Founding Editors FREDERICK SEITZ DAVID TURNBULL
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Contributors to Volume 61 Numbers in parentheses indicate the pages on which the authors’ contributions begin
J. ERLEBACHER (77) Department of Materials Science and Engineering, Johns Hopkins University, Baltimore, MD 21218 ROBERT C. CAMMARATA (1) Department of Materials Science and Engineering, Johns Hopkins University, Baltimore, MD 21218 H. T. JOHNSON (143) Department of Mechanical Science & Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 J. H. YOU (143) Department of Mechanical Science & Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801
Preface
This volume of the Solid State Physics Series contains three articles that span a range of topics, fundamental and applied, in materials physics. The article by R. C. Cammarata is a careful exposition of our current understanding of the thermodynamics of surfaces and interfaces. The foundations of this field were laid by J. W. Gibbs, whose work has been mined by later generations for rigorous insight into the stability of surfaces and their role in phase transitions. Important extensions of Gibbs’s work include the study surfaces and interfaces of solid, multi-component phases, and the thermodynamics of stressed solids, reviewed recently by P. W. Voorhees and W. C. Johnson in this Series (Volume 59). Cammarata pays particular attention to the concept of the surface stress and points out that the ‘‘availability’’ (also known as the ‘‘exergy’’) provides an elegant vehicle for the thermodynamic description of equilibrium surfaces. Since surfaces and interfaces play an increasing role in the stability and operation of devices as their dimensions approach the nanoscale, this article is a timely attempt to put precision and rigor into the analysis of these effects. At the same time, it is written in a highly pedagogical way and can serve as a self-contained text for an advanced course in this subject. The article by J. Erlebacher is an overview of our current scientific understanding of the catalysis process that is the basis of the hydrogen–oxygen fuel cell. These devices are of great current interest in the push to develop new and more efficient energy conversion devices. Of particular interest is the development of new cathodes that use much less, or no platinum. The scientific drivers for progress in this area, as described in the article, are new methods for controlling the microstructure of metals on the nanoscale (for example, by de-alloying), and more powerful computational methods of electronic structure that allow better assessment of the catalytic potential of new alloys. The article has also a self-contained description of the fuel cell and can therefore be usefully assigned in advanced courses on new energy technologies. The article by J. H. You and H. T. Johnson adresses the effect of dislocations on the electrical and optical properties of two important III–V compounds, ix
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PREFACE
GaAs and GaN. Dislocations in the tetrahedrally coordinated crystals Si and Ge were reviewed in Volume 22 of this Series by H. Alexander and P. Haasen, with emphasis on their roles in plastic deformation. These electronic effects of dislocations are generally deleterious to device performance, but, remarkably, much less in GaN than in GaAs. This article explains this difference through a detailed computational study of the electronic structure of the core structure of dislocations in both structures and attributes it to the larger deformation around the edge dislocations and a smaller electron effective mass in GaAs. HENRY EHRENREICH FRANS SPAEPEN
Editor’s Note
Since the publication of the previous volume of this Series, its two founding editors, Frederick Seitz (1911–2008; editor 1955–1984) and David Turnbull (1915–2007; editor 1955–1994), and my co-editor, Henry Ehrenreich (1928– 2008; editor 1967–2008) have passed away. The Solid State Physics Series traces its origin to Seitz’s seminal work ‘‘The Modern Theory of Solids,’’ published in 1940, which laid the foundation of our understanding of electronic band structure and its relation to the electronic and optical properties of materials. Seitz had the idea to update his book, not with new editions, but with a series of invited articles by experts in this rapidly developing field. He chose as his co-editor David Turnbull, the foremost authority in the study of the kinetics of phase transformations in materials. Henry Ehrenreich, a prominent electronic theorist, joined them in 1968. It was my privilege to succeed David Turnbull as editor. He had been my doctoral thesis advisor and an exceptional mentor on many aspects of materials science. Over the past 15 years, I very much enjoyed the collaboration with Henry Ehrenreich on editing the Series. He cared deeply about its quality and development, and I benefited greatly from his insights and judgment. The article by Erlebacher in this volume, the last one he solicited, is dedicated to his memory. The Series will continue: I am very pleased that Subir Sachdev, Professor of Physics at Harvard has agreed to join me as co-editor. His expertise is in theoretical condensed matter physics: collective dynamics of strongly interacting electrons and quantum phase transitions. The Series, therefore, will continue to have a judicious blend of articles that span condensed matter physics and materials science, as well as experiment and theory. Our predecessors set a high standard set for the Series in scholarship, relevance and pedagogy. We will make every effort to continue that tradition. FRANS SPAEPEN
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SOLID STATE PHYSICS, VOL. 61
Generalized Thermodynamics of Surfaces with Applications to Small Solid Systems R OBERT C. C AMMARATA Department of Materials Science and Engineering, Johns Hopkins University, Baltimore, MD 21218
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Laws of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Fundamental Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Euler and Gibbs–Duhem Equations . . . . . . . . . . . . . . . . . . . . . . . . . 4. Criteria for Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Thermodynamics of Bulk Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Bulk Solid–Fluid Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Availability and Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Thermodynamics of Systems with Surfaces . . . . . . . . . . . . . . . . . . . . . . . 9. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Gibbs Dividing Surface Construction for Fluid Interfaces . . . . . . . . . . . . 11. Gibbs Adsorption Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Stability of a Surface in a Fluid System . . . . . . . . . . . . . . . . . . . . . . 13. Mechanical Equilibrium Between a Solid and Fluid Including Capillary Effects 14. Gibbs’ Treatment of Capillary Effects Involving a Single Component Solid . . 15. Chemical Equilibrium for a Multicomponent Solid Including Capillary Effects 16. Surface Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. Surface Availability in Lagrangian Coordinates . . . . . . . . . . . . . . . . . . 18. Physical Origin of Surface Availability and Surface Stress . . . . . . . . . . . . 19. Layer Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. Adsorption Equation for a Solid Surface . . . . . . . . . . . . . . . . . . . . . . 21. Solid–Solid Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Physical Origin of the Solid–Solid Surface Availability and Interface Stresses . VI. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. Nucleation During Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . 24. Surface Stress Effects on Thin Films . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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VII. Appendix A: Stress and Strain in Solids. . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Appendix B: Effect of the Dividing Surface Location on the Curvature Contributions to the Fundamental Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Appendix C: Gibbs–Thomson Effects on Small Solids. . . . . . . . . . . . . . . . . . X. Appendix D: Critical Thickness for a Crystallographically Anisotropic Thin Film System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction
During the period 1873–1878, J. Willard Gibbs published three papers on thermodynamics.1 The first two2,3 were primarily concerned with geometrical methods for analyzing the behavior of fluids. In these papers Gibbs derived certain basic equations that described the interaction between a physically and chemically uniform fluid, identified as the thermodynamic ‘‘system,’’ and its surroundings. The system and surroundings could interact thermally by a flow of heat between them and interact mechanically through the performance of work, resulting in a change of the volume of the system. In his third paper4 Gibbs extended his analysis to include nonuniform systems that could contain solids, interact with external fields resulting in other forms of work (e.g., electrical, gravitational), and experience a change in chemical composition by processes such as chemical reactions or an exchange of matter with the surroundings. This paper, entitled ‘‘On the Equilibrium of Heterogeneous Substances,’’ was one of the great achievements in the history of theoretical physics. Gibbs showed how the classical laws of thermodynamics as formulated by Clausius, Kelvin, and others to describe dynamic processes involving the conversion of heat into work5–12 could be used as the basis of a general deductive theory of equilibrium. He presented a method for obtaining what he called the ‘‘fundamental equation’’ of a system and then showed how it could be employed to characterize the 1
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961). J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 1. 3 J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 33. 4 J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 55. 5 J. H. Keenan, Thermodynamics, Wiley, New York (1941). 6 E. Fermi, Thermodynamics, Dover, New York (1956). 7 A. B. Pippard, The Elements of Classical Thermodynamics for Advanced Students of Physics, Cambridge University Press, Cambridge (1957). 8 C. J. Adkins, Equilibrium Thermodynamics, Cambridge University Press, Cambridge (1983). 9 C. H. P. Lupis, Chemical Thermodynamics of Materials, North Holland, New York (1983). 10 C. B. P. Finn, Thermal Physics, Chapman and Hall, London (1993). 11 R. T. DeHoff, Thermodynamics in Materials Science, McGraw-Hill, New York (1993). 12 M. Baylin, Survey of Thermodynamics, AIP, New York (1994). 2
GENERALIZED THERMODYNAMICS OF SURFACES
3
equilibrium state of gases, liquids, and solids, as well as multiphase systems, phase transitions, mixtures, chemical reactions, surfaces, and electrochemical cells. It can be claimed that this paper, which ran over 300 pages and was published in two parts, was the first treatise on physical chemistry. Gibbs’ published work has been recognized for its extreme precision and its condensed and often difficult style. As a result, there have been many who have engaged in Gibbsian hermeneutics to extract all of the insights his papers provide.13 One area that has involved a large amount of exegesis has been Gibbs’ discussion of surfaces (about one third of his 1876–1878 paper was devoted to surface thermodynamics). His fundamental equation approach is most straightforwardly employed when a system can be treated as involving one or more uniform phases and for which capillary effects can be ignored. If capillary effects are to be included, then it is necessary to consider the influence of physically and/or chemical nonuniform regions between uniform phases. To avoid difficulties associated with these nonuniform regions, Gibbs introduced his ‘‘dividing surface construction’’ that allowed him to develop a general thermodynamics of surfaces for completely fluid systems as long as the interfacial region was not too diffuse. When he characterized the surface between a fluid and solid, the solid was taken as a single component material although the fluid could be a multicomponent phase. There have been attempts to extend his approach to treat a surface between a multicomponent solid and fluid. However difficulties arise in these analyses that have not generally been appreciated that may require that they be modified to obtain a rigorous treatment. Gibbs had little to say regarding the surfaces between two crystals although others have also tried to characterize them with the dividing surface method. Issues concerning the nature of general sold–solid surfaces may make these attempts problematic except for certain restricted cases. In this chapter Gibbs’ formulation of equilibrium thermodynamics of material systems will be reviewed and modern attempts to extend his method to multicomponent solids will also be discussed. This will be followed by a consideration of Gibbs’ thermodynamics of surfaces and his dividing surface construction. It will be shown that this method allows for a general and comprehensive treatment of nondiffuse fluid surfaces. Gibbs’ extension of his method to the case of a surface between a single component solid and multicomponent fluid will then be examined as well as the difficulties of generalizing his approach when the solid can have multiple substitutional components. It will be proposed that these difficulties can be addressed if the concept of thermodynamic ‘‘availability’’ is employed to describe the equilibrium state of a surface. In this way a generalized thermodynamics of
13
F. G. Donnan and A. Haas, eds. Commentary on the Scientific Writings of J. Willard Gibbs, Volume I: Thermodynamics, Yale University Press, New Haven (1936).
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R. C. CAMMARATA
surfaces can be formulated that may even obviate the need to use the dividing surface construction. Attempts to extend Gibbs’ dividing surface analysis to solid– solid interfaces will be discussed. It will be suggested that these will be generally feasible only for simple systems, such as one involving a planar interface between a thin sold film on a solid substrate, and that ‘‘gradient thermodynamics’’ methods are needed to treat the more complex cases. The final section will present examples of how Gibbsian surface thermodynamics can be applied to characterize systems for which capillary effects play a central role in the thermodynamic behavior.
II. Background
1. L AWS
OF
T HERMODYNAMICS
Gibbs used the laws of classical thermodynamics as the basis of his theory of equilibrium. Consider a material system that can interact thermally with its surroundings through a flow of heat q and that can interact physically through the performance of work w. Let q be positive when heat flows into the system and let w be positive when the surroundings perform work on the system. An example of positive work would involve an applied pressure exerted on a gas resulting in the compression of the gas. (This sign convention for w is opposite the one used by Gibbs.) Suppose the system is in an initial state A and undergoes a process that results in it evolving to a new state B. In general there can be a multitude of processes that can effect this change in state, and the resulting amount of heat flow q and work w will depend on the nature of the particular process employed. However, according to the first law of thermodynamics, the sum of q and w (measured in the same units) for the change in state A to B will be the same, independent of the ‘‘path’’ (series of intermediate system states) of the process. As a consequence it is possible to define a function of state U, called by Gibbs the energy of the system and in modern texts called the internal energy, whose change DU for a given change in state can be expressed as5–12 DU ¼ q þ w:
ð2:1Þ
Equation (2.1) can be taken as the mathematical statement of the first law. For an infinitesimal change in state the first law can be expressed in differential form as dU ¼ 2q þ 2w:
ð2:2Þ
It is notedRthat dU is an exact differential, meaning that for a change in state, B the integral A dU ¼ DU ¼ UB UA can be evaluated independent of the path from A to B, while 2q and 2w are inexact differentials, which means that it is
GENERALIZED THERMODYNAMICS OF SURFACES
5
generally necessary to know the path of the process to integrate them. Equation (2.1) or (2.2) can be taken as a statement of the principle of energy conservation. If a system is totally isolated from its surroundings so that there is no interaction between them, then according to Eq. (2.2) dU ¼ 0
ð2:3Þ
and for any process resulting in a change in state of an isolated system the change in internal energy DU ¼ 0. Consider a system in which mechanical work is performed on it by the surroundings that changes the volume V of the system. Letting the pressure of the surroundings be denoted as Po, then by simple mechanical considerations the infinitesimal work can be expressed as ð2:4Þ
2w ¼ Po dV:
In his discussion of processes, Gibbs restricted his attention almost exclusively to reversible processes; that is, those conducted extremely slowly (‘‘quasistatically’’) and without hysteresis.5–12 Owing to the quasi-static nature of a reversible process the system has sufficient time to equilibrate with the surroundings during each infinitesimal step of the process. As a result, a reversible process can be considered one in which the system evolves through a series of equilibrium states and that if it is reversed at any point by an infinitesimal change in conditions the system will evolve back through the same equilibrium states. Thus, for a system undergoing a reversible change in volume, mechanical equilibrium dictates that the system pressure P will equal the pressure of the surroundings Po so that the infinitesimal reversible work can be expressed as 2wrev ¼ PdV:
ð2:5Þ
Attention is now given to a system that can interact thermally with its surroundings. As noted previously, 2q will in general be an inexact differential, and this will also be true for the case of reversible heat flow 2qrev. According to the second law of thermodynamics, it is possible to define a function of state called the thermodynamic temperature T (which is an absolute temperature) such that the differential 2qrev/T is exact and therefore can be equated with the differential of another function of state called the entropy S: dS ¼ 2qrev =T:
ð2:6Þ relation.5–12
Associated with the second law is Clausius’ Suppose the system absorbs an infinitesimal amount of heat 2q from its surroundings. Let the temperature of the surroundings where the heat exchange is taking place be To. The Clausius relation can be expressed as dS 2q=T o
ð2:7Þ
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R. C. CAMMARATA
where the equality is given by Eq. (2.6), corresponding to a reversible flow of heat with T ¼ To. For a system that is thermally isolated from its surroundings so that no heat flow between the system and surroundings is allowed (e.g., owing to the system being enclosed by thermally insulating walls), Clausius’ inequality becomes dS 0
ð2:8Þ
and this can be integrated for a process to give DS 0. This result is the law of entropy increase which can be interpreted in the following way: the entropy of a thermally isolated system cannot decrease and that as the system evolves toward equilibrium, the entropy will continually increase until it reaches the maximum value it can have for the given system constraints at equilibrium. 2. F UNDAMENTAL E QUATION Consider a system under a hydrostatic pressure that can interact thermally and mechanically with its surroundings. Suppose a uniform region of the system undergoes a reversible infinitesimal change. According to the first law, Eq. (2.2), dU ¼ 2qrev þ 2wrev :
ð2:9Þ
Inserting Eqs. (2.5) and (2.6) into Eq. (2.9) gives dU ¼ TdS PdV
ð2:10Þ
Gibbs interpreted Eq. (2.10) as expressing the internal energy variation for a uniform region of the system owing to a variation in the entropy and volume in that region, keeping the chemical composition fixed.14 He also noted3,15 that Eq. (2.10) implied that the internal energy could be considered a function of the entropy and volume, U ¼ U(S, V). The full differential of the internal energy is @U @U dS þ dV ð2:11Þ dU ¼ @S V @V S which leads to Eq. (2.10) with T ¼ (@U/@S)V and P ¼ (@U/@V)S. Gibbs showed how this approach could be extended to include variations in the amount of the different chemical components.16–18 For the present it will be assumed that the 14
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 62. J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 20. 16 J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 92. 17 H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, Wiley, New York (1985). 18 N. W. Tschoegl, Fundamentals of Equilibrium and Steady-State Thermodynamics, Elsevier, Amsterdam (2000). 15
7
GENERALIZED THERMODYNAMICS OF SURFACES
amount of each component can be varied independently of the others. Letting ni denote the number of moles of component i, the internal energy can be now taken as a function of the entropy, volume, and number of moles of each component, and taking the full differential of U ¼ U(S, V, n1, n2, . . .) leads to X @U @U @U dS þ dV þ dni , ð2:12Þ dU ¼ @S V , ni @V S, ni @V S, V , nj6¼i i which can be written as dU ¼ T dS P dV þ
X
mi dni ,
ð2:13Þ
i
where T ¼ (@U/@S)v,ni, P ¼ (@U/@V)S,ni and mi ¼ (@U/@ni)s,v,nj6¼i. The parameter mi was called by Gibbs the potential of component i and in modern texts is called the chemical potential of component i. The term Simi dni can be considered the reversible chemical work to vary the number of moles of the components. Gibbs referred to Eq. (2.13) as the ‘‘fundamental equation’’ and it occupied a central role in his theory of equilibrium. The parameters T, P, and mi are defined locally within a system and are called intensive variables, while the parameters U, S, V, and ni are called extensive variables that scale with the mass of the system keeping the intensive variables fixed. 3. E ULER
AND
G IBBS –D UHEM E QUATIONS
Two important relations can be readily obtained from Eq. (2.13). Consider a process involving the formation of a physically and chemically uniform phase. Equation (2.13) can integrated for this process to obtain an expression for the internal energy of this phase19 X U ¼ TS PV þ mi ni : ð2:14Þ i
Equation (2.14) is called the Euler equation.7,8,12,17,18 Taking the full differential of Eq. (2.14) and subtracting Eq. (2.13) leads to the Gibbs–Duhem equation10–12,17,18,20 X 0 ¼ S dT V dP þ ni dmi : ð2:15Þ i
19 20
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 74. J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 85.
8
R. C. CAMMARATA
The Gibbs–Duhem equation relates how changes in the intensive state variables of a phase are coupled during a change in state. For a system composed of multiple phases, each phase will have an associated Gibbs– Duhem equation. 4. C RITERIA
FOR
E QUILIBRIUM
Since the law of entropy increase associated with the second law holds for a thermally isolated system it will necessarily apply to a completely isolated system. Gibbs used this fact and the result from the first law that the internal energy of an isolated system does not change to state the following criterion21: ‘‘For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations in the state of the system that do not alter its energy, the variation of its entropy shall vanish or be positive.’’ This equilibrium criterion can be expressed mathematically as ðdSÞU 0:
ð2:16Þ
Gibbs then stated the following alternate criterion that he showed was equivalent to the first: ‘‘For the equilibrium of any isolated system it is necessary and sufficient that in all possible variations in the state of the system that do not alter its entropy, the variation of its energy shall vanish or be positive.’’ This criterion can be written as ðdUÞS 0:
ð2:17Þ
Gibbs demonstrated that these two criteria were equivalent in the following way. It is possible to increase (decrease) both the energy and entropy of the system by a flow of heat into (out of) the system. If condition (2.16) is not satisfied, there must be a variation in the state of the system for which dS > 0 and dU ¼ 0, and therefore, by reducing both the energy and entropy of the system in this varied state, a new state would result that could have been obtained from the original state with the variations dS ¼ 0 and dU < 0, in violation of condition (2.17). Conversely, if condition (2.17) is not satisfied, there must be a variation in state of the system for which dU < 0 and dS ¼ 0, and then by increasing both the energy and entropy of the system in this varied state, a new state would result that could have been obtained from the original one with the variations dU ¼ 0 and dS > 0, in violation of condition (2.16).
21
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 56.
