisotropy controls the ratio between the two relevant shears in a martensitic transformation. From the expression of C, (eq. 22) it is easily obtained that 1

c s =1 + - K ( A

C’

2

-

1).

Analysis of different alloy families shows that K is practically independent of alloy composition. Therefore, according to eq. (23) the “scaling” of A with

192

ANTONI PLANES AND L L U ~ SM A ~ O S A

0

100

200

300

FIG. V.4. Elastic anisotropy as a function of the transition temperature:''7 (a) for different families of Cu-based alloys (circles: Cu-Al-Be; triangles: Cu-Al-Ni; squares: Cu-Zn-Al), (b) for Ni-A]. Open symbols correspond to values at room temperature and filled symbols, at the transition temperature. Data for Cu-based systems are from A. Planes, L1. Mafiosa, and E. Vives, Phys. Rev. B. 53, 3039 (1996), and for Ni-A1,the anisotropy is calculated from the data given in T. Davenport, L. Zhou, and J. Trivisonno, Phys. Rev. B. 59, 3421 (1999).

the transition temperature found for a number of shape alloys reflects a common behavior for the ratio of the shear constants associated with the shears relevant for the martensitic transition of bcc shape-memory alloys. This point will be discussed in more detail when dealing with the temperature dependence of the elastic constants. b. Temperature dependence With regard to the temperature dependence of the elastic constants, B and C,, conform to the predictions of standard anharmonic theories for solids; they increase linearly as temperature is reduced. The behavior for C' is anomalous; this shear modulus decreases on cooling. Such a behavior is common for shape-memory alloys, and reflects the decrease in the mechanical stability against the {llO}(lTO) shears as the alloy approaches the transition temperature. The change of C' with temperature has been found to be linear for most alloys.

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

193

In a number of experiment^,'^^*'^^ a deviation from the linear decrease of C' has been reported close to the transition temperature. Although this could be related to some kind of premonitory effect, attention must be paid along these lines because ultrasonic measurements are usually preformed on large single crystals. Small composition inhomogeneities are likely to be present in these samples. Furthermore, they can have retained internal stresses during prior heat treatments. Both effects can give rise to the transformation of a small fraction of the sample at temperatures above the nominal TM temperature, as has been shown for quenched Cu-Zn-A1 crystal^.'^' Actually, a partial transformation of the sample without a disappearance of the ultrasonic echoes (caused by the surface relief associated with the self-accommodating martensitic structure) can result in an apparent change of the slope of the C' vs. temperature curve. This effect is illustrated for Cu-Zn-A1 in the lower inset of Figure V.5, extracted from Ref. [lOS]. On cooling, data points start to deviate from the linear behavior (apparent extra softening) at 280K, but on heating, the linear behavior is recovered at -288K. The existence of such a thermal hysteresis is clear evidence of the beginning of the martensitic transformation at around 280 K. For Cu-Al-Be'" and Cu-Al-Nil l o crystals, which have been subjected to well-controlled heat treatments, ultrasonic and calorimetric measurements conducted on exactly the same samples have shown that the decrease of C' is linear with temperature down to the start of the martensitic transformation (This is shown in Fig. V.5.) Nevertheless, Ni-A1 and Au-based alloys do seem to exhibit a precursor softening a few degrees above the nominal transition temperature. Such behavior is clearly illustrated in the measurements by Nagasawa et al.lo3These authors measured Ni-A1 and Au-based single crystals subjected to different heat treatments. They showed that even for the slowly cooled samples there was an enhancement of the softening of C' around 20K above T,. Furthermore, at this point the temperature dependence of C,, deviated from the linear behavior. It must be mentioned that in all these alloys, significant pretransitional effects (central peak, tweed, etc.) have also been reported from neutron diffraction The temperature at which such a deviation occurs coincides (within experimental errors) with the temperature at which tweed pattern is observed.

-

M. Suezawa and K. Sumino, Scripta metall. 10, 789 (1976). G. Hausch and E. Torok, J. Phys (Paris) 42, C5-1031 (1981). "'5. L. Macqueron, LI. Maiiosa, and G. Guenin, Phys. Stat. Sol. ( a ) 117, 113 (1990). G. Guenin, PhD thesis, INSA, Lyon (1979). '09 A. Planes, LI. Maiiosa, D. Rios-Jara, and J. Ortin, Phys. Rev. B 45, 7633 (1992). ' l o LI. Maiiosa, M. Jurado, A. Planes, J. Zarestky, T. Lograsso, and C. Stassis, Phys. Rev. B. lo'

lo6

49, 9969 (1994).

M. Mori, Y. Yamada, and G. Shirane, Solid State Commun. 17, 127 (1975).

ANTONI PLANES AND L L U ~ SMAGOSA

194

0

-E Y

0

-LO

\

-0.01

7

+ TI

-80

t> -0.02

\

2 c!

-120 - 0.03

-160

2 20

-0.04 2 LO

260 280 Temperature ( K )

300

FIG. V.5. Relative variation as a function of temperature of the elastic constant C' in a C U , , , , , A I , ~ ~ , ~ N ~ , ,alloy, , , , and detail of the thermal effect recorded at the beginning of the martensitic transition (full thermogram is depicted in the upper inset). The anomalous (non-linear) behavior of C' vs T, marked with an arrow in the figure, coincides with the first thermal effect detected calorimetrically. Results from LI. Maiiosa, M. Jurado, A. Planes, J. Zarestky, T. Lograsso, and C. Stassis, Phys. Rev. B. 49,9969 (1994). The lower inset shows the relative change of C' with temperature for a Cu-Zn-A1 alloy. Data points start to deviate from linearity at different temperatures on cooling and heating. This hysteresis effect must be attributed to the transformation of a small fraction of the sample. From G. Grenin, PhD thesis, INSA Lyon (France), 1979.

We now turn our attention to the other soft long-wavelength acoustic mode. For Au-Ag-Cd and Ni-A1 alloys, Nagasawa et al.lo3 reported softening of C, at temperatures close to the nominal transition temperature, in which this softening is more pronounced for quenched samples. This is shown in Figure V.6 (upper and lower curves). The behavior found in Cu-based alloys is different; C, increases as temperature is lowered (middle curve in Figure V.6). In Cu-based systems the analysis of the (107) and (001) cross sections of the velocity surfaces at different temperatures demonstrates that the only modes exhibiting temperature softening are the slow transverse modes around the C l O l ] direction. This is shown in Figure V.3(a) and (b) for a Cu-Zn-A1 alloy. It has been shown"' that the measured increase of C , in Cu-based alloys arises from an increase in C,,, which is not completely compensated for by a decrease in C'. It should be taken into account that special mode softening can only occur after the C' mode becomes consider-

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

35‘7 38.5

1

195

TM

1

d

- 26.1

0” 2 5.9

J

7’50 7.42

150

2 50

350

Temperature ( K ) FIG. V.6. Temperature dependence of C , close to the martensitic transition for three different alloy systems: (upper curve)Ni-Al; (lower curve) Ag-Cu-Zn (data from A. Nagasawa, N. Nakanishi, and K. Enami, Philos. Mag. A 43, 1345 (1981); and (middle curve) Cuo,,,7AIo,,,7Beo,o,,, from A. Planes, L1. Maiiosa, and E. Vives, Phys. Reu. B53,3039 (1996)

ably soft. That is, according to Nagasawa et al.lo3 the softening of C’ enhances the two-dimensional lattice behavior along [l lo] and in these “two-dimensional lattices,” distortions toward a close-packed layer (as the mode associated to C, is) will be favored. Therefore, with no extra softening of C’ at any temperature above T’, in Cu-based alloys softening of C, is not observed and this elastic constant increases linearly until the transition temperature is reached. There are several crystallographic schemes to describe the martensitic transformation from a geometrical point of vie^.''.^^.^^ The temperature softening of both C’ and C, seems to provide experimental support to the Burgers’ mechanism,12 although in a reverse shear sequence to the one originally proposed by Burgers; the parent phase undergoes the (1 lo)[ 1101 shear first and then the (li2)[1ii] shear. This could be the correct picture for Ni-A1 and Au-based alloys; nevertheless, the absence of softening in C, in Cu-based alloys could be indicative that in these alloys the transformation mechanism is different from that proposed by Burgers. For all shape-memory alloys elastic constant softening is not complete: The elastic constant C’ has a finite (different from zero) value at the

196

ANTONI PLANES AND LLUIS MAROSA

transition temperature. The fact that there is no complete softening at the transition point makes the classical soft-mode theories inadequate to describe martensitic transformations. Alternative Landau-type theories have been proposed,' ' - ' which include coupling between the homogeneous and inhomogeneous deformations. The suitability of these theories will be discussed in a separate section in this review. In Cu-based alloys, the relative decrease of C' is larger than the increase of C,, and the elastic anisotropy increases as temperature is reduced. A remarkable experimental finding is that, within experimental errors, elastic anisotropy always reaches a fixed value at the transition temperature. This value ( A N 13.5) has been shown to be independent of the concentration of the alloy, and of the alloy system.'" To illustrate this result, Figure V.4(a) plots the elastic anisotropy for different families of Cu-based alloys, as a function of their transition temperatures. Very recently, Davenport et al.' have measured the elastic constants of Ni-A1 crystals with different compositions. We have used their room temperature data for those compositions that exhibit a martensitic transformation and, assuming that the temperature dependence will be linear with a similar slope for the different alloys, we have computed the extrapolated values of the elastic anisotropy at room temperature and at the transition temperature. Results are shown in Figure V.4(b). It is apparent from the figure that Ni-A1 exhibits the same behavior as Cu-based alloys; the elastic anisotropy at the transition temperature is independent of the concentration of the alloy, in this case at the transition point A N 34. Although up to now there has been no microscopic theory that predicts a constant value of A, from an experimental point of view it seems to be clear that the martensitic transition occurs at a fixed value of the ratio between the two elastic constants associated with the relevant shears. This can be interpreted in the sense that long-wavelength vibrations in the (110) planes are localized in the (110) directions for the transition to occur. This localization effect could arise from a nonlinear interaction of the strain modes associated with C' and C,, respectively. As previously mentioned, the elastic behavior of Ti-Ni alloys near equiatomic composition turns out to be completely different. The elastic anisotropy of these alloys is quite low ( ~ 2 owing ) to the low value of the C,, elastic constant ( Z35 GPa at room temperature). On cooling, C,, softens in such a way that the elastic anisotropy does not increase, but remains constant within experimental errors.' l6 An explanation for such a P. A. Lindgird and 0. G. Mouritsen, Phys. Rev. Left.57, 2458 (1986). R. J. Gooding and J. A. Krumhansl, Phys. Rev. B. 38, 1695 (1988). R. J. Gooding and J. A. Krumhansl, Phys. Rev. B. 39, 1535 (1989). T. Davenport, L. Zhou, and J. Tnvisonno, Phys. Rev. B. 59, 3421 (1999).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

197

different behavior has very recently been put forward by Ren and Otsuka."' According to these authors, the low value of C,, is related to the crystallographic structure of the martensitic phase in this alloy system. In Ti-Ni, the martensite is monoclinic B19', which can be viewed as a basal plane orthorhombic B19 structure being distorted by a large monoclinic shear. Such a monoclinic shear is on the (001) plane and [IT01 direction of the parent (cubic) phase. These authors showed that the existence of this distortion was made possible by the softening of the C,, elastic constant (which is associated with the {001}( 1TO) mode). They presented a phenomenological model based on a Landau free energy expressed in terms of the homogeneous shear associated with C' (this was the primary order parameter), the shear associated with C,,, and the amplitude of a phonon on the T A , branch. In that model when anharmonic coupling between the two homogeneous shears (this is the case of high anisotropy) is negligible, the resultant martensite is solely determined by basal plane shear and shuffle, while strong anharmonic coupling between the two homogeneous shears favors the formation of martensite with a monoclinic distortion. Finally, it is worth comparing the behavior of the elastic constants of the bcc phase with that of the elastic constants in the martensitic phase. There are very few experimental data available in the literature; the room temperature SOEC have been measured for orthorhombic Cu-Al-Nil" and Cu-ZnAl." The measurement of the temperature dependence of a restricted number of acoustic modes in Cu-Zn-A112' has recently been completed by the measurement of the temperature dependence of the complete set of SOEC in this alloy system.12' The structure of the martensite in Cu-based alloys is monoclinic (see Table 11.1). However, instead of using the monoclinic unit cell (which is the primitive cell), the martensitic structure is usually described using a larger unit cell containing 18 close-packed layers along the c direction (see Figure 11.4). The degree of monoclinicity of such a large unit cell (28R structure) is small. By measuring the 13 independent SOEC of an 28R Cu-Zn-Al, Rodriguez et al.l19 have shown that the values of the SOEC which appear due to the monoclinicity (C,,, C25, C , , and C,,, 11' T. M. Brill, S . Mittelbach, W. Assmus, M. Miilner, and B. Liithi, J. Phys: Condens. Matter. 3, 9621 (1991). 11' X. Ren and K. Otsuka, Scripta mater. 38, 1669 (1998). M. Yasunaga, Y. Funatsu, S. Kojima, and K. Otsuka, Scripta metall. 17, 1091 (1983). 11' P. L. Rodriguez, F. C. Lovey, G . Guenin, J. L. Pelegrina, M. Sade, and M. Morin, Acta metall. mater. 41, 3307 (1993). lZo G. Gutnin, D. Rios-Jara, Y. Murakami, L. Delaey, and P. F. Gobin, Scripta metall. 13,289 (1979). 12' A. Gonzdez-Comas, L1. Maiiosa, A. Planes, F. C. Lovey, J. L. Pelegrina, and G . Gutnin, Phys. Rev. B. 56, 5200 (1997).

ANTONI PLANES AND L L U ~ SMAROSA

198

2 50

c33

--_____

-

200 -

.----c,

c 11 c

---__I

23

2

1 5 0 - c*z

(3

-

0

w

5:

100-

---___

c12

.---

--____. -_____------.

C L4

50 -ebb

c13

----_-_

- css 0

c44

I

I

1

.---

I

I

C'

FIG. V.7. Comparison of the set of SOEC in the fi and 18R martensitic phase, given as a function of the temperature difference T - TM.Dashed lines correspond to extrapolated values in the two-phase coexistence region. From A. Gonzalez-Comas, LI. Mafiosa, A. Planes, F. C. Lovey, J. L. Pelegrina, and G. GuCnin, Phys. Rev. B56,5200 (1997).

see Ref. [89]) are much smaller that those of the remaining SOEC. This finding shows that an appropriate description of the elastic properties of the martensite in Cu-based alloys can be given by considering an orthorhombic symmetry with only nine independent SOEC. Under this approximation Gonzalez-Comas et al.' have measured the temperature dependence of all SOEC in Cu-Zn-Al. Results are compared in Figure V.7 to those for a fl Cu-Zn-A1 crystal. From the figure it is apparent that, in each phase, there is an elastic constant that exhibits significantly lower values than the remaining constants: C' and C , , for the fl and martensitic phases respectively. In addition, the relative temperature variation of these constants is larger than for the other elastic constants, and both soften on approaching the transition. It is important to note that C , , quantifies the elastic response for a shear in the (OOl), plane in the [loo], direction, which originate respectively from the (110), plane and [lTO], direction of the bcc structure. In this sense, the role played by C , , in the martensitic phase is equivalent

''

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

199

to that played by C’ in the bcc phase. The discontinuity in this particular elastic response is quantified by the difference between C , , and C‘ at the transition point. Evaluation of these constants at the transition temperature renders C’ N 6GPa (for C U . ~ ~ ~ Z and ~ . C,,~ 2~: 26GPa ~ A ~(for~ ~ ~ ~ ) Cu,696Zn,1z8A1.176),which results in a discontinuity of 20 GPa. For the martensitic phase it is also interesting to investigate the velocity surfaces of the elastic waves. A section on the (loo),, (OOl),, and (lOT), planes is shown in Figures V.~(C), (d), and (e). The surface velocities in the martensitic phase exhibit qualitative features similar to those in the bcc phase. A striking feature arising from the surface in Figure V.3(e) is the dip for the (111),[1?1], mode (marked with the C2 arrow in the figure). This mode corresponds to a complicated combination of SOEC yielding a value lower than C,,, and it derives from the C’ (llO),[lTO], mode in the p phase. (Notice that the symmetry change at the phase transition breaks the degeneracy of the (1 lo), { lTO}, shears.) The relative temperature variation (softening) of this mode is larger than that of all other modes (including Cs5) in the martensitic phase. For this particular mode, the discontinuity in the elastic response is only -4GPa.

-

7. Phonon Dispersion

In this section we will review a number of relevant experimental results concerning the phonon dispersion curves in shape-memory alloys. In all cases, the data have been obtained from inelastic neutron scattering experiments. A review of neutron scattering studies in bcc-based metals and alloys has been published by A. Nagasawa and Y.Morii.122The phonon spectrum of a B2-ordered Cu-Zn alloy, which did not undergo a martensitic transition, was reported long ago by G. Gilat and G . D ~ l l i n g . ” The ~ phonon dispersion curves of different shape-memory alloys exhibit similar characteri s t i c ~ . ~As~an, example, ~ ~ ~ *in~Figure ~ ~ V.8 we show the phonon dispersion curves for a DO3-ordered Cu-Al-Be alloy, along the main symmetry directions. Solid curves correspond to a Born von Karma, fit up to five nearestneighbor force constants. Two striking features show up from the phonon spectrum of shapememory alloys: (i) All the phonons propagating along the [1101 direction with [lTO] polarization (TA, phonon branch) have anomalously low 122

lZ3 lZ4 lZ5

A. Nagasawa and Y. Morii, Mater. Trans. JIM 34, 855 (1993). G. Guilat and G. Dolling, Phys. Rev. 138, A1053 (1965). R. A. Robinson, G. L. Squires and R. Pynn, J. Phys Fc M e t Phys. 14, 1061 (1984). L1. Mafiosa, J. Zarestky, M. Bullock, and C. Stassis, Phys. Rev. B. 59, 9239 (1999).

200

ANTONI PLANES AND L L U ~ SM A ~ ~ O S A 1001) 1111)

(112.112.1)

(1/2,1/2,0)

1000)

(001) (111)

3d000) 25

-% 20 -E

h

!?

15

0)

15

10

5

0 Reduced wave number FIG. V.8. Phonon dispersion curves at room temperature along high-symmetry directions for crystal in the extended zone scheme. The solid lines correspond a D0,Cuo,7,73AIo,2272Beo,03ss to a fit of the data to a fifth nearest-neighbor force constant bcc model. From L1. Mafiosa, J. Zarestky, M. Bullock, and C. Stassis, Phys. Rev. B59,9239 (1999).

energies, and (ii) there is a pronounced dip in the longitudinal [555] branch at 5 = 3. Actually, these two characteristics are common to most bcc solids.This behavior is illustrated, for instance, in the inelastic neutron scattering studies undertaken by Petry and co-workers in group I11 and IV bcc metals. -' The low-lying T A , branch has attracted the attention of many scientists and has been the subject of numerous studies in a number of shape-memory alloys.81,'26-'33 The low value of the whole T A , branch plus the low value

' '' '

lZ6 W. Petry, A. Heiming, J. Trampenau, M. Alba, H. R. Schober, and G. Vogl, Phys. Rev. B. 43, 10933 (1991). 12' J. Trampenau, A. Heiming, W. Petry, M. Alba, C. Herzig, W. Mikeley, and H. R. Schober, Phys. Rev. B. 43, 10963 (1991). 12* F. Guthoff, W. Petry, C. Stassis, A. Heiming, B. Hennion, C. Herzing, and J. Trampeneau, Phys. Rev. B. 47, 2563 (1993). 129 S. Hoshino, G. Shirane, M. Suezava, and T. Kajitani, Jpn. J. Appl. Phys. 14, 1233 (1975). G. Guhin, S. Hautecler, R. Pynn, P. F. Gobin, and L. Delaey, Scripta metall. 13, 429 (1979).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

20 1

of the corresponding elastic constant C' show that the bcc structure has a low dynamical stability for deformations on the { 110) planes along the (170) direction with wavevectors spanning over the entire Brillouin zone. Such an incipient dynamical instability becomes more pronounced as the alloy approaches the transition temperature; in this case the whole TA, branch softens upon cooling. However, as occurs with the elastic constant C', the temperature softening is not complete in the sense that there is no frequency that reaches zero value at the transition temperature which, as already mentioned, is clear evidence of the first-order character of the martensitic transition. It is instructive to compare the low-lying phonon branches for different alloy families. For Cu-based alloys, it has been shown'32 that the energies of the TA, branches are very similar, with a value at the zone boundary of approximately 5 meV. In Figure V.9 we have plotted the effective force constant (Mo')computed using the atomic mass ( M ) and the T A , branches for Aus,,sCd49,s,1 3 3 Au,3Cu30Zn47,'24 Ni 6 2. SA13 7.5' 61 and cu0.68 SA10.2 76 Ni, 0 3 8 . 1 3 2 Such a representation enables a suitable comparison between the different alloy systems. There are some significant peculiarities of each curve. With the exception of Ni-AI, the curves are very flat. As will be discussed later, such a flattening has significant consequences in the thermodynamic stability of the bcc structure in these alloys. The effective force constants for C U ~ . ~ ~ ~andA Au,,Cu,,Zn,, ~ ~ . ~ ~are~rather N ~similar ~ . and ~ they ~ ~are slightly larger than those corresponding to Au,,,,Cd,,,,. In contrast, the Ni62.5A137.5alloy system has larger values (except those modes around 5: N 6). There is an anomaly in all branches at some specific reduced wave number (with lattice parameter a* of the B2 structure): for Ni62.sA137.5 there is a pronounced, dip at 5 2: &, a less pronounced but still clearly visible dip exists at 5 2: 3 for A u ~ ~ . and ~ CAu,,Cu,,Zn,,, ~ ~ ~ . ~ and for C U ~ . ~ ~ ~ there A ~ is~only . ~a very ~ ~tinyN kink ~ ~at 5. 2~3. ~ ~ , The existence of anomalous (with very low energy) phonons at specific wave numbers on the T A , branch has been related to the structure of the martensitic phase. It is important to recall that in order to achieve the adequate stacking sequence of the close-packed planes in the martensitic phase, intracell displacements (shuffles) are necessary. As indicated in Section 11, these displacements correspond to phonons on the T A , branch with specific wave numbers. In Ni-A1 alloys it is interesting to mention that the location of the dip moves from 5 N 4 in the case of NiAl (x = 0.5) to M. Zolliker, W. Biihrer, and B. Schonfeld, Physica B 180 & 181, 303 (1992). LI. MaAosa, J. Zarestky, T. Lograsso, D. W. Delaney, and C. Stassis, Phys. Rev. B. 48, 15708 (1993). l J 3 T. Ohba, S. M. Shapiro, S. Aoki, and K. Otsuka, Jpn. J. Appl. Phys. 33, L1631 (1994).

131

lJZ

ANTONI PLANES AND L L U ~ SMAROSA

202

-

4000 -

c

I

E

0 cy

3000 -

>,

-E e c

2000 -

In C

0 0)

1000 -

2 0

LL

0-

A I

S I

I

I

1

I

FIG. V.9. Effective force constants of the phonons of the T A , branch for Ni-A1 at 85 K (T, = 80K) (circles) from S. M. Shapiro, B. X. Yang, Y.Noda, L. E. Tanner, and D. Schryvers, Phys. Rev. B. 44, 9301 (1991), Cu-Al-Ni at 294 K (T, = 260 K) (triangles) from L1. Maiiosa; J. Zarestky, T. Lograsso, D. W. Delaney, and C . Stassis, Phys. Reo. B . 48, 15708 (1993), Au-Cu-Zn at 298 K ( T , = 22%) (diamonds) from R. A. Robinson, G. L. Squires, and R. Pynn, J . Phys. F: Met. Phys. 14, 1061 (1984), and Au-Cd at 306 K (T, = 304 K) (squares) from T. Ohba, S. M. Shapiro, S. Aoki, and K. Otsuka, Jpn. J . Appl. Phys. 33, L1631 (1994). Notice that the lattice parameter of the bcc structure (a", see Figure 11.2) has been used in all cases for better comparison of the different alloy systems.

5 2: & for Ni65.5A137,5.76 This change has been explained from first-principle calculations6' from which a linear relation of the dip position with the where n electron concentration is deduced. Actually, one expects that 5 N i, is an integer, will lead to reduced phonon frequencies because these values maximize the degree to which the (011) planes move out of phase with each other. For the bcc -+ hcp transition, n = 2, and for the bcc -+ 9R, n = 3. Hence, it is argued that if the Ni-A1 system did undergo a martensitic transition, this reasoning would suggest that it proceeded to the 4H phase. The actual situation is, however, more complex. Ni62.5A135.5 displays a 7 R structure for which n = 7 would be expected instead of the n N 6 observed. In fact such a discrepancy seems to be a consequence of the interplay of the anomalous phonon with the shear strain necessary to obtain the martensitic phase.

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

203

Au-Cd and Cu-based alloys do not seem to conform to the above scheme. For Cu-based alloys, the differences in the T A , b r a n c h e ~ ' ~of~alloys ~'~~ transforming to the 2H or 18R structures are very small; both have the same small kink at 5 = 3 (lattice parameter a of the L2,/D03 structure) and the values of the zone boundary frequencies are similar. An equivalent situation occurs for Au-Cd: the T A , branch for A u ~ ~ . ~which C ~ transforms ~ ~ . ~ ,to the 5' martensite, is quite similar to that of A u ~ ~ . ~which C ~ transforms ~ ~ . ~ , to y ' (2H).' 33*13 5 However, for this alloy system the temperature softening of the zone boundary phonon is slightly larger than for A u ~ ~ . ~It C is also ~ ~ ~ . ~ . worth mentioning that detailed studies on Cu-Zn-A1 alloys showed that the zone boundary frequency for an alloy transforming to 2H was slightly lower than that for an alloy transforming to 18R.'34 For most shape-memory alloys, the anomaly on the T A , branch becomes more pronounced as the alloy approaches T,. This is not the case with Cu-based alloys; any phonon in the branch has a particular temperature dependence and the temperature softening in these alloys is very small in comparison to that of other shape-memory alloys. For instance, the value of 0 ' for 5 = 3 at the transition temperature is -20meV2, which is in contrast to the values of the equivalent phonons in A u ~ ~ . ~ C ~ ~ ( - 0.6meV2) and Ni62.sA137.5( - 2 meV2). This difference is illustrated in Figure V.10, where 0 ' is given as a function of temperature for a c~o.666zno.189A10.145 system and for Ni62.sA137.5.By comparing these data with the behavior of long-wavelength phonons (previous section), it is observed that the relative softening of the anomalous phonon in Ni-A1 is larger than that of C'. However, for Cu-based alloys, the amount of softening of C' is comparable to that of 0'. The significant distinctions between these alloy systems can probably explain their different pretransitonal behavior; in contrast to Ni62.5A137.5,no central peak or tweed patterns have been detected in Cu-based systems. Actually, these precursor effects could be related to the existence of short wavelength modes exhibiting particularly intense softening and giving rise to the marked dip in the T A , branch. Moreover, the amount of shear (long and short wavelength) required for a transformation to the 7R phase is lower than for a 9R (or 18R) structure. Hence, in Ni-Al, above the martensitic transition it costs less energy to get displacements that mimic the martensitic phase, specially when local variations of composition are present in the system. The results presented above suggest that in Cu-based alloys, the selection of the modulation of the low-temperature phase cannot be exclusively 134 G. Guenin, D. Rios-Jara, M. Morin, L. Delaey, R. Pynn, and P. F. Gobin, J. Phys. (France) 12,C4-597(1982). 13' T. Ohba, S. Raymond, S. M. Shapiro, and K. Otsuka, Jpn. J. Appl. Phys. 37, L64 (1998).

204

ANTONI PLANES AND L L U ~ SM A ~ O S A

20

6 ? ! n

k

-E 4

%

5 c

10

- 2

100

200

300 Temperature ( K )

FIG.V.10. ( h ~versus ) ~ temperature for the T A , phonon mode corresponding to the position of the dip. Filled symbols: Ni-A1 (the dip occurs at a reduced wave number -0.16), open symbols: Cu-Zn-A1 (the dip occurs at a reduced wave number -0.66). The extrapolation at T = 0 is indicated by dashed lines. Notice the different vertical scale for the two curves. Data for Ni-A1 from S. M. Shapiro, B. X. Yang, Y. Noda, L.E. Tanner, and D. Schryvers, Phys. Rev. 8.44, 9301 (1991); for Cu-Zn-A1 from LI. Maiiosa, J. Zarestky, T. Lograsso, D. W. Delaney, and C. Stassis, Phys. Reo. 8.48, 15708 (1993).

determined by the anomalies in the T A , branch; rather, the final structure selected by the alloy depends on small details of the internal energy. An analysis of the interplanar force constant of the T A , branch for a number of shape-memory alloys has been undertaken by Nagasawa and M ~ r i i . ' ~In~ this * ' ~analysis, ~ the cubic lattice is expressed by a linear chain model along symmetry axes. The particles constituting the chain represent atomic planes perpendicular to the chain, while interatomic forces are replaced by effective interplanar forces between the atomic planes. The dispersion relation is then expressed as

where

4,, is the effective interplanar force constant between the n-th

1 3 6 A. Nagasawa, Proc. Int. Conf on Martensitic Transformations, ed. The Japan Inst. of Metals, Nara, Japan (1986). p. 95.

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

205

neighbor atomic planes. This expression has also been used by Ye et al.137 to reproduce the pseudoelastic properties of martensitic crystals. Nagasawa and Morii' 22 classified the martensitic materials into three groups depending on the values of 4, and c # ~ ~materials ; with 4, > 0 and c$3 < 0 transform to a 2H structure (shuffling each 2 planes was easily accomplished for these materials). For materials with 4, < 0 and 43> 0, shuffling on the third plane must occur and the resulting martensitic structure will be 9R (or 18R). Finally, when 4, and 43are positive, with similar values, the system exhibits a dual lattice instability toward both 2H and 9R (or 18R) structures. Cu-based alloys belong to this latter class of materials. For Ni-A1 alloys, the dip in the T A , branch cannot be simulated by the interplanar force constant method. As indicated in Section 111, electronic band structure calculations62 showed that such a dip has its origin on strong electron-phonon interactions and Fermi surface nesting. The effect of a uniaxial stress applied along the [Ool] direction on a Cu-Al-Pd alloy was also studied by Nagasawa et al.13*The changes on the T A , curve were extremely small (almost undetectable). From their interplanar force analysis they found that 4, decreased and 43increased as stress increased. This finding indicates that the instability toward the 18R structure is enhanced by stress. This result is consistent with the fact that, as mentioned in Section 11, the 2H phase is always the stable phase at a low stress level. There are also measurements of uniaxial stress dependence of the T A , branch in Ni-A1 alloys.'39914oIn this alloy system, changes in the phonon curve were quite significant; on increasing the stress along the [Ool] direction, there is a decrease of the whole branch, the dip becomes more pronounced and moves toward higher wavevectors. Such a shift is also indicative that the stress tends to stabilize the 3R phase, which is consistent with stress-strain phase diagrams141 in this alloy system. Let us now turn our attention to the dip in the L(((5) branch at 5 = 3. This phonon is involved in the transformation to the o-phase. Such a phase is obtained by the freezing-in of the compression of two adjacent (11 1) planes leaving the third plane unaltered. Kelly and Stobbs14, formulated a Landau theory for /?-phase alloys in which [111] charge density waves would couple to the [1113 longitudinal phonons, resulting in a soft-mode Y. Y. Ye, C. T. Chan, and K. M. Ho, Phys. Rev. Lett. 66, 2018 (1991). A. Nagasawa, A. Kiwabara, Y. Morii, K. Fuchizaki, and S. Funahashi, Muter. Trans. JIM 33, 203 (1992). 1 3 9 S. M. Shapiro, E. C. Svenson, C. Vettier, and B. Hennion, Phys. Rev. B. 48, 13223 (1993). L. Ye, S. M. Shapiro, and H. Chow, Scripta metall. 31, 203 (1994). l4I V. V. Martynov, K. Enami, L. G. Khandros, S. Nenno, and A. V. Tkachenko, Phys. Met. Metallogr. 55, 136 (1983); Scripta metall. 17, 1167 (1983). '41 M. J. Kelly and W. M. Stobbs, Phys. Rev. Lett. 45, 922 (1980). 13'

138

206

ANTONI PLANES AND L L U ~ SMAROSA

instability. This mechanism assumes that the dispersion surface has a minimum at the point 5 = 3[lll]. It is worth noting that the w transformation can also be understood as a transverse displacement wave propagating along the [112] direction with [lll] polarization. The slope at the origin of this branch is associated with the elastic constant C,, which as shown in the previous section has a rather low value. Neutron measurements of the TA[5525] branch in c ~ - Z n - A l and ' ~ ~in Ni-A16' proved the existence of a saddle point at 5 = :[ 1111 instead of a genuine minimum, thus showing that the theory proposed by Kelly and Stobbs was not applicable to shapememory alloys. A similar conclusion was also reached for A u - C U - Z ~ . ' ~ ~ From purely geometrical arguments and assuming an effective ion potential composed of a Coulomb interaction plus an electronic part, Falter'43 showed that, due to the Coulomb interaction between ions, L[+$*] phonons in all bcc structures will always exhibit energies lower than the remaining phonons in the branch. Whether this general weakness of the bcc lattice toward the w-structure is enhanced or not depends on the details of the electronic structure of the solid.83 Actually, according to Guenin et al.,'34 bcc structures with low elastic anisotropy would transform to the w-phase while those with high elastic anisotropy transform to a martensitic structure. It is worth comparing the slope of the phonon dispersion curves at the origin of the Brillouin zone with that computed using the ultrasonically measured SOEC. These slopes coincide except for the T A , branch. Figure V.ll shows this branch for many P-phase alloys at low wave numbers, together with the slope computed from ultrasonically measured values of C'. It is clear that the slope of the branch is always larger than that given by ultrasonic measurements. This result indicates that, for shape-memory alloys, the long wavelength acoustic T A , modes are softer than all other modes. For proper ferroelastic martensites, Barsch and Krumhanslg7 have proposed that such a discrepancy arises from possible modulated microstructures (tweed) appearing above the structural transition. They have put forward a Landau-Ginzburg free energy functional for a nonlinear, nonlocal anisotropic elastic continuum. The dispersion relation along a symmetry direction q can be approximated by the truncated series wf

= Aj(q)

sin2

()'

+ Bj(q)sin4 ('),

where j denotes the direction of polarization. In the long wavelength limit, A j is related to the usual combination of SOEC for a given symmetry direction, and B j to the strain gradient components of the Ginzburg-Landau 143

C. Falter, Phys. Rep. 164, 1 (1988).

207

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

(a1 5 ,

51

3 c

> 1

w

E

0

Y

0.2

>5

0.4

0.6

0

0.2

(dl

5: C

W

3

t 0

0.4

0.6

0.4

0.6

(f)

5 7

. *

0.4

0.2

(el

: 0.2

0

0.6

3

0

0.2

0.4

0.6

1

0

0.2

0.4

0.6

Reduced wave number FIG. V.ll. T A , phonon dispersion curves at room temperature for (a) Cq27.63 at% Al, 3.85 Ni; (b) Cu, 28.05 at% Al, 3.85 at% Ni, (c) Cu, 23.13 at% Al, 2.80 at% Be; (d) Cu, 27.00 at% Al, 5.50 at% Pd; (e) Cu, 19.25 at% Zn, 13.00 at% Al; (f) Au; 30.00 at% Cu, 47.00 at% Zn. at%

The lines are the slopes at the origin computed using ultrasonically measured values of C‘. In all cases the slope of the branch is larger than that computed from the ultrasonic measurement. From L1. Maiiosa, M. Jurado, A. Planes, J. Zarestky, T. Lograsso, and C. Stassis, Phys. Rev. B. 49, 9969 (1994).

free energy. These terms can be estimated from experimental data by plotting (o,/~in(n
208

ANTONI PLANES AND L L U ~ SM A ~ ~ O S A

0.2

0.4

0.6

0.8

sin2 ( ~ c 5 / 2 ) FIG. V.12. Fourier expansion of phonon dispersion data v2/sin2(n(/2)versus sin2(n(/2) for Cuo,,,,Alo,zslBeo,o,,.(0):data obtained from neutron experiments; (0) obtained from ultrasonic measurements. From L1. Mafiosa, J. Zarestky, T. Lograsso, D. W. Delaney, and C . Stassis, Phys. Rev. B.48, 15708 (1993).