GENERALIZED THERMODYNAMICS OF SURFACES
9
Gibbs employed the second criterion, which can be taken as a minimum energy principle, as the cornerstone of his theory of equilibrium. The minimum energy criterion for the equilibrium of an isolated system can be written as dU 0
ð2:18Þ
subject to what Gibbs called the ‘‘equations of condition’’ which express the constraints of holding the total entropy, volume, and number of moles of each component of the system fixed: dS ¼ 0
ð2:19Þ
dV ¼ 0
ð2:20Þ
dni ¼ 0:
ð2:21Þ
For a multicomponent system, there will be a separate condition of equilibrium, Eq. (2.21), for each component; this condition also assumes no chemical reactions take place. Consider an isolated system composed of two different bulk fluid phases, denoted as a and b, where the term phase will refer to a subsystem that is physically and chemically uniform. Superscripts a and b on thermodynamic quantities will denote values of those quantities for the respective phases. Gibbs’ minimum energy criterion will now be used to obtain relationships between the intensive variables of the two phases at equilibrium. Since the system composed of two fluid phases a and b is isolated from its surroundings, the only thermal and mechanical interactions the phases can have are with each other. In addition, the phases can interact chemically by an exchange of matter. The fundamental equations for a and b are X mai dnai ð2:22Þ dU a ¼ T a dSa Pa dV a þ i
dUb ¼ T b dSb Pb dV b þ
X
mbi dnbi :
ð2:23Þ
i
The variation in the total internal energy of the system is given by the sum of Eqs. (2.22) and (2.23): dU ¼ dU a þ dU b X X ¼ T a dSa þ T b dSb Pa dV a Pb dV b mai d nai þ mai dnai : i
ð2:24Þ
i
It is recalled that in Eq. (2.24) the terms of the form TdS denote reversible heat flows, the terms of the form PdV represent reversible mechanical works, and the terms of the form midni denote reversible chemical works, where the term
10
R. C. CAMMARATA
reversible means it occurs quasi-statically and without hysteresis. When Gibbs used the term reversible variation,22 he imposed a stronger condition: the variation had to be one that could be of either sign. For the two-phase fluid system under consideration, as long as both of the phases initially have finite volumes and entropies, the variations dVa, dVb, dSa, and dSb can be termed Gibbs-reversible as they can be positive or negative. Gibbs considered the possibility that at equilibrium under certain conditions, one or more components could be present in one phase but not the other, although they may be present in both phases under different equilibrium conditions. He called a component that is, for example, present in a but not in b an actual component of a but only a possible component of b. Since a variation in the amount of that component can only be negative for a and can only be positive for b it cannot be conducted in a Gibbs-reversible manner. As another example, consider a system that is initially composed of just the single phase a that experiences a variation in state involving the formation of the new phase b. Since b was not present in the initial system, none of the variations in Eq. (2.24) can be conducted in a Gibbs-reversible manner. The symbol d will be used to denote a Gibbs-reversible variation, so that the fundamental equation for a two-phase fluid system that undergoes Gibbsreversible variations can be expressed as: X dU ¼ dU a þ dU b ¼ T a dSa þ T b dSb Pa dV a Pb d V b þ mai dnai þ mbi dnbi : i
ð2:25Þ In Eq. (2.25), if a particular component i is not an actual component of a and b, then dnai ¼ dnbi ¼ 0. At equilibrium, the criterion dU 0 will apply, employing variations in the state of the phases consistent with the constraints given by Eqs. (2.19)–(2.21). Since the variations on the right-hand side of Eq. (2.25) are all Gibbs-reversible, it is possible to employ an alternate set of variations obtained by giving the original set a change in sign. This would cause a change in sign in the inequality for the internal energy so that it becomes dU 0. However, since the new variations are being applied to a system initially in equilibrium, the condition dU 0 would also have to hold. The only way dU 0 and dU 0 can be true simultaneously is if dU ¼ 0. Thus, by considering only Gibbs-reversible variations, it is possible to use the equality as a criterion for equilibrium, subject to the constraints given by the equations of condition using Gibbs-reversible variations: dU ¼ dUa þ dUb ¼ 0
22
ð2:26Þ
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 222 (footnote).
GENERALIZED THERMODYNAMICS OF SURFACES
11
dS ¼ dSa þ dSb ¼ 0
ð2:27Þ
dV ¼ dV a þ dV b ¼ 0
ð2:28Þ
dni ¼ dnai þ dnbi ¼ 0:
ð2:29Þ
Equations (2.27)–(2.29) can be rewritten as dSa ¼ dSb
ð2:30Þ
dV a ¼ dV b
ð2:31Þ
dnai ¼ dnbi :
ð2:32Þ
Substituting these into Eq. (2.25) and setting dU ¼ 0 gives the following general criterion for equilibrium: X b mi mai dnbi : ð2:33Þ 0 ¼ T b T a dSb Pb Pa dV b þ i
Equation (2.33) can be used for any set of physically meaningful variations {dSb, dVb, dnib}. Consider a system variation involving reversible heat flow between the phases resulting in a variation in the entropy of the phases, keeping Vb and nib for all of the components fixed. Equation (2.33) for this system variation simplifies to 0 ¼ (Tb – Ta) dSb resulting in the following condition for thermal equilibrium: Ta ¼ Tb:
ð2:34Þ
Now consider a system variation in which the volume Vb is varied but where and nib for all of the components are fixed. In this case Eq. (2.33) becomes 0 ¼ (Pb – Pa)dVb which yields the condition for mechanical equilibrium: Sb
Pa ¼ P b :
ð2:35Þ nbi
for a particular component i is Finally, consider a system variation where varied, but Sb, Vb, and nbi for all of the components j 6¼ i are fixed. Then Eq. (2.33) for this variation is 0 ¼ mbi mai dnbi leading to the condition for chemical equilibrium: mai ¼ mbi :
ð2:36Þ
This condition will apply to any component that is an actual component in both phases. For a component j that is, for example, an actual component in a but not in b, a variation in the number of moles of that component, dnbi , is not
12
R. C. CAMMARATA
Gibbs-reversible. Thus the equilibrium criterion dU 0 would have to be used, b which, when combined with the constraints that Sb, Vb, and nk for the compob b nents k 6¼ j are held fixed results in dU ¼ mj maj dnj 0, leading to the condition maj mbj :
ð2:37Þ
III. Thermodynamics of Bulk Solids
5. I NTRODUCTION Solids differ thermodynamically from fluids in certain important respects. Unlike a fluid, a solid can maintain a nonhydrostatic stress state in equilibrium. Therefore, for a solid phase b, the mechanical work term PbdVb that can be used for the special case of a solid constrained to maintain a hydrostatic pressure is replaced when the solid can experience a nonhydrostatic stress state with an elastic work term of the form V b Tijb dXijb , where Xijb is a tensor that describes a general deformation state, Tijb is the corresponding stress tensor, and summation over the repeated indices is implied.23,24 A particular choice for the deformation tensor is the small strain tensor ebij referred to the actual state of the solid that can be used along with the Cauchy stress tensor sbij to express the mechanical work as V b sbij debij (see Appendix A). In many elasticity problems, it is sometimes convenient to use a reference state, such as the stress-free state, different from the actual state whose equilibrium behavior is being analyzed. In the most general case, large strains may be needed to relate the reference state to the current state. Nevertheless, use of the small strain tensor to describe variations in the equilibrium state involves no loss of generality since no restriction is being imposed on the magnitude of the deformations from the reference state to the equilibrium configuration if it is recognized that all thermodynamics variables refer to the actual equilibrium state of the solid. It is noted that at equilibrium the stress and strain tensors may be nonuniform; i.e., there may be stress and strain fields within the solid. Other important differences between solids and fluids concern the atomic or molecular arrangements within the phases and the related issues of the mechanism by which matter exchange takes place and how that affects the physical meaning of the chemical potential. Gibbs assumed that a variation in the amount of
23
L. E. Malvern, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Englewood Cliffs, NJ (1969). 24 P. W. Voorhees and W. C. Johnson, Solid State Phy. 59, 2 (2004).
GENERALIZED THERMODYNAMICS OF SURFACES
13
a particular component i in a fluid could be conducted independently of the variations of the other components. Implicit in this assumption is that when an interior atom or molecule of a particular component is removed from the interior of a fluid phase during a variation in the amount of that component that keeps the volume of the phase fixed, the resulting atomic-scale or molecular-scale ‘‘void’’ completely relaxes so that its volume becomes distributed as part of the local ‘‘free volume.’’ In a crystalline solid, atoms reside on lattice sites and the total number of lattice sites in a crystal is an additional extensive variable that characterizes the state of the system. Removal of an atom in the interior of a crystal keeping the number of lattice sites fixed will create a void in the form of a lattice vacancy. In the case of an amorphous material, although there is no long range structural order, there can be significant short range order including coordination of atomic clusters, and removal of an atom may result in an incompletely relaxed ‘‘void.’’ Such an amorphous material can be considered as displaying solid-like behavior with regard to the thermodynamics of chemical work. Crystal vacancies (or atomic-size ‘‘voids’’ in amorphous solids) can be treated as a separate component of zero mass. As a result, a system variation that involves removal of interior atoms of a particular component in a crystal, keeping the number of lattice sites fixed, cannot be conducted independently of the variations of the other components as the removal increases the amount of vacancies. When Gibbs was developing his thermodynamics of equilibrium, the molecular hypothesis for describing matter was still controversial and the understanding of atomic arrangements in crystals had not been completely formulated. Gibbs rarely appealed to atoms or molecules in his papers on thermodynamics, and in fact when he expressed the chemical work to vary the amount of a component in the form of midni, he referred to the variations dni in terms of the mass of the component rather than in terms of the number of moles or molecules.16 Therefore his approach was a completely continuum one that did not allow for ‘‘voids,’’ and when invoking a system variation that involved matter transfer to or from a solid he did not consider it in terms of a transfer of atoms but rather as resulting from accretion or dissolution of a layer of material. Instead of removing atoms from a crystal and creating lattice vacancies, another way to exchange matter between a crystal and a fluid that keeps the total number of lattice sites of the crystal fixed is to have a direct exchange of atoms, where every interior atom of component j that is removed from the crystal is replaced with an atom supplied by the fluid of another component i. Larche´ and Cahn25,26 have proposed that the contribution to the change in internal energy of the crystal associated with this process can be expressed as Mijb dnbij , where dnbij is
25 26
F. C. Larche´ and J. W. Cahn, Acta Metall. 26, 1579 (1978). F. C. Larche´ and J. W. Cahn, Acta Metall. 33, 120 (1985).
14
R. C. CAMMARATA
the number of moles of j removed and replaced with i, and Mijb is called the ‘‘diffusion potential’’ associated with this exchange. Consider a crystal b in chemical equilibrium with a fluid a. A condition for chemical equilibrium between the two phases is obtained by setting equal to zero the internal energy variation associated with a Gibbs-reversible transferring of atoms of component j, originally residing in lattice sites of the solid, to the fluid and replacing them with atoms i from the fluid, keeping the entropy, volume, and number of moles of all the other components for both phases fixed. This can be expressed as dU ¼ mai dnai þ maj dnaj þ Mijb dnbij ¼ 0:
ð3:1Þ
It is assumed that i and j are both actual components in both phases at equilibrium (otherwise dni ¼ dnj ¼ dnij ¼ 0). Since dnaj ¼ dnai ¼ dnbij ; ð3:2Þ mai maj Mijb dnai ¼ 0 so that the condition of chemical equilibrium is mai maj ¼ Mijb :
ð3:3Þ
If component j is taken as a crystal vacancy denoted as v, then mai ¼ Mivb ;
ð3:4Þ
since mav ¼ 0. This is because vacancies absorbed into a fluid completely relax to become part of the free volume so that a fluid acts as a perfect source or sink for vacancies. Although a diffusion potential approach has been applied only to crystals, it would appear that it could also be used for amorphous solids where removal of atoms creates partially relaxed ‘‘voids.’’ It has been suggested that Mivb can be taken as the ‘‘chemical potential’’ for the components i in a crystal when the number of lattice sites is conserved.27 An advantage of this choice is that the condition of chemical equilibrium as expressed in Eq. (3.4) retains a formal similarity to the condition of chemical equilibrium given in Eq. (2.36). One practical difficulty with this approach is that the concentration of vacancies is generally so small that it is difficult to measure their thermodynamic properties, making them inconvenient as a reference component for the diffusion potential.26 In addition, when treating small crystals at low temperatures, it is conceivable that there are no vacancies present at equilibrium24 and therefore variations in the number of vacancies would not be Gibbsreversible. Finally, exclusive use of chemical work terms involving diffusion
27
W. W. Mullins and R. F. Sekerka, J. Chem. Phys. 82, 5192 (1985).
GENERALIZED THERMODYNAMICS OF SURFACES
15
potentials, whether or not one of the components involved in the atomic exchange is a vacancy, keeps the total number of lattice sites fixed and therefore is not convenient when considering processes where the number of lattice sites is changed, such as accretion or dissolution of a crystal in a fluid or nucleation of a crystal during solidification. In his treatment of a solid, Gibbs considered it to be either a single element or a stoichiometric compound that could be taken as a single component system.28 When Gibbs analyzed the chemical equilibrium between a solid and a fluid, variations in the amount of the components of the solid occurred only by dissolution or accretion at the solid–fluid boundary and therefore would have effectively involved a change in the number of lattice sites if the solid was a crystal. When he employed a variation involving the addition of material to a solid that had a different composition, he treated the new material as a new phase. In addition to accretion or dissolution, Gibbs allowed a solid to change its volume by elastic deformation. In all cases, the solid maintained macroscopic connectivity. The principal component of the solid was not allowed to engage in long range diffusion although other components present in a contacting fluid could diffuse into the solid and elastically distort it. Gibbs referred to these secondary components as ‘‘mobile’’ components within the solid.26,27,29 Polymer fibers that can absorb solvent molecules,26 a network silicate glass with mobile modifier ions,26 and a solid solution composed of a crystal lattice with sites mostly occupied by one component and with mobile dilute impurities occupying interstitial sites26,30 have been cited as examples of Gibbs-model solids. 6. B ULK S OLID –F LUID E QUILIBRIUM Consider an isolated system composed of a bulk solid phase b under a hydrostatic pressure that is in equilibrium with a fluid a. When employing the minimum energy criterion to this system, the same fundamental equation as that for the fluid–fluid case, Eq. (2.25), can be used. In addition, the same equations of condition, Eqs. (2.27)–(2.29), will apply, leading to the constraints given by Eqs. (2.30)–(2.32) The analyses used to arrive at the conditions for thermal and mechanical equilibrium for the fluid–fluid case can also be employed for the solid–fluid case, leading to the same condition for thermal equilibrium, T a ¼ T b, and the same condition for mechanical equilibrium, Pa ¼ Pb. As was discussed in the previous section, the conditions for chemical equilibrium for a crystal–fluid system can be expressed by Eqs. (3.3) and (3.4) using diffusion
28
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 193. J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 215. 30 J. C. M. Li, R. A. Oriani, and L. S. Darken, Z. Phys. Chem. Neue Folge 49, 271 (1966). 29
16
R. C. CAMMARATA
potentials. Attention will now be given to the condition of chemical equilibrium associated with the stability of the solid with respect to accretion or dissolution at the solid–fluid interface. This will lead to a stronger condition with regard to local chemical equilibrium at the boundary. When Gibbs considered the accretion or dissolution of a single component solid at a solid–fluid boundary involving a variation in the number of moles dnb1 , he expressed the chemical work as mb1 dnb1 , where mb1 was identified as the chemical potential of the solid component. In contrast b , which can be interpreted as the ‘‘chemical potential’’ of component 1 to M1v keeping the number of lattice sites fixed,27 Gibbs’ chemical potential for a crystal is associated with a variation that does not keep the number of lattice sites fixed. Gibbs’ approach is easily extended for a multicomponent crystal composed of substitutional and dilute interstitial components. One way to define an interstitial component is as one whose atoms reside on a secondary sublattice that has mostly vacant sites, in contrast to a substitutional component whose atoms reside on the primary sublattice where the sites are mostly occupied.31 Consider an isolated system in equilibrium composed of a multicomponent fluid a and a crystal b that is under a hydrostatic pressure. Suppose a Gibbs-reversible variation occurs involving, in addition to heat exchange, dissolution of a layer of the crystal at the crystal–fluid boundary corresponding to the removal of a monolayer of the primary lattice sites plus adjacent sites on the secondary lattice containing the interstitial components. Let N denote the number of primary lattice sites removed from the crystal. It is assumed that almost all of the atoms in the monolayer of the crystal at the initial crystal–fluid boundary were actual components of both phases, and therefore those components could be varied in a Gibbsreversible manner. If the monolayer had a relatively small number of atoms corresponding to possible but not actual components of the fluid, then those atoms will remain at the newly created surface monolayer of the solid at the boundary after dissolution, occupying lattice sites vacated by atoms of components that could be Gibbs-reversibly varied and have been dissolved to make the reduction in the total number of primary lattice sites equal to N. The same fundamental equation used for the fluid–fluid system, Eq. (2.25), can then be used to express the variation in the internal energy for this process. Setting dU in Eq. (2.25) equal to zero and substituting the constraints given by Eqs. (2.30)– (2.32) leads to Eq. (2.33), and substituting the conditions for thermal equilibrium Ta ¼ Tb and for mechanical equilibrium, Pa ¼ Pb leads to X b mi mai dnbi : ð3:5Þ 0¼ i
31
M. Hillert, quoted in Ref. [26].
GENERALIZED THERMODYNAMICS OF SURFACES
17
Unlike for the case of a completely fluid system, the variations dnbi cannot all be conducted independently if vacancies are taken as a component since the number of lattice sites is being varied and also because dnbi for each component will be exactly or approximately proportional to the mole fraction of that component in the solid. To simplify Eq. (3.5), consideration is given to the condition for chemical equilibrium as given in Eq. (3.3). Assuming that chemical potentials for components in the solid are defined at the solid–fluid boundary, then the diffusion potential Mijb of the crystal can be related to these chemical potentials by Mijb ¼ mbi mbj
ð3:6Þ
mai maj ¼ mbi mbj
ð3:7Þ
so that Eq. (3.3) can rewritten as
when i and j are actual components in both a and b. Therefore, mbi mai ¼ mbj maj , leading to the conclusion that mbi mai is the same for all components i that are actual components in both phases. Thus, Eq. (3.5) can be rewritten as X ð3:8Þ dni ¼ 0 mbi mai resulting in the condition for chemical equilibrium that for all components i that are actual components in a and b, mai ¼ mbi :
ð3:9Þ
This condition is the same as for the completely fluid system, Eq. (2.36) and applies to both substitutional and interstitial components. A comparison of Eqs. (3.4) and (3.9) shows that Mivb ¼ mbi , and therefore mbv ¼ 0. Thus, the two definitions for the chemical potential, the diffusion potential Mivb (associated with variations that keep the number of lattice sites fixed) and Gibbs’ chemical potential mbi (associated with variations that change the number of lattice sites), will be numerically equal for bulk solids. It will be shown later that this will no longer be the case for a small crystal immersed in a fluid when capillary effects are important. In the above discussion, it was only necessary to assume that the chemical potentials mbi for components in the solid were defined at the solid–fluid boundary. It has been argued that for a hydrostatically stressed solid, mbi is defined and is uniform throughout the solid26,28 so that the expression Mijb ¼ mbi mbj would also be valid throughout the solid. However, this will not hold if the solid is nonhydrostatically stressed. Gibbs illustrated this by considering the case of a
18
R. C. CAMMARATA
single component solid in the shape of a rectangular parallelepiped in contact with three separate multicomponent fluids27,32,33 (see Figure III.1). The solid and fluids are assumed to be in thermal equilibrium. Let the component of the solid be denoted as 1, and consistent with Gibbs’ model for a solid, it is constrained not to engage in long range diffusion. Each fluid makes contact with the solid at two opposite faces. The fluids have different pressures so that the solid that is in mechanical equilibrium with these fluids is under a uniform but nonhydrostatic stress state with principal stresses given by the negative of the three fluid pressures. Each fluid contains two or more components including component 1. The condition for local chemical equilibrium at a particular boundary is obtained in the same manner as that used to derive Eq. (3.9) with i taken as component 1 and with Pb denoting the negative of the principal stress acting on the solid at that boundary. Since the chemical potential for component 1 can be different in each fluid, the chemical potential in the solid can be different at the different set of boundaries. Therefore the chemical potential of the solid in equilibrium is not uniform and is in fact not well defined for the interior of the nonhydrostatically stressed solid.26,34 The constraint that component 1 cannot
Fluid 2 of pressure −sy
Fluid 1 of pressure −sx
y
z
Solid with principal stresses s x, s y, s z
Fluid 1 of pressure −sx
Fluid 2 of pressure −sy x
FIG. III.1. Schematic diagram of a solid in the shape of a rectangular parallelepiped in contact with separate fluids. Each fluid makes contact with the solid at two opposite faces.
32
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 196. J. W. Cahn, Acta Metall. 28, 1333 (1980). 34 A. I. Rusanov, Surf. Sci. Rep. 58, 111 (2005). Rusanov has proposed that a ‘‘chemical potential tensor’’ can be employed that is well defined within the interior of a nonhydrostatically stressed sold. 33
19
GENERALIZED THERMODYNAMICS OF SURFACES
engage in long range diffusion within the solid makes each boundary chemically isolated from the others with respect to that component. Allowing component 1 to engage in solid state diffusion would result in chemical ‘‘communication’’ among the boundaries for this component, and the solid would creep in response to the nonhydrostatic stress state. IV. Availability and Free Energy
7. A VAILABILITY Consider a system f that is immersed in and interacts thermally and mechanically with a much larger fluid phase R (see Figure IV.1a). For the present f will be considered a closed system, meaning that it is constrained not to exchange matter with R. The composite system of f and R can be taken as an isolated system. It is supposed that R is much larger and more massive than f, so that any process involving thermal and/or mechanical interactions between them results in negligible changes in the temperature TR and pressure PR of R so that they can be considered invariant. It is common to refer to R as a thermal and mechanical reservoir for the system f. Suppose a process occurs involving a flow of heat q from R into f as well as a performance of work w that changes the volume of f by an amount DVf so that w ¼ PRDVf. Substituting this expression for w into the first law, Eq. (2.1), and rearranging gives q ¼ DU f þ PR DV f :
ð4:1Þ
According to the Clausius relation (2.7) DSf q=T R :
(a)
ð4:2Þ
(b) R
R f
f
W*
FIG. IV.1. System f immersed in a reservoir R. (a) f and R form an isolated composite system. (b) f can interact with the surroundings outside of R through a performance of work w*.