1.2 c

ul

c. .-

C 3

1

G

2 0.8 ul

u

c

0

0.6

II

.c

0

1

3 0.4

d

0.2

Energy (meV) FIG. V.13. Phonon density of states for Cu; 22.72 at% Al; 3.55 at% Be, computed from the Born-von-Karman model up to five nearest-neighbors force constants, which are fit to the phonon dispersion curves shown in Figure V.8. From L1. Maiiosa, J. Zarestky, M. Bullock, and C. Stassis, Phys. Rev. B. 59, 9239 (1999).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

209

peaks in this density of states. They correspond to the high phonon density regions of the dispersion curves shown in Figure V.8, except those in the vicinity of 12 meV. These peaks correspond to high phonon density regions of the TA(5525) dispersion curve (not shown in Figure V.8). An interesting peak to be considered is the one located at 7 meV. It corresponds to the low-lying T A , phonons. As will be discussed later, it is related to the high vibrational entropy of the bcc phase. With regard to the lattice dynamics of the martensitic phase, experimental data are very scarce. Tanahashi et al.I4, performed inelastic neutron scattering experiments on a Cu-Al-Ni alloy. By application of a uniaxial stress, they managed to measure the [001],[100], branch in the 3R, 18R, and 2H martensitic structures. They found that the phonon curves for these three structures were very similar, no energy gaps in the Brillouin zone were observed, and there was no anomaly (such as a dip) in any of the measured curves. The phonon frequencies did not shift by application of the uniaxial stress except for the 2H structure where a slight softening with stress was observed as the temperature approached the transition point. The phonon dispersion curves along the [loo],, [OlO],, and [OOl], symmetry directions of a 18R Cu-Zn-A1 crystal were measured at different temperatures by Guenin et al.I4'. No temperature softening on approaching the transition was measured for any of the phonons. When comparing the slopes at the origin with the values computed using the measured ultrasonic velocities,'20*'21excellent agreement is obtained for all modes. An interesting result arising from these measurements is that the phonons propagating along the [OOl], direction with [1001, polarization have lower energies than the remaining phonons in the martensite. Furthermore, they have lower energies than the phonons in the P-phase, with the exception of those on the low-lying T A , branch. By recalling that in the distortions is long wavelength limit the elastic response to the [001],[100], given by the elastic constant CS5,we claim that we are dealing with the same situation as discussed for the SOEC; C,, in the martensite was lower than B and C,, of the P-phase. A possible way of quantifying the discontinuity in the lattice response at the transition point consists of the evaluation of the difference in the frequency of the zone boundary phonons of the [l 10],[liO]s and [~l],[lOO], branches in the P and martensitic phases, respectively. For martensite, there are no values for these quantities at the transition temperature; nevertheless, because the temperature softening is

-

144 H. Tanahashi, Y. Morii, M. Iizumi, T. Suzuki, and K. Otsuka, Proc. Znr. Con$ on Martensiric Transformations, ed. The Japan Inst. of Metals, Nara, (1986) p. 163. 14' G. Gdnin, R. Pynn, D. Rios-Jara, L. Delaey, and P. F. Gobin, Phys. Stat. Sol. ( a ) 59,553 (1980).

210

ANTONI PLANES AND L L U ~ SMAROSA

small (if any), the room temperature values will provide a good approximation. Using the data given in Ref. [132] for the fl phase and in Ref [I441 for the martensite, we obtain the following values: miB N 5meV and myB N 12mev. The results discussed above show that the vibrational behavior of shape-memory alloys for wave numbers spanning over the entire Brillouin zone is similar to that observed in the long wavelength limit. Extension of such a parallelism leads us to suggest that the phonons propagating along the [ill], direction with polarization along the [121IM direction will have lower energies than any other phonon in the martensitic phase. At present there are no measurements to confirm this assertion. 8. THIRD-ORDER ELASTIC CONSTANTS AND VIBRATIONAL ANHARMONICITY This section deals with the vibrational anharmonic behavior of shapememory alloys. Different theoretical approaches to the martensitic transformation have shown that vibrational anharmonicity plays an essential role 13,147*148 in the mechanisms leading to the martensitic tran~formation.'~~.' On the other hand, a clear experimental manifestation of strong anharmonic behavior is the pronounced temperature dependence of some phonon modes discussed in the previous sections. The Gruneisen parameters (see Section IV.5) have been obtained for several shape-memory alloy^'^^*'^^-'^^ at room temperature and close to the martensitic transition temperature. As an example, in Figure V. 14, the Gruneisen parameters (hydrostatic) of Cu-Al-Be (a) and Cu-Zn-A1 (b) are plotted as a function of mode propagation at different temperatures. At room temperature, the y,(N) show normal behavior and have magnitudes in the usual range for bcc metals, and in a given branch, do not vary substantially with the propagation direction. However, as the alloy approaches the transition temperature, the longitudinal mode gammas decrease significantly and the y,(N) for the mode associated with C' becomes substantially large, thus indicating that the soft-mode anharmonicity increases near the transition. 14' Y. Y. Ye, Y. Chen, K.-M. Ho, B. N. Harmon, and P.-A. Lindgird, Phys. Rev. Lett. 58, 1769 (1987). 14' J. R. Morris and R. J. Gooding, Phys. Rev. Lett. 65 (1990), p. 1769; Phys. Rev. B. 43, 6057 (1991). 14' W. C. Kerr, A. M. Hawthorne, R. J. Gooding, A. R. Bishop, and J. A. Krumhansl, Phys. Rev. B. 45, 7035 (1992). M. A. Jurado, M. Cankurtaran, LI. Maiiosa, and G. A. Saunders, Phys. Rev. B. 46, 14174 (1992). 150 A. Nagasawa and A. Yoshida, Muter. Trans. JIM 30,309 (1989). A. Gonzalez-Comas and L1. Mafiosa, Philos. Mug. A , 80, 1681 (2000).

21 1

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

-

2.5

0)

\(a)

295K

j - L6 5

........

....

--_ 1

I

6 u

2 2.0 F

}

L

Z3

I

55 -t

......

L -

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

I I /

/

/

I'

I:\

18jK

\ \\ \ \

/

II

1.5

0 0

a

1.0 I

[OOlI

I

I

I

[1111

[llOI

[loo]

-

2

[OOlI

[1111

(1101

[loo]

Wave -propagation direction

FIG. V.14. Long wavelength longitudinal (solid line) and shear (dashed and dotted lines) acoustic mode Griineisen parameters of (a) Cu-23.1 at% A1-2.8 at% Be (at 295 and 268 K), and (b) 01-20. 8 at% Zn-12.7 at% A1 (at 295 and 183 K), as a function of mode propagation direction. From M. A. Jurado, M. Cankurtaran, L1. Maiiosa, and G. A. Saunders, Phys. Rev. B. 46, 14174 (1992).

Complementary information on vibrational anharmonicity can be obtained by plotting the Gruneisen parameters as a function of the direction of the applied uniaxial stress (equation (17)). Such a representation has been used by Nagasawa and co-workers' 3 8 * 1 in the investigation of different alloys and also by Gonzalez-Comas and M a i i o ~ a for ' ~ ~Cu-based alloys. An example for Cu-Al-Ni is shown in Figure V.15. It is clear that the Gruneisen gamma for the [llO][lTO] mode* is very anisotropic; a large negative minimum is observed when the stress is applied along the [Ool] direction. Such a minimum is related to the anomalous value of the stress derivative of C' which will be discussed later in this section. This minimum becomes more pronounced as the alloy approaches the transition temperature, which reflects an increase in the vibrational anharmonicity of this mode. As shown in Figure V.15, there is also a small minimum for a stress applied along the [Tl2] direction, which is related to the special mode (CJ; it also becomes deeper as the alloy approaches the transition temperature. The behavior of these two minima is exemplified in Figure V.16, which shows the values obtained at selected temperatures for different Cu-based alloys. Remarkably, the extrapolated values at the transition temperature are very similar among

*

Nagasawa et al. denoted the [llO][liO]

modes as T A , instead of TA,.

212

ANTONI PLANES AND L L U ~ SM A ~ ~ O S A

k 15

4-

u

; 10

bp rW

5

g

o

(5

-5

.-

VI

:3

u

U

0

E -10 .-0

5 -15 3 0

2 -20[ o i o i

[ o i l 1 [0011

11111 [ i i o i [lo01 [1101

[on1 [?izi[Tor~

Direction of uniaxial stress FIG. V.15. [llO] acoustic mode Griineisen parameters of a Cu-28.1 at% Al- 3.8 at% Ni crystal as a function of the uniaxial stress direction at room temperature.

the different systems. This finding indicates a common anharmonic behavior of this alloy family. A complete description of nonlinear acoustic behavior requires knowledge of the complete set of TOEC. They can be obtained from the uniaxial stress dependence of the ultrasonic velocities along specific crystallographic direct i o n ~ TOEC . ~ ~ have been measured for Cu-Zn-A1 alloys with different c o r n p ~ s i t i o n s ,s~3~Cu-Au-Zn,' ~~' 53 c ~ - A l - P d , C ' ~U~ - A L N ~ ,and ' ~ ~ Cu-AlBe.lSs The room temperature values found for all Cu-based alloys do not exhibit marked differences. All TOEC are negative, indicating the usual behavior of an increase of the vibrational frequencies under stress, giving rise to an increase in the strain-free energy. The absolute value of TOEC is large and is an indication that these alloys are significantly anharmonic. The SOEC and TOEC are respectively the coefficients of the quadratic and cubic term in the strain free-energy expansion (see equation (12)). It has been s ~ o w ~that ' if~ one ~ considers * ~ ~ ~that the lattice distortion associated with the martensitic transition is a pure (llO)[lTO] shear (uz), the free 15' 15'

B. Verlinden, T. Suzuki, L. Delaey, and G. Guenin, Scripta Metall. 18, 975 (1984). A. Nagasawa, T. Makita, and Y. Takagi, J. Phys. SOC.Japan 51, 3876 (1982). A. Gonzalez-Comas and LI. Mafiosa, Phys. Rev. B. 54, 6007 (1996). A. Gonzalez-Comas, L1. Mafiosa, A. Planes, and M. Morin, Phys. Rev. B. 59, 246 (1999).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

213

50

60 70 80 24 28 32 36

0

40

80 T- TM

120

FIG. V.16. [llO] TA, acoustic mode Griineisen parameters for uniaxial applied stress along the (a) [Ool] and (b) [i12] directions, as a function of the reduced temperature T - T,. Circles: Cu-Al-Ni, triangles: Cu-Zn-Al, squares: Cu-Al-Be. Solid lines are linear fits to the data.

energy does not contain a cubic invariant. In principle, this is an indication that q 2 alone cannot be the only order parameter in describing the martensitic transition. The following simple approach is to consider the tetragonal distortion ql. Actually, a tetragonal distortion is very close to a pure Bain strain that brings the bcc structure toward a close-packed structure. On the other hand, the tetragonal distortion can be obtained as the result of two pure perpendicular shears. Indeed, Guenin and G ~ b i n ' ~ ~ showed that for small strains, two {llO}(l~O)shears are equivalent to a constant volume Bain transformation. Hence, using ql as the order parameter, the relevant third-order invariant (see eq. (14)) in the free energy is C,. For C , < 0 a first-order transition is possible at a temperature for which the Landau theory predicts an incomplete softening of the elastic constant C'. This is in agreement with the experimental observalS6

G. Gutnin and P. F. Gobin, J. Phys. (France) 12, C4-57 (1982).

214

ANTONI PLANES AND L L U ~ SM A I ~ O S A

60

-a-

55

0

0

m

50

0 45

40

0

40

80

120

FIG. V.17. Absolute value of the TOEC combination C , as a function of the reduced temperature T- 7'. Cu-Al-Ni (circles), Cu-Al-Be (squares), and Cu-Zn-A1 (triangles). Dashed 1 GPa. From A. lines are linear fits to the data. The extrapolated value at 7' is IC,I=60 Gonzalez-Comas, L1. Mafiosa, A. Planes, and M. Morin Phys. Rev. B. 59,246 (1999).

tion that C' decreases on cooling, but keeps to a finite value at the transition temperature. For Cu-based alloys, the temperature dependence of the cubic invariant C, has been shown'55 to be nonnegligible. In Figure V.17. this TOEC combination is plotted as a function of the reduced temperature for different alloy families. It is clear that IC,I exhibits an unambiguous increase as the temperature lowers towards the transition. This behavior is closely related to the enhancement of the minimum in the T A , Gruneisen parameter discussed preceding (see Figures V. 15 and V. 16). An important experimental finding is that the extrapolated value C, = - 60 1GPa at T, coincides for the different alloy families. At present, there is no microscopic theory able to account for this finding, although, as will be discussed in Section VII, the peculiar temperature dependence of C, is important with regard to the mechanisms of the martensitic transformation (lattice instability and nucleation).

VIBRATIONAL PROPERTIES O F SHAPE-MEMORY ALLOYS

'O

215

I

5-

C

b

-0 \

iU, o -5

Direction of uniaxial s t r e s s FIG.V.18. Stress derivative of the shear elastic constant C' in the different shear planes as a function of the angle between COO11 and the uniaxial compression direction for a Cu; 20.8 at% Zn; 12.7 at% A1 alloy at room temperature. (a) (Oil) and (701) shear planes; (b) (011) and (101); (c) (110) and (d) (110). From B. Verlinden and L. Delaey, Metall. Zl-ans. A 19, 207 (1988).

The TOEC also enable computation of the dependence of the SOEC with the uniaxial stress, applied along different crystallographic directions.' 5 7 In Figure V.18 we show an example of the pressure derivatives of C' as a function of the angle between [OOl] and the uniaxial compression direction, for a Cu-Zn-A1 alloy, extracted from Ref. [l58]. The high anisotropy of the stress dependence of C' is remarkable; W / a a changes drastically as the direction of the stress varies. This behavior has been shown to be common for other Cu-based alloy^.'^^^'^^ The curves for the (llO)[lTO] and (liO)[llO] shears have a negative minimum when the stress is applied along the [Ool] direction. Such a minimum is related to the minimum found in the Gruneisen parameter of the T A , modes when the stress was applied along the [Ool] direction (see Figure V.15). Such a coincidence is not surprising because the expressions for y,(N) and W / a a both involve the same combination of TOEC. 15'

R. N. Thurston and K. Brugger, Phys. Rev. 133, 1604 (1964). B. Verlinden and L. Delaey, Metall. Trans. 19A, 207 (1988).

ANTONI PLANES AND L L U ~ SMAROSA

216

L

L

W

I

[OOll

I

[1111

,

I

(1101

[loo]

Direction of mode propagation FIG. V.19. Long-wavelength longitudinal (solid line) and shear (dashed and dotted lines) acoustic mode Griineisen parameters of Cu; 12.8 at% Zn; 17.6 at% A1 in the 18R martensitic phase as a function of mode propagation direction at 293 K. From A. Gonzalez-Comas, L1. Maiiosa, M. Cankurtaran, G. A. Saunders, and F. C. Lovey, J . Phys: Condens. Matter 10,9137 (1998).

With regard to the acoustic anharmonicity of the martensitic phase, there has been a lack of experimental data for many years. Very recently Gonzalez-Comas et al.ls9 have measured the hydrostatic pressure dependence of the ultrasonic velocity in an 18R Cu-Zn-A1 single crystal. From these measurements the Griineisen parameters of the long wavelength acoustic modes shown in Figure V.19 have been obtained. The overall values are lower than in the corresponding /?-phase (see Figure V. 14), which indicates that, in the acoustic long wavelength limit, the martensitic phase is less anharmonic than the /?-phase. There is a large anisotropy for the gamma corresponding to the slow shear mode. In particular, a pronounced negative minimum occurs in a direction close to the (lll)M[l?l]M mode. As already discussed in Section V.6.b, this mode is derived from the (llO)[lTO] mode of the /? phase, and the corresponding elastic modulus exhibited the larger temperature softening. Therefore, this proves that the martensitic phase inherits the marked anharmonic behavior of the (llO)[lTO] modes in the /?-phase.

1 5 9 A. Gonzalez-Comas, L1. Mafiosa, M. Cankurtaran, G. A. Saunders, and F. C. Lovey, J. Phys: Condens. Mutter 10,9737 (1998).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

217

VI. Phase Stability

In this section we analyse, from a thermodynamic point of view, the origin of the stability of the /?-phase at high temperature. Because metallic bonding is essentially nondirectional, to a first approximation, the greatest bonding will occur for those crystallographic arrangements that maximize the coordination of each atom.'60 From this argument, it is immediately concluded that the bcc phase is energetically unstable when compared with close-packed structures. Actually, this simple picture has been corroborated from first-principle calculations in nonmagnetic metallic elements. For most of these systems a close-packed structure is the lowest energy structure at zero temperature. Exceptions are only some alkali metals and group V transition metals for which the bcc phase is favored by electronic degrees of freedom.' At high temperature, the larger entropy of the bcc phase results in a low free energy which is, consequently, responsible for the stability of this phase. For instance, very recent total energy calculations undertaken by the group of Duisburg16' in Cu-based alloys have shown that the close-packed phase has lower energy than the bcc one. However, the energy difference between these two phases is low and thus at finite temperatures it is expected that entropy plays a major role in phase stability. The martensitic transition will take place when the entropic contribution to the free energy is compensated by the energy difference between both the open and close-packed phases. Fifty years ago Zenerlo4 predicted that the large entropy of the P-phase mostly has a vibrational origin. In what follows we will see that this is actually the case for shape-memory alloys and we will show that such an excess of entropy of the /?-phase has its origin in the low-energy TA, phonons.

9. TOTALENTROPY CHANGE In general, sufficiently accurate measurements of the energy and entropy differences between solid phases are difficult due to their small values. For the systems studied here, the existence of a martensitic transition represents a considerable advantage. This advantage rests on the following points: (i) the martensitic transition occurs at low temperature with little hysteresis (in temperature and stress cycling) between forward and reverse transition and A. Zangwill and R. Bruinsma, Comments Condens. Mater. Phys. 13, 1 (1987). H. L. Skriver, Phys. Rev. B. 31, 1909 (1985). 1 6 2 E. F. Wassermann, Proc. Int. Conference on Solid-Solid Phase Transformations. Ed. M. Koriwa, K. Otsuka and T. Miyazaki, The Japan Institute of Metals, Sendai, Japan 1999, p. 807. 160 16'

218

ANTONI PLANES AND L L U ~ SMAAOSA

therefore energy and entropy differences between p and martensitic phases can be determined to high accuracy; (ii) by suitable heat treatment an atomic distribution very close to the one corresponding to the ground state at T = OK can be obtained; (iii) its diffusionless nature ensures that the atom distribution in the high-temperature bcc phase will be inherited by the lowtemperature close-packed phase, and hence the measured entropy change only contains vibrational and electronic contributions. That is,

Typical experimental determination of AS is usually performed from calorimetric or stress-strain experiments. In the latter case the resolved transformation stress oM necessary to induce the martensitic phase is measured as a function of temperature. AS is then obtained through the Clausius-Clapeyron equation as' 6 3

where E is the transformation strain and I/ is the molar volume. E can be determined experimentally or calculated once the lattice structures are determined for /? and martensitic phases. In the preceding equation a volume change associated with the transformation is neglected. Actually, this is a very good approximation in the case of shape-memory alloys. Calorimetrically AS is usually determined using the differential scanning calorimetry (DSC) technique. AS is obtained as

where dQ/dt is the thermal power which, after proper correction of the base-line, is proportional to the calorimetric signal (the proportionality factor depends on temperature and is determined by previous calibration). and T, are temperatures located, respectively, above (below) the starting and finishing transition temperatures on cooling (heating). For thermoelastic transitions, because the heat capacity difference between the highand low-temperature phases is small, and the entropy production arising from irreversible effects negligible,' 6 4 no systematic differences are usually reported for the absolute values of AS measured during forward (cooling) M. Kato and H. R. Pak, Phys. Stat. Sol. ( b ) 123, 415 (1984). J. Ortin and A. Planes, Acta metall. 36, 1873 (1988).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

219

and reverse (heating) transitions. Both methods have their limitations because AS in martensitic transitions is small (the order of magnitude is 1J/K mol). The experimental uncertainty is around 10% for tensile experiments and is reduced to around 5% in the case of calorimetric measurements.' 6 5 In Cu-based shape-memory alloys a large amount of data are available. Measured values of AS range between 1.20J/Kmol and 1.70J/Kmol. Ahlers'66 has analyzed data of AS collected by Romero and Pellegrina'67 for many different alloys. He has inferred that AS increases linearly with the electron-to-atom ratio e/a according to AS = (2.87 -t O.lO)e/a - 2.80 in J/Kmol. A different scenario has been contemplated by Obrado et al.31 AS is supposed to depend only on the sequence of close-packed planes in the martensitic phase. This is suggested after noticing that when e/a increases, the martensitic structure changes from the 18R to the 2H at a value close to (ela),, corresponding to the eutectoid point in the equilibrium phase diagram, which is slightly dependent on the specific alloy system considered. For instance, (ela),, = 1.46 for Cu-Al-Mn, 1.48 for Cu-Zn-Al, 1.49 for Cu-Al-Be, and 1.53 for Cu-Al-Ni. In Figure VI.l, AS values for different Cu-based alloys of different compositions are plotted as a function of z = e/a - (e/u),,,. Within the experimental errors, all points collapse on a single step function that changes from 1.31 k 0.10 J/Kmol to 1.60 f 0.10 J/Kmol at z = 0. Systems with negative z transform to the 18R structure, while systems with positive z transform to the 2H structure. At z = 0 data are spread over a large range of values; this must be associated with the errors in the determination of the composition and the fact that samples with compositions close to the eutectoid may transform to a mixture of the two phases.16* The apparent discrepancy between both the linear and the step function representation can be explained by taking into account the fact that (ela),, is alloy dependent. This masks the scaling proposed for AS in a step function because the jump occurs over a broad range of e/a values. In any case the scatter of the data cannot discard a slight e/a dependence superimposed to the step function. An interesting point to be stressed is that for Cu-Zn, the high temperature entropy difference between B and CI (fcc) phases, as estimated by Ahlers,'66 is comparable (apart from short-range order effects) to the entropy change at the martensitic transition (B + 9R). This is consistent with the fact that

-

A. Planes, R. Romero, and M. Ahlers, Scripta metal/. 23 , 989 (1989). M. Ahlers, Z . Phys B, 99,491 (1996). 16' R. Romero and J. L. Pelegrina, Phys. Rev. B. 50,9046 (1994). 16' J. van Humbeeck, D. van Hulle, L. Delaey, J. Ortin, C. Segui, and V. Torra, Trans. Jpn. Ins?. Met. 28, 383 (1987). 166

ANTONI PLANES AND L L U ~ SMAGOSA

220

2.0 1.8 1.6 1.4

1.2

-0.12

- 0.08

- 0.04

0

0.04

0.08

(e/al-(e/al,, FIG. VI.1. Entropy differences between and martensitic phases for different Cu-based alloys A:Cu-Al-Ni, V:Cu-Al-Be; and . :Cu-Al-Mn) as a function of the reduced electron-to-atom ratio e/a - (e/a)eu;(e/a).. is the value corresponding to the eutectoid point in the equilibrium phase diagram (see Figure 11.1). Solid lines are averages on each region, and dashed lines indicate, in each case, the standard deviation. The inset shows results for Cu-Al-Mn alloys. In this case all measurements are performed under the same experimental conditions. The jump at e/a Y 1.46 is clearly seen. Results from E. Obrad6, L1. Maiiosa, and A. Planes, Phys. Rev. B. 56, 20 (1997).

(+: Cu-Zn-Al, 0 Cu-Zn;

the martensitic transition line in the phase diagram is approximately parallel to the LX/Pboundary equilibrium line (see Figure 11.1).26 This confirms the fact that entropy change data for the martensitic transition turn out to be useful in the study of the relative phase stability between equilibrium phases in Cu-based alloys. The results presented above indicate that an entropy difference exists between the 2H and 18R phases. The entropy for the 18R phase is larger than in the 2H phase. The difference amounts to 0.3 i-0.2 J/Kmol irrespective of the alloy system. Several works have been devoted to determining the entropy difference between martensitic close-packed structures from stressstrain experiments in a number of Cu-based systems. In this case the difference is deduced by stress inducing the 2H and 18R structures in a given alloy. Values reported in the literature for the entropy change between 2H

VIBRATIONAL PROPERTIES O F SHAPE-MEMORY ALLOYS

22 1

and 18R are 0.2 J/Kmo13' and 0.4 J/Krn01'~~ for Cu-Al-Ni, 0.2 J/Krn01'~' for Cu-Sn and 0.3 J/Kmol"' for Cu-Al-Mn. These values are in very good agreement with those determined from the analysis of entropy change between bcc and martensite for different alloy systems. 10. ELECTRONIC AND

VIBRATIONAL

CONTRIBUTION

A better comprehension of the stability between different crystallographic phases is gained by quantifying the different contributions to the overall entropy difference.' 7' Accurate band-structure calculations provide the electronic contribution (SeJ as a function of temperature and density, but these kinds of calculations are not available for most shape-memory alloys. The entropy change at the phase transition can be determined from the heat capacity difference between both the B and the martensitic phases as

where Ce, and Cvibare, respectively, the electronic and vibrational contributions to the heat capacity. An estimation of the electronic contribution has been undertaken for Cu-based alloys.'72 Most of these alloys are diamagn e t i ~ , * * '7~3 ~and * ' in the nearly-free electron approximation the entropy change is related to the change in the magnetic susceptibility as

where k, is the Boltzman constant, p is the intrinsic magnetic moment of electrons, and p' = (m/m')p, with m/m' the ratio between free and effective electron masses. A good assumption is to consider that the susceptibility change Ax is only associated with a change in the Fermi level between /3 and martensitic phases. This is indeed a reasonable approximation because the diffusionless nature of the martensitic transition ensures that no change in ion diamagnetism is expected. In some respects this is equivalent to assuming that the alloy behaves as a pure metal. 16'

J. Ortin, L1.Maiiosa, C. M. Friend, A. Planes, and M. Yoshikawa, Philos. Mug. A. 65,461

(1992). ''O

71

17* 173

H. Kato and S. Miura, Actu metall. muter. 43, 351 (1995). G. Grimvall, The electron-phonon interaction in metals, North-Holland, Amsterdam (198 1). L1. Mafiosa, A. Planes, J. Ortin, and B. Martinez, Phys. Rev. E. 48, 3611 (1993). D. Abbe, R. Caudron, and R. Pynn, J. Phys. F: Met. Phys. 14, 1117 (1984).

ANTONI PLANES AND L L U ~ SMAFJOSA

222

-8

-

c

d

0

E

=

E

-10

0)

I Y

% -1 2

220

250

280

Temperature ( K ) FIG. VI.2. Magnetic susceptibility (solid line) y, and transformed fraction x (dashed line) obtained from calorimetric measurements as a function of temperature, through the martensitic transition for Cu; 23.1 at% Zn; 7.2 at% Al. The inset shows the normalized change in magnetic susceptibility as a function of transformed fraction for four Cu-Zn-A1 alloys of different composition. From L1. Mafiosa, A. Planes, J. Ortin, and B. Martinez, Phys. Rev. B. 48, 3611 (1993).

emu/mol, and increases For Cu-based alloys x is of the order of in passing from the fl to the martensitic phase. In Figure VI.2 we show this change measured through the martensitic transition in Cu-Zn-A1 alloys. It is worth noting that the change of x in the two phase region is proportional to the transformed fraction as shown in Fig. VI.2. Reported values of Ax = xa - xM for different Cu-based systems are of the order of - 2 x 10-6emu/mol. This small value results in quite a small value for the electronic contribution to the entropy change. This contribution has been estimated for Cu-Zn-Al, Cu-Al-Be and Cu-Al-Ni alloys of different compos i t i o n ~7.2~Within experimental errors it is quite independent of the alloy composition and amounts to AS,,, = - 0.08 f0.02 J/Kmol. This represents about 5% of the whole entropy change. The striking feature is that the electronic entropy of the close-packed phase is larger than that of the /? phase. This means that in Cu-based alloys, the electronic degrees of freedom are a stabilizing factor for the martensitic phase. This result is in contrast to that found for group IV pure metals.126 For paramagnetic alloys, the preceding description is not adequate. Nevertheless, there are a number of experimental results that can be considered in order to obtain some insight into the amount of the electronic contribution. First, it must be considered that in Cu-based alloys containing Mn, in the range of compositions at which Cu-Al-Mn shows a martensitic

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

223

transition, the system is paramagnetic instead of diamagnetic, which is the standard situation in Cu-based alloys. The whole entropy change measured for this material3’ coincides with that measured for other Cu-based families. Therefore, it is not expected that the electronic and vibrational contributions differ markedly from those estimated for diamagnetic alloys. Second, low-temperature specific heat measurements are available for some alloys. For Ni-Al,’74 the coefficient of the linear term in the C vs T curves is slightly larger (1:1.9 rnJ/K2 mol) than the values measured for c ~ - Z n - A l ’ ~ ~ (0.7-0.8 mJ/K2rnol); this can be indicative that in Ni-A1 the electronic contribution may be slightly larger than in Cu-based alloys. For Ti-Ni alloys, Kuentzler’ 7 s measured the low-temperature specific heat for compositions over a broad range. He found the interesting result that the electronic and vibrational contributions showed relative maximum and minimum respectively at the compositions for which there were martensitic transitions. For these alloys the electronic contribution (4-5 mJ/K2rnol) is almost one order of magnitude larger than in Cu-based alloys. These experimental findings led Kuentzler to suggest that in Ti-Ni alloys the driving force for the martensitic transition has a dominant electronic origin. The vibrational entropy change ASuibcan be determined from the vibrational contribution of the heat capacity difference ACuibbetween the bcc and the martensitic phases. To our knowledge the most complete set of data has been reported by AbbC’73 in Cu-Zn-A1 alloys of different compositions. Especially interesting are the low-temperature heat capacity measurements in alloys remaining bcc down to OK. Debye theory is usually used as a reference model for data reduction. As an illustration, Figure VI.3 shows, as a function of the reduced temperature T / 8 D (8, is the Debye temperature), the ratio 9 = C,,ib/Cfebbetween the vibrational contribution of the measured heat capacity (obtained after substraction of the electronic contribution) and the Debye heat capacity C F b = 3Nk,F,(x), where F,(x) = ( 3 / x o ) f ~ [ x 4 e z / ( e-x 1 ) 2 ] dx, with x o = B,/k,T. At very low temperatures (TG0.018,) the Debye law is satisfied. However, W shows a “peak anomaly” at temperatures close to 0.058,. The obtained behavior of 9 is quite similar to that observed in some (commonly glassy) covalent nonmetal systems. In such a context the anomaly is called the Boson peak in current discussion^.'^^ It has been related to a strong anisotropy and/or 74 S. Rubini, C. Dimitropoulos, S. Aldrovandi, F. Borsa, D. R. Torgeson, and J. Ziolo, Phys. Rev. B. 46, 10563 (1992). 17’ R. Kuentzler, Solid State Commun. 83, 989 (1992). 1 7 6 B. Frick and D. Richter, Science 262, 1939 (1995).

224

ANTONI PLANES AND L L U ~ SMAROSA

0.01

0.1

FIG.VI.3. Ratio between the vibrational contribution to the heat capacity ( C f b ) ,obtained after subtraction of the electronic contribution to the experimental data, and the Debye heat capacity (CP’), as a function of the dimensionless temperature T/B,. C f b has been determined from experimental data given in D. Abbe, R. Caudron, R. Pynn, J . Phys. F: Metal Phys. 14, 1117 (1984).

to the noncentral character of the interaction force^.'^' The marked anisotropy, as discussed in the previous section, is also typical of systems undergoing a martensitic transition, while the importance of bonding directionality has been recognized recently for bcc transition metals such as Zr,178which exhibit a martensitic transition at very high temperature. It is generally accepted that in these highly anisotropic materials there are various “characteristic temperatures,” which give rise to the “anomalous” heat capacity. In many cases the observed behavior can be accounted for by adding an Einstein contribution to the Debye heat capacity function.’ 79 This has also been assumed for the heat capacity C of the bcc phase in Cu-based alloys.’73*180 That is,

c = 3Nk,{aF,(x) + (1 - a)F,(x,)}, 177

J. A. Krumhansl, Phys. Rev. Lett. 56, 2696 (1986).

179

N. Bilir and W. A. Philips, Philos. Mug. 32, 113 (1975). R. Romero and M. Ahlers, J. Phys. Condens. Matter 1, 3191 (1989).

’” M. Porta and T. Castan, Phys. Rev. B. submitted.

(32)

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

225

where a is the fraction of Einstein modes and F,(x) is the Einstein function given by FE(x)= [xzex/(ex- l)’], with x = O,/T, and O,, the Einstein temperature. In this case, in the 9 versus T representation, the position of the maximum determines the Einstein temperature O,, and the value of W at this point is related to the fraction u of Einstein modes. For the system considered in Figure VI.3 (Cu; 27.2 at% Zn, 13.2 at% Al) the following values are obtained: O D = 290K, 8, N 65K, and u N 0.03, which are quite representative values for Cu-based alloys. The observed behavior of the heat capacity of /? Cu-based alloys is qualitatively consistent with the characteristics of the phonons in these systems. Indeed, the combination of unusual low energy and “flatness” of the T A , acoustic branch (see, for instance, Figure V.9), originating a low-frequency (small) peak in the phonon density of states (see Figure V.13), produces a relatively large contribution to the heat capacity. Therefore the low-temperature heat capacity anomaly has its origin in the vibrational modes related to the instability leading to the martensitic transition. The zone-boundary frequency wzB for the T A , phonon branch, in the studied class of materials, is close to 5meV This corresponds to a temperature hulk, 1: 60K, which is perfectly comparable with 8, deduced from heat capacity measurements. For the P-phase the following phonon density of states is a simple representation consistent with the ideas discussed above:

where g,(w) and g E ( o ) are Debye and Einstein density of states, satisfying g E ( o ) d o = 3Na respectively. That is, = 3N(1 - a) and g,(o)= 9N(1 - a)(02/o%)if O < W , (0,is the Debye frequency given by kBO,/h) and 0 if w ~ w , ) ;and g E ( o ) = 3Nad(w - oE)(wE is the Einstein frequency taken to be ozB). This rough approximation does not give a very realistic description of the actual phonon spectra (as given for instance in Figure V. 13 for the Cu-Al-Be alloy). Nevertheless, the description is thought to be reasonable for high-temperature ( T > 6,) thermodynamic calculations. In the close-packed phase the characteristic frequency of the P-phase low-energy modes increases as a consequence of181(i) the symmetry change (which implies a change of coordination numbers) and (ii) a change in the strength of the bonds between atoms. Apart from these low-energy phonon states, the remaining modes are reasonably well characterized by the same Debye temperature of the P-phase. At high enough temperatures a plausible

s$ g,(w)do

J. Friedel, J. Phys. Lett. 35, L35 (1974).

226

ANTONI PLANES AND LLUIS

MAROSA

estimation of the excess of vibrational entropy of the /?-phase is

where w' is a characteristic frequency for those modes in the close-packed phase that derive from the bcc TA, modes. From this point of view the excess of entropy of the /?-phase is considered to have its origin in the low-lying TA, modes. This assertion is supported by the fact that AS amounts to the same value for different samples transforming over a broad range of temperatures, even at very low temperatures (well below the Debye temperature) when the only modes likely to be excited are those having very low energies (those in the TA,-branch). This scenario is also consistent with the fact that TA, branches are rather similar for different Cu-based alloys, regardless of their composition and their transition temperature. Romero and PelegrinaI6' have assumed that the frequency w' coincides with the Debye frequency in the P-phase. Within this approximation, taking c1 = 0.03, wE = 60K, and w' = wD N 290K, it is seen that the vibrational excess of entropy of the /?-phase amounts to about 1.1 J/Kmol, which is not too far from the value measured in Cu-based systems. While rendering reasonable results, this is probably too crude an approximation because, as discussed in Section V.7, these modes (which correspond to the transverse [lOO][OOl] phonon branch of the martensitic phase) have an energy of 12meK Low-temperature heat capacity data and measured phonon dispersion curves are only available for a very reduced number of systems. On the contrary, elastic constants have been reported for numerous shape-memory alloy systems of different compositions. Therefore, it is of interest to relate the ratio between the frequencies wE and w' to the elastic constants of the /?-phase. Hence, let C, and C, be the bond strength between nearest and next-nearest neighbor pairs, then in the central force approximation it is wE and 0 ' - &, where C , / reasonable to assume that,"' C, C,,/C' = A (A is the elastic anisotropy of the /?-phase). It is then obtained that

-

-

-

ASvib= 2 NkBuIn mA,

(35)

where m is a factor depending mainly on the symmetry of both phases. Romero and PelegrinaI6' have calculated this quantity assuming that all phonon modes in the close-packed phase can be described within the Debye theory and that the TA, phonons in the P-phase have a sinusoidal

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

227

dispersion relation. They have found that for Cu-based alloys m N 1.2. For Cu-based alloys, taking for the elastic anisotropy A = 13.5, which is a universal value at the martensitic transition in Cu-based alloys'00 (see Section V.6.b), ASvibN 1.2 JfKmol is obtained in good agreement with the previous estimation given using the characteristic frequencies. Owing to the small value of the electronic contribution, to a very good approximation, the excess of entropy of the B-phase, responsible for its high-temperature stability, has a vibrational origin that arises from the low-energy T A , phonons and is practically independent of composition. Therefore, the martensitic transition in Cu-based alloys can be considered as a purely vibrational entropy-driven transition. Within this context, Morris and G ~ o d i n g ' ~ ha ~ .ve' ~proposed ~ a model in which the hightemperature phase is stabilized by its large phonon-entropy. This is achieved by means of an anharmonic interparticle coupling that has practically no effect at high temperature, but, at low T, confines the particles in such a way that large-amplitude vibrations are inhibited. The change of effective interaction leads to a change in the phonon dispersion relations. The model reproduces an entropy-driven first-order transition and has been used to test the quasiharmonic approach in determining the entropy change between both phases. It is seen that the system behaves essentially harmonically, except in a very small temperature region near the transition. Exact calculations' 8 2 show the presence in this region of large-amplitude fluctuations connecting high- and low-temperature phases. These results permit the conclusion that an estimation of the entropy change within the quasiharmonic approximation, as given preceding, is adequate. A restriction of the model arises from the local character of the symmetry breaking term. In order to circumvent this limitation, Vives et a l l a 3 have proposed a pair interaction spin model. The model is a modified version of the Blume-Emery-Griffith (BEG) model.'84 It is suitable to describe a first-order transition between two ordered phases H (high temperature) and L (low temperature). In the original BEG model, the H-phase is not degenerate and the L-phase is twofold degenerate. The authors have generalized the degeneracy of the H-phase to any value p 2 1. This parameter p is supposed to effectively describe the larger number of vibrational degrees of freedom of the bcc phase. The main results of the model are: (i) the H-phase is stabilized for increasing values of p, (ii) the first-order transition region is increased for increasing p, and (iii) for p > 1, the entropy change at the first-order transition is found to be independent of the J. Morris and R. J. Gooding, Phys. Rev. B. 43, 6057 (1991). E. Vives, T. Castan, and P. A. Lindgird, Phys. Rev. B. 53, 8915 (1996). M. Blume, V. J. Emery, and R. B. Griffiths, Phys. Rev. A. 4, 1071 (1971).