20
R. C. CAMMARATA
Substituting Eq. (4.2) into the inequality (4.1) and rearranging gives DU f T R DSf þ PR DV f 0: ‘‘availability’’5,7,8,10,12,35,36
Let the (also called the system f immersed in the reservoir R be defined as Z f U f T R S f þ PR V f :
ð4:3Þ ‘‘exergy’’37–40)
of a ð4:4Þ
Since the reservoir temperature TR and pressure PR are constant, the change in availability for the process is DZf ¼ DU f T R DSf þ PR DV f ;
ð4:5Þ
and comparing Eq. (4.5) with (4.3) it is seen that DZ f 0:
ð4:6Þ
The condition (4.6) is equivalent to the law of entropy increase for the composite isolated system composed of the closed system f immersed in the reservoir R. It can be interpreted as stating that the availability of f cannot increase, and that as the system evolves to an equilibrium state the availability will decrease until it is minimized at equilibrium. Consideration is now given to the case where the composite system containing f and R has the same thermal and mechanical interactions between them as above, but where there is in additional another performance of work w* resulting from an interaction between f and the surroundings outside the composite system (see Figure IV.1b). The composite system f and R is now only thermally isolated from its surroundings. The work w* is often referred to as the ‘‘useful work’’5,7,8,10,12,35–37 that can be extracted from the system f. The first law for the reservoir is DU R ¼ q þ PR DV f :
35
ð4:7Þ
J. H. Keenan, Brit. J. Appl. Phys. 2, 183 (1951). K. C. Rolle, Thermodynamics and Heat Power, Pearson Prentice Hall, Upper Saddle River, NJ (2005). 37 J. R. Howell and R. O. Buckius, Fundamentals of Engineering Thermodynamics, McGraw-Hill, New York (1993). 38 J. E. Ahern, The Exergy Method of Energy Systems Analysis, Wiley, New York (1980). 39 A. Bejan, Entropy Generation Through Heat and Fluid Flow, Wiley-Interscience, New York (1982). 40 N. Sato, Chemical Exergy and Energy, Elsevier, Amsterdam (2004). 36
GENERALIZED THERMODYNAMICS OF SURFACES
21
The first law for the composite system f and R supplying work w* to its surroundings can be expressed as DU f ¼ DU R w:
ð4:8Þ
Substituting Eq. (4.7) into Eq. (4.8) gives DU f ¼ q PR DV f w:
ð4:9Þ
Since the composite system of f plus R is thermally isolated from its surroundings, the inequality (4.1) still holds and substituting Eq. (4.9) into that inequality and rearranging yields ð4:10Þ w DU f T R DSf þ PR DV f : Substituting Eq. (4.5) into Eq. (4.10) gives w DZ f:
ð4:11Þ
DZf
is negative, and the maximum useful work It is seen that for positive w*, w* that can be extracted from f is equal to |DZf|. In the discussion given above, the only interactions between f and R involved heat flow and mechanical work resulting in a change of volume of f. Consideration is now given to the case where f is an open system, meaning that in addition to the thermal and mechanical interactions it is allowed to exchange matter with R, so that R is a thermal, mechanical, and material reservoir for f. As before R is taken to be much larger and more massive than f so that any interaction between it and f results in a negligible change not only in the reservoir temperature TR and pressure PR but also in the chemical potentials mRi . In addition to the mechanical work PRDVf, the reservoir will perform chemical work equal to Si mRi Dnfi . The derivations as given above where f was taken as a closed system can be modified for f as an open system by including the chemical work term Si mRi dnfi along with the mechanical work term PRDVf. For the case where the composite system of f and R is completely isolated from its surroundings as in Figure IV.1a, instead of the inequailty (4.3) the result is X mRi Dnfi 0: ð4:12Þ DU f T R DSf þ PR DV f i
Defining the availability for an open system f immersed in a reservoir R as X mRj nfj ð4:13Þ Bf U f T R Sf þ PR V f j
it is seen that since the temperature, pressure, and chemical potentials of the reservoir are constant, the inequality (4.12) can be rewritten as DBf 0:
ð4:14Þ
22
R. C. CAMMARATA
When the composite system f and R is only thermally isolated from its surroundings as in Figure IV.1b, in addition to the mechanical work PRDVf and chemical work Si mRi dnfi associated with interactions between f and R there is an additional amount of work w* that can be extracted from f equal to ! X f mRj Dnj ¼ DBf ð4:15Þ w* DU f T R DSf þ PR DV f j
and the maximum useful work w* is equal to |DBf|. Zf defined in Eq. (4.4) will be referred to as the closed system availability and f B defined in Eq. (4.13) will be referred to as the open system availability. These functions were first given by Gibbs41,42 in connection with the ‘‘stability’’ of a phase with respect to the formation of a new phase and have generally been applied to analyze the energetics of nonequilibrium flow processes.5,35–40 As discussed below, these functions can also be usefully applied to describe the equilibrium thermodynamics of surfaces.43 8. F REE E NERGY The differential form of the closed system availability for a closed system f immersed in a reservoir R is dZf ¼ dU f T R dSf þ PR dV f :
ð4:16Þ
Suppose the system is enclosed in rigid walls so that the volume Vf remains fixed, and that as f undergoes a process that causes it to evolve from one equilibrium state to another, f and R remain in thermal equilibrium. The reservoir keeps the system temperature Tf fixed at the value of the reservoir temperature TR. For these constraints, Eq. (4.16) can be rewritten as dZ f ¼ dU f T f dSf :
ð4:17Þ
The Helmholtz free energy for the system f can be defined as Ff U f T f Sf :
ð4:18Þ
The full differential of the Helmholtz free energy can be written as dFf dU f T f dSf Sf dT f :
41
ð4:19Þ
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 40. J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 105. 43 R. C. Cammarata, Phil. Mag. 88, 927 (2008). 42
GENERALIZED THERMODYNAMICS OF SURFACES
23
Since the temperature of the system f is fixed, dFf dU f T f dSf :
ð4:20Þ
Comparing Eqs. (4.17) with (4.20) shows that dFf ¼ dZf
ð4:21Þ
DFf ¼ DZ f :
ð4:22Þ
which can be integrated to give
Thus, for a closed system f of fixed volume and fixed temperature equal to the temperature of the reservoir R, the change in Helmholtz free energy of f is equal to the change in availability, and minimizing Ff is equivalent to minimizing the availability and therefore also equivalent to the law of entropy increase for the composite isolated system of f and R. Consideration is now given to a system f that is constrained to remain in thermal and mechanical equilibrium with the reservoir so that Pf ¼ PR as well as Tf ¼ TR. For these constraints, Eq. (4.16) can be rewritten as dZf ¼ dU f T f dSf þ Pf dV f :
ð4:23Þ
The Gibbs free energy of the system f can be defined as G f U f T f S f þ Pf V f :
ð4:24Þ
The full differential of Gf is dGf ¼ dU f T f dSf Sf dT f þ Pf dV f þ V f dPf
ð4:25Þ
and since Tf and Pf are fixed, dGf ¼ dU f T f dSf þ Pf dV f :
ð4:26Þ
Comparing Eq. (4.26) with (4.23) shows that dGf ¼ dZf
ð4:27Þ
DGf ¼ DZ f :
ð4:28Þ
and therefore
Thus, for a closed system f of fixed pressure and temperature equal to the pressure and temperature of the reservoir R, the change in Gibbs free energy is equal to the change in availability. Furthermore, minimizing Gf is equivalent to minimizing the availability and therefore equivalent to the law of entropy increase. When dealing with a closed system that has its temperature fixed owing to it being in continual thermal equilibrium with a thermal reservoir, it is possible to obtain the equilibrium conditions by minimizing the Helmholtz free energy of the
24
R. C. CAMMARATA
system keeping the system volume fixed. Similarly, for a closed system that has its temperature and pressure fixed owing to it being in thermal and mechanical equilibrium with a reservoir, it is possible to obtain the equilibrium conditions by minimizing the Gibbs free energy of the system. Gbbs44 noted that although it is sometimes less cumbersome to use these minimum free energy criteria, the minimum internal energy criterion is the most general and often the most transparent, and Gibbs used it almost exclusively in his paper of 1876–1878. A relation that will be of use later is that for a uniform system X f f mi ni ð4:29Þ Gf ¼ i
which can be obtained by comparing Eq. (4.24) with the Euler equation, Eq. (2.14). V. Thermodynamics of Systems with Surfaces
9. I NTRODUCTION Consider a system composed of two uniform phases a and b separated by a nonuniform interfacial region, denoted as S, as shown schematically in Figure V.1a. A–A and B–B in the figure represent, respectively, boundaries separating a and S and b and S, and are placed within regions of the system that are physically and chemically homogeneous (i.e., within the uniform phases) so that there is no influence of the nonuniform interfacial region on their local thermodynamic state. There is some arbitrariness to the position of the boundaries A–A and B–B and therefore also to the thickness of S.
(a)
(b)
α A
α A
S B
s B
β
β
FIG. V.1. (a) Curved interfacial region S between two phases s and b. (b) Dividing surface s placed within the interfacial region.
44
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 90 (footnote).
25
GENERALIZED THERMODYNAMICS OF SURFACES
Consider a variation involving only reversible heat between the phases and the interfacial region. The variation in the system internal energy will be dU ¼ dU a þ dU b þ dUS ¼ T a dSa þ T b dSb þ T S dSS :
ð5:1Þ
The condition for thermal equilibrium is obtained by setting dU ¼ 0, so that T a dSa þ T b dSb þ T S dSS ¼ 0
ð5:2Þ
subject to the constraint dS ¼ dSa þ dSb þ dSS ¼ 0. The only way these can both be true simultaneously is if Ta ¼ Tb ¼ TS:
ð5:3Þ
Thus, the equality of the temperatures of the two bulk phases a and b as the condition of thermal equilibrium obtained previously, Eq. (2.34), does not change when the presence of the interface is taken into account. Since reversible heat flow can occur independently of any work interactions, the condition that a system in complete thermal equilibrium has a uniform temperature will be true in general, irrespective of other system conditions and constraints (e.g., when the system is exposed to external force fields). This result can be taken as a statement of what has been called the ‘‘zeroth law’’ of thermodynamics.7,8,10,12 Attention is now given to the particular case of a system where the phases a and b and the interfacial region S are in general multicomponent fluids. Gibbs45 assumed that it is possible to define locally the chemical potentials for the components within S. For a Gibbs-reversible variation in the state of the system, keeping the volumes and entropies of a, b and S fixed, the variation in the internal energy of the system is X mai dnai þ mbi dnbi þ mSi dnSi : ð5:4Þ dU ¼ dU a þ dU b þ dU S ¼ i
The criterion of equilibrium dU ¼ mai dnai þ mbi dnbi þ mSi dnSi ¼ 0 subject to the constraint dni ¼ dnai þ
dnbi
ð5:5Þ
þ dnSi ¼ 0 leads to the condition
mxi ¼ myi
ð5:6Þ
where x, y ¼ a, b, S, and i is an actual component in x and y. Thus, the condition for chemical equilibrium in bulk fluid systems, the uniformity of mi within
45
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 219.
26
R. C. CAMMARATA
regions where i is an actual component, also holds for a fluid system with a nonuniform interfacial region. Mechanical equilibrium requires that in the immediate vicinity of the boundary A–A, Pa ¼ PS, and in the vicinity of B–B, Pb ¼ PS, where PS denotes the pressure acting normal to the boundary. However there is no a priori reason to conclude that PS is the same at both boundaries or that it is uniform within the interfacial region; in fact, the pressure is neither uniform nor hydrostatic where S is nonuniform,46 so that PS will not be well defined in that region. 10. G IBBS D IVIDING S URFACE C ONSTRUCTION
FOR
F LUID I NTERFACES
For systems where the thickness of the interfacial region S is much smaller than the size of the phases a and b, Gibbs was able to finesse the problem associated with the lack of a well defined interfacial pressure PS by utilizing the device of a dividing surface9,11,12,34,43,47–54 that allowed him to retain many of the general features he developed for bulk equilibrium thermodynamics. This approach involves calculating extensive quantities for an actual system composed of two physically and/or chemically distinct phases with a relatively narrow interfacial region separating them using an artificial system where the phases are taken to be uniform up to a two-dimensional boundary s called the dividing surface (see Figure V.1B). Once the extensive quantities associated with the uniform phases, such as Ua and Ub, are calculated in this manner, ‘‘excess’’ amounts of these quantities associated with the surfaces can then be defined as Us ¼ U Ua Ub
46
ð5:7Þ
D. M. Anderson, G. B. McFadden, and A. A. Wheeler, Annu. Rev. Fluid Mech. 30, 139 (1998). J. R. Rice, in Commentary on the Scientific Writings of J. Willard Gibbs, Volume I: Thermodynamics, eds. F. G. Donnan and A. Haas, Yale University Press, New Haven (1936), p. 505. 48 C. Herring, in The Physics of Powder Metallurgy, ed. W. E. Kingston, McGraw-Hill, New York (1951). 49 C. Herring, in Structure and Properties of Solid Surfaces, ed. R. Gomer and C. S. Smith, University of Chicago, Chicago (1953), p. 5. 50 W. W. Mullins, in Metal Surfaces Structure, Energetics and Kinetics, American Society for Metals, Metals Park, OH (1963), p. 17. 51 A. Zangwill, Physics at Surfaces, Cambridge University Press, Cambridge (1988). 52 B. Widom, in Proceedings of the Gibbs Symposium, eds. D. G. Caldi and G. D. Mostow, American Mathematical Society, Providence, RI (1990), p, 73. 53 A. W. Adamson and A. P. Gast, Physical Chemistry of Surfaces, Wiley-Interscience, New York (1997). 54 J. B. Hudson, Surface Science: An Introduction, Wiley-Interscience, New York (1998). 47
GENERALIZED THERMODYNAMICS OF SURFACES
27
where U is the (actual) internal energy of the entire system. Expressions for the surface excess amounts of the other extensive properties are determined in a similar manner: S s ¼ S Sa S b
ð5:8Þ
Vs ¼ V Va Vb ¼ 0
ð5:9Þ
nsi ¼ ni nai nbi :
ð5:10Þ
It is important to note that the excess surface quantities do not refer to the actual values these quantities have in the finite-thickness interfacial region S. It is also seen from Eq. (5.9) that the ‘‘excess’’ volume of the two-dimensional dividing surface is zero so that no mechanical work terms of the form –PsdVs need be considered. In this way Gibbs avoided the need of a well-defined surface pressure. Consideration is given to how the dividing surface is to be constructed. The recipe given by Gibbs45 was ‘‘to take some point in or very near the physical surface of discontinuity [nonuniform interfacial region] and imagine a geometrical surface to pass through this point and all other points which are similarly situated with respect to conditions of the adjacent matter.’’ He further noted that although the position of the dividing surface has a certain degree of arbitrariness, directions normal to the surface are everywhere determined by this method. For a completely fluid system composed of two phases a and b separated by a dividing surface s that can in general be nonplanar and of variable area, the fundamental equation can be written as X mi dni þ 2ws ð5:11Þ dU ¼ dU a þ dU b þ dU s ¼ TdS PdV þ i
2ws
denotes the reversible surface mechanical work to form new surface where area. Gibbs noted that 2ws will in general have contributions associated with variations in the surface area and in the principal curvatures. He presented an analysis showing that if the thickness of the interfacial region is much smaller than the size of the phases, there is always a particular location for the dividing surface where the curvature terms are vanishingly small (see Appendix B). He also asserted that the assumption that the width of the interfacial region is small compared to the size of the uniform phases ensures that for any reasonable choice for the dividing surface location, the curvatures terms can be ignored compared to the area term. Hence the surface work can be taken as just the reversible work to change the surface area. Gibbs’ thermodynamics of surfaces is limited to systems where the interfacial region is not too diffuse and in this article attention will be restricted to such
28
R. C. CAMMARATA
systems. However, there are many cases of interest where this will not be valid, such as a system composed of a liquid and its vapor at a temperature T near the critical point Tc, for which the interface width grows without limit as T ! Tc. Diffuse interface methods,46 also referred to as gradient thermodynamic methods,55 have been developed to characterize this type of system. Examples include the Kortewegvan der Waals model for the liquid–vapor interface near the critical point46 and the the Cahn–Hilliard model for compositionally diffuse interfaces.11,55–57 Consider a process that creates new area of a fluid surface by increasing the number of atoms or molecules at the surface, keeping the excess surface density of the components fixed. Letting the reversible mechanical work for creating new surface area dA by this process be expressed as 2ws ¼ gdA, where g is called the surface energy, and inserting this term into Eq. (5.11) gives X dU ¼ dU a þ dU b þ dU s ¼ TdS PdV þ mi dni þ gdA: ð5:12Þ i
Substituting the fundamental equations for a and b, Eqs. (2.22) and (2.23), and the definitions of the surface excess quantities given in Eqs. (5.8)–(5.10), into Eq. (5.12) yields X msi dnsi þ gdA ð5:13Þ dUs ¼ T s dSs þ i
which can be integrated and rearranged to give X 1 msi nsi U s T s Ss g¼ A i
! ð5:14Þ
a result obtained by Gibbs58 that can be taken as the definition of g. It is now supposed that instead of creating new surface by adding or removing atoms or molecules to the interfacial region, the interfacial area can be changed by stretching a preexisting surface keeping the number of moles of the components at the interface fixed. Using the dividing surface construction, this mechanical work of stretching the surface by an amount dA be denoted as 2ws ¼ fdA,
55 J. W. Cahn, in Interfacial Segregation, ed. W. C. Johnson and J. M. Blakely, American Society for Metals, Metals Park, OH (1979), p. 3. 56 J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958). 57 J. E. Hilliard, in Phase Transformations, ed. H. I. Aaronson, ASM, Materials Park, OH (1970). 58 J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 229.
29
GENERALIZED THERMODYNAMICS OF SURFACES
where f is called the surface stress.24,33,34,43,48–51,54,55,59–63 Although the processes of creating new surface and stretching a preexisting surface are physically distinct, Gibbs58,64 asserted without explicitly proving that f ¼ g when he formulated the condition for mechanical equilibrium of a completely fluid system with a curved surface. The equality f ¼ g for a fluid surface can be shown in the following manner.43,63 Suppose an isolated system composed of two bulk fluid phases separated by a curved fluid surface undergoes a variation in which the volumes of the phases, the area of the surface, the entropies of the phases, and the excess surface entropy are varied while the number of moles of the components of the phases remain fixed and therefore the surface excess number of moles nsi also remains fixed. For this variation, which can be conducted in a Gibbs-reversible manner, the change in internal energy can be expressed as dU ¼ T a dSa þ T b dSb þ T s dSs Pa dV a Pb dV b þ f dA:
ð5:15Þ
The minimum energy criterion for equilibrium is dU ¼ 0 for dV ¼ 0 and dS ¼ 0. Since the entropy variations can be conducted independently (and in the absence) of the other types of variations, it is necessary that TadSa þ TbdSb þ TsdSs ¼ 0. Substituting this into Eq. (5.15) gives 0 ¼ Pa dV a Pb dV b þ f dA
ð5:16Þ
which can be rewritten as Pb Pa ¼ f ð c 1 þ c 2 Þ
ð5:17Þ
where c1 and c2 are the principal curvatures of the surface so that c1 þ c2 ¼ (dVb/ dA) ¼ (dVb/dA). The pressure difference Pb – Pa ¼ f(c1 þ c2) is often referred to as the Laplace pressure owing to the surface stress. Referring to Figure V.1a, since PS ¼ Pa at A–A and PS ¼ Pb at B–B, it is seen that the pressure PS is not uniform within S when the surface is curved. Gibbsian thermodynamics can reveal nothing about the nature of the pressure within the interfacial region, including the case of a planar surface. However, the Korteweg–van der Waals diffuse interface model46 shows that for both planar and curved fluid surfaces, the pressure in the interfacial region is neither uniform nor hydrostatic.
59
R. Shuttleworth, Proc. Roy Soc. A63, 444 (1950). R. C. Cammarata, Prog. Surf. Sci. 46, 1 (1994). 61 F. Spaepen, Acta Mater. 48, 31 (2000). 62 W. Haiss, Rep. Prog. Phys. 64, 591 (2001). 63 F. D. Fischer, T. Waitz, D. Vollath, and N. K. Simha, Prog. Mater. Sci. 53, 481 (2008). 64 J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 314, 60
30
R. C. CAMMARATA
Suppose the previously considered system now undergoes a variation during which heat and matter are exchanged in a Gibbs-reversible manner. The matter exchanges are performed keeping the atomic densities of the phases fixed, resulting in changes in the volumes of the phases and the area of the curved surface. The variation in internal energy is given by þ T b dSb þ T s dSs Pa dVa Pb dV b dU ¼ T a dSa X þ mai dnai þ mbi dnbi þ msi dnsi þ gdA:
ð5:18Þ
i
At equilibrium, the conditions for equilibrium are obtained by setting dU ¼ 0 subject to the constraints dS ¼ 0, dV ¼ 0, and dni ¼ 0. Once again it is noted that since the entropy variations can be conducted independently of the other variations, T adSa þ TbdSb þ T sdSs ¼ 0. Therefore X mai dnai þ mbi dnbi þ msi dnsi þ gdA: 0 ¼ Pa dV a Pb dV b þ ð5:19Þ i
As Gibbs58 alluded to in his discussion when he asserted the numerical equivalence of f and g for a completely fluid system, the key point is that the chemical potential of a particular component will be uniform for every part of the system where it is an actual component. Thus, for each component i, mai dnai þ b b s s a mi dni þ mi dni ¼ mi dni þ dnbi þ dnsi , where mi is the value of the chemical potential of component i for those regions where it is anactual component.Therefore the summation in Eq. (5.19) can be written as Si mi dnai þ dnbi þ dnsi and subb stituting the constraint dnai þ dni þ dnsi ¼ 0 shows that the chemical work b a s Si mi dni þ dni þ dni ¼ 0. As a result, Eq. (5.17) can be simplified to 0 ¼ Pa dV a Pb dV b þ gdA:
ð5:20Þ
Comparison of Eq. (5.20) with Eq. (5.17) shows that for a fluid surface, f ¼ g, and therefore the Laplace pressure can also be expressed as Pb – Pa ¼ g(c1 þ c2). 11. G IBBS A DSORPTION E QUATION Gibbs showed that taking the full differential of Eq. (5.14) and then substituting Eq. (5.13) leads to the following expression58: X Adg ¼ Ss dT s nsi dmsi : ð5:21Þ i
Equation (5.21) is the called the Gibbs adsorption equation43,48–51,54,55 and can be considered the analogue of the Gibbs–Duhem equation for a fluid surface that couples changes in the surface intensive variables. In particular it can be used to determine how the surface energy changes in response to surface segregation and adsorption.
31
GENERALIZED THERMODYNAMICS OF SURFACES
12. S TABILITY
OF A
S URFACE
IN A
F LUID S YSTEM
Attention is now given to the stability of a fluid surface s separating two fluid phases a and b that can thermally, mechanically, and chemically interact. One condition of stability of the surface is that the surface energy g has to be greater than zero, as otherwise it would be thermodynamically favorable for b to break up into finer droplets to increase the surface area. Also, in the absence of external fields, it is expected that the surface would be spherical as this minimizes the surface area for a given volume of the phase b. Consider a system composed of two fluid phases a and b where b is a sphere of radius r and let a be much larger and more massive than b so that it acts as a reservoir for b. For a system to be in stable equilibrium, the minimum energy principle requires that in addition to the criterion dU ¼ 0 at fixed S, V, and ni, it is also necessary that d2U > 0. For a Gibbs-reversible variation in the volumes of the phases and the area of the surface, keeping b spherical and keeping the entropies and number of moles of the components in each phase and the excess number of moles of the surface fixed, the variation in internal energy is dU ¼ PbdVb – PadVa þ fdA, and since dVb ¼ rdA/2 ¼ dVa, this leads to 2f dU ¼ Pb Pa þ : ð5:22Þ dV b r Setting (dU/dVb) ¼ 0 results in Eq. (5.17), where c1 þ c2 ¼ 2/r for a sphere, and taking (d2U/dVb2) > 0 as a condition for stable equilibrium leads to the inequality 0 > r
dPb b df P Pa þ 2 : dr dr
ð5:23Þ
Since f ¼ g, this condition can also be expressed as 0 > r
dPb b dg P Pa þ 2 dr dr
ð5:24Þ
which is the way Gibbs expressed the stability condition.65 Since g > 0, the Laplace pressure Pb – Pa ¼ 2g/r > 0. Gibbs noted that if the actual components of b and of the surface are also actual components of a fluid reservoir a, their chemical potentials are then fixed. According to the Gibbs–Duhem equation for b the pressure Pb cannot vary and according to the Gibbs adsorption equation the
65
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 242.
32
R. C. CAMMARATA
surface energy g cannot vary, so that the above inequality does not hold and the system is unstable to the dissolution and/or growth of b. However, if b or the surface contain a component (call it component 2) that is not an actual component of a, then the condition of stable equilibrium can be expressed as b dP df dm2 Pb Pa þ ð5:25Þ 0 > r dm2 dm2 dr and the equilibrium may be stable. Gibbs gave this criterion in terms of g rather than f and cited the example of a droplet of water that in its vapor is unstable but may be made stable by the addition of a small amount of salt.66-67 13. M ECHANICAL E QUILIBRIUM B ETWEEN I NCLUDING C APILLARY E FFECTS
A
S OLID
AND
F LUID
Consideration is now given to a system composed of a finite-size solid b in equilibrium with a large fluid a using a Gibbs dividing surface as shown in Figure V.1b. Both a and b can be multicomponent phases. The zeroth law can be invoked to assert the thermal equilibrium condition T a ¼ Tb ¼ T s. Suppose this system undergoes a variation in state in which the volumes of the phases, the area of the surface, the entropies of the phases, and the excess surface entropy are Gibbs-reversibly varied while the number of moles of the components of the phases and the excess number of moles of the surface are constrained to remain fixed. For this variation the change in internal energy can be expressed as Eq. (5.15), where f is the surface stress of the solid–fluid surface. As discussed by Gibbs58 and others,24,33,34,43,48–51,54,55,59–63 in contrast to the case of the completely fluid surface where the surface stress is a scalar, the surface stress for a crystalline solid–fluid interface is a tensor property and therefore it cannot be asserted that the surface stress is numerically equal to the surface energy. For a solid surface with a threefold or higher rotational symmetry, the surface stress is isotropic and can be taken as a scalar, but in general it is expected that it will be different in value from the surface energy. For the case of an isotropic surface so that the surface stress is a scalar, the analysis to obtain the condition for mechanical equilibrium between the solid and fluid phases proceeds in the same manner as for the completely fluid system, starting with Eq. (5.15) and leading to Eq. (5.17), Pb – Pa ¼ f (c1 þ c2). As for a
66 67
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 367. J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 317.
GENERALIZED THERMODYNAMICS OF SURFACES
33
completely fluid interface, there is a Laplace pressure in the solid–fluid case. One major difference between the two types of interfaces is that the solid surface stress f can, in principle, be negative as well as positive (see Ref. [43]), so that for c1 þ c2 > 0, the Laplace pressure can be negative, while for a fluid surface for which f ¼ g, the Laplace pressure must be positive owing to the stability condition g > 0. In addition, only Eq. (5.23) and not (5.24) can be used as a surface stability criterion. 14. G IBBS ’ T REATMENT OF C APILLARY E FFECTS I NVOLVING A S INGLE C OMPONENT S OLID When considering the condition of chemical equilibrium of a finite-size solid phase b immersed in a fluid phase a, Gibbs64 treated b as a solid containing a single immobile component with a defined and single-valued chemical potential, at least near the solid–fluid interface. Let this component be denoted as component 1 and assume that it is an actual component in the fluid phase. Although Gibbs constrained the solid not to contain any components of the mobile kind, the fluid phase a and surface s could be composed of multiple components. Without explanation he placed the dividing surface position so that there was no surface excess of component 1; that is, the dividing surface was located such that ns1 ¼ 0 (see Figure V.2). In addition Gibbs asserted in a footnote, without elaboration, that for components j 6¼ 1 that are actual components in both a and s, maj ¼ msj . Gibbs then defined the surface energy as ð5:26Þ g U s T s Ss ms2 ns2 ms3 ns3 . . . =A: This definition was consistent with Eq. (5.14) since ns1 ¼ 0:
Density of component 1
Dividing surface β
α
Spatial coordinate FIG. V.2. Dividing surface location where the excess amount of component 1 is zero. The areas enclosed by the dashed and solid lines within the interfacial region are equal.
34
R. C. CAMMARATA
Starting with these definitions and constraints, Gibbs proceeded to show that at equilibrium the chemical potential for component 1 has a value different in the fluid phase a from what it is in the solid phase b unless f ¼ g or c1 þ c2 ¼ 0. Although Gibbs referred multiple times in his text to the surface stress of a solid– fluid interface, which he called the true surface tension of the solid, he never used it in any equations. As a result his demonstration that in general for a system with a curved interface mb1 6¼ ma1 is less straightforward than his other derivations and somewhat obscured by the fact that the result is briefly stated in words rather than given in equation form. An alternate but more direct derivation of the condition of chemical equilibrium using equations that explicitly involve the surface stress will now be given. Consider a variation in the system involving the dissolution or accretion of a layer of the solid in a Gibbs-reversible manner. The variation in the internal energy of the system is given by dU ¼ dU a þ dU b þ dU s X ¼ T a dSa Pa dV a þ ma1 dna1 þ maj dnaj þ T b dSb Pb dV b þ mb1 dnb1 þ dUs : j>2
ð5:27Þ The equilibrium conditions are obtained by setting dU ¼ 0 subject to the constraints dS ¼ 0, dV ¼ 0, and dni ¼ 0. These constraints can be rewritten as dSa ¼ dSb dSs
ð5:28Þ
dV a ¼ dV b
ð5:29Þ
dnai ¼ dnbi dnsi
ð5:30Þ
Equation (5.30) implies dna1 ¼ dnb1 since the location of the dividing surface was chosen so that dns1 ¼ 0. Also, dnaj ¼ dnsj for a component j 6¼ 1 since it is not an actual component of b. Substituting these relations as well as Eq. (5.28) and (5.29) into Eq. (5.27), setting dU ¼ 0, and rearranging leads to 0 ¼ T b T a dSb Pb Pa dV b þ mb1 ma1 dnb1 þ
(
dU T dS s
a
s
X
)
maj dnsj
:
j>2
ð5:31Þ The first term on the right-hand side of Eq. (5.31) vanishes when the condition of thermal equilibrium T a ¼ Tb is substituted. The second term contains the Laplace pressure Pb Pa ¼ f(c1 þ c2). For the terms in braces, the condition of thermal equilibrium allows Tb to be replaced by T s, and Gibbs’ assertion that
35
GENERALIZED THERMODYNAMICS OF SURFACES
maj ¼ msj for components j > 1 that are actual components in both a and s, as otherwise dnsj ¼ 0, allows the terms maj dnsj to be replaced by msj dnsj . Making all of these substitutions into Eq. (5.31) yields ( ) X b b msj dnsj : 0 ¼ ½ f ðc1 þ c2 ÞdV b þ m1 ma1 dn1 þ dU s T s dSs j>2
ð5:32Þ Using the definition of the surface energy g given in Eq. (5.26), the term in the braces on the right-hand side of Eq. (5.32) can be written as gdA ¼ g (c1 þ c2) dVb. Substituting this into Eq. (5.32) and rearranging leads to the condition of chemical equilibrium for component 1 mb1 ma1 ¼ ½ð f gÞðc1 þ c2 ÞOb ;
ð5:33Þ
=nb1
where O V is the molar volume of b. It is seen from Eq. (5.33) that the chemical potential for component 1 is different in a and b at equilibrium unless f ¼ g or c1 þ c2 ¼ 0. Since b is a single component solid, the Euler equation can be used to express the chemical potential for the solid as mb1 ¼ Ub T b Sb þ Pb V b Þ=nb1 . Combining this with the mechanical equilibrium condition Pb ¼ Pa þ f(c1 þ c2) gives ð5:34Þ mb1 ¼ U b T b Sb þ ½Pa þ f ðc1 þ c2 ÞV b =nb1 b
b
and substituting this into Eq. (5.33) yields the following expression for the chemical potential of component 1 in the fluid a: ð5:35Þ ma1 ¼ U b T b Sb þ ½Pa þ gðc1 þ c2 ÞV b =nb1 : Gibbs derived Eq. (5.35) in a different manner32 in which he did not employ equations involving the surface stress, and therefore he did not explicitly present Eq. (5.33) or (5.34). He gave an extended discussion indicating that Eq. (5.35) would also express the chemical potential for component 1 in the solid, mb1 , if g equaled the surface stress f of the solid. However, he remarked at the end of that discussion that if f 6¼ g, as he previously asserted would be the case in general, then ma1 6¼ mb1 unless c1 þ c2 ¼ 0. This result shows that for a curved solid–fluid surface, the chemical potential of component 1 for the interfacial transition region of Figure V.1a is not uniform and therefore not single-valued as it equals ma1 at the a/S boundary and equals mb1 at the b/S boundary. This also leads to the conclusion that the chemical potential ms1 for the dividing surface also does not have a well-defined value. This would seem to explain why Gibbs used the dividing surface location where ns1 ¼ 0 as this removed the problematic term
36
R. C. CAMMARATA
ms1 dms1 from expressions involving a variation in component 1 as well as the term ms1 ns1 in the definition of g. Let it be supposed that the crystal is allowed to have a dilute concentration of vacancies. Since mbv ¼ mb1 ma1 where mbv is the chemical potential of vacancies in b, then according to Eq. (5.33) mbv ¼ ½ð f gÞðc1 þ c2 ÞOb
ð5:36Þ
6¼ 0 unless f ¼ g or c1 þ c2 ¼ 0. This result shows why the and therefore argument given in Section V.9 used to demonstrate that ma1 ¼ mb1 in a completely fluid system with an interface cannot be applied to a system with a nonplanar solid–fluid interface: vacancies with a nonzero chemical potential are created when atoms are transferred from the crystal to the fluid, keeping the volume and number of lattice sites of the crystal fixed. mbv
15. C HEMICAL E QUILIBRIUM FOR A M ULTICOMPONENT S OLID I NCLUDING C APILLARY E FFECTS Consider a system composed of a crystal b that is a multicomponent solid solution immersed in a multicomponent fluid a. As noted earlier, for most cases of interest the crystal components can be classified as substitutional components if they reside on the principal lattice whose sites are almost completely occupied (or possibly completely occupied in the case of a very small crystal) and as dilute interstitial components if they reside on a secondary lattice whose sites are almost completely unoccupied. As Cahn has argued,33 a variation in the number of moles of a dilute interstitial component in a crystal immersed in a fluid can be performed without changing the surface area so that no surface work is involved. Therefore the result map ¼ mbp for a component p that is an actual interstitial component in b and an actual component in a, obtained from bulk thermodynamics, also holds for a system with a finite-size crystal,33 a result first given by Herring.50 This means that mbv ¼ 0 for vacancies on the secondary sublattice containing the interstitial components. Since the above arguments would apply to the crystalline (or amorphous) parts of the interfacial transition region, then the chemical potential for vacancies (or voids) in this region, mSv , can also be taken as equal to zero, so that the general chemical equilibrium criterion would be mxp ¼ myp , where x, y ¼ a, b, s, and where p is an actual component in x and y. This result can be used to justify Gibbs’ assertion that for the secondary mobile components j in his model of a solid, maj ¼ msj . A generalization of Gibbs’ results for the case of a multicomponent solid b immersed in a multicomponent fluid a will now be presented43 by considering a variation involving the Gibbs-reversible dissolution or accretion of a monolayer
GENERALIZED THERMODYNAMICS OF SURFACES
37
solid at the surface conducted in the same manner as described in Section III.6. The variation in the internal energy of the system is given by P dU ¼ dUa þ dUb þ dU s ¼ T a dSa Pa dV a þ j maj dnaj P P ð5:37Þ þ T b dSb Pb dV b þ k mbk dnbk þ p mbp dnbp þ dU s where the summation index j denotes an actual component in a, the summation index k denotes an actual substitutional component (including vacancies) in b, and the summation index p denotes an actual dilute interstitial component in b. Setting dU ¼ 0, substituting the constraints (5.27)–(5.29), and rearranging gives P 0 ¼ T b T a dSb Pb Pa dV b þ k mbk mak dnbk n o ð5:38Þ P P þ p mbp map dnbp þ dU s T a dSs j maj dnsj : On the right-hand side of Eq. (5.38), inserting the condition of thermal equilibrium Ta ¼ Tb causes the first term to vanish. In the second term, Pb Pa can be replaced by the Laplace pressure f (c1 þ c2). With regard to the third term that involves a summation over the substitutional components, it is noted that a direct exchange of atoms between the fluid and solid can be conducted without involving any surface work so that Eq. (3.3) is still valid, and can be written as mbi mai ¼ mbj maj :
ð5:39Þ
mbk
mak is the same for all actual substitutional components k in b Therefore that are also actual components in a,so that the third term on the right-hand side of Eq. (5.37) can be rewritten as mbk mak dnb where dnb ¼ Sdnbk and can be taken as the total number of primary lattice sites that have been dissolved. The fourth term on the right-hand side involving the summation over the dilute interstitial components in b can be set equal to zero given that if p is an actual component of a and b then mbp ¼ map , and if p is not an actual component of both phases then the amount of that component cannot be varied in a Gibbsreversible manner so that dnbp is zero. Inserting all of the substitutions into Eq. (5.38) gives X maj dnsj : ð5:40Þ 0 ¼ f ðc1 þ c2 ÞdV b þ mbk mak dnb þ dU s T a dSs j
Let the quantity s be defined by the expression X sdA dU s T a dSs maj dnsj j
ð5:41Þ
38
R. C. CAMMARATA
where j refers to actual components in a. Using this definition in Eq. (5.41) and rearranging gives mbk mak ¼ ðf sÞðc1 þ c2 ÞOb
ð5:42Þ
and definition were used. where the relation dA ¼ (c1 þ c2) Since the chemical potential for vacancies in b is mbv ¼ mbk mak then, according to Eq. (5.42), mbv ¼ ðf sÞðc1 þ c2 ÞOb for vacancies on the principal sublattice containing the substitutional components. Equation (5.42) is the chemical equilibrium condition for a finite-size multicomponent solid that generalizes Eq. (5.33) for a single component solid and shows that at equilibrium, unless f ¼ s or c1 þ c2 ¼ 0, the chemical potential for a substitutional component k in b is different from the chemical potential of that component in the fluid a even if the component is an actual one for both phases. As a result the chemical potential for the interfacial region mSk is nonuniform and therefore the surface chemical potential msk is not defined. Since the definition for the surface energy as given in Eq. (5.14) or (5.26) involves the surface chemical potentials, it is not well defined for a solid–fluid surface. A dividing surface can be chosen to make msk ¼ 0 for one of the substitutional components k (as long as the density of k in a and b is different and this difference is large enough to allow a reasonable location for the dividing surface) but will not in general result in msj ¼ 0 for the other components j 6¼ k. Cahn33 has presented an analysis in which he started with Eq. (5.33) as obtained by Gibbs and then attempted to generalize it for the case of a multicomponent solid. He obtained an expression identical in form to Eq. (5.42) except with s replaced by what he called the surface free energy g (using the notation of this article). Although he stated that g was associated with the work to create new surface, he did not give an expression for it in terms of the other state variables of the system and ostensibly it was to be taken as the surface energy g as defined by Gibbs (Eq. (5.14) or (5.26)), so that his derivation would be valid only if mak ¼ msk for all components k at the surface. (This has also been explicitly or implicitly assumed in other studies.24,68,69) This equality will hold only for interstitial components but is not applicable to substitutional components. Equilibrium conditions for a system composed of two phases separated by a surface are summarized in Table V.1 and are illustrated in Figure V.3. dVb
68 69
Ob
J. I. Alexander and W. C. Johnson, J. Appl. Phys. 58, 816 (1985). P. H. Leo and R. F. Sekerka, Acta Metall. 37, 3119 (1989).
(dVb/dnb)
GENERALIZED THERMODYNAMICS OF SURFACES
39
TABLE V.1. EQUILIBRIUM CONDITIONS FOR SYSTEM WITH TO PHASES a AND b
Bulk phases (no capillary effects): Tb ¼ Ta P b ¼ Pa mbi ¼ mai Liquid b immersed in much larger fluid phase a Tb ¼ Ta P b ¼ Pa þ f ð c 1 þ c 2 Þ mbi ¼ mai f¼g Solid b immersed in much larger fluid phase a Tb ¼ Ta P b ¼ Pa þ f ð c 1 þ c 2 Þ mbk ¼ mak þ ðf sÞðc1 þ c2 ÞOb (k ¼ substitutional component) mbp ¼ map (p ¼ dilute interstitial component) f 6¼ s Components assumed to be actual components of both phases.
16. S URFACE A VAILABILITY The definition of the surface quantity s can be obtained in the following manner.43 If all of the components i at the surface are also actual components in a, then Eq. (5.41) can be expressed as X sdA ¼ dU s T a dSs mai dnsi ð5:43Þ i
which can be directly integrated to give U s T a Ss s¼
P i
A
mai nsi
:
ð5:44Þ
Taking a as a reservoir for the surface as well as for b, sA can be identified with the total open system surface availability Bs and is a measure of the maximum extractable work obtainable from the surface or equivalently the minimum work needed to create the surface. If there are components r that are actual components for the surface but not for a, then Eq. (5.43) can be integrated keeping the number of moles of these components fixed to give
40
R. C. CAMMARATA
β
Bulk phases (no capillary effects)
(a)
α
b
a
T
μi
P
b
c
a
b
c
a
(b) Fluid-fluid phases (with capillary effects)
c
c
a
c
a
b
P
T
? b
b
a
c
μi
b
a
c
Solid b -fluid a (with capillary effects)
(c) P
T
? b
c
a
μk
b
c
a
b
c
a
μp
? b
c
a
FIG. V.3. Schematic illustration of the spatial dependence of the temperature, pressure, and chemical potentials for a system composed of two phases a and b separated by an interfacial region between points a and b. Point c is the location of the dividing surface. (a) System with no capillary effects. (b) Capillary effects for system with a and b fluid phases. Component i is an actual component in the phases and the interfacial region. (c) Capillary effects for system with small solid b immersed in a much larger fluid a. Substitutional component k and dilute interstitial component p are actual components in the phases and the interfacial region, and the difference f s is taken to be positive.