228

ANTONI PLANES AND L L U ~ SM A ~ O S A

transition temperature, although the latter is strongly dependent on the model parameters. These results are, qualitatively, in remarkably good agreement with the experimental results for Cu-based alloys. Especially interesting is point (iii), which means that entropy change is expected to be independent of composition as far as composition changes do not suppose any modification of the symmetries of the problem. The different entropy change for the fl+ 18R and B -+ 2H transitions must, in this context, be attributed to a different fraction CI of low-energy T A , modes in both cases. Recalling equation (35), this conclusion is consistent with the constant value of A at the transition. 11. ENERGY CHANGE From DSC measurements, the enthalpy change between high- and lowtemperature phases is obtained as

Taking into account the very small transition volume change, the enthalpy change coincides with the energy change at the phase transition, that is AH = AE. For a given martensitic crystallographic structure, AS has been shown to be independent of composition and therefore, it is expected that AE and TMfollow the same composition dependence. This is illustrated in Figure VI.4, where AE is represented as a function of T, for Cu-Zn-A1 and Cu-Al-Be alloys that transform to the 18R structure, and for Cu-Al-Ni alloys that transform to the 2H structure. It is expected that AE is dominated by a structural contribution originating from the change of coordination and bonding at the transition. Nevertheless, a vibrational contribution AEvib cannot be neglected. This contribution can be roughly estimated within the same approximations as considered in the evaluation of the corresponding contribution to the entropy change. That is, AEvib = foT A C d T ,

(37)

where the difference of heat capacities A C between both phases can be approximated by A C = 3Nk,a{FE(x) - F E ( x ' ) ) ,where F E ( x ) is the Einstein function and x = ho,/k,T= 6 E / T and X I = hw'/k,T= WIT. After integration the obtained result is

VIBRATIONAL PROPERTIES O F SHAPE-MEMORY ALLOYS

229

500

400

- 300

c

z

1,

-> ;200 100

0

FIG. VI.4. Energy difference between martensitic and /l structures for different Cu-based alloys: Cu-Zn-A1 (filled circles); Cu-Al-Be (open squares), Cu-Al-Ni (open circles). Cu-Zn-A1 and Cu-Al-Be alloys transform to a 18R structure, and Cu-Al-Ni to 2H.Continuous lines are linear fits with slopes of 1.30 J / K mol and 1.59 J / K mol. These values compare well with the measured values of AS for /l + 18R and /l + 2H transitions respectively. From A. Planes, L1. Maiiosa, D. Rios-Jara, and J. Ortin, Phys. Rev. B.45, 7633 (1992).

This is a positive contribution (0, < 6') that favors the appearance of a close-packed phase which, to within a good approximation, is expected to be composition independent. At the transition point the following thermodynamic relation is satisfied:

Therefore, taking into account equations (34) and (38), it turns out that the transition temperature can be expressed in the following form:

230

ANTON1 PLANES AND LLUiS MAROSA

A2

800'L

-2

a

Y

600

3

c

2

400

e 200

0

2

4

6 6 at% Be

1

0

FIG. VI.5.Phase diagram of metastable phases of Cu-Al-Be alloys close to the eutectoid composition, showing the order-disorder (solid symbols) and martensitic (open symbols) lines. The order-disorder line shows very little composition dependence when compared with the martensitic transition line. Compare this phase diagram with that given in Figure 11.1 for CU-ZD-AI.

where the composition dependence is contained in the term AEc. If this term is supposed to be purely structural, a reasonable evaluation can be performed from ordering energies which, in turn, are related to order-disorder transition temperatures. In Cu-Zn-A1 the martensitic transition temperature and the temperature of the B2 er L2, transition follow very similar composition dependence (see Figure 11.l), which supports the purely structural origin of AEc. This is not, however, a general situation and such a simple explanation does not hold even for other Cu-based alloys. This is, for instance, the case of the Cu-Al-Be alloy. For this alloy the addition of Be around the Cu,Al composition has a dramatic effect on the martensitic transition temperature, while the order-disorder transition is practically unchanged (see Figure VI.5). This has been interpreted from the fact that the effective interaction between Cu and A1 atoms is not modified by the addition of Be.'85 The strong effect of Be can be understood within the

VIBRATIONAL PROPERTIES O F SHAPE-MEMORY ALLOYS

23 1

mass-disorder effect proposed by LindgH~-d,’~ which has been mentioned in Section 11. This gives, as in the purely structural case, a quadratic dependence of T, on composition.* An interesting feature to be discussed here is the effect of configurational order on T M . This problem has been exhaustively studied in Cu-Zn-A1 and to a lesser extent in Cu-Al-Ni alloy. Changes of the ordering state at temperatures close to the martensitic transition can be accomplished by means of a suitable heat treatment. It consists, for instance, in quenching from a temperature at which the equilibrium ordering degree has been achieved. The effect of such a heat treatment depends on the atomic mobility properties in the considered system.’ 8 6 In metallic alloys, atomic mobility takes place through a vacancy mechanism. At each temperature, the time necessary to reach equilibrium after a small perturbation is determined by an Arrhenius law z = zo exp(E,/k,T), characterized by the pre-exponential factor zo and the activation energy EM. zo depends on the vibrational characteristics of the system and on the vacancy concentration which, in turn, is temperature dependent. For the above reasons the following must be considered: (i) below a given temperature TL, the low-temperature equilibrium will not be reached within a reasonable annealing time within the time scale of the laboratory; (ii) above another limit TH the hightemperature equilibrium state cannot be frozen-in for a given quenching rate. In Cu-Zn-Al, for typical available quenching rates, the value of THis of the order of 600K. The value of TL depends on the vacancy concentration, which is also modified by quenching. In any case, below room temperature the kinetics of evolution to equilibrium become very slow. The change of the martensitic transition temperature as a function of the quenching temperature T, in four Cu-Zn-A1 alloys is shown in Figure VI.6. The observed change of TM has been attributed to a modification of the next nearest-neighbor long-range ordering degree (L2,) induced by the quench.18’ Similar behavior has also been found for Cu-Al-Ni alloys.Is8 Concerning the results shown in the figure, it is interesting to note that for the alloys A and B, the B2 c* L2, transition temperature satisfies TLz,< TH and a monotonous increase of IATJ up to T, is obtained. Actually, x-ray’89 M. A. Jurado, T. Castan, L1. Maiiosa, A. Planes, J. Bassas, X. Alcobe, and M. Morin, Philos. Mug. A . 75, 1237 (1997). * The dependence of TM with composition cannot be explained taking into account a dependence of vibrational properties on composition. Actually, phonon dispersion curves in Cu-Al-Be are quite similar to those reported for other Cu-based alloys. l a 6 M-. C. Cadeville and J. L. Morh-Lopez, Phys. Reports 153, 331 (1987). la’ A. Planes, R. Romero, and M. Ahlers, Actu metull. muter. 38, 757 (1990). E. Cingolani, J. van Humbeeck, and M. Ahlers, Merull. Muter. Trans. A . 30,493 (1999). l a 9 T. Suzuki, Y. Fujii, and A. Nagasawa, Mat. Sci. Forum 56-58,481 (1990).

232

ANTONI PLANES AND L L U ~ SM A ~ ~ O S A

FIG. VI.6. Absolute value of the shift in the martensitic transition temperature TM measured just after a quench from q ,plotted as a function of TJT,,,, for four different Cu-Zn-A1 alloys with different ordering TLZ1 temperatures; alloy D, TLZ1 = 640 K (filled triangles), alloy C, TLZ1 = 600 K (filled squares), alloy B, TL2,= 500 K (open circles), and alloy A, TLzl= 460 K (filled circles). For the alloys with lower TL2,,the configurational state at T, can be frozen by the quench.

and neutron diffraction' 90 experiments confirm the proposed interpretation. It has been shown that the intensity of the superstructure peak associated with the L2, atomic order (proportional to the square of the order parameter) is modified by the quench. In addition, an excellent correlation between the intensity and the transition temperature has been found. These results indicate that in alloys A and B, the L2, order can be completely suppressed by the quench. In contrast, in alloys C and D, TLzl> TH and hence the equilibrium state at cannot be suppressed for T, 2 TH.In this case, !ATMIdisplays a maximum before the order-disorder transition. It has been shown that the quench does not modify the entropy change at the martensitic tran~ition.'~'This means that a modification in the atomic order only affects AEc. This result is consistent with the fact that AEc L1. Maiiosa. M. A. Jurado, A. Gonzalez-Comas, E. Obradb, A. Planes, J. Zarestky, C. Stassis, R. Romero, A. Somoza, and M. Morin, Actu muter. 46, 1045 (1998). 19' A. Planes, J. L. Macqueron, R. Rapacioli, and G. Guknin, Philos. Mug. A. 61, 221 (1990).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

233

is composition dependent (AS is not) in this class of alloys because it is expected (for Cu-Zn-Al) to be purely structural. An interesting problem to be mentioned concerns reordering effects in the martensitic phase. This is possible if martensite forms at a high enough temperature. In general, the global effect shows up through an increase in the temperature of the reverse transition to the parent phase.19' This phenomenon is usually known as stabilization of the martensite and is a consequence of the change in the free energy of martensite compared to that of the P-phase. More interesting is the fact that reordering taking place during an aging process affects the stability of distinct martensitic variants differently'93s194It is acknowledged that during aging the atomic rearrangements result in a modification of short-range ordering (ordering within the same sublattice), which conforms the martensitic symmetry. 9 5 This effect is at the origin of the rubber-like (or ferroelastic) behavior displayed by some martensitic materials. Indeed, aged martensites can be deformed like rubber, which means that large macroscopic deformations are recovered when stress is released. VII. Modelling

The most widely used model to describe structural phase transitions is the soft-mode model proposed independently by W. Cochran and P.W. Anderson (see Ref. [196]). In this approach, a phonon frequency (or elastic constant in the case of a q = 0 mode) is expected to reach a zero value at the transition temperature. Therefore, the phonon becomes harmonically unstable and consequently the lattice is spontaneously displaced in this mode to a finite amplitude, which is limited by higher-order nonlinear contributions to the free energy. Actually the predictions of the soft-mode theory have been verified for a restricted number of structural transitions (SrTiO, or K,SeS0,).58 As we have seen in the previous sections, the pretransitional behavior of shape-memory alloys is rather different. In this case the T A , branch reduces its energy with decreasing temperature but at the transition point the softening is far from complete for any of the modes. Furthermore, in most cases extrapolation to a zero value of a characteristic frequency or elastic A. Abu Arab and M. Ahlers, Acta metall. 36, 2627 (1988). M. Ahlers, G. Barcelo, and R. Rapacioli, Scripta metall. 12, 1075 (1978). 194 R. W. Cahn, Nature 374, 120 (1996). 195 X. Ren and K. Otsuka, Nature 389, 579 (1997). lg6 W. Cochran, Adv. Phys. 9, 387 (1969). lg2

193

234

ANTONI PLANES AND L L U ~ SM A ~ O S A

constant yields a negative This means that within the quasi-harmonic approximation the /?-phase is stable at all temperatures and therefore the soft-mode concept is inapplicable to this class of martensitic transitions. It follows that one must go beyond the quasiharmonic approximation in order to obtain information concerning the mechanism driving the distortion associated with the low-temperature phase.19’ With this objective in mind two main approaches have been proposed in the literature: strain-phonon coupling models’ l 2 and approaches based on the influence of defects (localized soft-mode models).198 Both can be considered as generalizations of the standard soft-mode theory incorporating, as an important ingredient, the relevant anharmonic effects. Before describing these theories, for the sake of completeness, we will briefly mention the main features of the soft-mode approach based on the Landau theory.’99 The starting point is thus the identification of the order parameter. The symmetry change of the crystal at a structural phase transition is associated with the displacement field of the atoms from their stable positions in the parent phase. In general, these displacements can be expressed in terms of a linear combination of normal coordinates of the phonons of the high-symmetry structure. When only q = 0 modes are involved, the transition is purely elastic and symmetry-adapted strain tensor components (see Section IV.4.a) are the order parameters. Usually, only a very limited number of normal modes belonging to the soft phonon branch are enough to describe the displacements. Thus, the order parameter in a displacive transition can be expressed by the amplitude of the frozen-in soft mode. If d is such an amplitude, the simplest Landau-type free-energy expansion reads

where the “spring constant” m * 0 2 for the undistorted phase changes linearly with temperature, i.e., m*w2 = a ( T - T,). T, > 0 is, approximately, the transition temperature. From this free energy it follows that: (i) above T,, the average value (d)= 0 (undistorted phase), (ii) below T,, ( d ) ( T - T,)” 2, i.e., the amplitude of the distortion increases “continuously” from zero (therefore corresponding to a second-order transition) and the lattice displaces smoothly into the new structure, and (iii) the susceptibility diverges at T,. The Landau theory does not include fluctuations; when these are considered and properly treated by renormalization group theory,58

-

J. A. Krumhansl and R. J. Gooding, Phys. Rev. B. 39, 3047 (1989). P. C. Clapp, Phys. Stat. Sol. ( B ) 57, 561 (1973). ‘ 9 9 J. C. Toltdano and P. Toledano, The Landau theory of phase transitions, World Scientific, Singapore, (1987). lg7 19’

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

235

there are modifications in the temperature dependence of the order parameter, and critical fluctuations show the proper universal class behavior. In contrast to the above, in martensitic transitions, phonon frequencies (and elastic constants) only partially soften, and do not indicate harmonic instability. There is a finite discontinuity in the order parameter at the transition; critical fluctuations are missing and susceptibility does not diverge. 12. STRAIN-PHONON COUPLING MODEL

The important feature to remember is that for martensitic transitions there is a lattice correspondence between the bcc and the martensitic phases. Crystallographic (phenomenological) theories provide an exact correspondence between bcc and martensitic structures, but they do not provide any information about the actual mechanism of the martensitic transition. In a general way, the set of displacements may be made up of a simultaneous combination of elastic strains with (static) lattice modulation waves (or shuffles). Phonon dispersion curves display, as shown in Section V.7, information concerning the specific path followed by the atoms at the transition; dips or flattening in the phonon spectra and low elastic constants. Moreover, in several simple cases, first-principle total energy calculations200 confirm the availability of the expected paths. This is, for instance, the case for bcc Zr, which transforms from a bcc to a hcp structure. For this system Ye et al.’46 have shown that both a zone boundary N-type [11O]TA, lattice wave and a uniform shear on the same phonon branch are concerned with the transformation path. Lindgird and Mouritsen’ have proposed that the first-order character of the transition is a consequence of the anharmonic coupling between the anomalous phonon and the homogeneous deformation. The simplest model containing the most significant physical features takes the amplitude d of the phonon as the primary order parameter and the magnitude of the homogeneous deformation E as a secondary order parameter.”’ Actually, this is the correct choice if the structure has a tendency to become unstable with respect to the shuffle rather than the homogeneous shear strain. From first-principle calculations this has been shown to be true in the case of the group-IV metals such as Ti and Zr.”’ However, there is no such

a

K. M. Ho and B. N. Harmon, Mat. Sci. Engng. A127, 155 (1990). J. A. Krumhansl, Solid State Comm. 84, 251 (1992). ’02 A. Saxena, M. Sanati, R. C. Albers, Proc. Int. Conference on Martensitic Transformations, Barniloche (Argentina), 1998. Mat. Sci. Engng. A, 273-275, 226 (1999). ’01

236

ANTONI PLANES AND L L U ~ SMAROSA

information for other systems that transform martensitically. Ren and Otsuka1I7have argued that E is the primary order parameter in the case of the Ti-Ni system. Their model also contains the adequate coupling term between E and d and thus the results obtained are comparable to those of the model developed following when the roles of E and d are exchanged. For a bcc to hcp structural change, symmetry arguments allow the following terms in the Landau expansion of the free energy in terms of these two-order parameters:

where m*w2 N a ( T - T,). Here T, is the limit of stability of the undistorted (d= 0, E = 0) phase, and C’ is the elastic constant associated with E. The parameter K anharmonically couples the uniform strain and the modulation. The terms b and c are not expected to vary rapidly with temperature and will be assumed to be positive constants. Minimization of F with respect to E gives 1

’ { ’c”:}

Re,-,= Z m * w 2 d 2 + - b-4

1

d4+-cd6. 6

(43)

The interesting feature in this free energy is that a small value of C’ may cause the effective fourth-order term 6 = b - ( ~ K ’ / C ’to) be negative, and consequently to give rise to a first-order phase transition. This transition will take place at finite values of w’ and C’, before w 2 + 0. In order to analyze the generic behavior of the model in a relatively easy way, it is convenient to rescale the free energy to a dimensionless form, whence a single parameter, containing relevant physics, controls the transition. We define F0 = l6I3c-’, d o= (Ib”lc-’)’/’, and a = m*w’6-’ and we rescale in the form proposed in:201,203 f = Fef,-/Fo and 2 = &/do. It is then found that

where it is seen that there is a universal behavior of f with 2, which is controlled by the single parameter a. The following has been obtained:”l

+

(i) When sgn 6 = 1 (positive), only a second-order transition is possible if a + 0. (ii) When sgn 6 = - 1 (negative), for a > $, only the parent phase 2 = 0

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

237

is stable. For < cc < 4 there is a stable minimum at 2 = 0 and two metastable minima at some finite values f2. (iii) When sgn 6= - 1 and cc = & , there is a first-order transition at 2 = $12. (iv) For cc < A, the product phase (2#O) is thermodynamically stable. (v) For cc < 0, the parent lattice becomes dynamically unstable. From condition (iii) the equilibrium temperature for the martensitic transition can be obtained as a function of the model parameters. It can be explicitly written in the form

w(T,)

=

( X- b).

1 6 ~C'(TM)

(45)

Taking into account the linear dependence of o2 with temperature, the following relationship is obtained:

A plot of the free energies given in equations (42) and (43) is shown in Figure VII.l. Recently, the suitability of this simple model for martensitic transitions in different materials has been a n a l y z ~ e d . ' In ~ ~Figure * ~ ~ ~VII.2 (a) and (b), it is shown that the dependence between o and C'-' at T, predicted by equation (45) is satisfied in Cu-based and Ni-A1 alloys. For Ni-A1 the choice of the frequency o is unambiguous; it corresponds to the phonon mode showing the dip in the T A , branch. In contrast, the corresponding choice for Cu-based systems is less evident. Actually, Figure VII.2 depicts, as a function of C ' - ' , the behavior of o at qZBand w at 3qze. It is apparent that results are rather independent of the position at which o is taken. Indeed, this is consistent with the fact mentioned before, that the selection of the martensitic modulation in Cu-based systems is not determined by a particular anomalous phonon of the TA,-branch, but rather influenced by specific features of the internal energy of the system. Figure VII.3 illustrates, for three different Cu-based alloys, that the predicted relationship between T, and C' (eq. (46)) is also reasonably well satisfied. These results show that the model presented is suitable to qualitatively provide a unified view of the martensitic transition in Cu-based and Ni-A1 alloy. 203 204

F. Falk, Z . Phys. B. 51, 177 (1983). A. Planes, L1. Mafiosa, and E. Vives, unpublished results.

238

ANTONI PLANES AND L L U ~ SMAROSA

FIG.VII.l. Schematic representation of the free energy for the martensitic transition (eq. (42) as a function of the primary and secondary order parameters .d and E at the equilibrium transition temperature. (a) 3-d representation, (b) projection to the (I, E ) plane, and (c) effective free energy corresponding to the path E (I) = (K/C’)&. This path is shown as a dashed line in (b). It is obtained after minimization of the free energy 9 with respect to E. Model parameters correspond to the fitted values for Cu-based alloys given in A. Planes, L1. Mafiosa, E. Vives, Phys. Reo. B. 53, 3039 (1996).

The preceding theory has been extended to describe the transition from a bcc to other specific close-packed structures. Following the development given in Ref. [113] by Gooding and Krumhansl, we explicitly consider the case of the transition from the B to the 9R structure. The transition path (see Figure 11.4) can be described by means of a linear combination of the symmetry-adapted elastic strains q , and q1 (see eq. (11)) and the static phonon displacement +{ 1lo} on the T A , branch. q, shears the (110) planes along the k[lTO] direction and q 1 contracts the (110) planes along the [OOl] direction and stretches them in the & [lTO] direction. They are both

VIBRATIONAL PROPERTIES O F SHAPE-MEMORY ALLOYS

g u

5

239

4.6t///, 4.4

3

0.138

0.140

0.142

0.144

LL

1.3 1.2 0.20

0.30

0.40

C’(r,I-’ ( G Pa-’ I FIG. VII.2. Frequency of the incipiently unstable phonon as a function of the inverse of the elastic constant C. (a) For Cu-based systems; Cu-Al-Be alloys transforming to a 18R structure (circles), Cu-Al-Ni transforming to a 2H structure (triangles). Filled symbols: the frequency w corresponds to q = qzs in the T A , phonon branch, open symbols: the frequency corresponds to q = iqzs. Both the frequency and C’ are evaluated at the transition temperature of each alloy. From A. Planes, L1. Maiiosa, and E. Vives, Phys. Rev. B. 53, 3039 (1996). (b) Same representation for Ni-Al. The frequency in this case corresponds to the phonon-mode showing the dip in the T A , branch. It is important to mention that both w and C have been measured in the same sample in the case of Cu-based alloys. Values of w and C for Ni-A1 have been collected from different references in the literature. The representation given in the figure (b) needs the estimation of some data by proper interpolation. Therefore, results are affected by larger errors than for figure (a). The error bars in figure (b) are estimated from the interpolation procedure.

associated with the elastic constant C‘ (see section IV.4.a). The static phonon displacements can be expressed as

A{ 1lO}TA,

where q = i(110). As usual, this mode may be expressed in terms of the

ANTONI PLANES AND L L U ~ SMAROSA

240

400

300

--

200

Y

3 100

0

-1 00 0.002

0.004

( b / 2 -~ l/C(T,)f ~

0.006 (GP6')

FIG. VII.3. Relationship between transition temperature TM and the inverse of the elastic constant C evaluated at the transition temperature for different families of Cu-based alloys. Circles: Cu-Al-Be, triangles: Cu-Al-Ni, squares: Cu-Zn-Al. The factor 6 / 2 ~ '= 0.08 GPa-' is the same for all families. Lines correspond to the predictions based on evaluations of model parameters. The fact that data for the different alloy families spread in different curves arises from their different linear stability limit (T,). From A. Planes, L1. Mafiosa, and E. Vives, Phys. Reo. B. 53,3039 (1996).

two-component order parameter

+ = _Qzeib.

(48)

The symmetry properties of the order parameters have been analyzed in detail in Ref. [113]. They provide the following terms in the free energy:

+ s,: + & s h ? - 3 d ) + bC,(S22 + s32 +1 * 2 2m w [+I2 + ~ U , I + I ~+ ivOl+l6 + b ~ , c ++~(+*Y1+ Qwol+ls + ~ s ~ +[ (+*)31, + ~

F = 4C"d

(49)

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

24 1

where, as in the previous model, the “spring constant” m*02 is assumed to depend linearly on temperature. It is interesting to note that only the shear strain q2 is directly coupled with the phonon. Minimizing with respect to q 1 and q2 gives C

‘I1 -- C‘3 ‘I: ‘I2

2K C‘

COS(34),

---&3

(50)

where C, is the combination of third-order elastic constants given in eq. (16). It is worth noticing the fact that C3 < 0 (see section V.8) implies that qI > 0 in the low-temperature phase, as a direct consequence of the symmetry-restricted anharmonic form of the interactions coupling q1 and ‘I2.

The effective free energy along the transformation path given by equation (50) is then obtained in terms of the two components d and 4 of the order parameter .)I 1 1 9=- m * 0 2 d 2+- u 0 d 4 6 4

“> ’>

+ (3+-6 {(3 6 C ‘ 3 C

cos(64)}d6

+ 1w 0 d 8 . -

8

(51)

Because the elastic constant C’ is small, it is reasonable to assume that (rc2/C’)> (vl/3). Then the free energy is minimized when the phase takes the following values: 4 + = 0, +(2n/3), 4- = k(n/3), 7c. The different values of 4 determine six possible variants and correspond to the 1 values taken by cos(34). We note that q2 changes sign if 4 + changes to 4-, while q 1 is not modified. We recall that the change of q2 corresponds to a shear in the (110) planes in the [ l i O ] direction. The following effective free energy is finally obtained in terms of the phonon amplitude: 1 1 1 1 Feff= m * 0 2 d 2 + u 0 d 4 + - v e f f d 6 + - wad*, 4 6 8

where v,

is given by V,ff

=v0+2v1-

12,.

K2

c

(53)

242

ANTONI PLANES AND L L U ~ SM A ~ O S A

Stability requires that wo > 0, and uo is taken to be positive because there is no reason for the contrary; u e f f can be negative due to the small value of C’. In particular, if u e f f < ucrif, where ucrit= - 3 d G , as in the simpler model presented above, a first-order transition is possible before o goes to zero. This transition has its origin in the anharmonic coupling between q 2 and $. At the transition point a relationship between o and f 2 - l must be satisfied.204.Compared with the expression obtained in the previous model (see eq. (45)), in the present case the relationship is nonlinear. For instance, in the strong coupling limit it is found that w Z i 3 - - u e f f . Notice that this behavior is not incompatible with the experimental results presented in Figure VII.2 (a) and (b), which correspond to restricted ranges of o and C‘ values and where nonlinear contributions may fall within experimental uncertainties. An interesting feature of the preceding model is the possibility of studying stacking faults. This needs, however, to include spatial correlations of the order parameter $. If [ is a coordinate in the direction of q([ 1lo] direction), this can be done by considering the gradient term l~3$/C3[1~. Taking only the phase terms, the relevant free-energy expansion that needs to be considered is

which is minimized by functions

~

4 = 4([),which are solutions of

d2c,b 1 = - sin(6c,b), d[’ 6d2

(55)

where d-’ = 36/g((2rcZ/C’)- (u1/3))d4.The equation obtained is precisely the time-independent Sine-Gordon equation whose solutions are known as domain walls, kinks, or solitons. A detailed study of the properties of such an equation can be found in Ref. [205]. As a simple example, we consider here the following solution:

where to is a constant. If the + sign is chosen, c,b([ -, - co) = 0 and 4([ + co) = 743, which means that for [ << i0 a ++ distortion is obtained 205

A. R. Bishop, J. A. Krumhansl, and S. E. Trullinger, Physicu D.1, 1 (1980).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

243

while for [ << loa 4 - distortion occurs. More complicated situations can be considered, but this is out of the scope of the present review. Gooding and Krumhansl have further formulated a model for the case of the B + 7R transition and the results have been applied to analyze the alloy system.' l4 peculiarities of the martensitic transition in the Ni,Al, -, In this case they considered two independent static phonon displacements = $(110)) and its first as primary order parameters: I)1= dlei(qn.r+dl)(qo where $, is not associated with any soft-mode harmonic I), = d,ei(2q~.'+4~', anomaly. The coupling to the uniform shear q, is given by a term q2$:$;. The reason was to consider if this coupling could explain the change of location of the dip in the T A , branch (explained in section V.7) from a wave vector 4: = $ in the stoichiometric NiAl system to a wave vector ( M 4 ( f $) in the alloy with x = 62.5%. The important conclusion was that to account for a martensitic structure with a wavevector modulation different from that of the precursor phonon anomaly, there must be a wavevector dependence of the coefficients of the Landau expansion. As far as we know, no specific formulation has been proposed along these lines until now. An interesting result of the strain-phonon model for the p -,9R transition is given by equation (50), which shows up the crucial role of the third-order invariant C , in determining the martensitic distortion. In order to relate the predicted change of structure with the actual one (which can be computed from the parameters of the unit cell of the martensitic structure), it is convenient to use the following unconventional cell for the cubic structure:'I4 a, =a(-$

+ 2)

a, = a2 A

a3 = az,

(57)

where a is the simple cubic lattice parameter. The homogeneous strains deform this unit cell according to the following displacement pattern:

This procedure has been followed for Ni-A1'I4 by making use of the martensitic unit cell determined by Martynov et all4' and also for Cu-Znby using the martensitic structure given in Ref. [119]. It was found that q1 = 0.011 and q, = -0.023 for Ni-A1 and qI = 0.025 and q, = -0.044 for Cu-Zn-A1 reproduce the lattice parameters and monoclinic angle of the 7R and 18R martensitic structures, respectively, to within a few percent.

244

ANTONI PLANES AND L L U ~ SMAROSA

Using these values it is interesting to compare the value of the TOEC combination C , predicted by equation (50) to the experimental one at T,. For Cu-Zn-Al, it is found that the predicted value C, = - 86 GPa is slightly larger (in absolute value) than the measured value C,= -60 GPa (see section V.8). Taking into account the approximations involved in the computation of q 1 and q2 the agreement turns out to be rather satisfactory. Another interesting feature of the model to be analyzed concerns the role played by distortions that do not affect the symmetry of the martensitic phase. This has been considered for the p -,9R transition in Ref [206]. The results state that the effect of the nonsymmetry breaking order parameters associated with the elastic constant C,, and with the bulk modulus B is the reduction of the energy barriers between both the parent and martensitic phases. A rigorous multi-order-parameter Landau theory has been recently , ~ ~ ~ can be viewed as a proposed by G. B. Barsch and c o l l a b ~ r a t o r s which generalization of the two-order-parameter models presented in this section. The theory is formulated in reciprocal space and the free energy is derived from group theoretical principles taking into account experimental data. Within this approach all possible phases (including martensitic variants) allowed by symmetry are, in principle, available. Actually, the path followed by the system is determined by the alloy composition and thermomechanical treatment (which affects the parameters of the free energy). This theory has been successfully applied to describe the specific features of the martensitic transitions in Ni-Ti and Au-Cd shape-memory alloys (see Table II.1).20* Finally, it is important to mention that an alternative to the Landau strategy for the study of these kind of structural transitions is the formulation of lattice spin models and the use of standard treatments of statistical mechanics (including mean field and Monte Carlo simulation techniques). Within this framework, Lindglrd and Mouritsen have proposed a 2D Heisenberg model for a square to triangular transition that mimics the , ~ ~ model ~ was later symmetry change at the p to 2H t r a n ~ i t i o n . " ~This simplified to that of the three-state Blume-Emery-Griffith model, retaining the essential physics of the original.'83 A different three-state spin model has been proposed by Yamada and collaborators,210*2'1which can also be regarded within the group of the '06

R. J. Gooding, Y. Y. Ye, C. T. Chan, K. M. Ho, and B. N. Harmon, Phys. Rev. B. 43, 13626

(1991).

A. Saxena, G. R. Barsch, and D. M. Hatch, Phase Transitions 46,89 (1994). G. R. Barsch, T. Ohba, and D. M. Hatch, Proc. Int. Con$ on Martensitic 7kansforrnations, Bariloche, Argentina (1998). Mat. Sci.Engng A , 273-275, 161 (1999). 209 P.-A. Lindglrd and 0. G. Mouritsen, Phys. Rev. B. 41, 688 (1990). 'lo K. Fuchizaki, Y. Noda, and Y. Yamada, Phys. Rev. B. 39, 9260 (1989). '07

'08

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

245

models containing a dominant strain-phonon coupling term. These authors postulated the existence of local embryonic fluctuations and the energy of the system is expressed in terms of an embryo creation energy and an embryo-embryo interaction. Although this model gives results comparable to those obtained with the Landau models previously described for the martensitic transition, it is especially adapted to describe a possible pretransitional state in which the local symmetry has already been broken, while the overall cubic symmetry (parent phase) is still preserved. 13. LOCALIZED SOFT-MODE MODELS

Localized soft-mode models were originally formulated as an alternative to classical nucleation theories for martensites and, at present, this approach is considered the most suitable to understand nucleation mechanism in martensitic transitions. Before going into the details of this approach, we briefly discuss a number of problems associated with the use of other nucleation theories applied to martensitic transitions. a. Nucleation theories

’

Classical homogeneous nucleation” supposes the thermal formation of critical nuclei of the product (stable) phase by means of heterophase fluctuations in the bulk of the metastable parent phase. Applied to martensitic transitions the theory considers the following three relevant contributions to the free energy AG.’13 (i) the driving force, i.e., the bulk free energy difference between the parent and the martensitic phases Ag, (ii) the interfacial free energy gS associated with the boundary between the two crystalline structures, and (iii) the strain energy e associated with the lattice deformation caused by the change in shape of the transformed region embedded in an untransformed matrix. That is, AG

= Yo,

+ Ve - VAg,

(59)

where Y and V are respectively the surface area and volume of the nucleus. Considering the nucleation of a thin ellipsoidal nucleus of radius r and semi-thickness c, which forms by shear process parallel to the plane of the disc, the following is obtained: AG

= 27cr20s

+ 4-nr’c(e 3

- Ag).