GENERALIZED THERMODYNAMICS OF SURFACES
U s T a Ss s¼
P j
A
maj nsj þ fsr ¼
Bs þ fsr A
41
ð5:45Þ
where j denotes actual components in a and the integration constant fsr represents the chemical work associated with the components r. If it is imagined that there is an external fluid material reservoir R containing these components that is in chemical equilibrium with the surface, then the integration constant can be expressed as fsr ¼ Sr mRr nsr . For a completely fluid surface in thermal and chemical equilibrium with a, the surface availability s is the same as the surface energy g as T a ¼ T s and the surface chemical potentials are defined and equal to the corresponding chemical potentials in the fluid for actual components in a and s. For a solid–fluid surface where the solid is a single component material, if the dividing surface is located so that the excess number of moles of that component is equal to zero, then s ¼ g. For a solid–fluid surface where the solid contains more than one substitutional component, g is not well defined so that s needs to be used to characterize the thermodynamics of the surface. Suppose that either all of the actual components at a surface are also actual components in a or that Bs fsr , so that Eq. (5.44) can be used to express the surface availability s. If it is assumed that a and s are in thermal equilibrium, then s ¼ Us T s Ss Si mai nsi =A. If it is also assumed that the chemical equilibrium condition mai ¼ msi msv ¼ Mivs holds for each component i at the surface, where msv and Mivs are, respectively, the surface chemical potential for a vacancy (or incompletely relaxed void) and diffusion potential associated with exchanging an atom of component i with a vacancy/void, then the surface availability can be expressed as s ¼ Fs Si msi msv nsi =A ¼ Fs Si Mivs nsiv =A, where Fs is the surface Helmholtz free energy. These expressions may seem to be more appealing ways of defining the surface availability than Eq. (5.44) as they involve only state variables of the surface; however the equilibrium concentration of vacancies may be zero in many cases for a surface associated with small crystals where capillary effects are important. As a result, the exchange of species i with v cannot be conducted in a Gibbs-reversible manner so that mai ¼ msi msv ¼ Mivs cannot be assumed, and Eq. (5.44) has to be used to express s. For completeness it is noted that the surface stress can be associated with the closed system availability of the surface.43 Referring to the system illustrated in Figure V.1a, consider a general variation that keeps the number of moles nai , nbi , and nsi of each component i fixed. The variation in the internal energy of the system is dU ¼ dU a þ dU b þ dU s ¼ T a dSa Pa dV a þ T b dSb Pb dV b þ dUs : ð5:46Þ
42
R. C. CAMMARATA
Setting dU ¼ 0, substituting the constraints (5.27) and (5.28), and rearranging gives ð5:47Þ 0 ¼ T b T a dSb Pb Pa dV b þ fdU s T a dSs g: The term in braces on the right-hand side denotes the work to stretch the surface by an amount dA and can be used to define the surface stress by fdA ¼ dUs Ta dSs. It can also be identified with the variation in the closed system surface availability (see Eq. (4.18)) since the surface volume Vs ¼ 0, so that dZs ¼ dUs Ta dSs. Thus, for a preexisting surface (so that dZs ¼ dZs), the surface stress f can be defined as s s dZ dZ ¼ ð5:48Þ f ¼ dA dA and can be interpreted as the change in the closed system availability for a given change in the surface area owing to a stretching process. Since Ts ¼ Ta at equilibrium, dZs ¼ dFs, and therefore the surface stress can also be expressed as f ¼ (@Fs/@A). For a general anisotropic surface, defining the two-dimensional surface strain esij in the standard way (see Appendix A), the surface stress tensor fij can be defined by setting the reversible work to introduce a surface strain desij as Afij desij ¼ dZ s , so that 1 @Zs : ð5:49Þ fij ¼ A @eij es kl6¼ij
The surface stress can be related to the surface availability s by noting that for fixed nsi , dZ s ¼ dBs ¼ dðsAÞ and that dA ¼ Adij desij , where dij is the Kronecker delta, so that ! ! 1 @ ðsAÞ @s ¼ sdij þ ð5:50Þ fij ¼ A @esij @esij where the partial derivatives are conducted keeping the excess number of moles of surface components nsi and surface strain components eskl6¼ij fixed. For an isotropic surface, f ¼ s þ (@s/@es). For a completely fluid surface, (@s/@es) ¼ 0 since f ¼ s, but this is not expected to be the case for a general solid–fluid surface. 17. S URFACE A VAILABILITY
IN
L AGRANGIAN C OORDINATES
As discussed in Appendix A, strains can be expressed in terms of Eulerian or Lagrangian coordinates. This article has used Eulerian coordinates as the strains have been referred to the actual state of the system. In some situations it is more
GENERALIZED THERMODYNAMICS OF SURFACES
43
convenient to instead use a reference state, such as the strain-free state.33 Let A0 denote the area of the reference state, a Lagrangian measure, which can be related to the surface elastic strain tensor esij and actual area A using linear elasticity by the relation 0 ð5:51Þ A ¼ A 1 þ esii where summation of the repeated index is implied so that esii is the trace of the surface strain tensor. The surface availability in Lagrangian coordinates s0 is related to the surface availability s in Eulerian coordinates by the relation 0 ð5:52Þ s ¼ s 1 þ esii so that the total open system availability of the surface is Bs ¼ sA ¼ s0 A0 . Substituting Eqs. (5.52) and (5.53) into Eq. (5.50) results in the following expression for the surface stress: ! 0 @s : ð5:53Þ fij ¼ @esij It is also possible to express the surface stress in terms of using a finite surface strain tensor rather than the small strain tensor.63,70 18. P HYSICAL O RIGIN
OF
S URFACE A VAILABILITY
AND
S URFACE S TRESS
Consider a brittle material that experiences the process of fracture creating two surfaces, each of area A. The total reversible work to form one of these surfaces is X wfrac ¼ sA þ mai nsi ð5:54Þ i
where a is the vapor above the surface. If the material is a single component solid, then by choosing the ‘‘ns1 ¼ 0’’ dividing surface location, the chemical term disappears and the surface availability s can be taken as the conventional surface energy g, so that the reversible work to form the surface can be written as gA, as it is often expressed in many texts and papers, and can be considered the work to break chemical bonds to form a surface with ‘‘dangling bonds.’’ However, if the material is a multicomponent solid with two or more substitutional components, then it is important to realize that the work of fracture wfrac will generally involve both the reversible mechanical work sA and reversible chemical work Si mai nsi ,
70
W. D. Nix and H. Gao, Scripta Mater. 39, 1653 (1998).
44
R. C. CAMMARATA
and that expressing that work as wfrac ¼ sA may be ignoring the chemical contribution or effectively renormalizing the surface availability to incorporate the chemical term. Recognizing the contribution of the chemical work can be of particular importance if there are significant surface segregation and/or adsorption effects. It has been sometimes argued that the reason the surface energy g and surface stress f for a fluid surface are the same is that when a fluid surface is stretched molecules in the interior will diffuse to the surface to accommodate the new area, keeping the surface density of molecules the same. However this is misleading as this ‘‘stretching’’ process is creating new surface area and by definition the work to do this is associated with the surface energy. The surface stress is associated with an elastic stretching process that involves a change in the surface molecular density of a preexisting surface keeping the number of molecules in the interfacial region fixed. If a fluid is compressible, then such a change would be expected in response to a change in the hydrostatic pressure and this is a physically distinct process different from creating new surface by adding molecules. Thus, asserting that f and g are the same is saying that they have the same value despite the fact that they involve two different forms of work. A phenomenological thermodynamic proof was given in Section V.10 that showed f ¼ g for a fluid surface. The line of argument can be summarized as follows. The starting assumption is that when an atom or molecule is removed from a fluid region the resulting void fully relaxes and disappears. Therefore, voids are not a stable component in a fluid and the number of moles of each component can be independently varied. The independence of these variations was then invoked to argue that the equilibrium condition that the chemical potential for a component i in a fluid system is uniform in all regions where i is an actual component holds even when the system has a nonuniform interfacial region. This uniformity of the chemical potentials was the key condition that led to the demonstration that f ¼ g. Herring49 has offered a macroscopic mechanical argument based on the inability of a fluid to withstand a shear stress to show f ¼ g, although it is not clear if his demonstration is just a recasting of the argument that stretching a fluid surface allows molecules to diffuse to the surface. If his argument is valid it suggests that the inability to maintain a shear stress, complete relaxation of voids, the uniformity of the chemical potentials even in a system with a surface, and f ¼ g are all interrelated manifestations of the physical nature of fluids. Statistical mechanical calculations71–73 have been presented that also show f ¼ g.
71
J. G. Kirkwood and F. P. Buff, J. Chem. Phys. 17, 338 (1949). A. G. MacLellan, Prod. Roy. Soc. A213, 274 (1952). 73 J. Lekner and J. R. Henderson, Mol. Phys. 34, 333 (1977). 72
GENERALIZED THERMODYNAMICS OF SURFACES
45
In the discussion of capillary effects in a system with a solid–fluid surface, it was asserted up front that f and g (and more generally f and the surface availability s) will have different values and that this leads to the equilibrium condition that the chemical potential of a substitutional component in the solid mbk and the value for that component in the fluid mak will be unequal at equilibrium as expressed in Eq. (5.42). This result in turn showed that removal of an atom in the crystal will create a vacancy on the principal lattice with a nonzero chemical potential equal to mbk mak . If Herring’s mechanical argument mentioned previously is valid, then it suggests that the ability to withstand a shear, that incompletely relaxed atomic-size voids such as crystal vacancies are stable, and that for a solid–fluid surface f 6¼ s are all interrelated manifestations of the physical nature of solids. A simple qualitative microscopic argument for the physical origin of the surface stress for a solid is that since the coordination number (i.e., number of nearest neighbors and therefore number of bonds) for an atom at the surface is different from that in the bulk, it is expected that the equilibrium interatomic distance will be different at the surface compared its value in the interior. As a result, the underlying lattice can be considered as exerting a stress on the surface atoms to keep them in atomic registry. In some cases, the strain energy associated with the bulk constraining the surface atoms to remain coherent with the rest of the lattice is so large that it is thermodynamically favorable for the surface monolayer to lose atomic registry with the underlying lattice. This type of surface reconstruction has been observed in certain metallic surfaces74–76 and simple thermodynamic models similar in form to that used to characterize thin film epitaxy (see Section VI.24) have been proposed to describe this behavior.48,60,77–79 Recently atomistic calculations for the surface stress f and surface energy g of low index surfaces of single component fcc metals have been performed using central force pair-wise and embedded atom method (EAM) potentials that showed that both the surface energy and surface stress could be approximately expressed in terms of the lattice spacing and certain macroscopic properties.80 For the pairwise potentials, the surface energy could be expressed as g ¼ aEc =ro2 þ bKro and
74
U. Harten, A. M. Lahee, J. P. Toennies, and Ch. Woll, Phys. Rev. Lett. 54, 2619 (1985). K. G. Huang, D. Gibbs, D. M. Zehner, and S. G. J. Mochrie, Phys. Rev. Lett. 65 3313 (1992). 76 A. R. Sandy, S. G. J. Mochrie, D. M. Zehner, G. Grubel, K. G. Huang, and D. Gibbs, Phys. Rev. Lett. 65 3313 (1992). 77 R. J. Needs, M. J. Godfrey, and M. Mansfield, Surf. Sci. 242, 235 (1991). 78 R. C. Cammarata, Surf. Sci. 279, 341 (1992). 79 T. M. Trimble, R. C. Cammarata, and K. Sieradzki, Surf. Sci. 531, 8 (2003). 80 T. M. Trimble and R. C. Cammarata, Surf. Sci. 602, 2339 (2008). 75
46
R. C. CAMMARATA
the surface stress as f ¼ cKro, where K is the bulk modulus, Ec is the cohesive energy, ro is the nearest neighbor spacing, and a, b, and c are constants that depend on the surface orientation. For the Lennard–Jones potential, a ¼ 0, so that the ratio f/g was material independent. Depending on the orientation, the surface stress could be positive or negative while g was of course always positive. When EAM potentials, which included many-body electron effects, were used the surface stress was always positive (consistent with experimental measurements and first principle calculations43,81) and could be approximately expressed as f ¼ p(1 – qm/K)Kro, where m is the shear modulus and p and q are positive constants. In addition, there was a linear relationship between f – g ¼ @g/@e and Kro. It is seen that both types of potentials allowed the surface stress to be expressed with terms involving the product of the lattice spacing and the elastic constants, the latter being measures of the stiffness of the materials and related to the ‘‘spring constant’’ of the chemical bonds. 19. L AYER Q UANTITIES Some surface thermodynamics formulations9,55,82–86 have treated the interface S between two homogeneous phases a and b as a region of finite volume rather than using a two-dimensional dividing surface construction. Cahn55 has developed this approach for multicomponent planar surfaces (see Figure V.4) where he defines state functions for the interfacial region. An ‘‘interfacial pressure’’ PS can be defined as the negative of the stress component acting normal to the two boundaries A–A and B–B where the interfacial layers meets the homogeneous phases and where (for a planar interface) these stresses are equal as a condition of mechanical equilibrium. In addition, although for an interfacial layer between a crystal and a fluid the chemical potentials of the substitutional components of the crystal are not single valued, if the surface is planar it is possible to define ‘‘interfacial chemical potentials’’ mSk equal to the values these quantities have at the A–A and B–B boundaries. At equilibrium, the interfacial chemical potential mSk defined in this way for a particular component k will be equal to 81
R. C. Cammarata and K. Sleradzki, Annu. Rev. Mater. Sci. 24, 215 (1994). J. D. van der Waals and Ph. Kohnstamm, Lehrbuch der Thermostatik, Teil I, Barth, Leipzig (1927). 83 G. Bakker, in Handbuch der Experimentalphysik, Vol. 6, eds. W. Wien and F. Harms, Akademische Verlagsgesellschaft, Leipzig (1928). 84 E. A. Guggenheim, Trans. Faraday Soc. 36, 397 (1940). 85 E. A. Guggenheim, Thermodynamics: An Advanced Treatment for Chemists and Physicists, NorthHolland, Amsterdam (1959). 86 A. I. Rusanov, in The Modern Theory of Capillarity, ed. F. C. Goodrich and A. I. Rusanov, Akademie-Verlag, Berlin (1981). 82
GENERALIZED THERMODYNAMICS OF SURFACES
47
a A
A
Σ B
b
B
FIG. V.4. Schematic diagram of a system with a planar interfacial region treated as a surface layer of finite volume.
the corresponding chemical potentials in the uniform phases mak ¼ mbk assuming k is an actual component in a, b, and S. Using this approach Cahn was able to define the surface energy for the finite-thickness interface layer S as g ¼ U S T S SS þ PS V S Si mSi nSi =A and then derive results similar to those obtained previously using Gibbs’ dividing surface construction. Since PS does not have the same value at the A–A and B–B boundaries for a curved fluid interface, this approach cannot be used for such a nonplanar interface. This will also be the case for a curved solid–fluid interface since both PS and mSk for substitutional components k in the solid will have different values at the A–A and B–B boundaries. Therefore the Cahn layer approach is restricted to planar interfaces. It would appear that these features of a curved interface is what motivated Gibbs to invent his dividing surface construction as it avoided the need for a surface pressure since the excess surface volume was zero, and that by taking the solid of a solid–fluid system to be a single component phase, as Gibbs always assumed, the dividing surface could be located so that there was no excess surface amount of that component, thereby removing the need of a surface chemical potential for that component. Nevertheless, the surface layer approach can be used for curved surfaces if surface availabilities are employed. Referring to Figure V.1a, the closed and open availabilities for the interfacial region S can be determined using Eqs. (4.4) and (4.13): Z S ¼ U S T a S S þ Pa V S B S ¼ U S T a S S þ Pa V S
X
ð5:55Þ maj nSj
ð5:56Þ
j S andthe surface stress f are defined as before: s ¼ B =A availability s the surface [or BS þ fSr =A] and f ¼ dZ S =dA ¼ s þ ð@s=@eÞ, where the surface area A can be associated with the area of the boundary B–B. Since Eqs. (5.55) and (5.56) use the well-defined and single-valued pressure Pa and chemical potentials mak of
48
R. C. CAMMARATA
the uniform phase a rather than the ill-defined values for a curved interface, there is no longer a need to invoke a dividing surface construction to remove such terms or restrict surface layer thermodynamics to just planar surfaces When using surface layer quantities, the boundaries A–A and B–B in Figure V.1a are to be considered parallel with the dividing surface constructed in the manner described in Section V.10 and are placed within the uniform phases a and b but close to the transition region. The exact locations of these boundaries will affect the values of the extensive parameters on the right-hand side of Eqs. (5.55) and (5.56). However any change in the location of A–A boundary does not alter the value of the surface availability BS as long as the boundary remains within a since Ba ¼ U a T a Sa þ Pa V a Sj maj naj ¼ 0. Therefore a change in the location of the A–A boundary will leave the values of the s ¼ BS =A and S f ¼ s þ @s=@e unaffected. In principle, the values of s and f will depend on the location of the B–B boundary. However, if the thickness of the interfacial region is much smaller than the size scale of b, so that the distance between the location of the B–B boundary and the locations where a Gibbs dividing surface could be chosen is much smaller than the size scale of b, the variation in the values of s and f owing to shifts in location of the B–B boundary will be small, of the same order as the variation introduced by ignoring the contribution of the curvature terms to the surface energy g of a completely fluid system when using the dividing surface construction. 20. A DSORPTION E QUATION
FOR A
S OLID S URFACE
In Section V.11 the Gibbs adsorption equation, Eq. (5.21), was derived for a fluid surface. Since in general the surface chemical potentials in this equation are not well defined for a solid–fluid surface, this equation can no longer be used to describe the behavior of such a surface. An adsorption equation appropriate for a solid–fluid surface43 can be obtained by taking the full differential of Eq. (5.44) and substituting Eq. (5.43): X Ads ¼ Ss dT a nsi dmai : ð5:57Þ i
At fixed temperature, changes in the surface availability s owing, for example, to the adsorption of gases to the free solid surface can be associated with changes in the chemical potentials of the gases in the environment. If the layer quantities of Section V.19 are used, the adsorption equation is expressed as X nSi dmai : ð5:58Þ Ads ¼ SS dT a þ V S dPa i
GENERALIZED THERMODYNAMICS OF SURFACES
49
21. S OLID –S OLID I NTERFACES Consider a system composed of a small spherical crystal embedded in an infinite crystalline matrix. It can be imagined that the system was formed by inserting into a spherical void of the matrix a a sphere b with an initially different radius from that of the void. Let Rbo denote the stress-free radius of b before insertion, R the radius of the void in stress-free a, and e be a measure of the initial mismatch defined by Rbo ¼ ð1 þ eÞR:
ð5:59Þ
If e 6¼ 0 then inserting b into a will generate elastic strains and stresses in the crystalline phases. Mott and Nabarro87 analyzed the elasticity problem in the absence of capillary effects using linear elasticity and taking a and b as elastically isotropic. The result was that b will display a uniform hydrostatic strain state and that there would be a pure shear strain field in a where the strain ea at a distance r from the center of the inclusion was expressible as ea ðr Þ ¼
3K b e b 3 R =r : 3K b þ 4ma o
ð5:60Þ
In Eq. (5.60), Kb is the bulk modulus of b and ma is the shear modulus for a. Cahn and Larche´88 extended this analysis by introducing a boundary condition at the a/b interface that imposed a surface stress effect. This effectively placed the dividing surface at the chemical interface between a and b prior to any interdiffusion. Using this model Cahn and Larche´ derived equilibrium conditions that included capillary effects. Extensions of this approach to include cylindrical and ellipsoidal shapes for b have been offered.89,90 It is recalled that the dividing surface construction takes a and b to be uniform up to the dividing surface and it is assumed that the actual interfacial region has a thickness much smaller than the size of the phases so that the exact location of the dividing surface is not critical. However, it is seen from Eq. (5.60) that there is a long range strain (as well as stress) field that goes as 1/r3 within the matrix a. In addition it is expected that for many multicomponent systems interdiffusion will generate a composition field that couples with the strain field. As a result, there will be in general a physically and chemically nonuniform interfacial region that can be quite diffuse, significantly larger than the size of the inclusion b, so that there is no good way of locating the dividing surface. Since the strain field is a
87
N. Mott and F. R. N. Nabarro, Proc. Roy. Soc. 52, 86 (1940). J. W. Cahn and F. Larche´, Acat Metall. 30, 51 (1982). 89 P. Sharma and S. Ganti, J. Appl. Mech. 71, 663 (2004). 90 P. Sharma and L. T. Wheeler, J. Appl. Mech. 74, 447 (2007). 88
50
R. C. CAMMARATA
result of the surface between the two phases, the spirit of the Gibbs dividing surface method is to consider its strain energy as part of the ‘‘excess energy’’ owing to the surface rather than calculating it as a separate contribution to the overall energy of the system as is generally done. Referring to Eq. (5.60), the range of the strain field will be small only when Kb ! 0 or ma ! 1. The limiting case Kb ¼ 0 corresponds to b being a void and the limiting case ma ¼ 1 corresponds to a being an infinitely rigid matrix. Thus, except for these extreme cases, it is not clear that a dividing surface approach is the proper way of defining surface excess quantities for a diffuse interface such as that created by a finite-size inclusion with a curved interface and that a gradient thermodynamic model is more appropriate. Gradient thermodynamic approaches have been used to determine the surface stress for diffuse interfaces between two semi-infinite lattice-matched crystals.91,92 In this chapter, attention will be restricted to planar solid–solid surfaces associated with a physically and compositionally non-diffuse interfacial region. An example of a model system is one that has a surface separating a thin solid film b and a much thicker solid substrate w (see Figure V.5). Using a dividing surface construction let c denote the surface associated with the film b-substrate w interface. It will be assumed that a can be taken as a material reservoir for c (with surface diffusion at this interface allowing matter exchange between a and c) so that the surface availability for the c can be expressed as P U c T a Sc mai nci i ð5:61Þ sc ¼ A a
s
b
c
y
FIG. V.5. Schematic diagram of a thin film system where b is the solid film, w is the solid substrate, a is the surrounding vapor, s is the dividing surface for the film-vapor interface, and c is the dividing surface for the film-substrate interface.
91 92
J. W. Cahn, Acta Metall. 37, 713 (1986). W. C. Johnson, Acta Mater. 48, 443 (2000).
GENERALIZED THERMODYNAMICS OF SURFACES
51
where A is the area for c as well as for s. As was first pointed out by Brooks,93 there are two surface stresses associated with c. Following Cahn and Larche´,88 one of these stresses is associated with the reversible surface work resulting from straining one of the phases in the plane of the surface relative to the other and thereby changing the interfacial structure, and the other stress is associated with the reversible surface work when both phases are deformed in the plane of the surface by the same amount leaving the interfacial structure unaffected. The former stress will be called the interface stress tensor gij and the latter the interface stress tenosr hij. Letting A0 denote the area with respect to a particular reference state, the actual area A can be expressed in terms of the surface strain ecij associated with deforming b relative to w and the surface strain ecij associated with deforming both phases by the same amount by 0 ð5:62Þ A ¼ A 1 þ ecii þ ecii where ecii and ecii are the traces of the two-dimensional surface strain tensors. By analogy with Eq. (5.50), the interface stresses can be defined as60 ! @sc c gij ¼ s dij þ ð5:63aÞ @ecij @sc
hij ¼ sc dij þ
@ecij
! :
ð5:63bÞ
These expressions are referred to Eulerian coordinates. By analogy with Eq. (5.53), expressions for the interface stresses can be expressed in Lagrangian coordinates as88 ! 0 @sc gij ¼ ð5:64aÞ @ecij hij ¼
@sc
0
@ecij
! :
ð5:64bÞ
The interface between two crystalline phases can be characterized as coherent when there is perfect lattice matching at the interface, semi-coherent when there is substantial but not complete lattice matching, and incoherent when there is little or no lattice matching. Suppose the unstressed lattice spacings for the two phases are denoted as ab and aw. The lattice misfit m can be defined as
93
H. Brooks, in Metal Interfaces, American Society for Metals, Metals Park, OH (1963).
52
R. C. CAMMARATA
m ¼ ab aw =ab :
ð5:65Þ
When the interface is completely coherent, all of the misfit in a thin film b on a much thicker substrate are accommodated by elastic coherency strains in the film. For semi-coherent and incoherent interfaces, the part of the misfit not accommodated by coherency strains can be accommodated by misfit dislocations at the interfaces.94–99 Introduction of the surface strain ecij will change the interface structure by changing the density of misfit dislocations, while introduction of the surface strain ecij leaves the surface structure unaffected. Thus the interface stress gij is related to the reversible surface work that accompanies a change in the amount of coherency strain in the film and the interface stress hij is related to the surface work to stretch the surface leaving the coherency strain fixed. The conditions of thermodynamic equilibrium for the thin film system illustrated in Figure V.5 will be discussed in Section V.22. Consider a free-standing film b taken to be an elastically isotropic disk of thickness t, in-plane area A, and volume Vb ¼ At whose radius is much larger than t, and where the two planar surfaces have a surface stress f (see Figure V.6a). The (a)
(b) c
β f
χ
h
β t
χ β f FIG. V.6. (a) Thin disk with a radius much greater than the in-plane radius. The top and bottom circular surfaces both have a surface stress f. (b) Multilayered solid composed of alternating layers of b and w separated by surfaces that have an interface stress h associated with equal stretching of both phases in the plane of the surface.
94
F. C. Frank and J. H. van der Merwe, Proc. Roy. Soc. Lond. A 198, 205 (1949). J. W. Matthews, in Epitaixal Growth, ed. J. W. Matthews, Academic, New York (1975). 96 J. H. van der Merwe, J. Electron. Mater. 20, 793 (1991). 97 J. Y. Tsao, Materials Fundamentals of Molecular Beam Epitaxy, Academic, San Diego (1993). 98 J. Hu and P. H. Leo, J. Mech. Phys. Solids 45, 637 (1997). 99 R. C. Cammarata, K. Sieradzki, and F. Spaepen, J. Appl. Phys. 87, 1227 (2000). 95
GENERALIZED THERMODYNAMICS OF SURFACES
53
surface stress induces an in-plane strain eb relative to the strain-free state of a much thicker disk. There will also be a surface strain es ¼ eb. According to Hooke’s law the film will display an in-plane volume stress sb ¼ Ybeb, where Yb is modulus of the disk, resulting in a volume strain energy R bthe bbiaxialR elastic s dV ¼ 2 Vbsbdeb, where the factor of two takes into account the twodimensional nature of the R stress. There will also be surface strain energy for each surfaceR given by 2 Afdes so that total surface strain energy for the two surfacesRis 4 Afdes. The R total elastic Rstrain energy for the disk and the surfaces is wel ¼ 2 Vbsbdeb þ 4 Afdes ¼ 2A (tsb þ 2f)deb. The equilibrium stress and strain are those that minimize we and are obtained by setting dwel/deb ¼ 0: sb ¼ Y b eb ¼ 2f =t:
ð5:66Þ
Equation (5.66) is the analogue of the Laplace pressure for a large radius disk. Now consider the case of a multilayered solid composed of many layers of two solids b and w each of thickness t that are alternately stacked (see Figure V.6b). Assume for simplicity that the two materials have the same elastic modulus Y. The interface stress h will induce an in-plane strain e in the layers. The analysis given for a disk can be used to derive an expression for this strain by substituting h for 2f since there is now one interface per layer. The result is that the in-plane strain is e ¼ h=Yt
ð5:67Þ
that induces an in-plane stress s given by Hooke’s law as s ¼ Ye ¼ h=t
ð5:68Þ
Using similar but more sophisticated analyses the interface stress h in crystalline polymers and organic crystals100–102 as well as in metallic multilayered thin films103–110 have been obtained from measurements of the in-plane strain as a function of layer thickness. A typical value for the interface stress h for a metallic interface is of order 1 N/m. Substituting this value for h and Y ¼ 1011 N/m2 into 100
R. Cammarata and R. K. Eby, J. Mater. Res. 6, 888 (1991). H. P. Fisher, R. K. Eby, and R. C. Cammarata, Polymer 35, 1923 (1994). 102 M. Hu¨tter, P. J. in’t Veld, and G. C. Rutledge, Polymer 47, 5494 (2006). 103 Yu. A. Kosevich and A. M. Kosevich, Solid State Comm. 70, 541 (1989). 104 R. C. Cammarata and K. Sieradzki, Phys. Rev. Lett. 62, 2005 (1989). 105 J. A. Ruud, A. Witrouw, and F. Spaepen, J. Appl. Phys. 74, 2517 (1993). 106 J. Weissmu¨ller and J. W. Cahn, Acta Mater. 45, 1899 (1997). 107 D. Josell, J. E. Bonevich, I. Shao, and R. C. Cammarata, J. Mater. Res. 14, 4358 (1999). 108 K. O. Schweitz, J. Bttiger, J. Chevallier, R. Feidenhans’l, M. M. Nielsen, and F. B. Rasmussen, J. Appl. Phys. 88, 1401 (2000). 109 R. Birringer and P. Zimmer, Scripta Mater. 58, 639 (2008). 110 F. H. Streitz, R. C. Cammarata, and K. Sieradzki, Phys. Rev. B. 49, 10707 (1994). 101
54
R. C. CAMMARATA
Eq. (5.67) leads to a strain e of order 1% when the layer thickness is 1 nm. Strains of these magnitude as well as variations in the elastic moduli of order 10% relative to bulk properties have been observed in multilayered thin films with very thin layers and it has been proposed that the changes in elastic properties are due at least in part to higher order elastic effects owing to the large strains induced by the interface stress h.60,104 22. P HYSICAL O RIGIN OF THE S OLID –S OLID S URFACE A VAILABILITY AND I NTERFACE S TRESSES As with the case of the solid–fluid surface availability, the total reversible work to form a non-diffuse planar solid–solid surface c can be partitioned into a mechanical work term scA and chemical work term Si mai nci . In the case of a semicoherent or incoherent interface between two crystals most of the reversible surface work to create the interface will be the mechanical work associated with the formation of the interfacial defect structure. Returning to the system containing a film-substrate interface c separating a thin film b from a much thicker substrate w, as shown schematically in Figure V.5, consider a reference state in which a semi-coherent film-substrate interface is formed between a fully relaxed film with its bulk in-plane lattice spacing and a substrate that also has its bulk lattice spacing. In this initial reference state all of the misfit m as defined in Eq. (5.65) is accommodated by structural defects at the film-substrate interface c. Suppose that atoms are added to the film keeping the thickness and in-plane area fixed so that an in-plane elastic strain e, whose value is in the range 0 < e/m 1, is introduced into the initially relaxed film. This reduces the amount of strain associated with the defect structure to m – e. A simple model given by Matthews95 has this strain accommodated by a two-dimensional grid of misfit edge dislocations at the interface with their Burgers vectors in the plane of the interface, so that the surface work scA is the total elastic self-energy of these dislocations which to first order in strain will be approximately proportional to m – e ¼ m(1 – e/m). Therefore sc can be expressed as h i sc ¼ sco 1 ec11 þ ec22 =2m ð5:69Þ where ec11 and ec22 are the in-plane strain components equal to the coherency strain ec and sco is the surface availability of the film-substrate interface when the film is fully relaxed (ec ¼ 0) and can be expressed as sco ¼ kMb
ð5:70Þ
where b is the Burgers vector of an in-plane misfit edge dislocation, M is the elastic modulus associated with the strain state generated by an edge dislocation,
GENERALIZED THERMODYNAMICS OF SURFACES
55
and k is a dimensionless quantity that takes into account terms describing the selfenergy of a dislocation. Using Eqs. (5.69) and (5.70), simple model expressions can be given for the interface stresses.99 The interface stress h associated with straining both phases by the same amount e11 and therefore straining the interface by the amount ec11 ¼ e11 is given by h ¼ sc þ (@sc/e11c). Substituting Eq. (5.69) and 0 0 0 Eq. (5.70) gives h ¼ 1 þ k =k þ M =M þ b =b sco where a prime indicates a c partial derivative with respect to e11 . It is expected that the term in parentheses on the right-hand side will be dominated by the modulus term so that the interface stress can be approximately expressed as 0 ð5:71Þ h M =M sco : M0 can be taken as an effective third-order elastic constant associated with the second-order elastic modulus M. Experimental measurements and theoretical calculations99 indicate that for metals M0 /M is of order 10 so that h is of order 10sco. This model for h suggests that the physical origin of this interface stress is based on higher order elastic effects (which are related to the anharmonicity of the interatomic potential) when straining an interface and predicts that h is negative, consistent with many experimental measurements.105,107,108 The interface stress g is associated with straining the film keeping the substrate fixed, so that there is a change in the coherency strain e11 ¼ ec11 . Substituting Eq. (5.70) into the expression g ¼ sc þ @sc =ec11 0 0 0 gives g ¼ 1 þ k =k þ M =M þ b =b 1=2m sco where a prime now indicates a partial derivative with respect to ec11 . In this case it is expected that the modulus and misfit terms will dominate so that g can be approximately expressed as 0 ð5:72Þ g M =M 1=2m sco : If the misfit m is of order 5%, which would be a relatively large amount, both terms M0 /M and 1/2m are expected to be of the same order. If the misfit is less than or of order 1%, then the term 1/2m will dominate and g can be approximately expressed as g sco =2m:
ð5:73Þ
It is seen that g is opposite in sign to the misfit m and if jmj 1 then jgj sco . In this model the interface stress g is related to the surface work to change the misfit dislocation density at the interface owing to a change in coherency strain. The surface work to introduce a biaxial coherency strain e ¼ e11 ¼ e22 into an initially fully relaxed film, keeping the interface area A, fixed is AcDsc ¼ 2Ac
56
R. C. CAMMARATA
(@sc/@e11) ¼ 2Ac (g sc)ec where the factor of 2 takes into account the twodimensional nature of the strain state, and substituting Eq. (5.72) gives Ac Dsc sco ec =m.
VI. Applications
When the dimensions of a crystal b immersed in a fluid reservoir a are reduced enough so that capillary effects become important, certain properties such as the melting temperature and vapor pressure of the components of the solid can display a significant dependence on the surface thermodynamic parameters and size of the crystal. These behaviors are known collectively as Gibbs–Thomson–Freundlich effects. To correctly derive the expressions that describe these effects it is necessary to use the proper equilibrium condition for a system involving a solid–fluid surface as given in Table V.1. In many derivations that ostensibly apply to such a system, the equilibrium conditions for a completely fluid surface are used that effectively assume f ¼ s which in general is not to be expected. Cahn33 has shown how to use the proper equilibrium conditions to obtain a variety of Gibbs–Thomson–Freundlich equations (see Appendix C). However, in his examples for multicomponent systems, it is necessary to interpret what he calls the surface free energy as the surface availability. The concept of surface availability also has application in describing the thermodynamics of nonequilibrium processes. For example, consider a system composed of a vapor a and a single component solid b that has a free surface exhibiting undulations. Owing to these undulations there can be a driving force for surface diffusion that results in a relaxation of the surface.111 Let the component of the solid be denoted as component 1. The driving force for relaxation is often taken to be the gradient of a potential of the form f ¼ fo þ g(c1 þ c2)Ob where c1 and c2 are the curvatures of the undulations. This potential, which can depend on position if the curvatures vary along the surface, is generally referred to as the ‘‘surface chemical potential’’ but as has been discussed at length the actual surface chemical potential ms1 is not well defined. The correct identification of f is as the value the chemical potential of component 1 would have in the vapor were it in local equilibrium with the surface (see Appendix C).
111
D. Margetis, Phys. Rev. N 76, 193403 (2007).
GENERALIZED THERMODYNAMICS OF SURFACES
57
23. N UCLEATION D URING S OLIDIFICATION Another example of a process that involves surface thermodynamics is a change in phase by nucleation and growth.112–121 As discussed by Gibbs121 and others115,119 the reversible work to form a nucleus b by homogeneous nucleation from a parent phase a that can be taken as a thermal, mechanical, and chemical reservoir for the nucleating phase is equal to the change in open system availability of the system. There is a nucleus of critical size that is in unstable equilibrium with the reservoir. The activation barrier for nucleation is the reversible work to form this critical nucleus. In his derivation of this reversible work Gibbs121 only considered the case of fluid nucleus formed in a fluid parent phase and obtained the result that the reversible work to form a spherical critical nucleus of surface area A and pressure Pb was 16pg3/3(Pb – Pa)2 ¼ gA/3. Using the concept of surface availability, it is possible to generalize his result for the case of a multicomponent solid nucleus b formed in a multicomponent fluid a43. Let the nucleus b and the surface s taken together be considered an open system f immersed in the reservoir a. The change in availability owing to the formation of the nucleus and the surface can be expressed as DBf ¼ DBb þ DBs. For the surface the change in availability is DBs ¼ Bs ¼ sA, assuming that all the components at the surface are also actual components of the reservoir a, and for the nucleus the change in availability is DBb ¼ Bb. As a result the change in availability of the system DBf can be expressed as DBf ¼ Bf ¼ Bb þ Bs ¼ bbV V b þ sA
ð6:1Þ
where bV is the availability per unit volume of b. Since ¼ sA is positive, in order for the formation of the b phase to be thermodynamically favorable b
112
Bs
D. Turnbull, Solid State Phys. 3, 226 (1956). F. Abraham, J. Atmos. Sci. 25, 47 (1968). 114 K. F. Kelton, A. L. Greer, and C. V. Thompson, J. Chem. Phys. 79, 6261 (1983). 115 R. H. Doremus, Rates of Phase Transformations, Academic, Orlando (1985). 116 K. F. Kelton, Solid State Phys. 45, 75 (1991). 117 D. Kaschiev, Nucleation: Basic Theory With Applications, Butterworths-Heinemann, Oxford (2000). 118 J. W. Christian, Theory of Transformation in Metals and Alloys, Part I, Pergamon, Amsterdam (2002). 119 D. T. Wu, L. Gra´na´sy, and F. Spaepen, MRS Bull. 29, 945 (2004). 120 M. Hillert, Phase Equilibirum, Phase Diagrams, and Phase Transformations: Their Thermodynamic Basis, Cambridge University Press, Cambridge (2008). 121 J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 254. 113
58
R. C. CAMMARATA
Bb ¼ bVb Vb has to be negative. Substituting the Euler equation, Eq. (2.14), for the b phase into the expression Bb ¼ U b T a Sb þ Pa V b Si mai nbi leads to Bb ¼ T b T a S b P b Pa V b X b X ð6:2Þ þ mk mak nbk þ mbp mak nbp k
p
where the summation over k is for substitutional components and the summation over p is for the interstitial components. Let the critical nucleus have the shape of a sphere of radius r. Since the critical nucleus is in unstable equilibrium with the fluid reservoir, the following equilibrium conditions can be used: T a ¼ Tb, Pa ¼ Pb2f/r, mak ¼ mbk 2ð f sÞOb =r for substitutional components k that are actual components in a and b, and map ¼ mbp for interstitial components p that are actual components in a and b. Substituting these conditions into Eq. (6.2) and noting Ob ¼ V b =Sk nbk gives Bb ¼ 2f V b =r þ 2ð f sÞV b =r ¼ 2sV b =r so that the radius of a critical nucleus can be expressed as " # X b b mk mak nk =Ob : r ¼ 2s=bb ¼ 2s= Pb Pa
ð6:3Þ
ð6:4Þ
k
Substituting this result into Eq. (6.1) gives the reversible work to form a critical nucleus: 2 DBf ¼ 16ps3 =3 bbV " #2 b b X b 1 b 3 a a ð6:5Þ ¼ 16ps =3 P P V mk mk nk ¼ sA: 3 k The derivation of Eqs. (6.4) and (6.5) would also apply to the case of a liquid nucleus with mbk ¼ mak and f ¼ s ¼ g, so that the reversible work to form a critical nucleus in this case can be expressed as 16pg3/3(Pb – Pa)2 ¼ gA/3 as obtained by Gibbs.121 In many texts and papers, it is claimed that the reversible isothermal work to form either a liquid or solid nucleus b in a fluid a is equal to the change in the Gibbs free energy of the system f, DGf ¼ DGb þ DGs. But this is incorrect as has been shown113,119 for the case of condensation (vapor a to liquid b) by homogeneous nucleation in a single component system. The Gibbs free energy change for this process is
ð6:6Þ DGf ¼ DGb þ DGs ¼ Gb þ Gs ¼ mb ðr Þ ma nb þ Gs
GENERALIZED THERMODYNAMICS OF SURFACES
59
where mb(r) is the chemical potential of the liquid critical nucleus of radius r. Since a and b are in unstable equilibrium, mb(r) ¼ ma, so that DGf ¼ Gs. Choosing a dividing surface location such that ns ¼ 0 and recalling that Vs ¼ 0 leads to DGf ¼ Gs ¼ Us T sSs ¼ sA ¼ gA. Thus the Gibbs free energy change for the formation of a critical nucleus during condensation is just the reversible work to form the surface and is not equal to the total work DBf. It is also easy to show43 that DGf 6¼ DBf for the formation of a critical nucleus during solidification:
DGf ¼ DGb þ DGs ¼ Gb þ Gs ¼ mb ðr Þ ma nb þ Gs ¼ 2ð f sÞOb =r þ sA ð6:7Þ which for a spherical critical nucleus is equal to (2f þ s)A/3, a result with no important physical meaning. The reason the change in Gibbs free energy does not equal the reversible work of nucleation is that DGf is equal to DBf for an isothermal process only when the pressure as well as the temperature remain fixed and uniform within the composite system of a, b, and s. Owing to the Laplace pressure, the pressure within the composite system during nucleation is not uniform and the pressure of the solid nucleus changes as it grows. For the case of solidification of an incompressible solid b in a fluid reservoir a, the reversible work W to form a critical nucleus of b by homogeneous nucleation can be expressed as W ¼ DGV V b þ sA
ð6:8Þ
where DGV is the difference in the Gibbs free energies per unit volume of the bulk phases. This will be demonstrated by showing that DGV ¼ bVb and therefore W ¼ DBf when b is incompressible. Referring to Eq. (6.2), if the system is in thermal equilibrium so that Tb ¼ Ta, then X b mi mbi Þnbi : ð6:9Þ Bb ¼ ðPb Pa ÞV b þ i
Keeping the temperature of b fixed, the Gibbs–Duhem equation for b can be rewritten as X b b xi mi ¼ Ob dPb ð6:10Þ i
xbi
nbi =Si nbi
is the mole fraction of component i. Since b is incompresswhere ¼ ible and therefore the molar volume Ob ¼ V b =Si nbi is fixed, Eq. (6.10) can be integrated to give i X bh b
xi mi ðr Þ mb ð1Þ ¼ Ob Pb ðr Þ Pb ð1Þ ð6:11Þ i
60
R. C. CAMMARATA
where Pb(r) and mb(r) denote the pressure and chemical potential of component i, respectively, for a nucleus of radius r, and Pb(1) and mb(1) refer to the bulk solid phase. Substituting Eq. (6.11) into Eq. (6.9) yields the following expression for the open system availability of b: Xh b
mi ð1Þ mai nbi : ð6:12Þ Bb ¼ Pb ð1Þ Pa V b þ i
Si mbi ð1Þnbi
is the Gibbs free energy of the Referring to Eq. (4.29) it is seen that bulk solid and invoking the condition for mechanical equilibrium of the bulk phases, Pb(1) ¼ Pa, leads to Bb ¼ Gb(1) Ga ¼ DGVVb, where DGV is the difference in the Gibbs free energy per unit volume of the bulk phases and therefore also the open system availability of the nucleus b per unit volume bbV . Thus for an incompressible solid the total change in availability DBf to form a nucleus is equal to W of Eq. (6.8). In many texts Eq. (6.8) is presented, using the conventionally defined surface energy g in place of s, as the reversible work to form a nucleus but with W often incorrectly identified as the total Gibbs free energy change of the system. 24. S URFACE S TRESS E FFECTS
ON
T HIN F ILMS
a. Intrinsic Stress A thin solid film grown on a solid substrate is generally deposited in a state of stress.122–124 Such a stress develops owing to one or more processes that would change in the in-plane dimensions of the film were it not attached to the substrate. Stresses that result from processes that occur within the film during growth are called intrinsic stresses. A variety of mechanisms have been proposed for these stresses, most of which would cause the film to densify were it detached from the substrate but which instead induce a tensile film stress owing to the constraint of the substrate. It has been suggested43,61,125–128 that surface stress effects can produce an intrinsic stress in thin films that in many cases would be compressive. As an example, suppose during the initial stage of growth the deposited material forms islands that eventually coalesce to form a continuous film. This mode of 122
M. F. Doerner and W. D. Nix, CRC Critical Rev. Solid State Mater. Sci.14, 244 (1988). M. Ohring, The Materials Science of Thin Films, Academic, San Diego (2001). 124 J. A. Floro, E. Chason, R. C. Cammarata, and D. J. Srolovitz, MRS Bull. 27, 19 (2002). 125 R. Abermann, R. Kramer, and J. Ma¨ser, Thin Solid Films 52, 215 (1978). 126 M. Laugier, Vacuum 31, 155 (1981). 127 R. Koch, J. Phys.-Cond. Matter 6, 9519 (1994). 128 R. C. Cammarata, T. M. Trimble, and D. J. Srolovitz, J. Mater. Res. 15, 2468 (2000). 123
GENERALIZED THERMODYNAMICS OF SURFACES
61
growth is called the Volmer–Weber growth mode and is commonly observed in the case of a crystalline metallic film deposited on an amorphous substrate for which a compressive stress is often observed prior to coalescence.123,124 An equilibrium in-plane strain relative to the bulk will be generated in an island owing to the surface stresses. Modeling an island as a disk, and modifying the analysis given to derive Eq. (5.67) for a free-standing film by noting that one surface is the free solid surface of the island displaying a surface stress f and the other is the island-substrate interface with an interface stress g, leads to the following expression for the equilibrium strain at a thickness t: e ¼ ð f þ gÞY=t
ð6:13Þ
where Y is the biaxial elastic modulus of the island. If f þ g > 0 then the equilibrium strain is negative. Assume that an island becomes anchored to the substrate at a certain thickness to. Letting a denote the bulk in-plane lattice spacing of the island material, the lattice spacing for an island of thickness to will be ‘‘locked in’’ at a value a(1 þ eo). Up to this thickness, no intrinsic stress has been developed as the substrate has not yet constrained the in-plane dimension of the island. With further growth, the equilibrium strain will change according to Eq. (6.13) so that the equilibrium lattice spacing will also change. Assuming that atoms arriving on the island after the lock-in event deposit epitaxially, that is, maintain atomic registry with the island and deposit with a lattice spacing of a(1 þ eo), an intrinsic stress will be generated as the growing island is now constrained to maintain a nonequilibrium spacing. The difference between the equilibrium strain e(t) and the locked-in strain eo is De ¼ [e(t) - eo] ¼ ( f þ g)(1/t – 1/to)/Y, and by Hooke’s law the intrinsic stress generated in the island is43,128 sðtÞ ¼ YDe ¼ ð f þ gÞð1=t 1=to Þ:
ð6:14Þ
If f þ g > 0, the stress will be compressive and will reach a maximum value of ( f þ g)/to when the thickness t to : Owing to relaxation effects at the edges of the island, the magnitude of the stress generated by the surface stress can be less than that given in Eq. (6.14), and a more complete analysis would also include effects of the curved surfaces perpendicular to the plane of the film.128 The larger the ratio of island radius to thickness the closer the induced stress in the island is given by Eq. (6.14).