’”Y. Yamada, Y. Noda, and F. Fuchizaki, Phys. Rev. B. 42, 10405 (1990) ”’ D. T. Wu,Solid State Physics 50, 37 (1996). ’I3

L. Kauffmann and M. Cohen, Prog. Met. Phys. 7 , 165 (1958).

(60)

246

ANTONI PLANES AND L L U ~ SM A ~ O S A

Assuming complete coherence, the strain energy can be written as214 C

eNE-, a

where E is a function of geometrical parameters, elastic moduli, and the shear strain. The critical nucleus dimensions r, and c, and the nucleation barrier to be overcome by thermal fluctuations AGc are obtained from the variational extremum condition: aAG/ar = dAG/ac = 0. The result is

In general, martensitic transitions proceed with small driving force and large strains and thus the free energy increase caused by the gain of strain energy upon nucleation of one martensitic domain largely exceeds the reduction arising from the driving force term. Therefore, nucleation of this type is thermodynamically impossible. In other words, the critical nucleus tends to be much larger than accounted for by thermal fluctuations.215For instance, using available data for Fe-Ni (T, N 230K, Ag = 1300J/mol, E N 2000 J/cm3, and as N 200 mJ/m2), the critical dimensions of an ellipsoidal nucleus are rc 500 %, and c, 20 A, which presuppose a free energy barrier AG, 3 x lo5k,T,. These values are completely unrealistic. An alternative mechanism of homogeneous nucleation is the formation of a multivariant nucleus.’16 This is based on the fact that a suitable arrangement of martensitic variants eliminates the volume-dependent elastic energy created by a single martensitic domain embedded within the parent phase. In this case the strain field is short ranged and is located in the vicinity of the habit plane (or invariant plane) interface. The corresponding energy is proportional to the interface area and can be interpreted as strain-induced interfacial energy. For instance, for a nucleus consisting of a plate ( L x L x D) composed of twin-related lamellae of two martensitic variants, the sum of the surface energy of the interlamellar boundaries and the elastic energy stored in the short-range strain field is given by217

-

-

-

€(A)

=elY

Y D + arw1 ‘

J. D. Eshelby, Proc. R. SOC.London, Ser. A 241, 376 (1957). P. F. Gobin and G. Guenin, in Solid State Transformations in Metals and Alloys. Les Editions de Physique, Orsay, (1978) ’16 A. G. Khachaturyan, S. M. Shapiro, and S. Semenovskaya, Phys. Rev. B. 43, 10832 (1991). A. G. Khachaturyan, Trans. Mar. Res. SOC.Jpn. 18B,799 (1994). ’I4

’I5

’”

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

247

where Y is now the total area of the habit plane interface, D the thickness of the plate, otw the surface energy of the twin boundaries, and I the inhomogeneity characteristic length (typical distance of nearest-neighbor pair of twin-related lamellae).* Minimization with respect to I renders the optimal period I , = -./, Under these conditions, the free energy change associated with the nucleation of a martensitic plate of volume V can be expressed as

The condition of equilibrium of such a plate with the parent phase provides the inhomogeneous parameter IT of the nucleus.

Taking otw31 10rnJ/rn2 (which is a reasonable value for Fe-Ni), the theory renders a twinned critical nucleus with modulation at the atomic scale with a length less than or comparable to the characteristic modulation of the martensitic phase. Moreover, in this model I increases during the growth of the plate, and this raises serious doubts regarding the plausibility of this mechanism because, commonly, the modulation is an intrinsic component of the whole martensitic distortion. Nonclassical nucleation permits the homogeneous nucleation through a path involving a nucleus of varying structure.218 However, the theory has to assume that the average stress-free transformation strain of a nucleus is considerably less than that corresponding to the equilibrium system. Actually, this mechanism is only likely to occur near a mechanical instability. Hence it is appropriate for a quasi-martensitic (nearly second-order) transition. b. Defect-assisted nucleation For a very long time, there has been some experimental evidence that martensitic transitions are heterogeneously nucleated.2 More recently the influence of dislocations on the nucleation process in a Cu-Zn-A1 alloy has

* IY is assumed to be the volume where the short-range strain field is appreciable, and Y D / A is the total surface of twin boundaries (=area of one interlamellar boundary DL x the number of such boundaries L/d). * 1 8 G . B. Olson and M. Cohen, J. Phys. (France), 43, C4-75 (1975). '19 C. L. Magee, in Phase Transformations, AMS, Metals Park, OH (1970), p. 115.

248

ANTON1 PLANES AND LLUiS MAROSA

been demonstrated from synchrotron white beam x-ray topography experiments.220*221 Localized soft-mode models suggest that the transition is triggered by a strain-induced instability in special regions of the parent lattice. These regions are located around suitable defects, which provide a strain field with the appropriate tensor character and magnitude. In this case, the strain created by the defect locally modifies the elastic response of the system and makes the existence of localized soft-mode centers possible. This theory is, in a sense, complementary to the strain-phonon coupling model, which needs an adequate softening of co and C‘ for the first-order transition to occur. It is worth noting that as for such theories, in localized soft-mode models it is not necessarily expected that the local softening is complete. This idea was considered by clap^,'^^ who noticed the importance of the intrinsic anharmonicity of the system in reducing the nucleation barriers in specific strained regions. Based on this starting viewpoint, Guenin and Gobin222 applied the Born stability criteria’ to an elastic free energy including terms up to third order in the lattice strain. They showed that in a strained system the stability limit was determined by the value of the elastic constant C‘, renormalized by the TOEC combination C, (see equation (15)).

c: = C‘ f $C3E C: was referred to as the strained shear elastic constant.* Hence, the existence of a critical shear strain ( E J in a given plane (the so-called shear plane) would lead to the development of a mechanical instability in a certain direction (plane of instability) where C: < C‘. The zero value of this renormalized elastic constant will determine a critical shear strain for nucleation (EJ. Guenin and Gobin showed that the elastic strains created by dislocations with suitable Burgers’ vectors could locally reduce the value of C. Jourdan, S. Belkahla, G. Gutnin, P. Marzo, J. Gastaldi, and G. Grange, Appl. Phyx Lett. 59, 2527 (1991). C. Jourdan, J. Gastaldi, G. Grange, S . Belkahla, and G . Guenin, Acta metall. mater. 43, 4227 (1995). 222 G. Guenin and P. F. Gobin, Metall. Trans. 13A, 1127 (1988). * Actually, a stress modifies the equilibrium positions of the atoms. This effect is not considered in the localized soft-mode theory. Elgueta and G ~ t n i n have ~ ’ ~ shown that this effect yields the existence of harmonic terms in the expression of the renormalized SOEC, which increases the elastic stability of the bcc structure. Nevertheless, the same authors have shown that for the strains involved in the localized soft-mode theory, the anharmonic contribution is much larger and thus, the original expressions for the renormalized SOEC (which only include anharmonic terms) are accurate enough. 220

’”

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

249

the renormalized elastic constant and then a nucleus of the martensitic phase could develop without any significant increase of the lattice strain energy. Evaluation of these unstable regions gave elliptically shaped nuclei with realistic axes of 128, and 50 8, for Cu-Zn-Al, and the shear distortion necessary to induce the mechanical instability was E, 7 to 8%. The characteristic lengths of these elliptical nuclei 1 and 1 / f i can be expressed’55 as 1 [-u(C,/AF)]’/~, where a is the lattice parameter and AF is the elastic free energy barrier separating the two minima corresponding to the bcc and martensitic phases (AF cc (C”/C:)). The temperature decrease of C’ (Section V.6.b) and the temperature increase of ICJ (Section V.8) have a twofold effect; firstly, there is a significant decrease of the energy barrier on approaching T’, and second, there is an increase of the unstable region near the transition point. Both effects promote the nucleation of the martensitic phase. Later, Verlinden and Delaey’ 5 8 considered the combined effect of temperature and stress on the SOEC by assuming that these effects are additive. In analogy with the strained systems, the so-called stressed elastic constants can be expressed as

-

-

dC‘ da

C b= C’ + - a dC44 c24 = c44 + 0,

da

where the stress derivatives are a function of the orientation of the compression axes and the shear plane in which the shear constants are active.’57 Hence, the response of the solid to, for instance, a (llO)[lTO] shear will be given by the following renormalized elastic constant:

where 9 is given by

= 0. It is The instability limit of the bcc phase will be determined by Co, worth recalling that dC‘/da is very anisotropic (see Figure V.18) and so, the values of CAE will be very sensitive to the direction of application of the 223

J. Elgueta and G . GuCnin, J. Phys; Condens. Matter 2, 10235 (1990).

250

ANTONI PLANES AND L L U ~ SM A ~ ~ O S A

mechanical stress. By using SOEC and TOEC data for Cu-Zn-A1 at different temperatures, Verlinden and Delaey15* studied the mechanical stability of the p phase on approaching the martensitic transition. Their analysis led them to formulate a criterion for nucleation in terms of the elastic properties and the lattice instability of the p-phase. They concluded that for both thermally and stress-induced transformations, the martensitic transition would start when the value of the elastic anisotropy in the shear plane of the defect (A(o)) extrapolated at the transformation temperature and the critical shear strain necessary to produce an instability satisfied the condition

with n = 1.5 and K = 0.859. It is worth noticing that for all equivalent { 1lo}( 110) shears and for all the different directions of the applied stress, they obtained E, = 6.26% and A(a) = 13.45 at T = T,. Remarkably, this value for the elastic anisotropy coincides extremely well with the universal value of the bulk crystal at the transition point, as discussed in Section V.6.b. These authors suggested that this criterion could be extended to other Hume-Rothery alloys. Very recently, Gonzalez-Comas and Maiiosa15 have checked such a possibility by performing a similar analysis on a Cu-Al-Ni crystal. It has been shown that the nucleation criterion proposed for Cu-Zn-A1 cannot directly be applied to other Hume-Rothery alloys. Even so, the fact that a general vibrational behavior (universal values of A and C , at the transition point) has been shown to exist for Cu-based alloys (see Section V.6.b) suggests that the relationship proposed for Cu-Zn-A1 could be a particular case of a more general criterion for nucleation in Cu-based alloys. The main criticism to the localized soft-mode model arises from its homogeneous formulation. The theory, as presented above, should be considered as a first step that emphasizes the fact that the vibrational response of the system is very sensitive to particular strain fields. Efforts have been made to go beyond this crude approximation. Improved theories are based on Ginzburg-Landau free energy functionals, which incorporate the essential inhomogeneous character of the defect fields.224*225 Two different approaches have been proposed. In the model formulated by Cao et a1.,226the defect is represented by a stress field that is assumed to add an G. Guenin and P. C. Clapp, Proceedings of the International Conference on Martensitic Transformations, The Japan Institute of Metals, Nara, Japan (1986), p. 171. 2 2 s P. C. Clapp, Physica D . 66, 26 (1993). 224

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

25 1

additional term to the strain free energy density of the system. That is, in this model only the stress field produced by the defect is considered, but not the defect (“core”) itself. In contrast, in the model proposed by Guenin and ClappZz4the displacement configuration of the defect core is the essential ingredient of the model. They studied the stability of the defect strain configuration following the ideas of Cahn and Hilliard for spinodal decomposition.zz7If the possibility of a martensitic transition exists, some components of the defect strain field will expand as temperature is lowered until this “strain embryo” becomes unstable when the athermal instability limit is reached. It is worth noticing that in spite of the important differences between these formulations, they yield similar results. c. Remarks on the transition mechanism For many years nucleation has been acknowledged as the central issue in order to understand how martensitic transitions occur. A new perspective based on the existence of precursors in the form of tweed can be envisaged as a strategy developed by the system in order to overcome the difficulty of the nucleation process. This point of view leads to a reconsideration of the importance of nucleation on the martensitic transition mechanism. Tweed patterns represent a pseudo-periodic structure at an intermediate scale or mesoscale between the microscopic scale ( < 10A) and the macroscopic scale (> lOOOA) of the martensitic variants. Barsch et a1.,z2Sfirst suggested that this structure provides a pretransitional state in the natural evolution of the system from the homogeneous high-temperature phase toward a product phase (crystallographically homogeneous) constituted by twinned variants. This intermediate structure starts to develop within the parent phase as TM is approached. In these cases the formation of martensite can be understood as a coarsening process corresponding to the collapse and growth of the microdomains into martensitic domains. The preceding picture applies, for instance, to Ni-A1 alloy, but the situation is different in Cu-based alloys (and similar systems). No tweed is observed in these materials, and hence nucleation of martensite is necessary to activate the transition. From the discussion given in the previous subsection, it is evident that such a nucleation is assisted by suitable defects, characteristic of the highly anisotropic systems. It is worth noting however that, in spite of the fundamentally different behavior between Ni-A1 and Cu-based systems, the general thermodynamic condition for the transition to take place is the same in both cases: It is imposed by the anharmonic 226

W. Cao, J. A. Krumhansl, and R. J. Gooding, Phys. Rev. B. 41, 11319 (1990).

’” J. W. Cahn and J. E., Hilliard, J. Chem. Phys. 28,258 (1958). ’”G . R. Barsch, J. A. Krumhansl, L. E. Tanner, and M. Wuttig, Scripta metall. 21,1257(1987).

252

ANTONI PLANES AND L L U ~ SM A ~ ~ O S A

coupling of an incipiently unstable phonon and the homogeneous shear (see Figure VII.2 (a) and (b)), which is the main ingredient enabling the first-order transition to occur. The specific process followed by a given system (coarsening or heterogeneous nucleation) represents, from this point of view, the optimal mechanism that enables reduction down to the appropriate level (globally or locally) of the response function of the system to the vibrational modes related to the transition path. In both cases the process is essentially athermal and fluctuations are expected to play a minor role. This approach to the martensitic transition has been, in our opinion, underdeveloped. We think that research along these lines is thus necessary for a definite understanding of the problem. VIII. Magnetic Coupling

Magnetism, and therefore electronic properties, can play a decisive role in the stabilization of a specific crystallographic structure. In particular, magnetic entropy becomes relevant when electronic configurations with higher spin entropy, but with lower binding energy, are accessible. The re-entrant behavior of Fe with decreasing temperature is illustrative of this point. When cooling from the melt Fe crystallizes in a paramagnetic bcc phase, this bcc phase first transforms at T, (=1665K)to a weak or nonmagnetic fcc phase (the so-called y-phase). Actually, this is the expected behavior mainly controlled by vibrational entropy. Nevertheless, at a lower temperature T, (= 1184K), Fe transforms back to a bcc structure (the so-called a-phase) with a magnetic susceptibility that obeys the same Curie-Weiss law as the original bcc phase. Upon further cooling, at the Curie point ( T , = 1043K),the system orders ferromagnetically. The retransformation from the fcc to the bcc a-phase is acknowledged to be driven by magnetic degrees of freedom. Nevertheless, this transition occurs above the Curie point, which is indicative of the fact that magnetic long-range order is not the essential factor but rather, magnetic short-range correlations above T, can be strong enough to contribute to the stabilization of the a-phase. The addition of Ni (or other transition elements) to Fe stabilizes the y-phase down to lower temperatures. Most of these systems show a peculiar thermal expansion behavior related to Invar and anti-Invar proper tie^."^ These effects arise from the instability of the magnetic moment with respect to volume changes.230 In these alloys, the y + CI transition is classified as martensitic. This transition has been extensively studied in ferrous materials, 229

230

E. F. Wassermann, Advances in Solid State Physics 21, 85 (1987). M. Acet, H. Zahres, E. F. Wassermann, and W. Pepperhoff, Phys. Rev. B. 49, 6012 (1994).

VIBRATIONAL PROPERTIES O F SHAPE-MEMORY ALLOYS

253

including steels, because of its technological importance. In general the transition is highly irreversible* and hence these materials do not exhibit shape-memory properties.** Another interesting family of magnetic alloys are the Heusler alloys.231 These alloys are nearly stoichiometric X,YZ bcc-based intermetallic compounds with L2, configurational ordering. Among the alloys with this structural characteristic, the Heusler alloys display magnetic order (ferro- or antiferromagnetic). An important family is the one in which the Y element is Mn. Actually, the prototype Heusler alloy is Cu,MnAl. In these materials it is known that the entire magnetic moment must be attributed to the Mn atoms which are coupled via a RKKY interaction. Most shape-memory alloys, such as Cu-based alloys, can be considered as non-stoichiometric derivations of Heusler (or systems with the same structural characteristics) compounds. This suggests that it is reasonable to find Heusler alloys displaying a martensitic transition from an ordered bcc phase to a closepacked structure with related shape-memory properties. Strictly speaking, the only known Heusler system with these characteristics is the Ni,MnGa system.*** Notwithstanding, the Ni,Mn, -xAlx alloy must also be mentioned. Recently, it has been found that in a narrow range of compositions close to stoichiometry (x ranging from 0.76 to 0.84) Ni-Mn-A1 undergoes a martensitic transition from an antiferromagnetic B2 phase to a 10M martensitic structure.232At present, it is not clear whether this alloy system may exhibit ferromagnetic order. Actually, the antiferromagnetic order seems to be related to the absence, for kinetic reasons, of L2, ordering. It is worth noting that the great deal of interest devoted during recent years to these systems has not only to do with the fundamental understanding of the influence of magnetism in martensitic transitions, but also with the technologically appealing potential of magnetic control of the shapememory effect. Recently, several theoretical approaches have been put forward to account for the martensite domain redistribution caused by In this section the discussion magnetic fields in this class of

* Exceptionally, ordered Fe-Pt transforms thermoelastically, and hence with little hysteresis. ** A number of Fe-based systems (such as Fe-Mn-Si) display shape-memory properties. However, the mechanism involved is different from that of bcc-based shape-memory alloys. 2 3 1 P. J. Webster and K. R. A. Ziebeck, Landolt-Borntein New Series, ed. 0. Madelung, Springer-Verlag, Berlin (1988), Vol. 111/19c, p. 75. *** Although Cu-Al-Mn alloys are shape-memory systems, the range of compositions for which his alloy system exhibits a martensitic transformation is far from stoichiometric Cu,AIMn, and they are in the paramagnetic state. 2 3 2 S. Morito, T. Kakeshita, K. Hirata, and K. Otsuka, Acta metall. 46, 5377 (1998). 233 R. D. James and M. Wuttig, Philos. Mag. A 77, 1273 (1998). 2 3 4 R. C. OHandley, J. Appl. Phys. 83, 3263 (1998).

254

ANTONI PLANES AND L L U ~ SMAROSA

will concern the fundamental aspects based on the results found in Ni-MnGa alloys with a composition close to stoichiometry. 14. PROPERTIES OF Ni,MnGa

Around the stoichiometric composition, the Ni-Mn-Ga alloy system dis~' plays an ordered L2, structure (or /?-phase) at high t e m p e r a t ~ r e . ~The Curie temperature T, shows weak dependence on composition. For the stoichiometric system T, = 374K. This system undergoes a martensitic transition to a modulated tetragonal structure. At stoichiometry the transition takes place at T, = 202K toward a 5-layered modulated 5M structure. This temperature is very sensitive to composition. Figure VIII.l shows a representation of the phase diagram of this system. T, and T, are plotted as a function of the empirical parameter a obtained as the weighted composition, a = x at% G a + y at% Mn. The best fit corresponds to x = 0.6 and y = 0.4.* It is interesting to note that the periodicity of the modulation of the martensitic phase also depends on composition. It changes from 5M to 7 M and 8M when the composition moves away from stoichiometry (a < 25). In alloys undergoing the martensitic transition at a temperature above the Curie point, two successive transitions occur on cooling. The different sequences /? -+ 8M + T ( T denotes a non-modulated tetragonal structure) and /? -+ 10M -+ 7 M have been observed.237 The change of the period of modulation seems to occur in close relation to the onset of ferromagnetic ordering. In all cases, the reverse transition on heating has been found to take place in a single step. Furthermore, a sequence /? -+ 5M -+ 7 M -+ Tis obtained when the transition is stress induced.238 Ni2MnGa displays soft magnetic properties. The L 2 , phase is highly isotropic, while the martensitic phase is anisotropic with an easy magnetization axis along the (111) direction.235Closely related to such an anisotropy is the possibility of a magnetic-field induced deformation in this alloy system. While the strain-induced sensitivity of the /?-phase is very low and corresponds to a magnetostrictive effect, the behavior is more complex in P. J. Webster, K. R. A. Ziebeck, S. L. Town, and M. S. Peak, Philos. Mag. E. 49,295 (1984). V. A. Chernenko, Scripta muter. 40, 523 (1999). From the expression of e/a, x and y can be obtained as,

235

236

*

nNi - nMn

X = 2nNi

- nMu - nGo

and y =

nNn-nGo 2nNi

- nMn -

where n, is the contribution of i atoms to e/a. ChernenkoZJ6has assumed that for Ni-Mn-Ga, nNi = 10, nMn= 7, and nGa= 3, taken from the configurations of these elements in the periodic table. This yields x = 0.3 and y = 0.7, which are close to the values giving the best fit. 237 V. A. Chernenko, C. Segui, E. Cesari, J. Pons, and V. V. Kokorin, Phys. Rev. E. 57, 2659 (1998). 2 3 8 V. V. Martynov, . I Phys. (France) ZV, 5, C8-91. (1995)

255

VIBRATIONAL PROPERTIES O F SHAPE-MEMORY ALLOYS

600 PARAMAGNETIC L2,

L 00

2 00

15

0 I

.

20 25 aPh Mn ,

.

21

20

30 ,

.

,

.

23

22

,

24

.

,

.

25

,

26

.

27

a 260

, O 180

i

0FERROMAGNETIC t o \ MARTENSITE 24.2

24.6

a

25.0

25.4

i

FIG.VIII.1. Martensitic (solid circles) and Curie (solid up triangles), transition temperatures of the Ni-Mn-Ga system as a function of a weighted composition parameter. Data have been collected from S. J. Murray, M. Farinelli, C. Kantner, J. K. Huang, S. M. Allen, and R. C. OHandley, J. Appl. Phys. 83, 7297 (1998) and V. A. Chernenko, E. Cesari, V. V. Kokorin, and I. N. Vitenko, Scripta metall. mater. 33, 1239 (1995). The open diamond corresponds to the temperature of the change of the modulation of the martensitic phase. From V. A. Chernenko, C. Segui, E. Cesari, J. Pons,and V. V. Kokorin, Phys. Rev. B. 57, 2659 (1998). The hatched region in the inset shows the composition range from which data have been taken for this compact representation. The lower dashed line indicates the transition line separating the L2, phase from the intermediate modulated phase. (b) Enlarged view of the region where the intermediate modulated phase appears. We have only platted data corresponding to samples in which both the martesitic (solid circles) and premartensitic (open down triangles) have been reported. Additional data is from: F. Zuo, X. Su, P. Zhang, G. C. Alexandrakis, F. Young nd K. H. Wu, J. Phys.: Condens. Natter, 11, 2821 (1999) and LI. Manosa, A. Planes, J. Zarestky, T. Lograsso, D. L. Shlagel and C. Stassis, to be published.

the martensitic phase. In this structure the strain-induced sensitivity is of the order of 2.5 x lo-' O e - ' . More interesting is the fact that the induced deformation depends significantly on the direction of the applied field. For

256

ANTONI PLANES AND L L U ~ SMAROSA

instance, for a field strength of 10 kOe the induced strain changes by 0.2% by rotating the field from the [Ool] to the [l lo] direction.239This striking behavior is associated with a magnetic-field-induced reorientation of twinrelated martensitic variants and is at the origin of a magnetic control of the shape-memory effect in this system.* 15. PHONONS AND ELASTIC CONSTANTS

A complete neutron diffraction investigation of precursor phenomena in Ni,MnGa has been performed by Zheludev and collaborators.240*241 These authors have obtained the phonon dispersion curves along the main symmetry directions and have also performed elastic scattering measurements along a number of transverse directions. The general trends of the results reported are comparable to those presented in Section V for the Ni-A1 alloy. In particular, in the P-phase, a significant softening of the 4[110] T A , acoustic mode with decreasing temperature is observed (see Figure. VIII.2). Nevertheless, two important differences distinguish the behavior of Ni,MnGa with respect to non-ferromagnetic alloys. (i) The rate of change do2/dT of the square of frequency o of the anomalous phonon with temperature is much larger than in Ni-Al. Interestingly, Stuhr et al.242 have observed that this rate is enhanced when the system becomes ferromagnetically ordered at the Curie point (see inset of Figure VIII.3). Actually, in the paramagnetic phase d o 2 / d T is very similar to that measured in Ni-Al. (ii) At a temperature T, ( = 260K) > T, the softening is nearly complete and upon further cooling o2increases until the martensitic transition is reached (see Figure VIII.3). In addition, the broad and relatively weak elastic satellites at [440] become, below T,, narrow and Bragg-like with a substantial increase in their intensity. This is accompanied by the suppression of diffuse scattering along the [llO] direction, which, as in the case of Ni-Al, has its origin in the tweed pattern observed at high temperature.243 This 2 3 9 K. Ullako, J. K. Huang, C. Kantner, R. C. O’Handley, and V. V. Kokorin, Appl. Phys. Lett. 69, 1966 (1996). * At present deformations of -5.8% have already been obtained (see S. J. Murray, M. A. Marioni, A. M. Kukla, J. Robinson, R. C. OHandley and S. M. Allen, J. Appl. Phys., 87, 5774 (2000)). This deformation is close to the maximum achievable strain of 6.2%, estimated from the crystallographic change at the martensitic transition. 240 A. Zheludev, S. M. Shapiro, P. Wochner, A. Schwartz, M. Wall, and L. E. Tanner, Phys. Rev. B. 51, 11310 (1995). 241 A. Zheludev, S. M. Shapiro, P. Wochner, and L. E. Tanner, Phys. Rev. B. 54, 15045 (1996). 242 U. Stuhr, P. Vorderwisch, V. V. Kokorin, and P.-A. Lindgird, Phys. Rev. B. 56,14360 (1997). 243 V. V. Kokorin, V. A. Chernenko, J. Pons, C. Segui, and E. Cesari, Solid State Comm. 101, l(1997).

VIBRATIONAL PROPERTIES O F SHAPE-MEMORY ALLOYS

257

Reduced wave number FIG. VII1.2. TA, phonon dispersion curve of a nearly stoichiometric Ni-Mn-Ga alloy at different temperatures. From A. Zheludev, S . M. Shapiro, P. Wochner, A. Schwan, M. Wall, and L. E. Tanner, Phys. Rev. B. 51, 11310 (1995).

atypical behavior has been related to an intermediate phase transition at TI toward a modulated structure that seems to preserve cubic symmetry. This modulation has been characterized by x-ray244and electron microscopy245 to be transverse along the [llO] direction with wave number 3, which corroborates the fact that it results from the freezing of the anomalous $[llO] TA, phonon. It is important to mention that this intermediate transition has only been observed in a narrow range of compositions around the stoichiometry. In Figure VIII.l the boundary line separating the L2, from the intermediate modulated phase is indicated and in figure VIII.l(b) we give an enlarged view of the region where the premartensitic transiton has been observed. The existence of such an intermediate phase transition has been corroborated from calorimetric and magnetic measurements. The specific heat and magnetic susceptibility versus temperature curves exhibit small anomalies at In addition, a small entropy change (latent heat) of 0.04 JK-lmol-' G. Fritsch, V. V. Kokorin, and A. Kempf, J. Phys. Condens. Matter, 6, L107 (1994). E. Cesari, V. A. Chernenko, V. V. Kokorin, and J. Pons, Acta mater. 45,999 (1997). 246 L1. Maiiosa, A. Gonzilez-Comas, E. Obrado, A. Planes, V. A. Chernenko, V. V. Kokorin, and E. Cesari, Phys. Rev. E. 55, 11068 (1997). 244 245

258

ANTONI PLANES AND L L U ~ SM A ~ O S A

7-

(b)

6-

5N

>

-3-3 u

650

350

b c u

2-

7i

1-

a

0 200

2 50

I

I

300

350

Temperature (K) FIG. VIII.3. (ho)’as a function of temperature for the 1/3[110]TA2 acoustic mode corresponding to the dip of the branch. Data from: A. Zheludev, S. M. Shapiro, P. Wochner, A. Schwarz, M. Wall, and L. E. Tanner, Phys. Rev. B. 51, 11310 (1995). The inset shows the change of slope at T, obtained on a sample of slightly different composition. Data from U. Stuhr, P. Vorderwisch, V. V. Kokorin, and P. A. Lindgird, Phys. Rev. B. 56, 14360 (1997).

has been estimated by combining modulated and standard differential thermal analysis rnea~urements.~~’ These results not only confirm the existence of a phase transition to a modulated structure prior to the martensitic transition, but also assert its first-order character. Notwithstanding, no observable hysteresis has been reported for this transition. The elastic constants also exhibit an anomalous behavior when the intermediate transition is This is shown in Figure VIII.4. The anomaly is especially apparent in the elastic constant C’, which suffers a significant softening of 50% on cooling from 240K to 7’,. On further cooling from 7’,to T,, C‘ increases. Indeed, the behavior displayed by C’ parallels that of the frequency of the soft phonon mode. A less pronounced softening is shown by C44. This is rather surprising because the atomic

-

2 4 7 A. Planes, E. Obrad6, A. Gonzalez-Comas, and L1. Maiiosa, Phys. Rev. Lett. 79, 3926 (1997). 248 J. Worgull, E. Petti, and J. Trivisonno, Phys. Rev. B. 54, 15695 (1996).

VIBRATIONAL PROPERTIES O F SHAPE-MEMORY ALLOYS

..

I

c

z

0.01 CL

259

-"

0)

p 0.0 d

J=

u

-0.2

0)

.z -0.4 .w d

d

0)

t Y

-0.6 180

220

260

T(K) FIG. V111.4. Relative change of the elastic constants C,,, C,, C,,, and C as a function of temperature, through the intermediate phase transition in a nearly stoichiometric Ni-Mn-Ga alloy. From L1. Maiiosa, A. GonzHlez-Comas, E. Obrado, A. Planes, V. A. Chernenko, V. V. Kokorin, and E. Cesari, Phys. Reo. B. 55, 11068 (1997).

displacements determining the modulated structure are not related to vibrational modes associated with this elastic constant. Actually, with the exception of this effect found in C,,, the phonon branch T A , does not exhibit any observable peculiarity. The C,, softening could be a consequence of anharmonic coupling effects. It is worth mentioning that the behavior of C' and C,, leads to a remarkable maximum of the elastic anisotropy at TI. The effect of a uniaxial stress on the intermediate transition has also been investigated. Inelastic neutron scattering shows that the dip in the T A , phonon branch is stress-dependent and shifts from 5 = 0.33 at zero stress to 5 = 0.35 for an applied stress of 95 M P u . , , ~ Measurements of C,, as a function of temperature for different levels of uniaxial compressive stress along [OOl] and [lTO] indicate that the temperature of the forward intermediate transition is depressed, while that of the reverse transition 249

A. Zheludev and S. M. Shapiro, Solid State Commun. 98, 35 (1996).

ANTONI PLANES AND L L U ~ SMAROSA

260

increases.250These changes seem to reach a saturation value for stresses above 4MPa. Therefore, the intermediate transition under applied stress occurs with thermal hysteresis. All elastic constants increase with a uniaxial stress applied along the [OOl]. The increase is nonlinear with a tendency to reach a saturation value. It is worth noting that while the amount of change of C’ is of the order of those reported in other bcc-based shape-memory alloys (see Section V .8), the corresponding change of C, and C,, is unexpectedly large. This anomalous anharmonic behavior seems to be related to a change of magnetization induced by the stress. The existence of magnetoelastic effects has been unambiguously confirmed from the observed dependence of the elastic constants on an external magnetic field. Figure VIII.5 depicts the relative change of the elastic constants C, C,,, and C‘ with external magnetic field applied along the [lo01 direction. In both cases, the elastic constants increase up to a saturation value with increasing magnetic field. The saturation values are independent of the direction of the applied field. As could be expected, C‘ is the constant that exhibits the largest relative change. In view of the existence of magnetoelastic effects, it is worth considering the influence of a magnetic field on the intermediate and martensitic transition. Significant effects have been reported. For weak applied fields (up to 1 kOe), T, slightly decrease^,^,' while the decrease is significantly marked for intermediate values of the field (K/Oe). The change of T, is found to be linear with the square of the magnetization. For intense fields ( > 5kOe), when magnetic saturation is reached, the intermediate transition is inhibited.’ No magnetic field dependence has been found for the martensitic transition temperature T,.’51*’5’ This is consistent with an estimation based on entropy and magnetization changes at the martensitic transition (AS -0.5 J/Kmol, AM lO’emu/mol- ’) via the Clausius-Clapeyron equation, which gives dT/dH 21 2 K/kOe. This value shows that measurable changes in T, will only be possible for very strong magnetic fields.