b. Thin Film Epitaxy Consideration is now given to a crystalline film grown by layer-by-layer growth on a crystalline substrate in which there is substantial or complete lattice matching at the film-substrate interface.94–97 This type of growth, referred to as Frank–van
62
R. C. CAMMARATA
der Merwe growth, leads to coherency stresses, also called epitaxial stresses, generated by the amount of lattice mismatch taken up by elastic coherency strains in the film. If the misfit is not too large, there is a critical thickness for epitaxy below which a film in equilibrium has all of the misfit accommodated by coherency strains so that this film has a completely lattice-matched film-substrate interface. Above the critical thickness, it will be thermodynamically favorable for the film to display in-plane relaxations, resulting in a semi-coherent filmsubstrate interface (although a growing film may remain coherent above this thickness owing to kinetic limitations of the relaxation processes). Referring to the thin film system of Figure V.5, the vapor a will be taken as a thermal and material reservoir for the film b and the surfaces s and c. The system is assumed to be in complete thermal equilibrium so that T a ¼ Tb ¼ T w. Consideration of a system variation that involves a Gibbs-reversible evaporation of a layer of solid at the free solid surface parallel to the plane of the film would proceed in the same manner as performed in Section III.6 to derive the chemical equilibrium result mai ¼ mbi for either a substitutional or interstitial component i in b that is an actual component in both phases. The condition for mechanical equilibrium will now be investigated by considering a Gibbs-reversible process involving the addition of atoms around the edges of the film by condensation from the vapor keeping the in-plane area A and thickness t (and therefore also the volume) of the film fixed. This will introduce an in-plane (two-dimensional) elastic strain variation deb in the plane of the film. The resulting strain energy will be 2Vbsbdeb, where sb is the in-plane biaxial stress in the film and the factor of 2 takes into account the biaxial nature of the strain. The only interaction involving the substrate phase w is a reversible exchange of heat. The fundamental equation for this variation is (noting that none of the phases changes volume during the variation): X maj dnaj þ T b dSb dU ¼ dU a þ dU b þ dUw þ dUs þ dU c ¼ T a dSa þ þ 2V b sb deb þ
P k
mbk dnbk þ
P p
j
mbp dnbp þ T w dSw þ dU s þ dU c
ð6:15Þ
where the summations over k and p are for, respectively, substitutional and interstitial components in b. The chemical potentials of b in Eq. (6.15) are for the film surfaces perpendicular to the plane of the film. The criterion for equilibrium is dU ¼ 0 for dS ¼ dSa þ dSb þ dSw þ dSs þ dSc ¼ 0, dnak þ dnbk þ dnsk þ dnck ¼ 0, and dnap þ dnbp þ dnsp þ dncp ¼ 0. Substituting these into Eq. (6.15) gives
GENERALIZED THERMODYNAMICS OF SURFACES
X b 0 ¼ T b T a dSb þ T b T w dSX þ 2V b sb deb þ mk mak dnbk k ) ( X X þ mbp map dnbp þ dU s T a dSs þ mai dnsi p i ( ) X c þ dUc T a dSc þ mai dni
63
ð6:16Þ
i
where the summation over i is for all actual components in a. Inserting the thermal equilibrium condition T a ¼ Tb ¼ T w makes the first two terms on the right-hand side vanish. The condition given by Eq. (5.39) will hold so that mbk mak is the same for all substitutional components k in b that are also actual components in a. Therefore the term Sk ðmbk mak Þdnbk can be expressed as ðmbk mak Þdnb where dnb ¼ Sk dnbk . Since mbp ¼ map for interstitial components in b, the term Sp ðmbp map Þdnbp ¼ 0. The terms in the braces refer to variations in the surface availabilities Bs ¼ ssAs and Bc ¼ scAc keeping the areas fixed and therefore can be expressed as dBs ¼ Asdss and dBs ¼ Acdsc. Making all of these substitutions into Eq. (6.16) yields ð6:17Þ 0 ¼ 2V b sb deb þ mbk mak dnb þ As dss þ Ac dsc : Referring to Eqs. (5.50) and (5.63a), it is seen that Asdss ¼ 2As( f – ss)des and Acdsc ¼ 2Ac (g – sc)dec where f is the surface stress of the film-vapor surface s and g is the interface stress of the film-substrate surface c, and the factors of two reflect the two-dimensional nature of the strain states. Substituting these expressions into Eq. (6.17), noting that the in-plane film area A ¼ As ¼ Ac ¼ Vb/t and that the coherency strain of the film eb ¼ es ¼ ec, where es is the film-vapor surface strain and ec is the surface strain associated with straining the film keeping the substrate fixed, gives ð6:18Þ 0 ¼ mbk mak dnb þ 2A tsb þ f þ g ss sc deb : When atoms are added to the film keeping the area fixed this induces a compressive in-plane strain so that Obdnb ¼ 2Vbdeb where Ob Vb/nb ¼ At/nb, and inserting this into Eq. (6.18) and rearranging gives
ð6:19Þ mbk mak ¼ 2 sb þ f þ g ss sc =t Ob : The expression 2A(tsb þ f þ g ss sc)deb is the total strain energy owing to the variation in strain state of the film and surfaces and can be minimized for an arbitrary strain variation by setting tsb þ f þ g ss sc equal to zero. This leads to the chemical equilibrium condition at the film surfaces perpendicular to the plane of the film, mbk ¼ mak
ð6:20Þ
64
R. C. CAMMARATA
which is the same condition for components on the free solid surface parallel to the plane of the film, and to the mechanical equilibrium condition ð6:21Þ sb ¼ f þ g ss sc =t: The term (f þ g ss sc)/t in Eq. (6.21) can be interpreted99 as an ‘‘effective surface pressure’’ that at equilibrium gets balanced by the film coherency stress sb. In the limit t ! 1, this effective surface pressure goes to zero so that the coherency stress and strain also go to zero and the film displays its stress-free lattice spacing at equilibrium. For a finite thickness there will be an equilibrium coherency strain eb that induces a coherency stress sb that balances the effective surface pressure. Using Hooke’s law sb ¼ Ybeb, where Yb is the biaxial elastic modulus of the film, the equilibrium coherency strain can be expressed as eb ¼ f þ g ss sc =Yt: ð6:22Þ As the film thickness is reduced, the magnitude of the coherency strain will increase until a critical thickness tc is reached when this strain equals the misfit m and a completely lattice-matched film is produced. Letting sbc denote the coherency stress of a fully lattice-matched film then according to Eq. (6.21) the critical thickness is given by ð6:23Þ tc ¼ f þ g ss sc =sbc ¼ f þ g ss sc =Ym: Most analyses of the critical thickness for epitaxy only consider the surface work to change the structure of the film-substrate interface, which can be expressed using the interface stress and availability as ADsc ¼ 2A(g sc)ec, and ignore the contribution of the work of the film-vapor surface ADss ¼ 2A(g s s)es that can be significant, especially for large misfits and for very thin films.99 An analysis of the critical thickness for epitaxy of a single crystal film on a single crystal substrate taking into account crystallographic anisotropy is given in Appendix D. Acknowledgments
The author thanks M. J. Aziz, J. W. Cahn, J. D. Erlebacher, W. C. Johnson, P. H. Leo, S. M. Prokes, K. Sieradzki, F. Spaepen, K. M. Unruh, and P. W. Voorhees for useful discussions. Support from the U.S. National Science Foundation under grant number DMR 0706178 is gratefully acknowledged. VII. Appendix A: Stress and Strain in Solids23,24
To describe the work of deformation for a solid it is necessary to specify a reference state defined as the undeformed state. When the solid is in a different
65
GENERALIZED THERMODYNAMICS OF SURFACES
mechanical state, taken to be a deformed state, the amount of deformation is measured with respect to the reference state. The choice of the reference state is in general arbitrary and for many problems it is convenient to take it as the stress-free state. However, when using Gibbsian thermodynamics to determine equilibrium conditions, it may be easier to consider the reference state to be the equilibrium state that is not necessarily stress free and then impose an infinitesimal virtual variation to produce a deformed state. In Figure VII.1, a material body is shown in a reference state and in a deformed state. A vector X referred to a fixed Cartesian coordinate system identifies a material point (or volume element) within the solid in its reference state. A second material point located at X þ dX in the reference state is also shown. It is supposed that a change in the mechanical state has resulted in the body evolving from the reference state to the deformed state. Owing to this change, the material points identified by the vectors X and X þ dX have been displaced to new positions that have been identified by vectors denoted as x and x þ dx, respectively. In general, the change in the infinitesimal vector from dX to dx will involve translation, rotation, and stretching. The nature of the stretching can be quantified by relating dx to dX. One way to do this is to first express x as a function of X; employing indicial notation, x i ¼ x i ð X1 ; X 2 ; X 3 Þ
ð7:1Þ
dX u dx
X x X3
X2 X1 FIG. VII.1. Solid in its reference state and deformed state. The vector X in the reference state and the vector x in the deformed state locate the same material point (volume element) of the solid. The displacement vector u is the difference between these two locations. The vector element dX in the reference state is displaced, rotated, and deformed to become the vector element dx in the deformed state.
66
R. C. CAMMARATA
where i ¼ 1,2,3 and where Cartesian coordinates x ¼ x1, y ¼ x2, z ¼ x3 (similarly for Xi) have been used. The vectors dx and dX can be related by the tensor expression dxi ¼ Fij dXj
ð7:2Þ
where the second rank tensor Fij is called the deformation gradient defined as @xi ð7:3Þ Fij ¼ @Xj and where the standard convention that summation is to be taken over repeated indices (i.e., summation over j ¼ 1–3 in Eq. (7.2)) is assumed. Let dL denote the length (magnitude) of the vector dX in the reference state so that dL2 ¼ dXi dXi :
ð7:4Þ
Defining the Kronecker delta dij as dij ¼ 1 when i ¼ j and dij ¼ 0 when i 6¼ j, Eq. (7.4) can be written as dL2 ¼ dij dXi dXj :
ð7:5Þ
(It is noted that on the right-hand side of Eq. (7.5) summation over both of the indices i and j is to be taken.) Similarly, if the length of vector dx in the deformed state is denoted by dl then dl2 ¼ dij dxi dxj :
ð7:6Þ
Substituting Eq. (7.2) into Eq. (7.6) leads to dl2 ¼ Fij dXj Fik dXk : dl2
ð7:7Þ
dL2
– is a measure of the stretch that dX experiences as the The difference material evolves from the reference (undeformed) state to the deformed state. Using Eqs. (7.6) and (7.7), this difference can be expressed as ð7:8Þ dl2 dL2 ¼ Fij Fik dij dXj dXk ¼ 2Ljk dXj dXk where Ljk
1 Fij Fik dij : 2
ð7:9Þ
Ljk is called the Lagrangian strain tensor. Referring to Fig. VII.1, the displacement vector u is defined as u x – X, which can be written in indicial notation as u i ¼ x i Xi :
ð7:10Þ
GENERALIZED THERMODYNAMICS OF SURFACES
The deformation gradient can then be expressed in terms of u by @xi @Xi @ui @ui ¼ þ ¼ dij þ : Fij ¼ @Xj @Xj @Xj @Xj
67
ð7:11Þ
Substituting Eq. (7.11) into Eq. (7.9) and making the necessary changes in the indices allows the Lagrangian strain tensor to be written as 1 @uk @uk dkj þ dij : ð7:12Þ dki þ Lij ¼ @Xi @Xj 2 Multiplying out each term of Eq. (7.12), substituting the relation dikdjk ¼ dij, and simplifying leads to @uj @uj 1 @ui @ui þ þ : ð7:13Þ Lij ¼ @Xi @Xj @Xi 2 @Xj The strain tensor Lij in Eq. (7.13) has been expressed in terms of the material coordinates Xi, also called the Lagrangian coordinates. If it is assumed that the relation x ¼ x(X) can be inverted to give X ¼ X(x) then it is possible to define another strain tensor Eij in terms of the coordinates xi, called the Eulerian coordinates, by dl2 dL2 ¼ 2Eij dxi dxj :
ð7:14Þ
Proceeding in a manner analogous to that used to obtain Eq. (7.13) leads to the result that Eij, referred to as the Eulerian strain tensor can be expressed as @uj @uj 1 @ui @ui þ þ : ð7:15Þ Eij ¼ @xi @xj @xi 2 @xj Either Lij or Eij can be used to quantify the elastic deformation introduced into the reference state during a process that results in the deformed state. The small strain tensor can be defined as @uj 1 @ui þ : ð7:16Þ eij ¼ @xi 2 @xj If the derivatives of the displacement ui with respect to xj are small so that eij 1, then the derivatives of the displacement with respect to Xj will also be small and the small strain tensor can be written as 1 @ui @uj þ : ð7:17Þ eij 2 @Xj @Xi Furthermore, for this ‘‘small strain approximation,’’ the second order terms in Eqs. (7.13) and (7.15) can be neglected so that eij Lij Eij. Consider a strain state where the only nonzero component of Lij is L11. According to Eq. (7.9),
68
R. C. CAMMARATA
L11 ¼ 1/2 [(dl/dL)2 – 1], so that dl/dL ¼ (1 þ 2L11)1/2. Therefore, (dl – dL)/dL ¼ (1 þ 2L11)1/2 1, and in the small strain approximation where L11 e11 1, dl dL ð1 þ 2e11 Þ1=2 1 e11 : dL
ð7:18Þ
According to Eq. (7.18), e11 represents the change in the length per unit length of a line element in the x1-direction. This result can be generalized for a threedimensional small strain state to show that e11, e22, and e33 represent the change in length per unit length in the x1-, x2-, and x3-directions, respectively, and that for a volume element, the change in volume per unit volume will be (1 þ e11) (1 þ e22) (1 þ e33) – 1 e11 þ e22 þ e33. Similarly, for an area element in the x1–x2 plane, the change in area per unit area of an area will be e11 þ e22. (A corresponding twodimensional surface strain tensor can be defined using Eq. (7.16) where i, j ¼ 1,2.) 6 j are called the shear strain components and do The components of eij for which i ¼ not contribute to the volume and area changes but instead result in changes in shape. To express the work to impose an elastic deformation in a solid, it is necessary to define stress tensors conjugate to the measures of deformation used. This will be done from a thermodynamic point of view in a manner similar to how the pressure of a hydrostatically stressed body can be defined as P ¼ (@U/@V)S,ni (see Eq. (2.13)). Attention will first be given to the elastic work owing to an infinitesimal variation in the state of the deformed solid. Consider a volume element DV of the solid. The local densities of the internal energy, entropy, and number of moles of the components i can be defined, respectively, as u ¼ DU/DV, s ¼ DS/DV, and Gi ¼ Dni/DV. The fundamental equation associated with a variation in the state of the volume element that includes a variation in strain, deij, can be expressed as X du ¼ Tds þ sjk dejk þ mi dGi : ð7:19Þ i
where sjk is called the Cauchy stress tensor that can be defined as @u sjk : @ejk
ð7:20Þ
It is noted that in Eq. (7.19), summation over the repeated indices j and k is to be assumed. For a physically and chemically uniform solid, so that the strain state is uniform but not necessarily hydrostatic, both sides of Eq. (7.19) can be integrated over the volume of the solid to give X mi dni : ð7:21Þ dU ¼ TdS þ Vsjk dejk þ i
Another stress tensor can be defined by considering a variation in the reference state of solid. Let DV0 denote a volume element of the undeformed solid and
GENERALIZED THERMODYNAMICS OF SURFACES
69
define associated densities of the internal energy, entropy, and number of moles by u0 ¼ DU/DV0 , s0 ¼ DS/DV0 , and Gi0 ¼ Dni/DV0 . The fundamental equation associated with a variation in the state of this volume element that involves a variation in the deformation gradient, dFjk, can be expressed as X 0 0 0 0 du ¼ Tds þ V Tkj dFjk þ mi dG i : ð7:22Þ i
where Tkj is called the (first) Piola–Kirchoff stress tensor that can be defined as 0 @u Tkj : ð7:23Þ @ejk For a physically and chemically uniform solid both sides of Eq. (7.22) can be integrated over the volume of the solid to give X 0 dU ¼ TdS þ V Tkj dFjk þ mi dni : ð7:24Þ i
It is noted that ejk and sjk are symmetric tensors (i.e., ejk ¼ ekj) while Fjk and Tkj generally are not. Therefore it is important when using the latter to be clear about the sign conventions regarding the order of the indices. For a thermodynamic analysis involving a solid that requires the introduction of a finite deformation into the stress-free state to obtain the actual deformed state, Eqs. (7.22) and (7.24) are often used. However, if the amount of strain introduced into the stress-free state to create the actual state is small, then Eqs. (7.19) and (7.21) can be used. In addition, Eqs. (7.19) and (7.21) can be rigorously employed in Gibbsian thermodynamics to obtain the equilibrium conditions by taking the equilibrium state as the reference state and the deformed state as resulting from an infinitesimal variation imposed on the equilibrium state. VIII. Appendix B: Effect of the Dividing Surface Location on the Curvature Contributions to the Fundamental Equation
In his formulation of the dividing surface construction for a fluid interface, Gibbs129 expressed the general form for the fundamental equation for a surface of area A and principal curvatures c1 and c2 as X dU s ¼ T s dSs þ ms dnsi þ gdA þ C1 dc1 þ C2 dc2 : ð8:1Þ i
Gibbs noted that the curvature terms on the right-hand side of Eq. (8.1) can be rewritten as
129
J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Vol. 1, Dover, New York (1961), p. 225.