-

’

-

-

16. MODELS FOR THE INTERMEDIATE TRANSITION

The interplay between magnetic and vibrational properties showed up through the influence of a magnetic field on the the elastic constants and by LI. Mafiosa, A. Gonzalez-Comas, E. Obrad6, A. Planes, Proc. Int. Conf on Martensitic Transformations, Bariloche, Argentina, 1998. Mat. Sci. Engng. A, 273-215, 329 (1999). 2 5 1 F. Zuo, X. Su, and K. H. Wu, Phys. Rev. B. 58, 11127 (1998). 2 5 2 A. Gonzalez-Comas, E. Obrado, L1. Mafiosa, A. Planes, V. A. Chernenko, B. J. Hattink, and A. Labarta, Phys. Rev. B. 60,7085 (1999). 250

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

26 1

0 0°0

8

CL

c :

41

e

C44

o='

0

1

2

3

4

5

6

H(k0e) FIG. VIII.5. Relative change of the elastic constant C,, C,,, and C as a function of an applied DC magnetic applied field along the [Ool] direction. Solid symbols correspond to measurements with an increasing magnetic field, and open symbols, with a decreasing magnetic field. Notice the effect of the magnetization remanence.

the change of the slope dw2fdT at the Curie point. Such a coupling is the decisive ingredient that determines the unique features of Ni,MnGa, including the existence of the premartensitic transition to the modulated intermediate phase. Within this framework magnetism is assumed to cause the freezing of the incipiently unstable 4 [ l l O ] T A , phonon split from the symmetry breaking associated with the homogeneous strain. This means that the intermediate phase must be understood as a static precursor of the martensitic transition. The difficulty with this interpretation of the intermediate transition arises from the fact that the actual modulation of the martensite (5-layered modulation) seems to be different from that of the intermediate phase. From this point of view it has been claimed that both

262

ANTONI PLANES AND L L U ~ SMAAOSA

the intermediate and the martensitic transition are really independent transitions.245 To support the first scenario two considerations must be pointed out. The first deals with the fact that a possible effect of the homogeneous deformation associated with the martensitic transition may induce a change in the modulation. Actually, this is comparable to what occurs in Ni-Al, where the dip is observed at &, which slightly differs from the 7-layer modulation of the martensite. The observed influence of a uniaxial stress on the position of the dip of the T A , branch supports this point of view. A second consideration has been raised by Stuhr and collaborators.z4zThey argued that the wave vector [440] characterizing the intermediate phase becomes a C0.38 0.38 01 vector in the tetragonal phase, which is close to the [330] expected for a 5-layered modulated structure. The model proposedz4’ for the intermediate transition is based on ideas similar to the case of the strain-phonon coupling model developed in Section VII.12 for the martensitic transition. The amplitude d of the f[llO] T A , phonon is taken as the primary order parameter, and secondary fields are considered; ql, the homogeneous shear suitable for describing a cubic to tetragonal transition, and M , the magnetization, considered to be scalar (this is a reasonable approximation for the almost magnetically isotropic L2, phase). In terms of these three order parameters, the free energy can be assumed to have the following general form:

which includes a purely structural term, a magnetic term, and a contribution accounting for the magnetoelastic and spin-phonon interaction. The structural term must be appropriate to describe the martensitic transition in a nonmagnetic system. Indeed, a free energy expansion of the form given in eq. (42) can be adopted. The magnetic energy term can be taken to have the following form:

where M , is the magnetization of the L2, phase close to the intermediate transition. Considering the symmetries of the system, the following biquadratic coupling has been proposed: 1 F,(M, d, ql) = -IC,MZd2 2

+ 21 IC2Mzq:. -

(73)

It is reasonable to assume that the energy arising from the coupling between

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

263

d and q l is negligibly small compared with F,(M, d , q l ) . In this model, this assumption is the crucial condition that enables the condensation of the modulating phonon decoupled from the shear strain at a temperature above the martensitic transition. All the coefficients in this expansion are taken to be positive and only d,which is identified with the frequency of the anomalous phonon, is supposed to vary linearly with temperature. Minimization with respect to q1 and M yields an effective free energy along the path q l ( d ) = 0, M ( d ) = Mo(l + ( ~ , / 2 B ) d ' ) - ' and has the same form as the free energy given in eq. (43). In this case the effective coefficients of the free energy expansion are m*G2 = a( T - To)= a( T - [T, - ( K Mo/a)]); b = b - ( K ~ M ~ / BZ. = ) ; c + &c:M$/B2). The feature to be pointed out is that when the spin-phonon coupling is large enough, 8 can become negative, and in this case a first-order transition to a modulated structure can take place before the system becomes harmonically unstable at To. The finding that the intermediate phase has only been observed in alloys for which the martensitic transition is far enough from the Curie point (see Figure VIII. 1) supports the requirement of strong spin-phonon coupling for this transition to take place. This simple model predicts, consistently with experimental results, a decrease of the magnetic susceptibility and C' at the intermediate transition, and a decrease of TI with an applied magnetic field. However, a limitation of the model refers to the fact that a = d o 2 / d T is independent of the magnetic degrees of freedom. Furthermore, the effect of the coupling is just to induce a depletion of the temperature limit of stability from T, to T, - ( K ~ M , / Ubut ) , the model is unable to predict a change of a at the Curie point as observed experimentally. The Landau model just described belongs to the mean field class of models, which neglects any possible effect of fluctuations. A lattice model has been proposed aimed at considering such effect in the premartensitic state of the Ni2MnGa.253It constitutes a simplified statistical-mechanics model that incorporates the main ingredients of the physics of this system. ~ ~addition to the It is based on the p-degenerated BEG H a m i l t ~ n i a n . 'In structural local variables, the model is extended in order to include magnetic degrees of freedom. Lattice sites are defined at a mesoscopic level. The Hamiltonian of the system contains the following three terms; the purely structural and magnetic contributions, and the term arising from a coupling interplay. They correspond well to the three terms of the free energy (71). The model can exhibit three phase transitions associated respectively with the onset of increase of the following order parameters; the tetragonal deformation, the phonon amplitude, and the magnetization. The importance of the fluctuations in the premartensitic state shows up from Monte Carlo

-

253

T. Castan, E. Vives, and P.-A. LindgHrd, Phys. Rev. B.60 7071 (1999).

264

ANTONI PLANES AND L L U ~ SMAROSA

simulations. For appropriate values of the magnetoelastic coupling, a critical point between the magnetic and the martensitic transition is obtained. Below this critical point, and for a limited range of model parameters, a first-order premartensitic phase transition is possible. As occurs in the Landau model, the interplay between the phonon and the magnetic variables is the relevant ingredient enabling the existence of such an intermediate transition. Interestingly, the intermediate transition is preceded by anomalies in the specific heat and magnetic susceptibility originating from large-amplitude fluctuations. These anomalies may be the reason for the two-step premartensitic transition found by some authors in magnetic rnea~~rernent~.’~~ An important problem not contemplated by effective models deals with the microscopic origin of the phonon anomaly and its magnetic interplay, which gives rise to the intermediate transition. In the paramagnetic phase, the behavior of Ni,MnGa is similar to that of Ni-Al; therefore, the peculiarities of the electronic structure and electron-phonon interplay are at the origin of the incipient dip observed in the T A , branch, observed above T,. Nevertheless, the electronic properties of the system are modified when it becomes ferromagnetically ordered. How this may explain the existence of an intermediate phase is still an open question. With regards to the martensitic transition in cubic ferromagnetic materials, recent models have been proposed.’ s s*2s6 In these models the magnetoelastic coupling between the strain tensor and the (vector) magnetization is fully considered. Nevertheless, none of these models include terms related to the :[ 1lo] T A , anomalous phonon in the free energy expansion, and therefore cannot account for the existence of the intermediate phase. These models are especially suited to account for the change of the magnetic properties at the martensitic transition and, therefore, they represent a first step to realistically considering the magnetic shape-memory properties exhibited by these materials. From a microscopic point of view it has been suggested that the martensitic transition in Ni,MnGa is driven by a band Jahn-Teller e f f e ~ t . ” ~ This suggestion is based on a band structure calculation using the KorringaKohn-Rostocker method. In the paramagnetic state, the Fermi level is

254 F. Zuo, X. Su, P. Zhang, G. C. Alexandrakis, F. Yang, and K. H. Wu, J. Phys.: Condens. Mutter 11, 2821 (1999). 2 5 5 V. A. L‘vov, E. V. Gomanaj, and V. A. Chernenko, J. Phys.: Condens. Mutter. 10, 4587

(1998). 2 5 6 A. N. Vasil’ev, A. D. Bozhko, V. V. Khovailo, I. E. Dikshtein, V. G. Shavrov, V. D. Buchelnikov, M. Matsumoto, S. Suzuki, T. Takagi, and J. Tani, Phys. Rev. B. 59, 1113 (1999). 2 5 7 S . Fujii, S. Ishida, and S. Asano, J. Phys. SOC.Japan, 58, 3657 (1989).

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

265

located at a high peak of Mn d bands in both the cubic and martensitic (tetragonal) phases. The magnetic moment is found to be largely confined to the Mn sites, but with a small moment (<0.3 p B ) associated with the Ni sites. In the ferromagnetic state the DOS of Ni, which becomes larger than that of Mn near the Fermi level, splits into two peaks in the tetragonal structure. This splitting is a consequence of the interaction between electronic and vibrational degrees of freedom and causes a lowering of the energy of the system. Therefore, the tetragonal distortion is attributed to an electronic instability, supposed to be of the band Jahn-Teller type. Actually, the same mechanism has been proposed by Labbe and Friedel’’* to explain the cubic to tetragonal transition in A 15 compounds. The redistribution of electrons between 3d sub-bands in the martensitic phase is predicted to be accompanied by a small increase of the magnetic moment of Mn relative to that of Ni. This has been recently confirmed from polarized neutron scattering experiment^.^'^ Despite the fact that the experimental moments differ from those expected from band structure calculations, the change of symmetry of the magnetization distribution is consistent with the band Jahn-Teller origin of the transition.

IX. Conclusions

Traditionally, shape-memory alloys have been studied from numerous viewpoints and have attracted the interest of researchers in many different areas, ranging from applied mathematics (interested, for instance, in nonlinear elasticity) to materials engineering (concerned with the technological applications of shape-memory properties). The different features of this cross-disciplinary field of research have been briefly outlined in the introductory sections of this review. The core of the present chapter provides a unified view of the lattice dynamic behavior of these alloys. The mechanism that leads to the martensitic transition, and the stability of the different phases involved, is discussed in terms of this behavior. The chapter is expected to be quite comprehensive and an effort has been made aimed at being pedagogical so that it can be of interest for both condensed matter physicists and materials scientists. The main conclusions of the work can be summarized as follows. L. Labbt, and J. Friedel, J. Phys. (Paris), 27, 153 (1964).

259P. J. Brown, A. Y. Bargawi, J. Crangle, K. -U.Newmann, and K. R. A. Ziebeck, J. Phys. Condens. Matter, 11, 4715 (1999).

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ANTONI PLANES AND L L U ~ SM A ~ O S A

Lattice dynamics reveals that for all shape-memory alloys the parent P-phase offers weak restoring forces for long- and short-wavelength phonons of the T A , branch. Therefore, these systems show highly anisotropic vibrational properties. In particular, these T A , modes supply the excess of vibrational entropy of the /?-phase with respect to close-packed structures, which plays a major role on its high temperature stability. The contribution from conduction electrons depends on the particular alloy system; it is irrelevant for Hume-Rothery alloys but larger for Ni-based materials. The incipiently unstable modes are intimately related to the actual transition path that leads the parent phase toward the martensitic closepacked structure. In the martensitic phase the elastic moduli and phonon frequencies corresponding to the distortions derived from the soft modes in the /?-phase (via lattice correspondence) are higher. Even so, the instability of the P-phase is partially inherited; the lattice response to the modes derived from the soft TA,-branch are significantly lower than the other vibrational modes of the martensitic structure. The tendency to instability becomes more pronounced as the martensitic transition point is approached (lowering temperature or increasing external applied stress). This shows up by a softening of the whole T A , branch and of the corresponding elastic constant C’. In all shape-memory materials this softening is far from complete for any of the low-lying modes at the onset of the transition. Therefore, the following singular behavior is possible in these materials: while low-energy modes enable the existence of precursor effects, the transition has an evident first-order character. In all shapememory systems the selected distortion consists of homogeneous deformations (characteristic of martensitic transitions), combined with a short wavelength modulation determining the sequence of close-packed planes. In some systems the wavelength of this modulation is announced by the significant softening of the T A , phonon with a wave number close to that of the modulation. This is, for instance, the case of Ni-AI. The behavior is, however, quite different in other systems such as Cu-based alloys. In this case the T A , branch is very flat and all phonons soften slightly with decreasing temperature. The branch also displays a small dip anomaly at 3 of the zone boundary, but the tendency to instability of such a mode is not significantly different from that of other modes. Therefore, the selection of the modulation wavelength is very much influenced by small details of the internal energy of the system. The first-order character of the martensitic transition is a consequence of the interplay between an incipiently unstable mode to a secondary field. There is some experimental evidence that this interplay is related to the anharmonic coupling between the T A , phonon related to the modulation of the martensitic phase and the homogeneous shear necessary to obtain the

VIBRATIONAL PROPERTIES OF SHAPE-MEMORY ALLOYS

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martensitic phase. In shape-memory alloys, the martensitic transition is athermal. Thermal fluctuations are irrelevant and the effect of temperature is to change the elastic coefficients and phonon frequencies in such a way that the instability of the lattice increases, thus enabling the nucleation of the martensitic phase. The influence of magnetic degrees of freedom with anomalous phonon modes can induce a condensation of a soft phonon giving rise to the development of a modulated structure that can be understood as a static precursor of the martensitic transition. In this case magnetization is the relevant secondary field that couples to the incipiently unstable acoustic mode. This interplay enables the premartensitic transition to be first order. Acknowledgments We wish to express our gratitude to our colleagues of the Condensed Matter Group at the Departament d'Estructura i Constituents de la Materia (University of Barcelona), T. Castan, J. Ortin, and E. Vives for continuing collaboration and stimulating discussions, and also to our former students A. Gonzalez-Comas, M. A. Jurado, and E. Obrado who have actively participated in obtaining some of the results presented in this chapter. Thanks are also due to J. A. Krumhansl, L. Delaey, G. Guenin, and M. Ahlers who have read the manuscript and offered helpful comments and criticisms. During the time we have been working on the subject of the present paper, we have also benefited from enlightening discussions with many other friends, amongst them we would like to mention C. Frontera, P.-A. Lindgbrd, J. L. Macqueron, M. Morin, M. Porta, D. Rios-Jara, R. Romero, G. A. Saunders, and C. Stassis. This work has received financial support from the CICyT (Spain), projects MAT95-0504 and MAT98-0315 and CIRIT, projects 1996SGR119 and 1998SGR48.

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SOLIP STATE PHYSICS. VOL. 55

Grain Growth and Evolution of Other Cellular Structures CARLV . THOMPSON Dept . of Materials Science and Engineering M.1.T Cambridge, M A 02139

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. The Phenomenological Model of Burke and Turnbull . . . . . . . . . . . . . . . III . Grain Growth in Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . Application of Euler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Lewis’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . The Aboav-Weaire Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Switching Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Mullins-von Neumann Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Mean Field Models for Grain Growth . . . . . . . . . . . . . . . . . . . . . . . . 1 . Hillert’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Louat’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Hybrid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Mean Field Models Based on the Mullins-von Neumann Law . . . . . . . . V. Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Experiments on Two-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . VII. Experiments on Foils and Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . 1. Thick Polycrystalline Foils and Films . . . . . . . . . . . . . . . . . . . . . . 2. Polycrystalline Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Agglomeration and Grain Growth in Very Thin Films . . . . . . . . . . . . 4. Grain Rotation in Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Non-Idealities Associated with Alloy Additions . . . . . . . . . . . . . . . . . VIII . Three-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Modeling of 3D Grain Growth . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Simulations of 3D Grain Growth . . . . . . . . . . . . . . . . . . . . . . . . . 3. Experiments in 3D Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

270 270 272 273 274 275 276 271 279 280 282 283 283 286 290 293 293 296 303 304 304 306 306 309 312 313 314

269 ISBN 0-12-607755-X ISSN 0081-1947/01 $35.00

Copyright a> 2001 by Academic Press All rights 01 reproduction in any form reserved.

270

CARL V. THOMPSON

1. Introduction

The properties of polycrystalline materials are strongly affected by the average size of their grains, and by the nature of the distribution of their grain sizes. This is especially true of the mechanical properties of materials, but also holds true for the electrical, optical, and magnetic properties. While the initial grain size distribution of a material is controlled by the nucleation of its constituent crystals and by subsequent growth and impingement processes, further evolution often occurs through motion of the grain boundaries that form when crystals impinge. Grain boundary motion results in the growth of some grains, and the shrinkage and disappearance of others, with the average grain size increasing. This process, which occurs in fully crystalline materials, is generally referred to as grain growth. In contrast with recrystallization processes, grain growth does not involve the nucleation of new grains but instead involves the growth of preexisting grains at the expense of other preexisting grains, resulting in a reduction in the total number of grains in a fixed volume, and therefore a decrease in the total grain boundary area per unit volume, and an increase in the average grain size. Grain growth processes are often categorized as normal and abnormal. Normal grain growth is usually assumed to lead to an evolutionary process that falls into a scaling state in which the rate of increase of the average grain dimension, d , scales with a power of time as

d

- ta,

(1)

and for which the normalized grain-size distribution, f ( d / ( d ) , t), where ( d ) is the average grain diameter, is time-invariant. In abnormal grain growth, growth of a subpopulation of grains is favored, so that scaling behavior (normal grain growth) is not achieved until, or unless, only favored grains survive. Abnormal grain growth sometimes leads to transient states in which the grain sizes are clearly bimodally distributed. We will first focus on early phenomenological models for normal grain growth, and then, because of their relative simplicity and success, focus on models for normal and abnormal grain growth in two dimensions. This discussion will be followed by a review of what is known about grain growth in three-dimensional systems. II. The PhenomenologicalModel of Burke and Turnbull

Consider the schematic representations of grain structures shown in Figure 11.1. These can be thought of as images of the surface of a section from a

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271

FIG. 11.1. Grain structure evolution in a 2D simulation. Small grains shrink and disappear and large grains grow larger, so the average grain size increases and the total grain boundary energy decreases.

3D polycrystalline structure or as images of a 2D polycrystalline structure. In grain growth, some grains shrink and disappear, with the result that other grains grow and the average grain size increases over time. It is convenient to deal with a linear measure of size, referred to as a “diameter”, d, or “radius”, r, where the average diameter (d) can be taken to be, in 3D,

(0 =

(?)

6(V)

or, in 2D, 4(A)

1/2 9

and where ( V ) and ( A ) are the average volume and average area, respectively. The radius is defined such that ( r ) = (d)/2. Burke and Turnbull’ developed a model for the kinetics of grain growth in pure materials by first assuming that all boundaries move with velocities that are proportional to an average driving force for their motion, AF, such that u

= m(AF),

(3)

where m is the grain boundary mobility. They further assumed that the J. E., Burke and D. Turnbull, Prog. Met. Phys. 3, 220 (1952).

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CARL V. THOMPSON

driving force for motion of a grain boundary was proportional to its average so that eq. (3) energy per unit area, (ygb), and its average curvature, (IC), can be rewritten as

where

Burke and Turnbull next considered a grain of “radius” r, for which the proportional to l/r, so that the boundaries had an average curvature (IC) rate of growth of the grain is given by

where C is a constant and rn and (ygb) have implicitly been taken to be uniform for all boundaries. As a final step, Burke and Turnbull further assumed that a relationship of the form of eq. ( 6 ) applied to the average grain radius ( r ) for a population of grains, so that

where ( r o ) is the average radius of the ensemble of grains at time t = 0. When cast in the form of eq. (l), eq. (7) gives CI = 0.5. The boundary mobility m is assumed to have an Arrhenius temperature dependence such that = mo ~ X P-(Q / k T),

(8)

where rn,, is a temperature-independent constant, and Q is the activation energy for the rate-limiting atomic process involved in boundary motion. The behavior illustrated in Figure 11.2 therefore results, where, in the case of Figure II.2b, ( r ) >> (lo).As will be discussed later, the expected growth exponent CI = 0.5 and the Arrhenius behavior illustrated in Figure II.2b are consistent with data from a wide variety of experimental systems, as well as for more complex models for grain growth. 111. Grain Growth in Two Dimensions

To understand how the distribution of grain sizes evolves during grain growth it is necessary to go beyond the phenomenological model outlined

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273

t

l/T FIG. 11.2. (a) The average grain size as a function of time based on eq. (7), with the average area linear in time. (b) The grain growth rate as a function of temperature based on eqs. (7) and (8).

previously. This has been accomplished with greatest success for 2D systems, for which some relatively simple but powerful topological rules are known. Many of the concepts discussed here in the context of grain growth apply to the evolution of other cellular systems such as magnetic domains and froths or foams. The character of cellular structures such as the ones schematically illustrated in Figure 1.1, and how they evolve, is also of interest not only to materials scientists, but also to biologists, geographers, sociologists, and even cosmologists, and contributions to the understanding of cellular structures have come from scientists working in diverse fields. 1. APPLICATION OF EULER’S THEOREM First consider the relationship among the number of cells or grains, G, the number of vertices, and the number of grain boundaries, B, in a 2D cellular structure. From Euler’s theorem, on a Euclidean plane G+V‘-B=l.

(9)

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CARL V. THOMPSON

In single-phase 2D polycrystalline structures, three and only three grain boundaries meet at a vertex, given that a vertex with four boundaries is unstable with respect to generation of a new boundary and a second three-boundary vertex. With this constraint, V is the number of grain boundary triple junctions and

It is also the case that if G, is the number of grains with n boundaries, 00

nG,

= 2B.

n= 1

Defining g(n,t) as the fraction of grains with n boundaries, GJG, as a function of time, and noting that 1 in eq. (9) is small compared to the rest of the terms, we have that m

(n)

ng(n, t) = 6

= n= 1

at all times. Noting that Cg(n, t) = 1, from eq. (9) we have that, for large values of n, the average number of boundaries per grain has to be 6. (n)

= 6.

(13)

2. LEWIS’SLAW If it is assumed that both the grain-size and number-of-sides distributions remain self-similar during grain growth, it might be expected that the grain size and number of sides are related. L e ~ i s ’ , ~ first , ~ , ~proposed such a relationship based on empirical observations of biological cellular structures, such as cucumber skins, for which he found that

where (a,) is the average area of n-sided cells and A and no are constants F. T. Lewis, Anat. Rec. 38, 241 (1928). F. T.Lewis, Anat. Rec. 55, 323, (1928). F. T. Lewis, Am. J . Bot. 30, 74 (1943). F. T. Lewis, Am. J . Bot. 31, 619 (1944).

GRAIN GROWTH AND EVOLUTION OF CELLULAR STRUCTURES

275

to be empirically determined. This is often referred to as Lewis’ law. While a rigorous proof of Lewis’ law has not been provided, Rivier and LissowskP have argued that if the entropy of a cellular structure is defined as

the principle of maximum entropy requires that for a sufficiently large system, the structure will adopt a configuration that maximizes H . Rivier and Lissgowski found that when H is maximized, given the constraints that X g ( n ) = 1, (n) = 6, and that the total grain boundary area is fixed, g(n) is an exponential distribution function of the form g(n)cr exp[ - C , n], where C , is a constant. They further demonstrated that for an exponential g(n), Lewis’ law holds. 3. THEABOAV-WEAIRE LAW Another empirically derived topological characteristic of cellular structures is Aboav’s law7,*.Aboav originally analyzed the grains in polycrystalline MgO samples and empirically found that m, = 5

+

(a),

where m, is the average number of sides of the neighbors of an n-sided grain. Aboav’ later analyzed data from C. S. Smith’s experiments on froths” and found that

m, = (6 - a)

+

fi

p2))>

where p 2 is the second moment of the number-of-sides distribution and is given by

and where “a” was empirically found to be about 1.2. N. Rivier and A. Lissowski, J. Phys. A: Math. Gen. 15, L143 (1982).

’ D. A. Aboav and T. C. Langdon, Metallography 2, 171 (1969).

D. A. Aboav, Metallography, 3, 383 (1970). D. A. Aboav, Metallography 13,43 (1980). l o C. S. Smith, in Metal Interfaces, Cleveland, OH, American Society for Metals (1952), p.65.

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CARL V. THOMPSON

Weaire" later argued that "a" should be 1. The argument begins with the observation that ZC,m,lm,ng(n) is the expected value of nz because it sums the number of sides of all grains but counts each n-sided grain n times. Further noting that (n2) = (n)' + p z , from elementary statistics, and using eq. (12), we have that

which is satisfied by eq. (17) with a = 1 so that

m, = 5

+

(T).

Equation (20) is now commonly referred to as the Aboav-Weaire law. 4. SWITCHING EVENTS

A 2D cellular structure evolves through one of only two fundamentally different switching events (assuming that boundary disappearance is not possible). These have come to be known as T1 events and T2 events," the former referring to the neighbor switching depicted in Figure III.la and the latter referring to grain disappearances such as the ones depicted in Figure 1II.lb. A T1 event occurs when two triple junctions come into contact, resulting in an unstable configuration, which immediately results in the creation of new boundary. Two grains lose sides and two grains gain sides, so that the number of sides (and the number of grains) is conserved. In T2 events, grains disappear and the number of sides and number of grains are not conserved. When a three-sided grain disappears, its three neighbors lose one side each, and hence an n-sided neighbor switches to topological class n - 1. There are two ways in which a four-sided grain can disappear, but both ways result in two neighbors losing a side each, and the number of sides of the other two neighbors remains unchanged. Disappearance of a five-sided grain can occur in five ways, and results in two neighbors losing a side, one neighbor gaining a side, and two neighbors having an unchanged number of sides. It is expected that grains with six or more sides are unlikely to disappear before changing to lower-order topological classes. Mocellin and coworkers have investigated the effects of repeated random application

'' D. Weaire, Metallography 7, 157 (1974).

GRAIN GROWTH AND EVOLUTION OF CELLULAR STRUCTURES

277

FIG. 111.1. (a) A type I switching event. (b) Type I1 switching events (grain disappearances). The numbers indicate the change in the number of sides of grains after the switching event.

of these switching operations to cells within cellular stru~tures'~~' and characterized the resulting steady-state number-of-sides distribution.

5. MULLINS-VON NEUMANN LAW In considering the evolution of a soap froth, von NeumannI4 derived a simple but important relation between the number of sides of a 2D bubble and the rate of change in its area, A,:

M. Blanc and A. Mocellin, Acta metall. 27, 1231 (1979). E. Carnal and A. Mocellin, Acta metall. 29, 135 (1981). l4 J. von Neumann, Metal Interfaces, Cleveland, OH, American Society for Metals (1952) p. 108.

l2

I3

278

CARL V. THOMPSON

FIG. 111.2. Illustration for the derivation of the Mullins-von Neumann law [after J. Stavans, Rep. Prog. Phys. 56, 733 (1973)l.

where C , is a constant independent of n and time. Mullins'' later presented a more general derivation that applies to grains in a 2D system for which the velocity of motion of their boundaries is proportional to the local radius of curvature, and for which all boundaries meet at an angle of 2n/3. Rivier16 later argued that a similar relationship holds for the average area of grains in topological class n, ( A " ) . Following Stavans,' von Neumann's original result can be derived by considering an n-sided grain with boundaries that meet at angles of 2n/3 at triple points, such as the five-sided bubble/grain shown in Figure 111.2. We will consider the angle through which a vector tangent to the boundaries, v, rotates as it travels along the grain perimeter. As the vector is moved through a vertex it rotates by 71/3. Because the boundaries are circular arcs, as the vector is moved along the ith boundary, it rotates through an angle ui = 1JRi where li is the arc length of the boundary and Ri is its radius of curvature. As the vector completes a circuit around the grain perimeter, we find that

c a,

i= 1

LY1

+

(y)

= 2n.

In considering growth or shrinkage of bubbles in a froth, von Neumann assumed that the rate of gas transfer by diffusion from one bubble to its neighbor through their shared boundary is proportional to the boundary length Ii and the pressure difference across the boundary, Api. This pressure difference, through Laplace's equation, is proportional to the radius of l5 l6

W. W. Mullins, J. Appl. Phys. 27, 900 (1956). N. Rivier, Phil. Mag. B52, 795 (1985). J. A. Glazier and J. Stavans, Phys. Rev. A3, 306 (1989).

GRAIN GROWTH AND EVOLUTION OF CELLULAR STRUCTURES

279

curvature (Bpi cc K~ = l/Ri) so that the rate of change of the area of an n-sided bubble is given by

where C , is now seen to be a constant of proportionality that includes the diffusivity of the gas through the boundary wall. Use of eq. (22) with eq. (23) yields eq. (19). In the equivalent expression for grain boundaries, C, is replaced with the mobility p, which is defined so as to include the boundary energy,

r2)

=

(F)(n

-

6),

where it is assumed that all boundaries have the same mobility everywhere along their length. Glazier and Stavans” have shown that the result given in eq. (21) can be generalized for bubbles with internal vertex angles 8, not necessarily equal to 2nf3, such that

They also derived an expression for C, applicable when cell walls have a finite width.

IV. Mean Field Models for Grain Growth

While eq. (24) gives a simple model for the evolution of a single grain, a detailed model for the evolution of a collection of interacting grains, in which grains disappear and change topological classes, has been elusive. In most cases, so-called mean field models have been developed in which the behavior of individual grains is influenced by the average characteristics of the ensemble of grains. The goals of such models have generally been not only to recover the behavior embodied in eq. (l), but also to provide a description of the distribution of grain sizes, as well as, in many cases, descriptions of the topological characteristics of evolving grain structures.

280

CARL V. THOMPSON

1. HILLERT’S MODEL Noting that during grain growth, large grains tend to grow and small grains tend to shrink, Hillert began the development of his model by asserting that the average rate of growth for a grain of radius r is given by

’*

dr dt

1

1

where a‘ is a constant of proportionality, l/r is taken to be the average curvature of the grain ( ( K ) ) , and l/r, (or K , ) is a critical curvature defined such that grains with average curvature less than K , grow, and those with greater average curvature shrink. Hillert noted that for 2 D grain growth, this equation with the Mullins-von Nuemann relationship (eq. (24)) implies a relationship between the number of sides of a grain and its radius, specifically n=6

+ 6 a ’ ( r ~-, l),

(27)

which it is worth noting is not consistent with Lewis’ law. More recently, Abbruzzese et al.I9 began with a proposed linear relationship of the type given by eq. (27) to rederive Hillert’s model, and presented experimental evidence to support the assumed linear dependence of n on the average value of r. Given eq. (26), Hillert noted the similarities between grain growth and coarsening of spherical particles in a mean field, and, following Lifshitz and SlyozovZoand Wagner”, he required that there be a continuity for the flux of grains in size space such that

where f ( r ) is the grain-size distribution. Hillert then found a function f ( p ) , with p = r / ( r ) , which was time-independent and satisfied eqs. (27) and (28),

I* l9

21

M. Hillert, Acta metall.13, 227 (1965). G. Abbruzzese, I. Heckelmann, and K. Liicke, Acta Metall. Mater. 40,519 (1992). I. M. Lifshitz, and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961). C. Z. Wagner, Electrochem. 65, 581 (1961).

GRAIN GROWTH AND EVOLUTION OF CELLULAR STRUCTURES I

I

281

I

I

--.

=Hillert -Lognormal

--

-

Weibull I

I

I

Grain Size FIG. IV.1. Proposed grain-size distribution functions: the Rayleigh (dot-dash line), and lognormal (solid line) distributions; and the function proposed by Hillert (dotted line) for 2D normal grain growth.

where p = 2 in 2D and B = 3 in 3D. This is the same scaling distribution found for interface-limited coarsening of particles with zero volume in a mean field. Hillert also showed that for f ( p ) given by eq. (27), eq. (1) is satisfied with a = 0.5 (as in eq. (7)). Hillert further showed that for this function, with ( K ) defined as the average curvature for the ensemble of boundaries, K , = ( K ) for the 2 D case and K , = 9 ( ~ ) / 8for the 3 D case. The central new result of Hillert's model was the grain-size distribution function given by eq. (27), which is shown in Figure IV.l. It is characterized by a sharp cutoff at r = 2(r). In experiments, grain-size distributions are generally found to be well fit by 1 0 g n o r m a l ~or ~ *WeibullZ4 ~~ distribution functions. The lognormal and Weibull distribution functions are shown in Figure IV.l. The lognormal function is

22

23 24

H. V. Atkinson, Acta metall. 36, 469 (1988). C. V. Thompson, Annual Review of Materials Science 20, 245 (1990). W. Fayad, C. V. Thompson, and H. J. Frost, Scripta Mater 40,1199 (1999).

282

CARL V. THOMPSON

where d , , is the median value of d and n is the standard deviation of ln(d), and where ( d ) = d , , exp(a2/2). The Weibull distribution function is given by

where 6 is a time-dependent parameter and /3 is a time-independent constant. It is clear that data which is well fit by either of these functions will not be well fit by Hillert’s distribution function. Hillert’s model is therefore not a satisfactory model for grain growth.

2. LOUAT’S MODEL LouatZ5argued that Hillert’s analysis did not capture the statistical uncertainty of the grain growth process caused by what he called “grain collisions” or switching events. He argued that the rate of growth of a given grain varied in a random manner, with grains moving randomly through size space so that the flux is given by a diffusivity-like rate constant D, which is multiplied by af/ar. In this case, continuity requires

Solving this equation with f (0) = 0, Louat found the Rayleigh distribution function 2d 5 p 3 exp f R ( d ,t ) = -

(s) p

3

9

(33)

where 5 is a constant. Frost has discussed the failure of Louat’s solution to conserve volume, and also treated analogous models for diffusion in area or volume spacez6. In the former case, for the grain-area distribution f ’(A,t), eq. (32) obeys the equation:

25 26

N. P.Luoat, Acta metall. 22, 721 (1974). H. J. Frost, Materials. Sci. Forum 94-96,903 (1992).

GRAIN GROWTH AND EVOLUTION OF CELLULAR STRUCTURES

283

so that the “diffusion coefficient” D , has units of (length)4/time. The solution for f’has the form

where w’ and

t’ are constants. Equivalently, f “(d,t ) = w“dz [exp

where w“ and

($)/(t”tz/3)],

t” are appropriately redefined constants.

3. HYBRIDMODELS Hunderi and Ryum” first pointed out that Hillert’s model for grain growth was fully deterministic while Louat’s model was fully probabilistic, and argued that the actual process has components of both behaviors. They proposed a composite equation

(g)).

”=”$;(f at ar

(37)

Pandez8 later solved this equation (with D replaced by D/2), assuming, in contrast with eq. (26), that

with [ being a constant of proportionality. He found f to be a Rayleigh distribution function modified by a degenerate hypergeometric series. This distribution and the Rayleigh distribution cannot be usefully distinguished in comparisons with data. 4. MEANFIELDMODELSBASEDON

THE

MULLINS-VON NEUMANN LAW

Another approach to developing a mean field theory for grain growth, based directly on the Mullins-von Neumann relationship, can be developed by

’’ 0. Hunderi, and N. Ryum, J . Muter. Sci. 15, 1104 (1980). ’* C. S. Pande, Actu Metall. 35, 2671 (1987).

284

CARL V. THOMPSON

considering a time-dependent distribution function for the grain areas and the number of sides per grain, F(A, n, t). The time variation of F is then given by

so that with eq. (24) we have, aF - -

at

a -(% 5 aA

- 6)F)

+ I,,

where I , is defined by the second term on the right-hand side of eq. (39). Because n takes on only discrete values, I , describes the rate at which grains of an area A change topological class through switching events. F r a d k ~ vwas ~ ~the first to develop a model of this type. He considered three types of events for an n-sided grain:

(i) Switching to topological class n - 1 through loss of a side to a neighbor due to a T1 switching event. (ii) Switching to topological class n + 1 through gaining of a side from a neighbor due to a T1 switching event. (iii) Switching to topological class n - 1 through a T2 event involving disappearance of a neighboring grain. By assuming that switching events as well as vanishing events are uncorrelated, he argued that the rate at which an n-sided grain of area A gains a side is

where F , = F,(A, n,t ) is the number of grains per unit area in the nth topological class in the interval d A around A at time t. The rate at which an n-sided grain loses a side is given by

where B is a “switching intensity parameter,” related to the relative frequencies of side loss due to T1 and T2 events, and r is the frequency with which 29

V. E. Fradkov, Phil. Mag. Lett. 58, 271 (1988).

GRAIN GROWTH AND EVOLUTION OF CELLULAR STRUCTURES

285

T1 events lead to a side loss. In this case, I,

=

x,, + x;-l - x, - x;,

(43)

so that

and

since 2-sided grains cannot become 1-sided grains. Fradkov showed that

r = (-A ) 6

2

(n - 6)'F,(O, n, t),

n=2

but that p remains adjustable. Fradkov et alZ9 developed a computer simulation in which grains change area according to the Mullins-von Neumann relationship, and change toplogical class according to the model outlined above. They found scaling behavior obeying eq. (1) with c1 = 0.5 and an exponential area distribution for the ensemble of all grains,

as well as a number-of-sides distribution dependent only on fi and not on the initial state of the system. They further compared their simulation with experiments on A1 films3' to argue that has a value of about 0.5. Marderj' developed a similar model, taking n-1 I(A, n) = u(A) -F , - [u(A) AT

n n + l + d(A)] F, + d(A) -Fn AT AT

(48)

where AT is the total area, and u(A) and d(A) are the rates at which a grain of area A gains or loses a side, respectively. Marder then developed approximate expressions for u(A) and d(A) by treating the probability that 30 31

V. E. Fradkov, A. S. Kravchenko, and L. S . Shvindierman, Scripta Metall. 19, 1291 (1985). M. Marder, Phys. Rev. A36,438 (1987).

286

CARL V. THOMPSON

grains would neighbor disappearing 3-, 4,or 5-sided grains, and solved the resulting equations numerically, finding results similar to those of Fradkov, but without a corresponding floating parameter like p. Flyvbjerg and Jesspen3’, Stavans et al.33,Iglesias and de Almeida34,and Segel et al? have developed models based on different treatments of I , but find similar results, as seen in Figure IV.2, which shows predicted number-of-sides distributions for several treatments.

V. Simulations The most widely used approach to modeling grain growth is based on a Monte Carlo technique, usually formulated as a Potts model in which area in 2D or volume in 3D is divided into an array of cells corresponding to points, each of which is assigned a state corresponding to a grain identity (see Figure V.la-b).36*37Boundaries between regions with the same states are taken to be grain boundaries, and boundary migration is achieved by changing the state of cells that neighbor the boundary. Individual cells are considered in a random order, and their state is or is not changed, depending on the energy of interaction with neighboring cells. Having the same state as neighboring cells gives a lower energy, while having a different state gives a higher energy. The cell is then allowed to change its state so as to minimize energy, with a probability defined by a Boltzmann distribution of probable states, according to their energy levels. As cells at boundaries change states, the boundaries move. The energetics of grain growth are addressed through modification of interaction energies. Variations in kinetic effects, such as changes in grain boundary mobility, are modeled by changing the Boltzmann function that defines the probability of state transitions, or by changing the frequency with which states of cells at boundaries are treated. A recent variation on the Monte Carlo approach allows for a continuum of states so that grain centers have fixed state identities. The state changes over a number of cells at grain boundaries, simulating diffuse bounda r i e ~ . While ~ ~ . this ~ ~ provides a potentially improved model for magnetic 32

33 34

3s 36

37

’*

39

H. Flyvberg and C. Jeppesen, Phys. Scr. 38,49 (1991). J. Stavans, E. Domany, and D . Mukamel, Europhysics Letts. 15,479 (1991). J. R. Iglesis and R. M. C. de Alameida, Phys. Rev. A43, 2662 (1991). D. Segel, D. Mukamel, 0. Krichevsky, and J. Stavans, Phys. Rev. E47, 812 (1993). M. P. Anderson, D. J. Srolovitz, C. S. Crest, and P. S. Sahni, Acta Metall. 32,783, (1984). D. J. Srolovitz, M. P. Anderson, P. S. Sahni, and G. S. Crest, Acta Metall. 32,793 (1984). D. Fan, and L.-Q. Chen, Acta mater. 45, 611, (1997). D. Fan, C. Geng, and L.-Q. Chen, Acta mater. 45, 1115 (1997).

GRAIN GROWTH AND EVOLUTION OF CELLULAR STRUCTURES

0.5

I

I

I

I

287

I

0

0.4 0

0.3

p(n) 0.2 0.1 /

0-

!

I

I

I

FIG.IV.2. Comparison of the number-of-sides distribution predicted using the models of Flyvberg3’ (open circles), Marder3’ (full circles), Iglesisas and de A l ~ n e i d a(empty ~ ~ triangles), (full triangles), with experimental measurements on soap froths (Stavans and Stavans et et [from J. Stavans, Rep. Prog. Phys., 56, 733 (1993), reprinted with permission].

I

FIG. V.l. Illustration of two techniques for simulating grain growth through use of a Potts model, and through front tracking [H. J. Frost and C. V. Thompson, Current Opions in Solid State and Materials Science, 1, 361 (1996)l.

288

CARL V. THOMPSON

domain structures, its advantage in treating the grain growth process is less clear, though Fan et al. suggest that this technique might allow treatment of inhomogeneous compositional or stress states. As an alternative to this discrete element Monte Carlo approach, simulations of grain growth can be based on the tracking of the migration of the grain boundaries themselves (Figure V. lc-d). Boundary segments can be moved according to specific relationships between the boundary migration velocity and the local characteristics of the boundary, such as its curvature, the relative crystallographic orientation of neighboring grains, and the stress state of neighboring grains. For example, in simulations of normal grain growth, boundary points are moved according to u = PIC,and grain boundary junctions are moved to maintain the balance of grain boundary tensions. Such front-tracking algorithms are fairly easy to implement in two dimens i o n ~ . ~One ~ . advantage ~ ~ * ~ ~of the front-tracking approach is that it can relatively easily be made to be very accurate. Another is that it can handle arbitrary relationships between local curvature and local velocity, which is important for describing many physical The disadvantage of the front-tracking technique is that extension to three dimensions is difficult. While 3D front-tracking simulations have been developed45946,47948.49 results from large-scale grain growth simulations have not yet been reported. Another class of simulation techniques, first proposed by FullmanS0and generally referred to as vertex models, describes the evolution of polygranular structures in terms of the motion of the points where three boundaries meet in 2D, or where four boundaries meet in 3D. This technique has the advantage of computational convenience and, when properly developed, can be thought of as a first approximation to a complete numerical description of curvature-driven migration.’ H. J. Frost, C. V. Thompson, C. L. Howe, and J. Whang, Scripta Metall. 22, 65 (1988a). H. J. Frost and C. V. Thompson, J . Electronic Mat. 17,447 (1988b). 4 2 S. P. A. Gill and A. C. F. Cocks, Acta mater. 44,4777 (1996). 4 3 H. J. Frost, C. V. Thompson, and D. T. Walton, Acta Metall. et Mater. 38, 1455 (1990). 44 H. J. Frost, C. V. Thompson, and D. T. Walton, Acta Metall. et Mater. 40,779 (1991). 45 K. A. Brakke, 7he Motion of a Surface b y its Mean Curvature, Princeton University Press, Princeton, NJ (1978). 46 K. A. Brakke, Experimental Mathematics 1, 144 (1992). 47 K. A. Brakke, in Wdeo Proceedings of the Workshop on Computational Crystal Growing, ed. J.E. Taylor, Providence, RI: American Mathematical Society (1992). 48 A. Kuprat, to be published in SIAM J. Sci. Comp. (1999). 49 D. C. George, N. Carlson, J. T. Gammel, and A. Kuprat, Proceedings of Modeling and Simulation of Microsystems, San Juan, Puerto Rico, (April 1999), p. 463. 5 0 R. L. Fullman, in Metal Interfaces, American Society for Metals, Cleveland, OH, (1952), p. 179. ” K. Kawasaki and Y. Enomoto, Physica 150A,462 (1988).

40

41

GRAIN GROWTH A N D EVOLUTION OF CELLULAR STRUCTURES

289

There are several simulations of the evolution of soap froths, following the original work of Weaire and K e r m ~ d e . ~The ”~~ physical situation described by these models is inherently different from that of grain growth in polycrystals because the individual films (or boundaries) that separate the gas cells have constant curvature over their entire extent, and grain boundaries in polycrystals need not. There are also a few models that do not fit conveniently into the four categories just listed. These include simulations in which the evolution of size distributions of a population of grains is followed, without maintaining an explicit geometrical mode1.’9~54~55~56~57’58 Some of these models include an evolving representation of the neighbor relationships. Those that do not are basically numerical evaluations of analytic models. A few models describe grain growth through the coalescence of grains associated with the disappearance of boundaries as a result of grain rotation^,^^.^' a process that is not commonly observed in experiments. All of these simulation approaches lead to behavior consistent with eq. (l), with u = 0.5. In general, it is also found that the version of the Aboav-Weaire law given in eq. (20) is obeyed, and that the average area of n-sided grains obeys the Mullins-von Neumann Law. All of the simulations also appear to evolve to similar steady state behavior, resulting in similar time-invariant normalized grain size and number-of-sides distributions, shown for Potts, front-tracking, and vertex models in Figure V.2. Recent analysis of the steady-state grain-size distribution obtained using a front-tracking simulation indicates that it is well fit by a Weibull distribution function given by eq. (31),24 with constant B and with 6 scaling with t’”. Also,

where r, is the gamma function. The fit of the steady-state grain-size distribution is better for a Weibull function than for a lognormal function or a Rayleigh function.24 Figure V.2b shows the grain-size distribution from various simulations on a Weibull plot of ln(d) vs. In( - ln(1 - F)), where F 52

” 54 55

56 57 58

59 6o

D. W e a k and J. P. Kermode, Philos. Mag. B48,245 (1983). D. W e a k and J. P. Kermode, Philos. Mag. B50,379 (1984). 0. Hunderi, N. Ryum, and H. Westengen, Acta Metall. 65,161 (1979). 0. Hunderi, Acta Metall. 27, 167 (1979). K. Lucke, I. Heckelmann, and Abbruzzese, G., Acta Metall. et Mater. 40, 533 (1992). V. Y. Novikov, Acta Metall. 26, 1739 (1978). J. A. Floro and C. V. Thompson, Acta Metall. et Mater. 41, 1137 (1993). T. 0. Saetre and N. Ryum, Met. Pans. A22, 2257 (1991). T. 0. Saetre and N . Ryum, Met. Matl. Pans. A26, 1687 (1995).

290

CARL V. THOMPSON

is the cumulative fraction of grains with sizes less than or equal to d. In this plot, data that fit a Weibull distribution fall on a straight line, as seen for data from the simulations. Also shown in Figure V.3a, for comparison, are the lognormal and Rayleigh distributions that best fit the data. Note that for the Weibull distribution function, p for the best fit is 5/2.24 The reason for this is unknown. VI. Experiments on Two-Dimensional Systems

There have been extensive experiments on the evolution of soap froths, and these have been reviewed elsewhere.61,62 Froth evolution is accomplished through movements of cell walls at velocities proportional to their curvatures, and through the disappearance of cells, which results in an increase in the average cell size. Glazier, Stavans, and coworkers have carried out studies of the evolution of froths between glass plates, and followed the evolution after the average cell size becomes comparable to and larger than the spacing between the plates. It has been shown that under proper experimental conditions, a steady-state behavior is observed for which p2 (see eq. (18)) obtains a constant value (1.4 f 0.1). In this regime, the Aboav-Wearie law (eq. (20)) is obeyed, the average area of nsided grains obeys the Mullins-von Neumann relationship, eq. (1) is satisfied (with CI = 0.5) and time-invariant normalized grain-size and number-of-sides distributions are observed. The latter distributions are shown in Figure V.3b. It should be noted that froths are different from polycrystalline materials in several important ways. First, the cell boundaries have uniform energies per area. While this is assumed in the idealized models for grain growth previously outlined, it is not a valid assumption for real polygranular systems in which the grain boundary energy is a function of the relative crystallographic orientations of the grains on either side of the boundary, and of the orientation of the boundary itself. One consequence of this is that cell boundaries in froths adopt shapes of constant curvature while grain boundaries generally do not. All cell boundaries in froths also move with the same uniform mobility while grain boundaries do not. These characteristics in many ways make modeling the behavior of froths more straightforward than modeling that of polycrystalline materials. Froths also provide relatively easily studied experimental analogs for idealized grain growth, which serve as experimental tests for our basic understanding of the 61

J. Stavans, Rep. Prog. Phys. 56, 733 (1993). J. A. Glazier and D. Weaire, J. Phys.: Condens. Matter 4, 1867 (1992).

GRAIN GROWTH AND EVOLUTION O F CELLULAR STRUCTURES

291

FIG. V.2. Grain-size distributions found for simulations of 2D normal grain growth using different simulation techniques: (a) plotted on a linear scale [left-pointing triangles, Fayad et dotted line, Weibull distribution CI = 1.127, B = 2.5; asterisks, Fan et a1.38*39;x’s, Nagai et allo4; squares, Srolavitz et aLJ7; upward-pointing triangles, Gill and Cox42]; (b) plotted as the cumulative percent as a function of grain size, on axes for which a Weibull distribution function gives a straight line [solid line, Weibull distribution with a = 1.13 and p = 2.5; dashed line Rayleigh distribution, with CI = 1.13 and B = 2; circles, Fan et a1.38,39;diamonds, Gill and open triangles, Nagai et a1.104]; (c) number-of-sides filled triangles, Srolovitz et distributions for simulations of 2D normal grain growth [solid line, Fayad et al.24;circles, Gill and squares, Fan et a138*39;open diamonds, vertex model, Nagai et a1.lo4; filled diamonds, modified vertex model, Nagai et a1.’04].

evolution of cellular systems. However, the cell walls and the triple lines in froths have a finite thickness that can increase as the froth evolves, unless care is taken to drain fluid during the evolution so as to retain constant boundary and triple junction widths as the total boundary area of an experimental system is reduced. Also, a topological event that is possible for froths, disappearance (or “popping”) of cell boundaries without switching events, is not possible for polycrystalline materials. Froths between plates are not truly two dimensional, because disappearing grains can switch to 3D

292

CARL V. THOMPSON

L

QI

; 1.00 E

.a .-C 0

E

a

0.10

QI N

.0

E L a

z 0.0 1 1

10

Cumulative

50

Z

0.4

0.3 c

.-0 u 0.2 e LL U

0. I

0.0

Number of Sides FIG.V.2. Continued.

9099

GRAIN GROWTH A N D EVOLUTION OF CELLULAR STRUCTURES

293

shapes (e.g. from 2D three-sided cells with two plates providing two cell walls, to four-sided 3D shapes with a plate providing only one cell wall) before disappearing. Other examples of evolving quasi-2D experimental systems include evolving lipid monolayers on water-air i n t e r f a c e P ~and ~ ~ magnetic domains in In the former case, behavior similar to that observed for froths, as outlined above, has also been seen. Steady-state grain-size and numberof-sides distributions from these studies are included in Figure V.3. The evolution of magnetic domains is fundamentally different in that the end state is generally one of fixed and finite domain size. However, magnetic domain structures and domain structure evolution can be analyzed using some of the theoretical tools outlined earlier.67

VII.

Experiments on Foils and Thin Films

1. THICKPOLYCRYSTALLINE FOILS AND FILMS

When the grain size is similar to or larger than the thickness of a polycrystalline foil or film, most of the grain boundaries traverse the thickness of the foil so that the grain structure takes on a two-dimensional character. Grain growth in systems with this 2D-like character was first studied in relatively thick (> 100 pm) sheets of metallic alloy^.^**^^ It was usually found that grain growth stagnated when the in-plane grain size was about three times the film thickness. This phenomenon, called the specimen thickness effect, is also usually observed in thin films, and will be discussed in detail in the next section. Recrystallization has also been extensively studied in metallic sheets with comparable grain sizes and t h i c k n e ~ s e s The . ~ ~ term recrystallization refers to a process in which new grains grow at the expense of an otherwise static matrix of pre-existing grains. Recrystallization often occurs in sheets that have been heavily deformed through rolling. After cold working there is a 6 3 K. J. Stine, S. A. Raueso, B. G. Moore, J. A. Wise, and C. M. Knobler, Phys. Rev. A41,6884 (1 990). 64 B. Berge, A. J. Simon, and A. Libchaber, Phys. Rev. A61,6893 (1990). 6 5 K. L. Babcock and R. M. Westervelt, Phys. Rev. Letts. 63, 175 (1989). 6 6 K. L. Babcock, R. Seshadri, and R. M. Westervelt, Phys. Rev. A41, 1952 (1990). 6 7 D. Weaire, F. Bolton, P. Molho, and J. A. Glazier, J . Phys. Condens. Matter 3, 2101 (1991). 68 P. A. Beck, M. L. Holworth, and P. R. Sperry, Pans. Am. Inst. Min. (Metall.) Eng. 180, 163 (1949). 69 F. J. Humphreys and M. Hatherly, Recrystallization and Related Annealing Phenomenon, Pergamon Press, Oxford (1996).

294

CARL V. THOMPSON

L

u

1.00

E a

5 C ._

t!

a

0.10

u

.-

N

a

0.0 I 1

10

50

9099

Cumulotive %

Number of Sides per qroin FIG. V.3. (a) Grain-size distributions from experiments on nearly-2D normal grain or cell growth plotted as the cumulative percent as a function of cell size, on axes for which a Weibull distribution function gives a straight line [line with asterisks, simulation Fayad et al.24; solid line, Weibull distribution a = 1.13 and /3 = 2.5; dash-dot line, Rayleigh distribution ct = 1.13, /I= 2; dotted line, lognormal distribution d , , = 0.9, a = 0.48; circles, experiments on soap froths, StavanP; triangles, PDA layers, Stine et (b) Number-of-sides distributions for nearly-2D systems, compared to results of simulations of 2D normal grain growth [solid line, simulation, Fayad et aLZ4;filled circle, experiments on soap froths, Stavans et asterisks, PDA layers, Stine et pluses, lipid monolayers, Berge et

GRAIN GROWTH AND EVOLUTION OF CELLULAR STRUCTURES

295

high concentration of defects such as dislocations. The new grains that nucleate and grow in recrystallization processes have low defect densities, so that the energy of the rolling-induced defects significantly contributes to the driving force for grain boundary motion. Because new low-energy grains nucleate and grow at the expense of a static matrix of high-energy grains, recrystallization is a fundamentally different process from grain growth as described previously. Metallic sheets that have been extensively deformed through cold rolling often develop what is called a crystallographic texture. Grains deform in such a way that, after rolling, they tend to be aligned with specific crystallographic directions in the rolling direction and normal to the plain of the film. Asmus et observed a phenomenon in cold-rolled metallic sheets they called secondary recrystallization. Sheets of 3% Si-Fe that had already undergone defect-driven recrystallization (primary recrystallization) underwent a second transformation resulting in a change in the crystallographic texture of the sheet (the grain orientations relative to the normal to the plane of the sheet). Dunn and Walter69g71later showed that a third transformation process (tertiary recrystallization) could lead to different textures if the annealing ambient was changed, and proposed that in secondary and tertiary recrystallization, texture-related differences in the surface energy of the films significantly contributes to the driving force for recrystallization. This phenomenon is similar to, and difficult to distinguish from, surface-energy-driven grain growth that is commonly observed in thin films, and will be discussed in more detail in a later section. While grain growth driven solely by the elimination of grain boundary area has been shown to stagnate in most thick metallic sheets and thin films, there is one example in which behavior consistent with 2D normal grain growth has been observed in foils. Fradkov et al.” studied the evolution of grain structures in 120-pm-thick A1 sheets obtained by cold rolling of bulk Al. The A1 was doped with 0.01% Mg to obtain “boundaries with rather similar and isotropic properties.” Observation of evolution in these sheets showed a behavior consistent with eq. (1) (with c( = 0.5), and data consistent with the results of Figures V.2 and V.3 were obtained. The results were therefore in remarkably good agreement with the model proposed earlier by the same group. Behavior similarly consistent with idealized 2D normal grain growth has also been observed in polycrystalline 500- to 1000-pm-thick sheets of polycrystalline succinonitrile (SCN). These sheets were formed through solidification on glass plates and their thicknesses were defined by thermal

’’ F. Assmus, K. Detert, and G. Ibe, Zeitschrft fur Metallkunde 48, 344 (1957). J. L. Walter and C. G. Dunn, Trans. of the Metallurgical SOC.A I M E 215, 465 (1959). ’*V. E. Fradkov, A. S. Kravchenko, and L. S. Shvindlerman, S c r i p M e t . 19, 1291 (1985).

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gradients perpendicular to the glass plates.73,74,75*76 Grain structures were studied through optical imaging of grain boundary grooves at the top surface of the sheets. It remains unclear what, if any, effect these grooves had on the evolution of the structure.74 Behavior consistent with eq. (1) (a = 0.5) and steady-state grain-size and number-of-sides distributions were reported for this system with behavior similar to that observed earlier by Fradkov and coworkers in A1 foils. An important observation common to the A1 foils and SCN films studied by Fradkov and coworkers and froth experiments of the type described in the previous section is that significant numbers of fourand five-sided grains shrink and disappear without first reducing their numbers of sides. This also occurs in simulation^.^^ 2. POLYCRYSTALLINE THINFILMS Vapor-deposited polycrystalline thin films with thicknesses of order 1 pm or less also often have 2D-like structures for which most grain boundaries traverse the thickness of the film. Most materials deposited at temperatures of order 0.3 times their absolute melting temperatures or less have nonequiaxed grains with sizes in the plane of the film that are smaller than the film thickness, and develop 2D-like structures only when annealed. However, fcc metals such as Au, Al, and Cu can have 2D structures even when deposited at room temperature, while grain growth and 2D structures do not develop in undoped Si or Ge films unless they are annealed at 90% of their absolute melting temperature^.^^*^'*^^ Grain growth in thin films generally does not lead to the ideal behavior described in the modeling sections, or in the discussion of experiments on froths, monolayers, and foils. Grain growth stagnates once the average in-plane grain size is about three times the film thickness.'O The grain sizes in stagnant grain structures are also lognormally distributed. In some cases, further grain growth occurs through the "abnormal" or preferential growth 73 M. A. Palmer, V. E. Fradkov, M. E. Glicksman, and K. Rajan, in Modeling of Coarsening and Grain Growth ed. S.P. Marsh and C.S. Pande, p. 227 (TMS, Warrendale), p.227 (1993). 74 K. Rajan, M. E. Glicksman, V. E. Fradkov, M. A. Palmer, and J. Nordberg, in Modeling of Coarsening and Grain Growth ed. S.P. Marsh and C.S. Pande, TMS Warrendale p. 217 (1993). 7 5 V. E. Fradkov and D. Udler, Adu. in Phys. 43, 739 (1994). 7 6 M. Palmer, K. Rajan, M. Glicksman, V. Fradkov, and J. Nordberg, Met. and Mate. Pans A26, 1061 (1995). 77 V. E. Fradkov, M. E. Glicksman, M. Palmer, and K. Rajan, Acta Metull. Muter. 42, 2719, (1994). 7 8 C. V. Thompson and R. Carel, Mat. Sci. and Eng. B52, 211 (1995). 7 9 C. V. Thompson, Ann. Rev. of Materials Science (2000). J. E. Palmer, C. V. Thompson, and H. I. Smith, J. Appl. Phys. 62, 2492 (1987).

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of a few grains that usually have specific crystallographic orientation relationships with respect to the plane of the substrate surface. When the number of preferentially growing grains is small, and they grow into a “matrix” of grains with static boundaries, a bimodal grain-size distribution develops and the process is sometimes referred to as secondary grain growth.” This mode of grain growth is illustrated for Ge films in Figure VII.la and schematically in Figure VII.lb. The grains that grow abnormally often have a restricted set of, or the same set of, crystallographic planes parallel to the plane of the film. When this is the case, the grains are said to have a restricted or a uniform crystallographic fiber texture. When secondary grain growth occurs in films on single-crystal substrates, the orientations of growing grains can have three-dimensionally constrained epitaxial orientations.” Secondary grain growth in thin films therefore usually involves an evolution in the distribution of the crystallographic orientations or textures of grains, as well as an evolution in the grain-size distribution. During secondary grain growth, the normalized grain-size distribution never reaches a time-invariant scaling state. Once the grain-size distribution passes from bimodal back to monomodal, boundaries of impinging secondary grains are generally immobile so that the final grain-size distribution (which also appears to be lognormal) does not continue to evolve. The “non-ideal” grain growth seen in thin films results from them not being truly two-dimensional. Mullins (W. W. Mullins, Acta Metal. 6, 414 (1958)) argued in 1958 that the stagnation of grain growth observed in metallic sheets could result from grain boundary drag and eventual pinning due to the formation of grooves where grain boundaries intersect surfaces (Figure VII.2). This effect has been included in front-tracking simulations of grain growth as illustrated in Figure VII.3. In simulations of capillaritydriven 2D normal grain growth, the boundary mobility vs. local curvature relationship of eq. (4)and Figure VII.3a is used. However, if grain boundary grooves form at the film surface, boundaries become trapped. If it is assumed that only boundaries whose velocities fall below a critical velocity, corresponding to a critical curvature, IC,,,are trapped, the relation shown in Figure VII.3b results. Simulations carried out with this condition lead to grain growth and stagnation, with the stagnant structures having numberof-sides distributions that are significantly different from the distribution for scaling growth, as shown in Figure VII.4.83 The distributions of grain sizes are also different, with the stagnant structures having grain-size distributions that are well fit by lognormal distribution functions, which are readily

*’ *’

C. V. Thompson, J . Appl. Phys. 58, 763 (1985). C. V. Thompson, J. A. Floro, and H. I. Smith, J . Appl. Phys. 67,4099 (1990). S. P. Riege, C. V. Thompson, and H. J. Frost, Actu Muter. 47, 1879 (1999).

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FIG. VII. 1 (a) Transmission electron micrograph showing an intermediate stage of secondary grain growth in a 300A-thick Ge film on an amorphous SiO, membrane.80 (b) Schematic illustration of a film such as the one shown in (a).

distinguished from the Weibull distribution functions that apply to scaling Further, it was found in simulations that the average stagnant grain size, dstag, was 0.874/~,,. Mullins argued that the critical curvature for stagnation is approximately

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FIG. VII.2. Grooves form where grain boundaries intersect the free surface of a film, as a result of the balancing of surface and grain boundary forces [inset after W. W. Mullins, Acta Metall. 6, 414 (1958)l.

velocity

velocity

V

V

curvature U

curvature U

FIG. VII.3. (a) For front-tracking simulation of 2D normal grain growth, it is assumed that the velocity of boundary motion is proportional to the local boundary curvature. (b) To include the effects of surface grooves, which cause grain growth stagnation, it is assumed that when the boundary curvature falls below a critical level, a groove forms, and the velocity drops to

300

CARL V. THOMPSON

40

c

!I

0)

-

30-

e

1 8

20: 100--

2

4 6 8 10 12 Number of Neighboring Grains

'4

FIG. VII.4. Number-of-sides distributions for 2D normal grain growth compared with the number-of-sides distribution obtained when growth stagnates due to the presence of grain boundary grooves.83

FIG. VII.5. (a) In 2D grain growth, surface energy differences between grains displace the velocity-curvature relationship for the grain boundary at which the grains meet. (Compare to Figure VII.2a). (b) Velocity-curvature relationship including the effects of both surface energy differences and grain-boundary groove-induced stagnation, leading to secondary grain gro~th.~"

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301

where ys is the energy of the film surface and h is the film thickness, so that dstag is approximately 3h in the simulations as in most experiments in sheets and thin films. Mullins also argued that grains with anomalously low surface energies compared to their neighbors can escape stagnation, so that those grains that grow into a static matrix have crystallographic orientations with low film-surface energies. This can lead to the secondary grain growth observed in thin films, and described earlier, with the abnormally growing grains having surface energy minimizing crystallographic orientations. This phenomenon can also be accounted for through modifications of 2D fronttracking simulations as shown schematically in Figure VIISa, if the surface energies of grains are tracked and eq. (4) is modified to have the form

where

and where Ays,i is the difference in the sum of the energies of the film-surface ys and the film-substrate interface yi for the grains on either side of the boundary

This results in the simulated behavior shown in Figure VII.6, which matches the phenomenological characteristics of secondary grain growth in thin

FIG.VII.6. Simulation of secondary grain growth, with stagnation of grains with boundary curvatures less than a critical curvature, and the preferred growth of grains with surface-energy minimizing textures.44

302

CARL V. THOMPSON

films, including its increasing importance with decreasing film thickness h.23 In addition to affecting the evolution of the grain-size distribution, surface energy effects also drive the grain orientation distribution toward favored sub-populations with surface-energy minimizing orientations. As discussed earlier, surface-energy minimization has also been proposed as the driving force for texture evolution during recrystallization of much thicker metallic sheets.84 It should be noted that even in the absence of groove-induced stagnation, surface- and interface-energy minimization favors the growth of a subpopulation of grains, so that true normal grain growth, characterized by time-invariant normalized grain-size and number-of-sides distributions, should not occur until or unless only equally favored grains remain. The general case in which a favored sub-population of grains exists can be characterized as abnormal grain growth. Abnormal grain growth should be expected for grain growth in thin films of any material with anisotropic surface and interface energies and a non-uniform grain texture. This does not necessarily lead to bimodal grain-size distributions, but always leads to a time evolution of the grain-size and grain-orientation distributions. This can be simulated by assuming the relationship schematically illustrated in Figure VIISa, in which eq. (51) is applied without a stagnation criterion. It should also be noted that when a film is heated on a substrate, unless the film and substrate have the same thermal expansion coefficient, the film develops a biaxial strain that can lead to grain-by-grain differences in strain-energy density. The magnitude of the strain energy of a grain depends on its crystallographic orientation and on the elastic constants of the material. Grain-by-grain variations in the strain-energy density can lead to an additional term within the parentheses in eq. (51), of the form

where AM is the difference in the appropriate biaxial moduli of the grains meeting at the boundary and E is the elastically-accomodated biaxial strain.85 Inclusion of this effect leads to similar phenomenology to that described as resulting from surface- and interface-energy minimization, including an evolution in the grain-orientation distribution during grain growth. However, strain-energy minimizing textures are generally not the same as surface- and interface-energy minimizing textures, so that strainenergy, interface-energy, and surface-energy minimization can compete in 84

85

C. G. Dunn and J. L. Walter. Pans. Am. Inst. Min. (Metall.) Eng. 224, 518 (1962). R. Carel, C. V. Thompson, and H. J. Frost, Acta Metall. Mater. 44, 2479 (1996).

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determining the texture favored during grain growth. Surface- and interfaceenergy minimization dominates in sufficiently thin films because these have high surface-to-volume ratios. Clearly the phenomenology observed in thin films is strikingly different from the idealized normal grain growth described in preceding sections. Effects such as those due to surface grooving, surface-energy anisotropy, and strain-energy anisotropy cannot be readily accounted for in existing models for grain growth that only capture the basic phenomenology of grain structure evolution driven by grain boundary energy minimization. Simulations have been developed that also correctly describe the evolution of simple cellular systems such as froths, and satisfy conditions such as those embodied in the Mullins-von Neumann and Aboave-Weaire relationships. However, the additional advantage of simulations is that they can relatively easily be modified to account for more complex phenomenology. For example, the assumptions that all grain boundaries have the same energies and mobilities can be relaxed in simulations. This leads to an evolution in the distributions of grain boundary energies toward what appears to be a steady-state distribution, corresponding to a steady-state behavior consistent with normal grain growth.86

3. AGGLOMERATION AND GRAINGROWTH IN VERYTHINFILMS The development of grain boundary grooves at surfaces is driven by a force balance among the surface energies and the grain boundary energy of the neighboring grains. Grooving can occur via surface self-diffusion or bulk diffusion, or by diffusion through the external phase. In thin films with oxide-free surfaces (e.g., as-deposited films in ultrahigh vacuum systems, or films of noble metals), surface self-diffusion is expected to dominate. Grooves are also expected to form at grain boundary triple points, and while there is an equilibrium depth for grooves at boundaries, there is no intrinsic limiting depth for grooves at triple junction^.^' Grooves at triple junctions can become significantly deeper than grooves at isolated boundaries, and in very thin films (< 100 nm), can reach the substrate and lead to film agglomeration, as has been demonstrated and studied in Au and Ag films.88989*90In very thin films that do not wet their substrate, this 8 6 H.J. Frost, Y. Hayashi, C. V. Thompson, and D. T. Walton, MRS Symp. Proc. 317, 485 (1994a). '' D. J. Srolovitz and S. A. Safran, J. Appl. Phys. 60,247 (1986). E. Jiran and C. V. Thompson, J. Electronic Materials 19, 1153 (1990). E. Jiran and C. V. Thompson, 7hin Solid Films 208, 23 (1992). R. Dannenberg, E. A. Stach, J. R. Groza, and B. J. Dresser, 7hin Solid Films 359 (2000).

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agglomeration process can compete with, and significantly modify, grain growth, as has been demonstrated in Ag films.” When triple junction groove depths are a significant fraction of the film thickness, they may cause drag and trapping effects that modify the simple treatment of grooving outlined in the preceeding section. 4. GRAIN ROTATIONIN THINFILMS

Harris et al. have recently reported in situ TEM observation of grain rotation in freestanding very thin (25-nm-thick) Au films that are in the process of agglomerating.” Grain rotation can, in principle, lead to the disappearance of low angle boundaries and therefore contribute to the evolution toward a larger average grain size.92In discussing their results on Au membranes, Harris et al.91 develop a model that assumes that the rate-limiting step for grain rotation is diffusion in the grain boundaries. Based on this model, they argue that the rotation rate should be inversely proportional to the fourth power of the grain size. Making the unreasonable assumption that grain rotation leads to the reduction of the energy of all a grain’s boundaries at the maximum possible rate (and that none of the energy-increasing processes need be considered), Harris et al. argue that their observations are consistent with grain-boundary diffusion-limited rotation. While the model of Harris et al. overestimates the energy driving grain rotation, it may underestimate the rate at which the required diffusion can occur. In very thin membranes, grain rotation can occur through diffusion on the surface of the membrane. In the experiments of Harris et al. for which boundary thicknesses were small (due to grooving in the already very thin films) and for which surface self-diffusion could occur at relatively high rates on both surfaces of the membrane, surface self-diffusion would seem to be the likely mechanism for the observed rotation of very small grains. Grain rotation is generally not expected to play an important role in grain structure evolution in thicker films or films on substrates (or in 3D systems). 5. NON-IDEALITIES ASSOCIATEDWITH ALLOYADDITIONS Impurities often interact with grain boundaries and can exert a drag force that can lead to departures from idealized normal grain growth. Impurities that segregate to boundaries tend to reduce their mobilities, because, unless 91

92

K. E. Harris, V. V. Singh, and A. H. King, Acta Materialia 46, 2623 (1998). J. C. M. Li, J. Appl. Phys. 33, 2958 (1962).

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FIG. VII.7. The velocity-curvature relationship used to simulate the effects of solute drag on grain

boundaries can break free of the impurities, their motion is limited by the bulk diffusivity of the i m p u r i t i e ~ .This ~ ~ effect has again been modeled through modification of a 2D front tracking model. As illustrated in Figure VII.7, it is assumed that boundaries moving below a critical velocity and with curvatures below a critical value, are subject to solute drag leading to a mobility p2 lower than that characteristic of boundaries which are not subject to drag and therefore have a higher mobility p l . These simulations showed that when the average value of K is large or small compared to K,,, behavior similar to steady-state normal grain growth, with a grain-size distribution fit by the Weibull distribution, is observed. However, for K similar to K,,, the grain-size distribution is well fit by a lognormal distribution function.94 Second phase precipitates can also impede the motion of grain boundaries, and lead to grain growth stagnation. Zener95was the first to treat this, and he showed that the average stagnant grain size ( d ) is inversely related to the volume fraction of the precipitates, f, such that

93 K. Lucke and H. P. Stuwe, in Recovery and Recrystallization, ed. L. Himmel, Interscience, New York, (1963),p. 171. 94 H. J. Frost, Y. Hayashi, C. V. Thompson, and D. T. Walton, MRS Syrnp. Proc. 317, 431 (1994b). 9 5 C. Zener, quoted by C.S. Smith, ll-ans. 7MS-AIME 175, 15 (1949).

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CARL V. THOMPSON

where 1 and n are constant, and where d, is the average diameter of the precipitate^.^^ The effects of precipitates on grain growth in 2D systems has been simulated, and the Zener relationship has been confirmed, with f being the area fraction occupied by precipitates and n = 0.5 (as opposed to 0.33 for strong pinning in 3D system^).^^^^' In thin films there is a competition between Zener pinning due to precipitates and pinning due to grain boundary grooves.83

VIII.

Three-Dimensional Systems

Models for 2D normal grain growth are advanced, but not complete. Simulations for idealized 2D normal grain growth do seem to be complete, in that different simulation techniques lead to similar behavior, and that the same or similar behavior is also found in experimental systems that meet the idealized conditions of the simulations (e.g., froths). These simulations can be extended to examine the impact of the assumptions made in simpler treatments (such as uniformity of boundary mobilities and energies), and to treat phenomenology arising from impurities and from surface effects in thin films. Modeling and simulations for 3D systems are not as advanced, and the experimental literature, at least on model systems (e.g., froths), is not as extensive. The experimental literature on grain growth in bulk polycrystalbut is almost always complicated by nonidealline systems is extensive,22s69 ities.

1. MODELING OF 3D GRAIN GROWTH The mean field models of Hillert and Louat, and related models based on combinations of Hillert’s and Louat’s approaches, can be applied to grain growth in three dimensions as readily as they can be applied in two dimensions, and were in fact developed in order to explain observations in bulk systems. However, Hillert’s analysis leads to a grain-size distribution with a sharp cut-off at a maximum grain size (eq. (25)) and with a shape that does not match experimental data. Louat’s analysis can produce grain-size distributions consistent with experimental results, but this consistency can be obtained even with significantly varying underlying assumptions, and so does not constitute convincing evidence in support of this approach.26 96

97

T. Nishizawa, I. Ohnuma, and K. Ishada, Muter. Trans. J I M 38, 950 (1997). E. A. Holm, J. A. Glazier, D. J. Srolovitz, and G. S. Grest, Phys. Rev. A43, 2662 (1991).

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As in two dimensions, it is likely that a complete description of the evolution of 3D cellular structures must account for the rate of change of grain volumes as a function of topological class (e.g., as a function of the number of faces a grain has) and for the rate at which grains enter and leave topological classes. In two dimensions, the former is provided through the Mullins-von Neumann relationship, and the latter can be treated approximately owing to the relatively limited number of possible transitions. In considering the topology of 3D grain structures it is useful to begin with Euler's law, which for three dimensions gives61

where C is the number of cells or grains, F the number of faces, E is the number of edges, and V is the number of vertices. From this it can be shown that

where ( f ) and ( e ) are the average numbers of faces and edges per cell, respectively. Already, this result points to the increased complexity of 3D systems. Comparison with the equivalent result in 2D (eq. (13)) shows that 3D structures have one more degree of freedom in defing the topological characteristics of the 3D cellular structure. Empirically, ( f ) z 14, though this relationship is not exact because there is no regular polyhedron with planar faces that fills space. The closest approach to such a polyhedron is a slightly distorted tetrakaidecahedron, which fills space when placed on a body-centered cubic lattice (forming the Wigner-Seitz cell for the bcc lattice) as in Figure VIII.1.22 Based on a maximum entropy analysis, R i ~ i e proposed r~~ that

where uf is the average volume of grains with f faces and ( f ) is the average number of faces for all grains. Glazier99 later used a 3D Potts simulation to establish empirically that

9 cc (Uf)"3(f dt 98 99

N. Rivier, Phil. Mag. B47, L45 (1983). J. A. Glazier, Phys., Rev. Letts. 70, 2170 (1993)

-

(59)

308

CARL V. THOMPSON

FIG. VIII.l. A space-filling packing of tetrakaidecahedra [from Atkinson", reprinted with permission from Elsevier Science].

where fo was 15.8 and distinguishable from (f), which had a value of 13.7, where the factor ( L I / ) ~is/ ~required for the proportionality constant to have the same units as mobility."' Weaire and Glazier"' have derived a relationship between fo and (f) in terms of the second moment of the number-of-faces distribution. f o = (/)(I

++).

Equation (59) provides a 3 D analog for the Mullins-von Neumann relationship with a constant of proportionality that scales with the grain boundary mobility. However, it is important to bear in mind that while the Mullinsvon Neumann relationship applies to individual grains, and has been rigoursly proven, eq. (59) applies to grain ensembles and is largely empirical. In considering topological transitions, there are two classes of 3D switching events, again in analogy with the 2D case (see Figure V111.2): T1 processes, in which two vertices combine to create a new face, and T2 processes, in which grains disappear directly. However, in three dimensions, there is a larger number of different types of T2 events and T1 events do not conserve the total number of faces.'" To date, there has been no attempt to develop a 3 D mean field model that accounts for volume evolution within topological classes as well as transitions among topological classes (in analogy with Fradkov's or Marder's models). loo lo' lo'

D. Weaire and S. McMurry, Solid State Physics 50,l (1997). D. Weaire, and J.A. Glazier, Phil. Mag. Letters 68, 363 (1993). M. P. Anderson, G. S. Grest, and D. J. Srolovitz, Philos. Mag. B 59, 293 (1989).

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@I FIG.VIII.2 3D switching events [from Stavans61, reprinted with permission].

2. SIMULATIONS OF 3 D GRAINGROWTH

As already indicated, 3 D Potts models have been used to simulate grain growth in three dimension^.^^-^^^ It was shown that a scaling state can be reached for which eq. (1) is satisfied with CI = 0.48 f 0.04102Both grain-size distributions and number-of-faces distributions have been empirically determined, and are shown in Figures VIII.3 and VIII.4, respectively. In experiments it is common to determine the distribution of grain radii or diameterslo3 on 2 D cuts through 3 D structures. This can also be done with 3 D simulations. The resulting distribution is shown for the Potts model in Figure VIII.5, which also shows experimental data for Fe for comparison. Experimental data for distributions of the number of faces per grain are also shown in Figure VIII.6 for comparison with the results of the Potts model. The results from experiments and simulation are in qualitative agreement, but do not show the same degree of agreement found in 2 D systems. This may be due, in part, to the fact that the experiments were carried out on real polycrystalline systems, in which the non-idealities discussed earlier apply. Also, 3 D Potts models are computationally intensive, so that the Io3

S. K. Kurtz and F. M. A. Carpay, J. Appl. Phys. 51, 5745 (1980).

310

CARL V . T H O M P S O N 0.12

n 0.10

0-07

b

1:o

2:o

FIG.VI11.3. Grain-size distributions for 3 D normal grain growth, as simulated using a Potts model. Also included are Rayleigh (dot-dash), lognormal (solid), and Hillert (dots) distribution functions [from Anderson et aL102. reprinted with permission of Taylor & Francis, http:// www.tandf.co.uk/journals].

evolution of fewer grains can be simulated and so that measured properties are subject to greater statistical variations. Nagai et al.'04 developed a 3D vertex model that incorporated interface curvature effects and found close agreement with the results obtained using the Potts model. Fuchizaki and K a w a ~ a k i ' ~ha~ve~ 'also ~ ~ developed a modified 3D vertex model that also matches the results of 3D Potts models, including the result given in eq. (59). Three-dimensional front-tracking simulations, in which grain boundaries are tesselated with triangles and triangle vertices are moved with velocities proportional to the local 3D curvature, have also been developed and incorporated into a program called the Surface Evolver, and used to explore grain growth behavior to a limited

'06

T. Nagai, S. Ohta, and K. Kawasaki, Materials Science Fortrm 94-96, 313 (1991). K. Fuchizaki and K. Kawasaki, Physica A221. 202 (1995). K. Fuchizaki and K. Kawasaki, Materials Science Forum 204-206, 267 (1996).

GRAIN GROWTH AND EVOLUTION OF CELLULAR STRUCTURES

31 1

012

0.08 0.10

Lm--

Y

0.04

0.02

-

0 0

' l l l l l l l l l l t l 2 4 6 8 10 12 14 16 18 20 22 24 26 : 3 NumkrofFacm

FIG. VIII.4. Number-of-faces distributions for 3D normal grain growth, as simulated using a Potts model. The points are simulation results, and the line is provided as a guide to the eye only [from Anderson et a1.102, reprinted with permission of Taylor & Francis, http:// www.tandf.co.uk/journals].

6

FIG. VIII.5. Grain-size distributions for grain structures in 2D slices from 3D grain structures: The points are a Potts-model-based simulation of 3D grain growth, and the histogram shows data from experiments on Fe [from Anderson et a1.'02, reprinted with permission of Taylor & Francis, http://www.tandf.co.uk/journals].

312

CARL V. THOMPSON

Number of Side8

FIG. VIII.6. Number-of-sides distributions for grain structures in 2D slices from 3D grain structures: Filled circles are from a Potts-model-based simulation of 3D grain growth, open circles are from 2D Potts model simulations, triangles are from experiments on Mg0,7 squares are from experiments on A1 [from P. A. Beck, Phil.Mag. Suppl. 3,245 (1954)], open circles are for experiments on Sn [from P. Feltham, Acta. Metall. 5, 97 (1957)], the solid line connects the 3D simulation results, and the dashed line shows the average for the experimental results [from Anderson et al.loz, reprinted with permission of Taylor & Francis, http://www.tandf.co.uk/ journals].

e ~ t e n t . Monnereau ~ ~ . ~ ~ et a1.1°7 have used Surface Evolver to track the evolution of 28 bubbles and found behavior consistent with eq. (59), and Wakai et al.loS have used Surface Evovler to track the evolution of 1000 cells, finding behavior consistent with eq. (59) with f,= 14.8. Surface Evolver has not been used to establish whether eq. (59) applies for individual cells, as it appears to for populations of cells. 3. EXPERIMENTS IN 3D SYSTEMS It is expected, and sometimes found, that grain growth in 3D polycrystalline systems obeys eq. (1) with c1 = 0.5. However, reported values range from lo7 lo*

C. Monnereau, M. Vignes-Adler, and N. Pittet, Phil. Mag. B79, 1213 (1999). F. Wakai, N. Enomoto, and H. Ogawa, Acta Mater. 48, 1297 (2000).

GRAIN GROWTH AND EVOLUTION OF CELLULAR STRUCTURES

313

0.05 to 0.5.'09 The distribution of grain sizes is often reported to be well fit by lognormal or Rayleigh distributions, though care must be taken in extracting the grain diameter distribution from observations made on 2D cross sections.'02.' l o Experiments on 3D froths are less common than on 2D froths, but Durian et al."' have used light scattering techniques to study the evolution of foams (shaving cream) and shown that the average bubble size obeys eq. (1) with a = 0.45 +_ 0.05. Monnereau et al."23"3 have used optical tomography to monitor the evolution of 28 cells in a 3D froth, and have determined the number-of-face and number-of edges-per-face distributions for this small population. They also demonstrated that the growth of individual grains with f = 9 and f = 11-16 was consistant with eq. (59).Glazier and Prause' l4 and Prause' ' have used magnetic resonance imaging during evolution of froths and have found behavior consistent with eq. (59) for populations of cells. Models, simulations, and experiments suggest that eq. (59) provides a relationship between volumetric growth rates and topological characteristics of cells in 3D structures. However, neither a formal theoretical nor an unambiguous empirical basis for this relationship has so far been provided. IX. Summary

The phenomenological model of Burke and Turnbull still gives the simplest description of the kinetics of ideal 2D and 3D grain growth. However, this model does not give statistical information about the characteristics of evolving grain structures. Such descriptions can, in principle, be provided by mean field models. However, the models of Hillert and Louat, and similar models that consider only grain fluxes in size space, have had limited success in matching detailed analyses of the characteristics of experimental systems that meet the idealizations implicit in the models. In 2D systems, mean field models that track grain size evolution within topological classes and also account for transitions to and from topological classes seem to be more successful. G. T. Higgins, Met Sci. J . 8, 143 (1974). E. E. Underwood, Quantitative Stereology, Addison Wesley, Boston (1970). '" D. J. Durian, D. A. Weitz, and D. J. Pine, Phys. Rev. A44,7902 (1991). C. Monnereau and M. Vignes-Adler, J. Colloid and Interface Science 202, 45 (1998). C. Monnereau and M. Vignes-Adler, Phys. Rev. Lett. 80, 5228 (1998). J. A. Glazier and P. Prause, to be published in the Proceedings of Eurofoam 2000, Delft (2cw. B. Prause, Magnetic Resonance Imaging Studies of Three-Dimensional Liquid Foams, Ph.D. thesis, U. of Notre Dame, Dept. of Physics (2000). log 'Io

314

CARL V. THOMPSON

All mean field models start with the assumption that all grain boundaries in the evolving ensemble are characterized by the same mobilities and energies, and that the velocity of a boundary is directly proportional to the boundary curvature. These conditions can be met in properly controlled evolving froths, and appear also to apply in experiments on evolving cellular structures in molecular monolayers. They may also sometimes be met in polycrystalline metallic sheets. While the complexly interrelated evolution of grains in 2D structures appears to be moderately well described in ideal 2D systems, the added complexity of 3D systems have so far led to the absence of similarly successful analytic models. While 2D and especially 3D grain growth models have proven difficult to develop, considerable success has been attained with computer simulations of grain growth. 2D Potts, vertex, and front-tracking simulations have provided both kinetic and structural statistical information that compares well with data on ideal experimental systems, especially froths. 3D versions of all three types of simulations have also been developed, but they have not yet been as extensively compared with each other and with experiments. Computer simulations of grain growth allow modification to account for the non-idealities that characterize most experimental systems of practical importance. This has been demonstrated by accounting for the effects of particle pinning, for groove-induced stagnation in thin films ("quasi 2 D systems), and for the effects of surface-energy and strain-energy anisotropy on grain growth in thin films. The effects of variable grain boundary energies and mobilities have also been examined. As simulations become more computationally efficient, and more powerful computation capabilities become available, computer simulations may provide tools that enable detailed analyses of non-ideal grain growth in 3D or bulk systems. Unless there is a significant breakthrough in the theory of grain growth, simulations are likely to provide the only tools for such analyses. X. Acknowledgements

This work was supported by the NSF and the Alexander von Humbodlt Foundation. The bulk of the preparation of this review was carried out while the author was a guest of Prof. Eduard Arzt and the Max Planck Institute fur Metallforschung in Stuttgart Germany. The author would also like to thank Frans Spaepen, Walid Fayad, and Steve See1 for critical reviews of the manuscript.

Author Index Numbers in parentheses are reference numbers and indicate that an author’s name is not cited in the text.

A Abakumov, V. N., 79(194) Abbe, D., 221(173), 223, 224 Abbruzzese, G., 280, 280(19), 289(56) Abeln, G. C., 124(25), 142(52) Abe, S., 18(83-85) Aboav, D. A,, 275, 275(7-9) Abu Arab, A., 233(192) Acet, M., 252(230) Adlrovandi, S., 167 Agari, Y., 5(39) Ahlers, M., 166(22), 168, 168(26, 30), 171(33), 172(40), 173(43), 219, 219(165-166), 224( 180), 23 1(187-188), 233(192-193). 267, 165 Ahmed, H., 124(24) Alam, M. A,, lO(73) Alba, M., 179(81), 200(126-127) Albers, R. C., 235(202) AlcoM, X., 231(185) Aldrovandi, S., 223(174) Alexandrakis, G. C., 264(254) Alivisatos, A. P., 120(4), 123(21) Alivisatos, P., 123(21), 124(26) Allen, S. M., 255, 256(239) Alonso, J. A,, 145(57) Alvarado, S. F., 36(134) Amengual, A,, 172(40) Anderegg, J., 9(61) Anderson, J. D., 3(16) Anderson, M. L., 44(142) Anderson, M. P., 286(36-37), 308(102), 310, 311 Anderson, P. W., 185(95), 233 Anderson, R. A,, 6q169) Andersson, M. R.,115(212) Angelopoulos, M., 9(58) Antoniadis, H., 3(10-31 l), 98(201) Aoki, S., 201(133), 202 ArceBrouchet, A., 4(28) Arias, A. C., 9(64)

Arkhipov, V. I., 78(188-190), 105(209) Armstrong, N. R., 3(16), 44(142) Arneodo, W., 171(33), 173(43) Arzt, Eduard, 3 14 Asano, S., 264(257) Ashcroft, N. W., 126(37) Ashoori, R. C., 120(2) Asmus, F., 295 Aspnes, D. E., 29(131) Assmus, W., 197(116) Athouel, L., 57(166) Atkinson, H. V., 281(22) Au, Y. K., 167 Avouris, Ph., 124(25), 125(30, 33), 142(50, 52) Awschalom, D. D., 125(34) Axe, J. D., 185(93)

B Babcock, K. L., 293(65-66) Baker, G., 20(106) Balbas, L. C., 145(57) Bal, C. A. W., 121(9) Baldo, M. A,, 3(17, 19, 20, 21) Baldwin, K., 4(31) Bao, Z., 4(25-27, 31), 115(229) Bao, Z. N., lO(72) Baram, J., 173(44) Barashkov, Nikolai, 117 Barashkov, N. N., 12(75), 19(102), 38(136), 49(146- 147), 88( 198) Baratoff, A., 142(49) Barcelo, G., 233(193) Bargawi, A. Y., 265(259) Barsch, G. R., 163(14), 185(96-97), 206, 244, 244(207-208), 251, 251(228) Barth, S., 9(65-66), 78(191) Bartkowiak, M., 123(20) Bassas, J., 231(185) Bassler, H., 9(65-66), 67, 67(174-175), 78(188-191), 105(209-210)

315

316

AUTHOR INDEX

Batlogg, B., 22(117), 57(168) Bechgaard, K., 4(24) Becker, H., 84(197) Beck, P. A., 293(68), 312 Becquart, C. S., 179(80) Beenakker, C. W., 12O(3) Beeson, P. M., 60(169) Beljonne, D., 9(67), 19(105) Belkahla, S., 248(220-221) Bellmann, E., 3(16), 98(202) Bennett, J. C., 156(73) Berge, B., 293(64), 294 Bergmann, T., 143(54) Berg, S., 22(117) Besselink, P. A., 164(17) Bhatt, R. N., 136(45) Bilir, N., 224(179) Bishop, A. R., 15(76), 18(80-81), 19(102), 117, 21O( 148), 242(205) Black, C. T., 121, 121(8), 124(22) Blanc, M., 277(12) Blase, X., 145(57) Blatchford, J. W., 105(206) Blau, W. J., 8(51) Bleyer, A,, 56(164) Blom, P. W. M., 55(157-159), 77(183) Blout, E. I., 185(95) Blume, M., 227(184) Blunt, J., 178(71) Bolton, F., 293(67) Born, M., 160(1) Borsa, F., 167, 223(174) Borsenberger, P. M., 2(4) Boswell, F. W., 156(73) Bouchriha, H., 77(186) Bourguiga, R., 77(186) Bozano, L., 55(160) Bozhko, A. D., 264(256) Bradley, D. D. C., 3(13), 4(35), 8(49-51), 19(104), 36(134), 54(151- 152), 55( 155156), 56(164), 98(201) Brakke, K. A., 288(45-47) Brassington, M. P., 185(94) Brauer, H. E., 156(74) Braun, D., 5(36) Brazovskii, S., 18(80-81), 19(86, 102) Brazovskii, S. A., 18(79) Bredas, J. L., 3(13), 7(44), 9(67), 18(78), 19(96,99, 103-105), 23(119-121), 25(125- 126)

Breedijk, S., 77(183) Brill, T. M., 197(116) Brockenbrough, R.,123(18) Brock, P. J., 9(58), 24(123), 55(153, 160) Brom, H. B., 55(159) Brown, A. R., 4(35) Brown, David, 117 Brown, P. J., 4(24), 265(259) Bruce, A. D., 175(58) Brugger, K., 187, 187(98-99), 215(157) Bruinsma, R., 217(160) Brus, L. E., 143(53) Bruyndoncx, V., 121(5) Bruynseraede, Y.,121(5) Buchelnikov, V. D., 264(256) Bhrer, W., 201(131) Bullock, M., 199(125), 200, 208 Bulovic, V., 2(6), 3(12), 7(43), 8(48), 9(53), 55( 154), 115(227-228) Burgers, W. G., 162(12), 195 Burke, J. E., 270, 271,271(1), 272, 313 Bums, P. L., 4(35), 8(50) Burns, S. E., 84(197) Burroughes, J. H., 3(13), 4(35) Burrows, P. E., 2(6), 3(12, 17), 7(43), 8(48), 9(53-54, 56), 55(154), 115(218, 227) Buyndoncx, V., 121(5) C

Cadeville, M-. C., 231(186) Cahill, P. A., 60(169) Cahn, J. W., 251, 251(227) Cahn, R. W., 233(194) Callegari, A., 4(32) Campbell, A. J., 8(49), 55(155-156) Campbell, D. K., 22( 114) Campbell, I. H., 1, 8(52), 12(75), 19(88), 21(113), 26(128, 130), 32(133), 38(136), 43( 141), 49( 146- 147), 54(150), 55( 163), 63(170), 65(171), 77(184), 88(198), 102(203204), 106(211) Campbell, P. M., 125(35) Cankurtaran, M., 210(149), 211,216,216(159) Cao, W., 251(226) Cao, Y.,26(129), 55(162) Capek, V., 25(127) Carel, R., 296(78), 302(85) Carlson, N., 288(49) Carnal, E., 277(13)

317

A U T H O R INDEX Carpay, F. M. A., 309(103) Carpinelli, J. M., 123(20) Carrillo, LI., 174(50) Carter, S. A,, 9(58), 55(160) Castan, T., 179(77-78), 224(178), 227(183), 231(185), 263(253), 267 Caudron, R., 221(173), 224 Celotta, R. J., 125(29) Cesari, E., 254(237), 256(243), 257(245-246), 259, 255 Chakrabarti, A,, 20(111) Chalin, F., 157(75) Cha, M., 2q106) Chan, C. T., 205(137), 244(206) Chandrasekhar, H . R.. 57(165) Chandrasekhar, M., 57(165) Chandross, M., 19(89-91), 2q110) Chen, C. H., 2(7). 4(34) Chen, D., 5(38) Chen, L.-Q., 286(38-39) Chen, W., 124(24) Chen, Y.,210(146) Chernenko, V. A,, 254(236-237), 255, 256(243), 257(245-246), 259, 260(252), 264(255) Choong, V., 84(196) Chou, M. Y., 143(56) Chow, H., 205(140) Christian, J. W., 162(9-10) Cingolani, E., 231(188) Clapp, P. C., 173(46), 179(80), 234(198), 248, 250(224-225), 251 Clarke, J., 123(17) Cleave, V., 4(22), 115(217) Cochran, W., 233, 233(196) Cohen, M., 162(10), 245(213), 247(218) Cohen, M. L., 143(56) Coleman, R. V., 123(16) Constant, M., lO(70) Conwell, E. M., 19(92-94, 101), 22(115), 68, 68(176), 78(192-193) Cornil, J., 19(103), 25(125-126) Cowley, R. A,, 175(58), 185(91) Cox, A. C. F., 288(42), 291 Crangle, J., 265(259) Crornrnie, M. F., 125(32) Crone, B., 115(229) Crone, B. K., 55(163), 65(171), 77(184) Crone, Brian, 117 Cronin, J. A., 55(154)

D DaCosta, P. G., 19(92), 22(115) Dahmen, K. A., 174(51-52) Dahmen, K. A. Dahrne, 174(53) Dai, L., 115(224-225) Dalsegg, I., 23(118) Dandrea. R. G., 22( 115) Dannenberg, R., 303(90) Dannetun, P., 23(120-121) Davenport, T., 188, 192, 196, 196(115) Davids, Paul, 117 Davids, P. S., 26(130), 55(163), 63(170), 65(171), 77(184), 88(198) Davis, P. S., 19(87) de Alrneida, R. M. C., 286, 286(34), 287 DeFontaine, D., 162(11) de Heer, W. A,, 143(55-56) DeJong, M. J. M., 55(157-158). 77(183) Dekker, C., 121(9), 157(75) Delaey, L., 160(6), 163(15), 164(18, 20), 165, 173(42), 189, 189(101-102), 190, 197(120), 200(130), 203(134), 209(145), 212(152), 215, 215(158), 219(168), 249, 250, 267 Delaney, D. W., 201(132), 202, 204, 208 deleeuw, D. M., 4(29) Demanze, F., 4(28) Dernuth, J. E., 142(47) Denton, G. J., 115(214-215) Derrida, B., 69, 69(182) Deruyttere, A,, 163(15) Deshpande, R. S., 115(228) Detert, K., 295(70) Dewar, H. J. S., 15(77) DiazGarcia, M. A., 55(162), 115(212-214) Dikshtein, I. E., 264(256) Dimitrakopoulos, C. D., q32) Dirnitropoulos, C., 223(174), 167 DiSalvo, F. J., 123, 123(14-15, 17) Dodabalapur, A,, 4(25-27, 31), 9(57), 1q68, 72-73), 77(185), 115(229) Dohrnen, M. A. H., 157(75) Dolling, G., 199, 199(123) Dornany, E., 286(33) Dorsinville, R., 7(42) dos Santos, D. A,, 3(13), 7(44), 9(67) Dragoset, R. A,, 125(29) Drake, B., 123(16) Dresser, B. J., 303(90) Drotleff, E., 8(51) Drury, C. J., 4(29)

318

AUTHOR INDEX

Dubovoy, T., 3(21) Duisberg, 217 Dunlap, D. H., 68, 68(177-181) Dunn, C. G., 295, 295(71), 302(84) Durian, D. J., 313, 313(111) Dutkiewicz, J., 171(34) Dynes, R. C., 155(68) E

Edwards, P. P., 178(71) Eigler, D. M., 125(28, 32). 142(48) Ekvall, I., 149(63), 156(74) Elgueta, J., 248(222), 249(223) EIKassmi, A,, 4(28) Emelianova, E. V., 78(188) Emery, V. J., 227(184) Enami, K.. 189(103), 195, 205(141) Enomoto, N., 312(108) Enomoto, Y., 288(51) Epstein, A. J., 7(47), lOS(206-207) Eriksson, O., 160(4) Eshelby, J. D., 246(214) Etemad, S., 18(82), 20(106, 108-109) F

Fahlman, M.. 23(120) Falk, F., 237(203) Falter, C., 206(143) Fan, D., 286(38-39), 291 Farinelli, M., 255 Fauquet, C., 23(120) Fayad. W., 281(24), 291, 294 Fayad, Walid, 314 Feldman, L. C., 5(40) Feltman, P., 312 Feng, Y., 4(27) Ferraris, John, 117 Ferraris, J. P., 8(52), 12(75). 19(88, 102). 26(128,130), 32(133), 38(136), 49(146-147), 54(150), 55(163), 88(198), 106(211) Fichou, D., lO(70-71) Fidt, R. F., 157(76) Filas, R. W., 1 15(229) Finlayson, T., 180 Fleischmann, P., 173(48) Floro, J. A., 289(58), 297(82) Flyvberg, H., 286(32), 287 Flyvbjerg, H., 286, 286(32), 287

Forrest, S. R., 2(6), 3(12, 17, 19-21), 7(43), 8(48), 9(53, 54-56), 55(154), 115(218, 227220 Fradkov, V. E., 284, 284(29), 285, 285(30), 286, 295, 295(72), 296, 296(73-77), 308 Frank, A. J., 57(167) Fredriksson, C., 23(119, 121) Freeman, A. J., 129(40) Freeman, P., 178(71) Frick, B., 223( 176) Friedel, J., 225(181), 265, 265(258) Friend, C. M., 171, 221(169) Friend, R. H., 2(2), 3(13), 4(22-24, 3 9 , 7(44-49, 9(63-64, 67), 19(104-105), 84(197), 105(208), 115(214-217) Frindt, R. F., 150(65) Fritsch, G., 257(244) Fritz, T. C., 187, 187(98) Frolov, S. V., 115(220-221) Frontera, C., 267 Frost, H. J., 281(24), 282, 282(26), 288(40-41, 43-44), 297(83), 302(85), 303(86), 305(94) Froyer, G., 57(166) Fuchizaki, K., 205(138), 244(210), 245(21l), 3 10, 3 1O( 105- 106) Fu, D. K., lOS(206-207) Fujii, S., 264(257) Fujii, T., 47(144) Fujii, Y., 231(189) Fujita, F., 178(73-74) Fujita, M., 47(144) Fuji, Y., 177(66) Fullman, R. L., 288, 288(50) Funahashi, S., 205(138) Funatsu, Y., 197(118) Fung, A. W. P., lO(68)

G Galvao, D. S., 20(107-109) Gammel, J. T., 288(49) Gammie, G., 123(18) Ganz, E., 123(17) Gao, J., 9(59), 26(129), 1IS(223-224) Gao, Y., 84(196) Garbuzov, D. 2.. 7(43), 8(48) Gamier, F., 4(28), lO(69-71) Gartstein, Y. N., 19(93, 100-101), 68,68(176), 78(193) Gastaldi, J., 248(220-221)

319

AUTHOR INDEX Gebler, D. D., lOS(206-207) Gelinck. G. H., 7(44) Gelius, U., 23( 118) Geng. C., 286(39) George, D. C., 288(49) Gherman, B., 19(95) Giambattista, B., 123(16) Giebeler, C., 98(201) Gielen, L., 121(5) Giersig, M., 143(53) Gilat. G.. 199 Gill, S. P. A,. 288(42), 291 Gill, W. D., 66(173) Gimzewski. J. K.. 124(27), 125(31) Glaser. R., 57(165) Glavao, D. S., 18(82) Glazier. J. A,, 278( 17). 279, 290, 290(62), 293(67), 306(97), 307, 307(99). 308, 308(101), 313, 313(114) Glicksman, M. E., 296(73-74, 76-77) Glundlach, D. J.. 4(30) Gntherodt, H.-J., 131(43) Gobin, P.F., 197(120), 200(130), 203(134), 209(145), 213. 213(156), 246(215), 248, 248(222) Goicoechea, J., 174(55) Gomanaj, E. V., 264(255) Gonzalez-Comas, A,, 250 Gonzalez-Comas, A,. 191, 197(121), 198, 210(151), 211, 212(154-155), 214. 216, 216(159), 232(190). 257(246), 258(247), 259. 260(250, 252). 267 Gooding. R. J., 196(113-114). 210(147-148). 227, 227(182), 234(197), 238, 243, 244(206), 25 l(226) Graeme-Barber, A,, 178(71) Graham, S. C.. 19(104) Grange, G., 248(220-221) Granstrom, M., 9(64) Graupner, W., 57(165), 1IS(219) Greenham, N. C., 2(2), 7(45) Grell, M.. 56( 164) Crest, G. S., 286(36-37), 306(97), 308(102) Grice, A,, 56(164) Griffiths, R. B., 227(184) Grimvall, G., 160(3), 221(171) Grohlich, H., 143(54) Groza, J. R., 303(90) Grubbs, R. H., 3(16), 98(202) Gruner, G., 122(11)

Gruner. J.. 6(41), 7(44) Guenin. G., 173(41-42), 174(57), 191, 193(107-108). 194, 197(119-121), 198, 200( 130). 203( 134). 206, 209, 209( 145), 212(152), 213. 213(156), 232(191), 246(215), 248, 248(220-222), 249(223), 250(224), 251, 267 Gu, G., 2(6), 3(12), 8(48). 9(53, 55-56) Guha, S., 57(165) Guilat, G., 199(123) Gundlach. D. J., 5(37) Gupta. R., llS(214) Gustafson, T. L., 7(47) Guthoff, F., 200(128) Gymer, R. W., 3(13)

H Haasen, P., 160(6) Hagler, Thomas, I 17 Hagler, T. W., 19(88). 26(130), 32(133) Haight, R., 47(145) Hajlaoui, M.. 77(186) Hajlaoui, R., 4(28), lO(69-70), 77(186) Halls, J. J. M., 9(63) Halperin, B. I., 177, 177(63) Hamada, Y.. 47(144) Hamaguchi, M., 19(90) Hanamura, E., 142(51) Hansam, P. K., 123(16) Han, W., lSO(65) Harmon. B. N., 177(62), 210(146), 235(200), 244(206) Harris, K. E.. 304, 304(91) Harrison, M. G.. 6(41) Harrison, N. T., 1 lS(216) Hart, C.M., 4(29) Hasegawa. T., 148(61) Hassen, P., 181(87) Hatch, D. M., 244(207-208) Hatherly, M., 293(69) Hattink, B. J., 260(252) Hausch, G., 193(106) Hautecler, S., 200(130) Hawthorne, A. M., 210(148) Hayashi, Y., 303(86). 305(94) Hayes, G. R., 7(45) Healy, E. F., 15(77) Heckelmann, I., 280(19), 289(56)

320

AUTHOR INDEX

Heeger, A. J., 2(1), 5(36), 9(59, 62). 19(103), 25( 125- 126). 26( 129), 38( 137- 138), 55(162), 115(212, 213-214, 222-224) Heike, S., 156(72) Helming, A., 179(81), 2 w 1 2 6 - 128) Heller, Christian, 117 Heller, C. M., 26(130), 38(136) Hennion, B., 200(128), 205(139) Hergenrother, J. M., 121(6) Hertel, D., 67(175) Herzig, C., 200(127) Herzing, C., 179(81), 200(128) Hess, H. F., 155(68-69) Heuer, A. H., 178(70) Hide, F.. 55(162), 115(212-214) Higgins, G. T., 313(109) Hillert, M., 280, 280(18), 281, 282, 306, 313 Hilliard, J. E., 251, 251(227) Hill, I. G., 41(140), 105(205) Hirata, K., 253(232) Ho, K. M., 205(137), 235(200), 244(206) Ho, K.-M., 210(146) Holm, E. A,, 306(97) Holmes, A. B., 3(13), 4(35), 7(45), 9(63, 67). 40( 139) Holworth, M. L., 293(68) Holzer, L., 115(224-225) Holzer, W., 8(51) Horhold, H. H., 9(65), 67(175), 105(209) Hori, F., 171(37) Horovitz, B., 163(14) Horowitz, G., 3(9), 4(28), lO(69-71). 77(186) Hoshino, S., 200(129) Howe, C. L., 288(40) Hsieh, B. R., 84(196) Huang, C. M., 177(64) Huang, H., 157(77) Huang, J. K., 255, 256(239) Huang, J. L., 124(23) Huang, J.-L., 157 Huang, K., 160(1) Hubacek, J. S., 123(18) Huber, A,, 7(46) Hueschen, M., 3(10) Hull, G. W., Jr., 157(76) Humphreys, F. J., 293(69) Hunderi, O., 283, 283(27), 289(54-55) Hung, L. S., 9(60), 98(199) Hunt, E. R., 150(65) Hu, Y., 41(140)

I Ibe, G., 295(70) Iglesis, J. R., 286, 286(34), 287 Iizumi, M., 209( 144) Iizumi. Y., 3( 15) Inbasekaran, M., 54(151-152) Ishada, K., 306(96) Ishida, S., 264(257) J Jabbour, G. E., 3(16) Jackson, T. N., 4(30), 5(37) Jacobs, S. J., 60(169) James, R. D., 253(233) Jannssen, R. A. J., 4(24) Jeglinski, S., 19(89) Jeglinski, S. A., 24(122) Jessen, S. W., 105(206) Jesspen, C., 286, 286(32) Jiran, E., 303(88-89) Joachim, C., 124(27), 125(31) Joensen, P., 157(76) Johansson, A., 23( 118) Johnson, A., 123(16) Jonckheere. R., 121(5) Jones, R., 178(71) Jordan, R. H., 9(57) Josephson, A. C., 122(10) Joswick, M. D., 12(75), 32(133), 102(203) Joswick, Michael, 117 Jourdan, C., 248(220-221) Jung, T. A,, 125(31) Jurado, M., 193(1lo), 194, 207 Jurado, M. A,, 210(149), 211, 231(185), 232(190), 267 K

Kachi, S., 171(36) Kaerijama, K., 23(120) Kahn, A,, 41(140), 44(143), 105(205) Kajitani, T., 200(129) Kajiwara, S., 168(28) Kakeshita, T., 253(232) Kanawa, Y., 178(75) Kanazaki, H., 142(51) Kantner, C., 255, 256(239) Karasz, F. E., 5(38) Karg, S., 9(58)

32 1

AUTHOR INDEX

Kartha, S., 174(51), 179, 179(77-79), 180 Kastner, M. A., 12q2) Katari, J. E. B., 123(21) Katisikas, L., 143(53) Kato, H., 171(34), 221(170) Kato, M., 218(163) Katz, H . E., 4(27, 31), lO(68, 72), 77(185), 115(229) Kauffmann, L., 245(213) Kawasaki, K., 288(51), 310, 310(104-106) Kelly, M. J., 168(29), 205, 205(142), 206 Kempf, A., 257(244) Kenkre, V. M., 68(177-181) Kent, A. D., 125(34) Kepler, R.G., 20(109), 60(169) Kermode, J. P., 289, 289(52-53) Kerr, W. C., 21q148) Kessener, Y., 7(45) Kevan, S. D., 148(58) Khachatruyan, A. G., 246(216-217) Khalfin, V., 9(54), 9(56) Khalfin, V. B., 8(48) Khandros, L. G., 205(141) Khovailo, V. V., 264(256) Kido, J., 3(15) Kim, 154 Kim, c. Y., 9(59) Kim, J., 148(61), 149(63) Kim, P., 124(23) Kim, Philip, 119 King, A. H., 304(91) Kippelen, B., 3(16), 98(202) Kirova, N., 18(79-81), 19(86, 102) Kitazawa, K., 148(61) Kivelson, S.,2( 1) Kiwabara, A,, 205(138) Klein, D. L., 123(21) Klimov, V., 19(102) Kloc, C., 57(168) Kloc, Ch., 22(117) Knight, M. D., 143(56) Knobler, C. M., 293(63) Kobayashi, K., 142(51) Koch, N., 57(166) Kohler, A., 9(67) Kojima, S.,197(118) Kokorin, V. V., 254(237), 255, 256(239, 242-243), 257(244-246), 258, 259 Kondo, S., 156(72) KoppingGrem, G., 7(46)

Kouki, F., 4(28), lO(69) Kozlov, V. G., 9(54), 115(218) Kragler, R., 185(92) Kramer, E. J., 181(87) Kraus, A. B., 9(67) Kravchenko, A. S.,285(30), 295(72) Kress, J. D., 21(113), 49(146-147) Kress, Joel, 117 Krichevsky, O., 286(35) Krishnan, R. V., 164(18) Krumhansl, J. A,, 163(14), 174(51), 175(60), 179(77-79), 180(82,85), 185(96-97), 196(113114), 206, 210(148), 224(177), 234(197), 235(201), 238,242(205), 243,251(226,228), 267 Kuentzler, R., 223, 223(175) Kugler, T., 23(118), 40(139) Kukla, A. M., 256(239) Kulkarni, S. D., 166(23) Kuprat, A., 288(48-49) Kurtz, S. K., 309(103) Kus, M., 68(179, 181) Kwock, E. W., 19(89) Kwong, R. C., 3(21) Kyllo, E. M., 7(47) Kymissis, J., 4(32)

L Labarta, A., 260(252) Labbb, L., 265, 265(258) LaDuca, A., 115(229) Laigre, L., 4(28) Laikos, J. K., 184, 184(90) Lamansky, S., 3(17) Lampert, M. A,, 55(161) Lane, P. A., 8(50), 19(90), 57(167) Langdon, T. C., 275(7) Langezaal, C., 121(9) Laquindanum, J. G., 4(27) Lareze, J. Z., 179(76) Latyshev, Y,I., 157(75) Laubender, J., 8(49) Laudise, R. A,, 57(168) Lauge, T., 143(54) Lawrence, B., 2q106) Lazzaroni, R., 23(120), 23(121) LeBarney, P., 4(22), 115(217) Lee, H. W. H., 5(40) Lee, J. Y., 9(59) Leising, G., 7(46), 57(165-166), 115(219,

322

AUTHOR INDEX

224-225) Leonard, W., 3(10) Lewis, F. T., 274,274(2-5), 275 Lhost, O., 19(104) Libchaber, A,, 293(64) Lidzey, D. G., 36(134), 55(155-156) Lieber, Charles M., 119 Lieber, C. M., 123(19), 124(23), 125(36), 130(41), 153(66) Liess, M., 19(90), 57(167) Lieth, R. M. A,, 155(71) Li, F., 9(61) Lifshitz, I. M., 280, 280(20) Li, J. C. M., 304(92) Lim, A. K. L., 123(21), 124(26) Lindgird, P. A,, 196(112), 227(183), 258 Lindgird, P. A.-, 168, 231, 235 Lindgird, P.-A,, 168(27), 210(146), 244, 244(209), 256(242), 263(253), 267 Lin, Y.-Y., 4(30-31), 5(37), 115(229) Lissowski, A,, 275, 275(6) Liu, J., 124(23), 148(62), 157 Li, W., 4(31), 115(229) Li, Y., 49(148) Li, Y. F., 26(129) Logdlund, M., 3(13) Lograsso, T., 193(110), 194, 201(132), 202, 204, 207, 208 Loram, J., 178(71) Louat, N. P., 282, 282(25), 306, 313 Louie, S. G., 22(116), 145(57) Lovey, F. C., 172(40), 191, 197(119, 121), 198, 216, 216(159) Lovinger, A. J., 2(5), 4(25), 4(26) Lucke, K., 280(19), 289(56), 305(93) Lu, J. G., 121(6) Lupton, J. M., SO(195) Liithi, B., 197(116) Lutwyche, M., 156(72) Lutz, C. P., 125(32) Lu, X., 157(77) L'vov, V. A,, 264(255) Lyding, J. W., 123(18), 124(25), 142(52) LYO,I.-W., 125(30, 33) Lyons, W. G., 123(18)

M MacDiarmid, A. G., 105(206) Mackay, K., 4(35)

Macqueron, F. J. L., 193(107) Macqueron, J. L., 173(41-42, 47-48), 232(191), 267 Magee, C. L., 247(219) Makhija, A,, 4(27) Makita, T., 212(153) Maliminek, J., 171(35) Malliaras, G. G., 24(123), 55(153, 160) Maniloff, E., 38(138) Mafiosa, LI., 168(31), 171, 173(47-49), 174(50), 188, 188(100), 191, 192, 193(107, 109-1 lo), 194, 195, 197(121), 198, 199(125), 200, 201(132), 202, 204, 207, 208, 210(149, 151), 211,212(154-155), 214,216,216(159), 220, 221(169, 172), 222, 229, 231(185), 232(190), 237(204), 238, 239, 240, 250, 257(246), 258(247), 259, 260(250, 252) Maosa, Llus, 159 Mantel, 0. C., 121(9), 157(75) Marder, M., 285, 285(31), 287, 308 Marder, S. R., 3(16), 98(202) Maree, C. H. M., 5(40) Marioni, M. A,, 256(239) Markovic, N., 157(75) Mark, P., 55(161) Marks, R. N., 4(35), 3(13) Martens, H. C. F., 55(159) Martnez, B., 221(172), 222 Martin, Richard, 117 Martin, R. L., 15(76), 21(113), 49(146-147) Martin, S. J., 8(50) Martin, T. P., 143(54) Martynov, V. V., 205(141), 243, 254(238) Marzo, P., 248(220) Mason, M. G., 9(60) Massalski, T. B., 161(8) Masse, M. A,, 5(38) Matsumoto, M., 264(256) Matsumoto, S., 3(18) Matters, M., 4(29) Mattes, B., 38(138) Mattheiss, L. F., 129(39) Mau, A. W. H., 115(224-225) Mazumdar, S., 19(89-91), 20(110-111) McBranch, D., 8(52), 19(102), 38(138) McBranch, Duncan, 117 McCarthy, D. M., 55(154) McEuen, P. L., 123(21), 124(26) McGehee, M. D., 115(214) McMillan, W. L., 136(44-45)

323

AUTHOR INDEX McMurry, S., 308(100) McNairy, W. W., 123(16) Meghdadi, F., 57(166) Meichle, M. E., 177(64-65) Meier, F., 121(7) Meijer, E. W., 4(24) Mellor, H., 8(50) Mermin, N. D., 126(37) Meth, J., 2q106) Mhajan, S., 123(15) Michaelson, H. B., 36(135) Mikeley, W., 200(127) Miller, E. K., 115(214) Miller, J., 3(10) Miller, T. M., 19(89), 98(200) Mirza, K., 178(71) Mittelbach, S., 197(116) Miura, S., 171(34, 36-37), 221(170) Mizutani, U., 161(8) Mocellin, A., 277(12-13) Molho, P., 293(67) Monceau, P., 157(75) Monnereau, C., 312, 312(107), 313,313(112113) Moon, R.,3(10) Moore, B. G., 293(63) Moore, E., 19(95, 98) Moore, E. E., 19(96-97) Moran-Lopez, J. L., 231(186) Moratti, S. C., 7(45), 9(63) Morii, Y., 199, 199(122), 204, 205, 205(138), 209(144) Mori, M., 193(111) Morin, M., 173(41-42, 48), 197(119), 203( 134), 212( 155), 214,231( 185), 232(1go), 267 Mori, T., 171(36) Morito, S., 253(232) Morris, J., 227, 227(182) Morris, J. R., 210(147) Morrison, S. R.,157(76) Moses, D., 115(214) Moshchalkov, V. V., 121(5) Moss, S. C., 179(76) Mouritsen, 0.G., 196(112),235,244,244(209) Mukamel, D., 286(33, 35) Mullen, K., 9(67), 84(196), 115(219, 224) Mullin, K., 19(104) Mullins, W. W., 278, 278(15), 297, 298, 299, 301

Mlner, M., 197(116) Murakami, Y., 165, 171(36), 197(120) Murphy, D. W., 157(76) Murray, J. L., 167 Murray, S. J., 255, 256(239) Muto, S., 178(73-74) Mutsaers, C. M. J., 4(29) Myron, H. W., 129(40) N Nagai, 291 Nagai, T., 31q104) Nagasawa, A,, 189(103), 194, 195, 199, 199(122), 204, 204(136), 205, 205(138), 210(150), 211, 212(153), 231(189) Nakai, K., 171(32) Nakanishi, N., 171(36), 171(37), 195 Nakanishi, N. N., 189(103) Nakazoto, K., 124(24) Nash, P., 167 Neef, Charles, 117 Neef, C. J., 54(150), 55(163), 106(211) Neher, D., 7(44) Nelson, S. F., 4(30), 5(37) Nemchuk, N., 157(77) Nenno, S., 205(141) Nessakh, B., lO(70) Newmann, K.-U., 265(259) News, D. M., 142(50) Nielsen, M. M., 4(24) Nikitenko, V. I., 105(209), 105(210) Nishio, Y., 47(144) Nishizawa, T., 306(96) Noda, Y., 176, 176(61), 177(66), 178, 179(76), 180(84), 204, 244(210), 245(211), 202 Nordberg, J., 296(74,76) Novak, V., 171(35) Novikov, S. V., 68(178) Novikov, V. Y., 289(57) Nuesch, F., 49(148-149) Nye, J. F., 181(88) 0

Obrado, E., 168(31), 219, 220, 232(190), 257(246), 258(247), 259, 26q250, 252), 267 OBrien, D. F., 3(19-20), 56(164) Ogawa, H., 312(108) OHandley, R. C., 253(234), 255, 256(239)

324

AUTHOR INDEX

Ohba, T., 201(133), 202, 203(135), 244(208) Ohnuma,I., 306(96) Ohta, S., 31q104) O h , H., 149(63), 156(74) Olson, G. B., 162(10), 247(218) Ortin, J., 171, 172(38-39), 173(49), 174(50, 5 3 , 188, 193(109), 218(164), 219(168), 221(169, 172), 222, 229, 267 Oshima, R., 178(73-74) Otsuka, K., 171, 171(32), 197, 197(117-118), 201(133), 202,203(135), 209(144), 233(195), 236, 253(232) P Pai, D. M., 66(172) Pakbaz, K., 5(40) Pak, H.R., 218(163) Palmer, J. E., 296(80) Palmer, M. A., 296(73-74,76-77) Pande, C. S., 283, 283(28) Pankratov, O., 15q65) Pannetier, B., 157(75) Papadimitrakopoulos, F., 98(200) Park, C., 148(61) Parker, I. D., 24(124), 32(133), 102(203) Park, H., 124(26) Park, J., 124(26) Park, Y.,84(196) Parris, P. E., 68(177-181) Partee, J., 57(167) Parthasarathy, G., 9(54-59, 115(218) Pasch, G.. 77(187) Pascual, R., 173(43) Pati, S. K.,19(99) Peak, M. S., 254(235) Peierls, R. E., 122, 122(12) Pei, Q. B., 115(212, 222, 226) Pelegrina, J. L., 191, 197(119, 121), 198, 219, 219(167), 226 Penzkofer, A., 8(51) Pepperhoff, W., 252(230) Perel, V. I., 79(194) Perenboom, J. A. A. J., 121(7) Pkrez-Magrank, R., 173(49) Perkovic, O., 174(52) Persson, B. N. J., 142(47,49) Petritsch, K., 9(64) Petty, W., 179(81), 180(83), 181(87), 200, 200( 126- 128)

Petti, E., 258(248) Pettifor, D., 164(21) Peyghambarian, N., 3(16), 98(202) Philips, W. A,, 224(179) Phillips, R. T., 7(45) Pichler, K., 9(63) Pichlmaier, M., 8(51) Pine, D. J., 313(111) Pinner, D. J., 105(208), 115(217) Pinto, M. R., lO(73) Pittet, N., 312(107) Planes, A,, 165, 168(31), 171, 172(38-39), 173(41-42.47-49), 174(50, 54-55), 188, 188(100), 191, 192, 193(109-110), 194, 195, 197(121), 198,207, 212(155), 214, 218(164), 219(165), 220, 221(169, 172), 222, 229, 231(185,187), 232(190-191), 237(204), 238, 239,240,257(246), 258(247), 259,260(250,252) Planes, Antoni, 159 Plummer, E. W.,123(20) Pommerehne, J., 105(209) Pons, J., 254(237), 255, 256(243) Pope, M., 2(3) Popovis, I. G., 143(53) Porta, M., 224(178), 267 Pourminaie, F., 3(11) Prause, B., 313(115) Prause, P., 313, 313(114) Predel, B., 167(24) Priestley, R., 7(42) Probst, M., 47(145) Purushothaman, S.,4(32) Putnis, A,, 178(71) Pynn, R., 199(124), 200(130), 202,203(134), 209(145), 221(173), 224

Q Qiu, X., 121(5) Quattrocchi, C., 7(44), 19(104)

R Rafols, I., 173(49) Rajagopal, A., 41(140), 44(143) Rajan, K., 296(73-74, 76-77) Raju, V. R., 4(27, 31) Ralph, D. C., 121(8), 124(22) Ramasesha, S., 18(82), 19(99), 2q107, 109) Rapacioli, R., 173(43), 232(191), 233(193)

AUTHOR INDEX Rauch, P., 123(17) Raueso, S. A., 293(63) Raymond, S., 203(135) Redecker, M., 54(151-152) Reed, M., 120(1) Remmers, M.. 7(44) Ren, X., 197, 197(117), 233(195), 236 Resel, R., 57(165-166) Rhule, M., 178(70) Rice, M. J., 19(93, 100-101) Rice, T. M., 123(14) Richter, D., 223(176) Riege, S. P., 297(83) Rifkin, J. A,, 179(80) Rios-Jara, D., 174(57), 188, 193(109), 197(120), 203(134), 209(145), 229, 267 Rivier, N., 275, 275(6), 278, 278(16), 307(98) Roberts, B. W., 174(51) Robertson, I. M., 178(72) Robinson, J., 256(239) Robinson, R. A,, 199(124), 202 Robinson, R. B., 155(68-69) Rodriguez, P. L., 197, 197(119) Rogers, J., 4(27) Rogers, J. A,, lO(72) Rohlfing, M., 22(116) Roitburd, A. L., 160(5) Roitman, D., 3(10) Roitman, D. B., 3(11) Romero, R., 165, 219,219(165, 167), 224(180), 226, 231(187), 232(190) Rosen, M., 173(44) Rost, H., 9(65), 40(139) Rothberg, L. J., 2(5), 9(57), 10(68), 49(148), 98(200) Roth, R., 123(21) Rotzinger, F., 49(149) Rouby, D., 173(47-48) Rouxel, J., 123(13) Rowe, J., 29(131) Rubini, S., 167, 223(174) Rubin, S., 49(146) Rubio, A., 145(57) Ruoff, R. S., 157(77) Ryum, N., 283, 283(27), 289(54, 59-60), 291 S

Saburi, T., 163(16), 167(24) Sade, M., 197(119)

325

Saetre, T. O., 289(59-60), 291 Safran, S. A,, 303(87) Sahni, P. S., 286(36-37) Sakamoto, H., 171, 171(32) Salamon, M. B., 177(64-65) Salanceck, W. R., 23( 118) Salaneck, W. R., 23(120-121), 40(139) Salem, J. R., 24(123), 55(153) Salje, E., 178(71) Sampietro, M., 115(224) Samuel, I. D. W., 7(45), 80( 195) Sanati, M., 235(202) Sano, T., 47(144) Sariciftci, N. S., 38(137) Sarpeshkar, R., 4(31), 115(229) Saunders, G. A., 184, 184(90), 185(94), 210(149), 211, 216, 216(159), 267 Saxena, A,, 15(76), 18(80-81), 19(87), 26(130), 235(202), 244(207) Saxena, Avadh, 117 Schafer, H., 131(42) Scheinert, S., 77(187) Schenk, R., 19(104) Scherf, U., 67(175), 115(219,224) Schimetta, M., 7(46) Schlittler, R. R., 125(31) Schlom, D. G., 5(37) Schluter, M., 155(70) Schmahl, W. W., 178(71) Schmidt, A,, 44(142) Schober, H. R., 179(81), 181(87), 200(126127) Scholte, P. M. L. O., 121(9) Scholz, G. A., 156(73) Schonfeld, B., 201(131) Schon, H. J., 57(168) Schon, J. H., 22( 117) Schrieffer, J. R., 2(1) Schryvers, D., 176, 176(61), 178, 202, 204 Schwartz, A,, 256(240), 257, 258 Schwartz, B. J., 115(212-213) Schweizer, E. K., 125(28) Scott, C., 55(153) Scott, J. C., 9(58), 24(123), 55(160) Seel, Steve, 314 Segel, D., 286, 286(35) Segui, C., 219(168), 254(237), 255, 256(243) Seidler, P. F., 36(134) Semenovskaya, S., 246(216) Seshadri, R., 293(66)

326

AUTHOR INDEX

Sethna, J. P., 174, 174(51-53), 179(77-79) Shaheen, S. E., 3(16), 98(202) Shapiro, S. M., 176, 176(61), 177(64, 66), 178, li9, 179(76), 180(84), 201(133), 202, 203(135), 204, 205(139-140), 246(216), 256(240-241). 257, 258, 259(249) Shavrov, V. G., 264(256) Shaw, J. M., 4(32) Shaw, T. M., 125(34) Sheats, J., 3( 11) Sheats, J. R., 3(10) Sheehan, P. E., 125(36) Shen, T. C., 124(25) Shen, T.-C., 142(52) Shen, Z., 2(6), 3(12), 55(154) Shen, Z. L., 115(227) Shibata, K., 47(144) Shiino, O., 148(61) Shi, J., 2(7) Shimada, M., 5(39) Shimitzu, K., 171, 171(32) Shimoi, Y.,18(83-85), 20(110) Shinar, J., 9(61), 57(167) Shirane, G., 180(84), 193(111), 200(129) Shirota, Y., 98(201) Shkunov, M., 1 15(220) Shore, J. D., 174(51) Shoustikov, A., 3(19) Shuai, Z., 9(67), 18(78), 19(96, 99, 105 Shvindlerman, L. S., 285(30), 295(72) SiAhmed, L., 49(149) Sibley, S., 3(19), 3(21) Silbey, R., 20(112) Silinsh, E. A,, 25(127) Simmons, J. A,, 173(45) Simon, A. J., 293(64) Sinclair, M. B., 60(169) Singh, K. K., 178(71) Singh, R. C., 165 Singh, V. V., 304(91) Singleton, M. F., 167 Sirringhaus, H., 4(23-24) Skala, S., 123(18) Skriver, H. L., 217(161) Slough, C. C., 123(16) Slusher, R. E., 9(57) Slyozov, V. V., 280, 280(20) Smith, C. S., 275, 275(10), 305(95) Smith, D. L., 1, 8(52), 15(76), 19(87-88), 21( 113), 26( 128- 130), 32( 133), 38( 136), 43(141), 49(146-147), 54(150), 55(163),

63(170), 65(171), 77(184), 88(198), 102(204), 106(211) Smith, H. I., 296(80), 297(82) Smits, A. B., 121(9) Snow, E. S., 125(35) Sokolowski, M., 8(49) Somoza, A., 232(190) Sonoda, Y., 23(120) SOOS,Z. G., 18(82), 20(107-109) Sotgui, R., 115(224) Spaepen, Frans, 314 Spencer, G. C. W., 6(41) Sperry. P. R., 293(68) Spiering, A. J. H., 4(24) Squires, G. L., 199(124), 202 Srolovitz, D. J., 286(36-37), 303(87), 306(97), 308( 102) Stach, E. A,, 303(90) Stafstrom, S., 23(121) Stassis, C., 181(86), 193(110), 194, 199(125), 200, 200(128), 201(132), 202, 204, 207, 208, 232(190), 267 Stavans, J., 278,278(17), 279, 286,286(33, 3 9 , 287, 290, 290(61), 294 Stegeman, G. I., 20(106) Steinfort, A. J., 121(9) Stelzer, F., 7(46) Stine, K. J., 293(63), 294 Stobbs, W. M., 205, 205(142), 206 Stocking, A,, 3(10) Stoscio, J. A., 142(48) Stroscio, J. A,, 125(29) Strukelj, M., 98(200) Strunk, C., 121(5) Stuhr, U., 256, 256(242), 258 Stumpf, R., 123(20) Stuwe, H. P., 305(93) Suezava, M., 200(129) Suezawa, M., 193(105) Sumino, K., 193(105) Sundheimer, M. L., 20(106) Sun, R. G., 7(47) Su, W. P., 2(1), 18(78), 19(99) Su, X., 260(251), 264(254) Suzuki, S., 264(256) Suzuki, T., 209(144), 212(152), 231(189) Svenson, E. C., 205(139) Swager, M., 105(206) Swager, T. M., lOS(207) Swenberg, C. E., 2(3) Sze, S. M., ll(74)

327

AUTHOR INDEX

T Tadic, B.. 174(56) Tajbakhsh, A. R., 56(164) Takagi, T., 264(256) Takagi, Y., 212(153) Takeda, S., 178(73) Tak, Y. H., 78(188), 105(209-210) Taliani, C., 3(13) Tanahashi, H., 209, 209(144) Tang, C. W., 2(7), 4(33-34), 9(60-61), 84(196), 98(199)

Tang, H., 125(31) Tani, J., 264(256) Tanner, L. E., 175(59), 176, 176(61), 178, 178(69), 179(76), 180(84), 202, 204, 251(228), 256(240), 256(241), 257, 258 Tasch, S., 115(219, 224-225) Tas, H., 163(15), 164(18) Tecklenburg, R., 77(187) Tessler, N., 4(22-23), 105(208), 115(214-217) Thomas, D. S., 115(217) Thompson, Carl V., 269 Thompson, C. V., 281(23-24) 288(40-41, 43-44), 289(58), 296(78-SO), 297(81-83). 302(85), 303(86, 88-89), 305(94) Thompson, M. E., 2(6), 3(12, 17, 19-21, 53), 55(154) 115(227) Thomson. M. E., 123(17) Thomson, R. E., 123(17) Thurston, R. N., 215(157) Tighe, T. S., 121(6) Tinkham, M., 121(6, 8), 124(22) Tkachenko, A. V., 205(141) Toledano, J. C., 234( 199) Toledano, P., 234(199) Torgeson, D. R., 223( 174) Torok, E., 193(106) Torra, V., 172(40), 219(168) Torruellas, W. E., 20(106) Torsi, L., 10(68), 77(185) Touminen, M. T., 121(6) Town, S. L., 254(235) Toyozawa, Y., 142(51) Trampenau, J., 179(81), 200(126-127), 200( 128) Trivisonno, J., 188, 192, 196(115), 258(248) Trullinger, S. E., 242(205) Tucker, J. R., 123(18), 124(25), 142(52) Turnbull, D., 270, 271, 271(1), 272, 313

U Udler, D., 296(75) Ueda, A., 5(39) Ueta, M., 142(51) Ullako, K., 256(239) Underwood, E. E., 313(110) V Vacar, D., 38(138) Valencia, V. S., 60(169) Valles, J. M., Jr., 155(68) van der Zant, H. S . J., 121(9), 157(75) Van Haesendonck, C., 121(5) van Houten, H., 120(3) van Hulk, D., 219(168) van Humbeeck, J., 219(168), 231(188) Van Look, L., 121(5) Van Munster, M. G., 55(158) Vannikov, A. V., 68(178) Van Slyke, S. A,, 4(33-34) Vardeny, Z. V., 19(89-90), 24(122), 57(167). 115(220-221)

Varma, C. M., 177, 177(63) Vasil'ev, A. N., 264(256) Verlinden, B., 189, 189(101-102), 190, 212(152), 215, 215(158), 249, 250

Verschuere, M., 121(5) Vettier, C., 205(139) Vignes-Adler, M., 312(107), 313(112-113) Vives, E., 173(49), 174(50, 54-55), 188(100), 192, 195, 227(183), 237(204), 238, 239, 263(253), 267, 240 Vleggaar, J. M., 55(157) Vogl, G., 179(81), 200(126) Vogl, P., 22(114) von Molnr, S., 125(34) von Neumann, J., 277, 277(14), 278 Vorderwisch, P., 256(242), 258 Voss, B. M. W. Langeveld, 4(24) Vossmeyer, T., 143(53)

W

Wada, Y., 156(72) Wadley, H. N. G., 173(45) Wagner, C. Z., 280, 280(21) Wahlstrom, E., 156(74) Wakai, F., 312(108) Walkup. R. E., 124(25), 142(50), 142(52) Wallace, D. C., 160(2, 4), 182, 183(89)

328

AUTHOR INDEX

Wall, M., 256(240), 257, 258 Walser, A. D., 7(42) Walter, J. L., 295, 295(71), 302(84) Walter, U., 123(17) Walton, D. T., 288(43-44), 303(86), 305(94) Wang, C., 124(25), 142(52) Wang, D. K., 7(47) Wang, H. L., 38(138) Wang, Y. Z., 7(47), 105(206-207) Warlimont, H., 164(18, 20) Warman, J. M., 7(44) Wasserman, E. F., 217(162), 252(229-230) Waszczak, J. V., 155(68-69) Wayman, C. M., 163(13, 16). 167, 171(32), 177(64-65). 178(72) Weaire, D., 276, 276(11), 289, 289(52-53), 290(62), 293(67), 308, 308(100-101) Weaver, M. S., 55(156) Webster, P. J., 253(231), 254(235) Wehrmeister, T., 84(196) Wei, S., 19(89) Weiss, D. S., 2(4) Weis, X.,57(167) Weitering, H. H., 123(20) Weitz, D. A,, 313(111) Wei, X., 24(122) Weller, H., 143(53) Weller, R. A,, 5(40) Wenzl, F. P., 115(224-225) Westengen, H., 289(54) Westervelt, R. M., 293(65-66) Whang, J., 288(40) Whitman, L. J., 125(29) Wickmann, L. K., 179(79) Wieting, T. J., 155(70) Williams, R., 31(132) Willis, J. M., 160(4) Wilson, J. A,, 123, 123(15) Winkler, B., 115(224-225) Winokur, M. J., 5(38), 6 Wise, J. A,, 293(63) Wisesendanger, R., 131(43) Withers, R. L., 123(15) Wochner, P., 256(240-241), 257, 258 Wolf, U., 78(189-191) WOO,E. P., 54(151-152) Woo, H. S., 19(104) Worgull, J., 258(248) Wu, C. I., 44(143)

Wu, D. T., 245(212) Wu, K. H., 260(251), 264(254) Wu, M. W., 78(192) Wuttig, M., 175(59), 251(228), 253(233) Wu, X. L., 123(19), 130(41), 153(66) Wyder, P., 121(7) Y

Yahioglu, G., 4(22), 115(217) Yamada, Y., 177(66), 178(67-68). 180(85), 185(93), 193(111), 244(210), 245(211) Yamaguchi, W., 148(61) Yang, B. X., 176, 176(61), 178, 180(84), 202, 204 Yang, C. Y.. 55(162) Yang, F., 264(254) Yang, Y., 115(222, 226) Yaron, D., 19(95-98), 20( Yassar, A,, lO(69, 71) Yassievich, I. N., 79(194) Yasunaga, M.. 197(118) Ye, L., 205(140) Ye, Y. Y., 205(137), 210(1 5), 235, 244(206) Yosada, K., 178(75) Yoshida, A,, 210(150) Yoshikawa, M., 171, 221(169) Yoshino, K., 19(90), 115(220-221) You, Y.. 3(19) Yu, G., 9(62), 26(129), 115(222-223) Yu, Z., 173(46) Yu, Z. G., 15(76), 18(80-81) Yu, Zhi Gang, 117

Z Zahres, H., 252(230) Zangwill, A,, 217(160) Zarestky, J., 193(110), 194, 199(125), 200, 201(132), 202, 204, 207, 208, 232(190) Zrubov, N., 171(35) Zawodzinski, T. A., 49(146) Zawodzinski, Thomas, 117 Zener, C., 160, 160(7), 190(104), 217, 305, 305(95) Zenz, C., 115(219) Zettl, A,, 123(17) Zhang, C., 115(222) Zhang, J., 124(23), 148(60)

AUTHOR INDEX Zhang, Jian, 119 Zhang, P., 264(254) Zhao, G. L., 177(62) Zheludev, A., 256, 256(240-241), 257, 258, 259(249) Zheng, Q. B., 7(47) Zhou, L., 188, 192, 196(115) Zhou, P., 123(19)

Ziebeck, K. R. A., 253(231), 254(235), 265(259) Ziolo, J., 223(174) Zoebisch, E. G., 15(77) Zolliker, M., 201(131) Zuo, F., 260(251), 264(254) Zuppiroli, L., 49(149)

329

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Subject Index

Cu-Zn-A1 alloy, martensitic transition, 165,

A

192,250

Aboav-Weaire law, two-dimensional cellular structures, 275-276 AFMs. See Atomic force microscopes Alloys addition to polycrystalline materials,

E

304-306 shape-memory alloys, 160,265-267 ferromagnetic, 161,252-265 lattice dynamics, 181-187,266 martensitic transition, 160,

162-164-181,187-267 Alq (tris-(8-hydroxyquinolate)-aluminuminum), 4, 5 carrier mobility measurements, 61,62,63 electronic properties, 46-48 exciton binding energy, 45-46 Schottky energy barrier, 43-45,49 Atomic force microscopes (AFMs), nanostructure creation, 125 Au alloys, martensitic transition, 170,194,201

Elastic constants shape-memory alloys ferromagnetic SMAs, 256-260 second-order, 187- 199 third-order, 210-216 Electrical transport properties, organic electronic materials, 54-77 Electronic structure metal/organic interface electronic structure,

24-54 x-conjugated organic materials, 13-16 Entropy, martensitic transition of shapememory alloys and, 217-221 Euler’s theorem three-dimensional cellular structures, 307 two-dimensional cellular structures,

273-274

B

F

Biphenyl, electronic structure, 14-16 Bipolarons, 19,21-22 Bipolar organic light-emitting diodes (LEDs),

Fermi surface nesting, nanocrystals, 143- 148 Ferromagnetic shape-memory alloys, 161,

93-98 Bloch reciprocal space, 120-121 C

Carrier mobilities, in organic electronic materials, 56-77 Carrier mobility, models, 66-77 Cell structure, two-dimensional, 272-279 Charge density waves (CDWs), 121 in bulk materials, 122 formation in bulk transition metal dichalcogenides, 126- 132 in nanoscale systems, 123-124 Christoffel equation, 182, 189 Cu-based alloys, martensitic transition, 165,

168-171, 188, 192-197,251

252-265 Field-effect transistors (FETs), organic, 2-4 device model, 108-113 device structure, 9-10 electronic transport properties, 55-56 Front-tracking technique, grain growth in polycrystalline materials, 288 Froth, evolution, 289,290-292

G Gaussian disorder model (GDM), carrier mobility, 67-68 Gold alloys, martensitic transition, 170,194,

201 Grain growth normal/abnormal, 270 polycrystalline materials, 270, 313-314

331

332

SUBJECT INDEX

Grain growth (Continued) alloy additions, 304-305 Burke-Turnbull model, 270-271 films, 293-304 foils, 293-296 mean field models, 279-286, 306 simulations, 286-290 three-dimensional systems, 305-313 two-dimensional systems, 271 -279, 290-293 Grain rotation, in thin films, 304 Griineisen parameters, shape-memory alloys, 187-188, 210-216

H Heusler alloys, 253 Hillert’s model, grain growth in polycrystalline materials, 280-282, 306 Hooke’s law, 181, 183 Huang d i h e scattering, 177 Hybrid models, grain growth in polycrystalline materials, 283 I(

Kinetics grain growth in polycrystalline materials, 27 1-272 martensitic transition in shape-memory alloys, 172-174

L Landau-type free-energy expansion, 234, 236, 244,262 Lattice dynamics, shape-memory alloys, 181187 Lewis’s law, two-dimensional cellular structures, 274-275 Light-emitting diodes (LEDs), organic, 2-3 bipolar devices, 93-98 device model, 79-84 device structure, 8-9 electronic transport properties, 55-56 high current density operation, 105- 108 multilayer devices, 98- 105 Schottky energy barriers, 24, 31, 32-45 singletarrier devices, 84-93 transient response, 105-108

Localized soft-mode models, martensitic transition in shape-memory alloys, 245-252 Louat’s model, grain growth in polycrystalline materials, 282-283, 306

M Magnetic coupling, shape-memory alloys, 253 Martensites, 162, 168 Martensitic transition shape-memory alloys, 160, 162-164, 266-267 experimental results, 187-216 Grneisen parameters, 186-187, 210-216 kinetics, 172-174 magnetic coupling, 252-265 modeling, 233-252 phase diagram, 164-171 phase stability, 217-233 phonon dispersion, 199-210, 256 precursor effects, 175-180, 256 second-order elastic constants, 187-199 thermodynamics, 172- 174, 217-221 third-order elastic constants, 210-216 vibrational anharmonicity, 186-187, 210-216 Mean field models, in polycrystalline materials, 279-286, 306 MEH-PPV (polyC2-methoxy, 5-(2’-ethylhexy1oxy)- 1,4-phenylene vinylene]), 5 carrier mobility, 58-60, 61, 63-65, 72-73 exciton binding energy, 45-46 Schottky energy barrier, 32-41,49, 50 Metal/organic interface, Schottky barriers, 24, 31, 32-45 Modeling grain growth in polycrystalline materials, 279-286,306-307 martensitic transition in shape-memory alloys, 233-252, 262-264 Monte Carlo technique, grain growth in polycrystalline materials, 286 Mullins-von Neumann law grain growth in polycrystalline materials, 283-286 two-dimensional cellular structures, 277-279 Multilayer organic lightemitting diodes (LEDs), 98-105

333

SUBJECT INDEX N Nanocrystals Fermi surface nesting, 143-148 formation in 2H-TaS2, 151-152 by STM tip, 132- 142 in H-layer of 4Hb-TaSe2, 148-151 to other TMD systems, 148-155 size-dependent effect, 142- 143 Nanostructures, fabrication, 124- 125 Neutral bipolaron, 19 Neutron scattering, as martensitic transition precursor effect, 175-176, 256 NitMnGa, properties, 254-256 Ni-A1 alloy, martensitic transition, 167, 176-179, 201, 251 Nucleation, martensitic transition in shapememory alloys, 245-251

0 Organic electronic devices, 1-4, 8-12, 113115 electrical transport properties, 54-77 field-effect transistors (FETs), 2-4, 9-12, 77-79, 108-112 light-emitting diodes (LEDs), 2-3, 8-11, 77-108 materials, 4-8, 12-24 metal/organic interface electronic structure, 24-54 Organic electronic materials, 4-8, 12-24 carrier mobilities in, 56-77 electrical transport properties, 54-77 electronic structure, 13-16 electron-ion and electron-electron interactions, 16-21 metal/organic interface electronic structure, 24-54 solid state properties, 21-24 Organic field-effect transistors (FETs), 2-4 device model, 108-1 13 device structure, 9-10 electronic transport properties, 55-56 Organic light-emitting diodes (LEDs), 2-3 bipolar devices, 93-98 device model, 79-84 device structure, 8-9 electronic transport properties, 55-56 high current density operation, 105- 108

multilayer devices, 98-105 Schottky energy barriers, 24, 31, 32-45 single-carrier devices, 84-93 transient response, 105-108

P Pentacene, 4, 5 carrier mobility measurements, 61, 63 exciton binding energy, 45-46 hole mobility, 77 Schottky energy barrier, 41 -43 PFO (poly(9,9-dioctylfluorene)), carrier mobility, 72-73 Phonon dispersion, shape-memory alloys, 199-210, 256 a-conjugated organic materials, 4-8, 12-24 electronic structure, 13- 16 electron-ion and electron-electron interactions, 16-2 1 metal/organic interface electronic structure, 24-54 solid state properties, 21-24 Polycrystalline materials films, 293-304 foils, 293-296 grain growth, 270, 313-314 alloy additions, 304-305 Burke-Turnbull model, 270-271 films, 293-304 foils, 293-296 mean field models, 279-286, 306 simulations, 286-290 three-dimensional systems, 305-3 13 two-dimensional systems, 271-279, 290-293 Poly(9,9-dioctylfluorene). See PFO Poly[2-methoxy, 5-(2’-ethyLhexyloxy)-l,4phenylene vinylene]. See MEH-PPV PPV (poly(p-phenylene vinylene), 4-6, 16-18 Primary recrystallization, 295

R Recrystallization, 270, 293, 295 S

SAM. See Self-assembled monolayer Scanning probe microscopes, nanostructure creation, 125-126, 132-142

334

SUBJECT INDEX

Scanning tunneling microscopes (STMs), nanostructure creation, 125 Schottky energy barrier Alq, 43-45 at metal/organic interface, 24. 31, 32-45 manipulating using dipole layers, 49-54 MEH-PPV, 32-41 pentacene, 41-43 self-assembled monolayer (SAM) and, 49-54 Second-order elastic constant (SOEC), 182 martensitic transition of shape-memory alloys, 187-199 Self-assembled monolayer (SAM), Schottky barrier and, 49-54 Shape-memory alloys, 160, 265-267 ferromagnetic, 161, 252-265 lattice dynamics, 181, 266 elastic behavior, 181-186 Gruneisen parameters, 186- 187 martensitic transition, 160, 162-164, 266-267 experimental results, 187-216 Gruneisen parameters, 186- 187, 2 10-21 6 kinetics, 172- 174 magnetic coupling, 252-265 modeling, 233-252, 262-264 phase diagram, 164-171 phase stability, 217-233 phonon dispersion, 199-210, 256 precursor effects, 175-180, 256 second-order elastic constants, 187-199 thermodynamics, 172- 174,2 17-22 1 third-order elastic constants, 210-216 vibrational anharmonicity, 186- 187, 2 10-21 6 Simulations, grain growth in polycrystalline materials, 286-290, 309-312 Single-carrier organic light-emitting diodes (LEDs), 84-93 Single-carrier SCL diodes, carrier mobility, 63-66 Soap froth, evolution, 289, 290-292 SOEC. See Second-order elastic constant “Special mode,” 189 STMs. See Scanning tunneling microscopes Strain-phonon coupling model, martensitic transition in shape-memory alloys, 235-245, 262

Succinonitrile, polycrystalline, grain growth, 295-296 Superconductivity, in confined mesoscopic system, 121-122 Switching events, two-dimensional cellular structures, 276-277 Switching intensity parameter, 284

T Tertiary recrystallization, 295 Thermodynamics, martensitic transition in shape-memory alloys, 172-174, 217-221 Thin films, polycrystalline, grain growth, 293-304 Third-order elastic constant (TOEC), 183184 martensitic transition of shape-memory alloys, 210-216 Time-of-flight, mobility measurements of organic electronic materials, 57-62 Ti-Ni alloy, martensitic transition, 167, 190 TOEC. See Third-order elastic constant Transition metal-dichalcogenide (TMD) compounds, 123 charge density waves in bulk TMDs, 126-132 crystal structure, 128-129 Tris-(8-hydroxyquinolate)-aIuminum see Alq Tweed patterns, as martensitic transition precursor effect, 178-179

V

Vertex models, grain growth in polycrystalline materials, 288 Vibrational anharmonicity, shape-memory alloys, 187-188, 210-216

W Wigner-Seitz cell, 307

X X-ray diffraction, as martensitic transition precursor effect, 177-178