70
R. C. CAMMARATA
1 1 C1 dc1 þ C2 dc2 ¼ ðC1 þ C2 Þdðc1 þ c2 Þ þ ðC1 C2 Þdðc1 c2 Þ: 2 2
ð8:2Þ
He then gave an extended mathematical analysis to show that there is a particular location for the surface within the actual interfacial region between two uniform phases so that C1 þ C2 ¼ 0. He further argued that if the interfacial region is thin enough so that the surface can be considered as composed of area elements that are initially planar and subsequently deform into nearly spherical forms of small curvature, then c1 – c2 ¼ 0. The conclusion was that for a sufficiently narrow interfacial region there is a dividing surface location where, to a good approximation, C1 dc1 þ C2 dc2 ¼ 0. Gibbs’ analysis is not easy to follow and a simplified and more transparent treatment as given by Rice47 will now be reviewed. As Rice pointed out, it is convenient to introduce at the very beginning the assumption that the surface can be taken as composed of initially planar area elements that deform into spherical forms so that c1 ¼ c2 and C1 ¼ C2. In Fig. VIII.1a, lmno represents a portion of the interfacial region in its initial planar form and let pq be the associated part of the dividing surface. Also let x ¼ np ¼ oq so that it represents a measure of the position of the dividing surface within the interfacial volume element lmno and let the thickness of this volume element be denoted as y ¼ nl ¼ om. In Fig. VIII.1b, LMNO illustrates the interfacial volume element after it has been deformed into a spherical form. The curvature of the surface has varied from zero to 1/r, where r is the radius of the sphere that has a spherical cap PQ. Following Gibbs, consider a variation that is conducted under the constraints that there is no change in surface area A (so that the area of PQ equals the area of pq), and that there is no change in the surface excess entropy and number of moles. Under these constraints Eq. (8.1) reduces to dU s ¼ 2Cdc ¼ 2C=r (a)
l
m
p
q
n
o
ð8:3Þ
(b) L
M
P
Q
N
O
FIG. VIII.1. (a) lmno is a portion of a planar interfacial region contain a portion of the dividing surface pq. (b) LMNO is the same portion after it has been deformed into a spherical form; PQ is the deformed dividing surface (based on Ref. [47]).
GENERALIZED THERMODYNAMICS OF SURFACES
71
where c c1 ¼ c2 and C C1 ¼ C2. Rice noted that the only physical cause for the variation dUs is the change in the volume going from lmno to LMNO. Letting this volume change be denoted as dVS, it is possible to relate the change in the excess energy of the surface to the change in interfacial volume by dU s ¼ uS dV S :
ð8:4Þ
A/R2
The solid angle subtended by PQ is so that the volume of the spherical region LMNO is (A/3R2)[(r þ y – x) 2 – (r – x)2], where r – x and r þ y – x are, respectively, the radii for the spheres associated with the spherical caps NO and LM. To lowest order in x/r and y/r, this volume is equal to (A/3r2)[3r2(y – x) þ 3r (y – x)2 þ (y – x)3 þ 3rx2 þ x3] ¼ A[y þ (y2 – 2xy)/r]. This means that the change in volume going from lmno to LMNO is dV S ¼ Aðy2 2xyÞ=r:
ð8:5Þ
Substituting this into Eq. (8.4) and comparing this with Eq. (8.3) leads to dU s ¼ 2C=r ¼ uS A y2 2xy =r: ð8:6Þ It is seen that C can be positive or negative depending on the sign of y2 – 2xy, and that C ¼ 0 when x ¼ y/2. For this ‘‘C ¼ 0’’ placement of the dividing surface, the fundamental equation for the surface, Eq. (8.1), can be expressed without the curvatures terms (c.f. Eq. (5.13)). When Gibbs discussed capillary effects owing to a curved surface in a fluid system he generally assumed that the dividing surface was located at the ‘‘C ¼ 0’’ position. However, as discussed in Section V.14, when dealing with a surface between a multicomponent fluid and a single component solid, he located the dividing surface so that the excess number of moles of the solid component vanished. Although it is not to be expected that this dividing surface location will be the same as ‘‘C ¼ 0’’ position, Gibbs nevertheless did not include curvature terms in his fundamental equation for the surface. This implies that Gibbs assumed that as long as the width of the actual interfacial region was much smaller than the size of the uniform phases and that the surface curvatures were not too large, it was acceptable to ignore the curvature terms for any reasonable dividing surface location within the interfacial region. IX. Appendix C: Gibbs–Thomson Effects on Small Solids
Examples of the capillarity effects on the equilibrium behavior of a small spherical solid b of radius r immersed in a much larger fluid a will be reviewed based on the analysis of Cahn.33 The equilibrium conditions are summarized below: Tb ¼ Ta T
ð9:1Þ
72
R. C. CAMMARATA
Pb ¼ Pa þ 2f =r
ð9:2Þ
mbk ¼ mak þ 2ðf sÞOb =r
ð9:3Þ
for k a substitutional component; and mbp ¼ map
ð9:4Þ
for p a dilute interstitial component. For simplicity, it will be assumed that gases are ideal, solutions are dilute, and the solid is incompressible so that the molar volume of the solid Ob is fixed. These assumptions allow the use of easily employed expressions33 that lead to results that can be expressed analytically. Consideration is first given to the vapor pressure of a single component solid. The chemical potential of an ideal monatomic vapor of pressure Pa in equilibrium with a solid of radius r can be expressed as9,11 ma ðr Þ ¼ ma ð1Þ þ RT a ln
Pa ð r Þ Pa ð 1 Þ
ð9:5Þ
where R is the gas constant. Substituting Eq. (9.2) into Eq. (6.11) (the integrated Gibbs–Duhem equation for a single component solid) leads to mb ðr Þ ¼ mb ð1Þ þ 2Ob f =r:
ð9:6Þ
Inserting Eqs. (9.1), (9.5), and (9.6) into Eq. (9.3), noting that mb(1) ¼ ma(1), gives RT ln
Pa ð r Þ ¼ 2sOb =r: Pa ð 1 Þ
ð9:7Þ
It is seen that the equilibrium vapor pressure depends on s and not on f. (Since the solid is a single component material, the substitution g ¼ s can be made if the dividing surface location for which the excess number of moles of the solid component is equal to zero is used.) If the vapor species, rather than being monatomic, consists instead of N-atom molecules, then ma ðr Þ ¼ ma ð1Þ þ
RT a Pa ð r Þ ln a N P ð 1Þ
ð9:8Þ
leading to the result RT P a ðr Þ ¼ 2sOb =r: ln a N P ð1Þ
ð9:9Þ
Attention is now given to the solubility of a single component solid in a multicomponent fluid solution, assuming that the solid component to be dilute
GENERALIZED THERMODYNAMICS OF SURFACES
73
and completely dissociated in the fluid solution. Letting ca(r) denote the equilibrium concentration of the solid component in the fluid and assuming ideal solution behavior, the chemical potential of the solid component in the fluid is9,11 ma ðr Þ ¼ ma ð1Þ þ RT a ln
ca ðr Þ c a ð 1Þ
ð9:10Þ
where ca(1) is the saturation concentration for a large solid particle. Substituting Eqs. (9.1) and (9.3) into Eq. (9.10) and again noting that mb(1) ¼ ma(1) leads to RT ln
ca ðr Þ ¼ 2sOb =r: ca ð1Þ
ð9:11Þ
The dependence of the equilibrium melting temperature Tm of a single component solid on r is now derived. Let sa and sb denote the molar entropies of a and b, respectively. Integrating the Gibbs–Duhem equation, Eq. (2.15), for each phase leads to
ð9:12Þ ma ðr Þ ma ð1Þ ¼ sa Tma ð1Þ Tma ðr Þ and
mb ðr Þ mb ð1Þ ¼ sb Tmb ð1Þ Tmb ðr Þ þ Ob Pb ðr Þ Pb ð1Þ :
ð9:13Þ
Subtracting Eq. (9.12) from Eq. (9.13) and substituting the conditions Eqs. (9.1)–(9.3) yields Tm ð1Þ Tm ðr Þ ¼ 2sOb = sa sb r: ð9:14Þ Attention is now given to the equilibrium vapor pressure of a dilute component in a multicomponent solid. It is assumed that the component forms an N-atom molecule (or reacts with a vapor species to form a molecule containing N atoms of the component) in the vapor. Consider first the vapor pressure associated with a dilute interstitial component p. Assuming ideal behavior the chemical potential of the component in the vapor can be expressed in a manner similar to that given in Eq. (9.8): map ðr Þ ¼ map ð1Þ þ
RT a Pa ð r Þ ln a N P ð 1Þ
ð9:15Þ
where Pa now denotes the partial pressure of the molecular species containing component p. The chemical potential of component p in the solid can be written in a manner similar to that given in Eq. (9.6): mbp ðr Þ ¼ mbp ð1Þ þ 2Obp f =r
ð9:16Þ
74
R. C. CAMMARATA
where Obp @V b =@nbi is the partial molar volume of p in the actual solid.33 Subtracting Eq. (9.15) from Eq. (9.16) and inserting Eqs. (9.1) and (9.4) gives RT ln
P a ðr Þ ¼ 2NObp f =r Pa ð1Þ
ð9:17Þ
indicating that it is f and not s that determines the change in vapor pressure. For the case of a dilute substitutional component k, Eqs. (9.16) and (9.17) will still apply (substituting the subscript k for p). However, the chemical equilibrium condition, Eq. (9.3), must be used instead of Eq. (9.4), resulting in P a ðr Þ ¼ 2NOb s=r þ 2N Obk Ob f =r: ð9:18Þ RT ln a P ð1Þ The first term on the right-hand side of Eq. (9.18) reflects the work to change the area of the surface by accreting or dissolving a layer of the surface while the second term130 is associated with the work to stretch the surface if the partial molar volume Obk of the component k is different from the molar volume Ob. X. Appendix D: Critical Thickness for a Crystallographically Anisotropic Thin Film System
Crystal anisotropy can be accounted for in the analysis of thin film epitaxy by expressing the mechanical work terms to strain the film and surfaces in tensor form.131 The elastic strain energy to change the two-dimensional strain state in the film can be expressed as Atsbij debij and the surface work terms can be expressed as As dss ¼ 2As fij ss dij desij and Ac dsc ¼ 2Ac gij sc dij decij , where i,j ¼ 1,2. Using these terms in the analysis of Section III.6 leads to the following expression that generalizes Eq. (6.21):
ð10:1Þ tsbij ¼ fij þ gij ss þ sc dij : A symmetric, second rank ‘‘surface force tensor’’ can be defined131 as Bij ¼ [ fij þ gij (ss þ sc)dij] so that the condition of mechanical equilibrium can be expressed as sij ¼ Bij =t:
ð10:2Þ
130 Equation (9.18) corrects a sign error in the second term on the right-hand side given in Ref. [33]. This was pointed out to the author by K. Sieradzki. 131 R. C. Cammarata and K. Sieradzki, J. Appl. Mech. 69, 416 (2002).
75
GENERALIZED THERMODYNAMICS OF SURFACES
(a)
(b)
X2
X2
X1
X1
FIG. X.1. Representation ellipses for the surface pressure Bcij =t tensor (dashed) and coherency stress tensor scij (solid) for a completely lattice-matched film. (a) General system where Bcij =t and scij do not share the same set a principal axes. (b) System where Bcij =t and scij do share the same set a principal axes (based on Ref. [131]).
The dependence on orientation of a two-dimensional symmetric tensor of second rank can be graphically illustrated using a representation ellipse.132 A representation ellipse is plotted on a two-dimensional Cartesian coordinate system such that the length of any radius vector is equal to the reciprocal of the square root of the magnitude of the tensor in the radial direction. The major and minor axes of the ellipse lie along what are called the principal axes for the tensor, and the values of the tensor in the direction of the principal axes called the principal values. Let the surface force tensor evaluated at the strain state for a completely lattice-matched (coherent) film be denoted as Bcij . The critical thickness for epitaxy will be the value of t when the representation ellipse for the ‘‘surface pressure’’ Bcij =t makes tangential contact with the representation ellipse for the in-plane stress of a completely coherent film, scij . This is illustrated in Fig. X.1. The general case when scij and Bcij do not display the same set of principal axes is illustrated in Fig. X.1a. When scij and Bcij have the same set of principal axes, tangential contact will be made along one of those axes, as illustrated in Fig. X.1b. In this case, the critical thickness can be determined from Eq. (10.1) using the principal values for the tensor components. If the thin film-substrate system displays threefold or higher rotational symmetry about the axis perpendicular to the plane of the film, the coherency stress and surface stresses can be taken as scalars and the critical thickness can be obtained using Eq. (6.23).
132
J. F. Nye, Physical Properties of Crystals, Oxford Scientific, Oxford (1985).
SOLID STATE PHYSICS, VOL. 61
Materials Science of Hydrogen/Oxygen Fuel Cell Catalysis J. E RLEBACHER Department of Materials Science and Engineering, Johns Hopkins University, Baltimore, MD 21218
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Thermodynamics and Kinetics of the Hydrogen/Oxygen PEM Fuel Cell . . . . . . 1. The Open Circuit Potential of the Hydrogen/Oxygen PEM Fuel Cell . . . . . 2. Kinetics of Electrode Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Kinetics of the Hydrogen Oxidation Reaction . . . . . . . . . . . . . . . . . . 4. Pt Catalyst Electrode Dynamics in Acidic Media in the Absence of Hydrogen 5. Pt Electrode Catalysis in Acidic Media the Presence of Hydrogen . . . . . . . 6. Kinetics of the Oxygen Reduction Reaction . . . . . . . . . . . . . . . . . . . III. The Catalyst Layer in PEM Fuel Cells . . . . . . . . . . . . . . . . . . . . . . . . . 7. General Description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. The Catalyst Layer in the Membrane Electrode Assembly . . . . . . . . . . . 9. Synthesis Methods of Catalysts for PEM Fuel Cells . . . . . . . . . . . . . . . 10. Degradation of Catalyst Layers . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Improving the Hydrogen Oxidation Reaction . . . . . . . . . . . . . . . . . . . . . 11. General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Materials for CO-Resistant HOR Catalysts . . . . . . . . . . . . . . . . . . . . V. Improving the Oxygen Reduction Reaction . . . . . . . . . . . . . . . . . . . . . . 13. New Insights into Oxygen Reduction from Density Functional Theory . . . . 14. Calculations Pertaining to the Oxygen Reduction Reaction . . . . . . . . . . . 15. Experimental Development of New Metallic Catalysts for ORR . . . . . . . . 16. Binary Alloy Electrocatalysts . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. Pt-Skin Electrocatalysts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. Dealloyed Catalysts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Synthesis and Outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2009 Elsevier Inc. (USA) All rights reserved.
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J. ERLEBACHER
I. Introduction
The fuel cell celebrates nearly 175 years since its invention. This clever electrochemical device tricks an exothermic chemical reaction into releasing its energy in the form of electrical work rather than as heat, thus avoiding the intrinsic Carnot inefficiency associated with heat engines. In theory, a motor run directly from a fuel cell can convert fuel energy to mechanical energy with much higher efficiency than an internal combustion engine. Fuel cells can be run from a wide variety of fuels, helping reduce dependency on foreign petroleum. Because high temperatures are not intrinsically part of their operation, fuel cells can be run in low or moderate temperature environments. Many reviews and books summarizing the promise (and challenges) of fuel cells have appeared over the last 15 years.1–4 Given their promise, why have not fuel cells become a mainstream technology? This question is loaded with political and economic concerns, but from the scientific standpoint, the basic answer is that fuel cells just do not work well enough—yet. On a pragmatic engineering level, power densities and fuel energy densities are not as high as burned petroleum, and fuel cell systems do not last as long as internal combustion engines. On a fundamental and microscopic level, we think most fuel cell researchers will agree that a part of the answer to why fuel cells do not work well enough is that while the fuel cell is easy to describe in the context of introductory thermodynamics, actual functioning fuel cells contain significant amounts of microstructural complexity, the chemical reactions occurring at the various catalytic electrodes are still being clarified, and the degradation mechanisms are still being debated. Here we focus on the ‘‘simplest’’ of all fuel cells, the acidic hydrogen/oxygen fuel cell using Pt-based catalysts. The overall chemical reaction 2H2 þ O2 ⇄ 2H2 O
ð1:1Þ
is highly exothermic with a reaction energy of 572 kJ mol1 (5.9 eV mol1). The fuel cell schematic shown in Figure I.1 shows the basic operating principle. Instead of mixing and burning hydrogen and oxygen, the fuel cell uses catalytic electrodes to drive two electrochemical half-reactions, hydrogen oxidation to protons at the anode (the hydrogen oxidation reaction, HOR), and oxygen reduction to water at the cathode (the oxygen reduction reaction, ORR). Electrons traveling through the external circuit perform work, perhaps driving a motor, and
1
G. W. Crabtree and M. S. Dresselhaus, MRS Bull. 33, 421 (2008). B. C. H. Steele and A. Heinzel, Nature 414, 345 (2001). 3 K. B. Prater, J. Power Sources 51, 129 (1994). 4 T. R. Ralph and M. P. Hogarth, Platinum Met. Rev. 46, 3 (2002). 2
MATERIALS SCIENCE OF H/O FUEL CELL CATALYSIS
PEM
Catalytic anode
79
Catalytic cathode Oxidant (O2) in
H+
O2 + 4H+ + 4e− → 2H2O
Fuel (hydrogen) in 2H2 → 4H+ + 4e−
Product (water) out e−
e−
To device (e.g., motor)
From device Membrane Electrode Assembly (MEA)
FIG. I.1. Schematic cartoon of the hydrogen/oxygen proton exchange membrane (PEM) fuel cell. Hydrogen is fed to the catalytic anode (left) where it is oxidized to protons. Electrons travel through a proton conducting membrane, and recombine with electrons and oxygen at a catalytic cathode (right) to form water. The combined electrodes plus PEM is the membrane electrode assembly (MEA).
they recombine with the protons that travel through a proton conducting membrane. The combination of the two catalytic electrodes and the proton conducting membrane is called the membrane electrode assembly (MEA). Section II.2 reviews the thermodynamics and electrochemistry of the H2/O2 fuel cell. Grove’s original 1837 fuel cell also used hydrogen and oxygen.5 One aspect of his experiment that does not receive much attention is the nature of the catalyst he used for each of the electrode reactions. That he used platinum in not the unusual part. Platinum remains the most common catalyst material for both HOR and ORR, and the detailed nature of each reaction over Pt remains a subject of active research. Much of the discussion here revolves around platinum. What is interesting is that Groves used ‘‘platinized platina,’’ that is, he used an electroless reduction of platinum (probably from a chloroplatinate solution) over a sheet of platinum. Presumably, as-received platinum did not work. This could have been
5
W. R. Grove, Phil. Mag. 21, 417 (1842).
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J. ERLEBACHER
due to a physical surface area effect, and a rough, chemically deposited platinum film increased the effective surface area for catalysis to occur, but we are free to speculate that the more likely explanation is that surface contaminants destroyed the reactivity of the ‘‘platina’’ sitting around the lab. The experimental subtlety encountered by Groves highlights the difficulty found even in modern fuel cell research and engineering. Small amounts of impurities can quickly degrade fuel cell performance, such as carbon monoxide in the fuel inlet stream which irreversibly binds to the anode Pt, and there are other effects that can quickly kill performance, such as an oversaturation of water in the catalyst environment condensing within the catalyst layer and blocking reaction sties. The fuel cell catalyst must thus be designed not only with efficiency in mind, but also with intrinsic environmental tolerance. This is a challenging Material Science problem which has not been completely solved. An important point to consider is that the H2/O2 fuel cell really does work impressively well. Generated power densities are high enough that vehicles and electronics can be powered by reasonable amounts of hydrogen. There are challenges in hydrogen storage and generation, but the rate of progress in these areas is great enough that optimism is warranted. In Section II.3, we review the microstructure and performance characteristics of state-of-the-art H2/O2 fuel cells, and the take-home lesson should be that there have been huge increases in the power density generated by these devices in the past 50 years. Some of this increase is attributable to ‘‘inspired intuition’’ ala Groves’ own intuition to use platinized platina—for instance, Wilson and Gottesfeld’s insight in the early 1990s to mix ionomer with the catalyst layer6,7—but much of the increase in fuel cell performance has been due to the hard work of characterizing fuel cell structure/properties relationships. Currently, we are at a stage where the problems and challenges in optimizing the H2/O2 fuel cell are describable at the atomic and mesoscales. Atomistic details of the surface reactions are being probed by numerous modern analytic approaches, and the microstructural details of the catalyst layer are being probed at ever finer scales. Simulation methods employing the most advanced molecular dynamics (MD) and density functional theory (DFT) methods are being used to attack these problems. Given this status, a review of fuel cell catalysis from the vantage point of surface physics is warranted, and we present this exercise here. We will attempt to distinguish this review from the many excellent reviews of fuel cell technology by a pedagogical focus on the physics of the catalytic
6 7
M. S. Wilson and S. Gottesfeld, J. Appl. Electrochem. 22, 1 (1992). M. S. Wilson and S. Gottesfeld, J. Electrochem. Soc. 139, L28 (1992).
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81
reactions in the H2/O2 fuel cell, and even more specifically on the so-called proton exchange membrane (PEM; also commonly called the polymer electrolyte membrane) fuel cell. We exclusively concentrate on metallic catalysts containing Pt, a transition metal, which are the best developed to date, and have a significant history. By keeping such a detailed focus, our goal is to highlight how the details of the surface science of catalysis have translated into high performance energy conversion devices, and also to highlight how rational materials synthesis helps to address a number of unanswered questions that may significantly impact the design of next-generation catalysts and perhaps bring fuel cells into the mainstream. The structure of this paper is as follows. In Section II.2, we discuss the thermodynamics and chemistry of the H2/O2 fuel cell, followed by a microstructural description of state-of-the-art PEM fuel cells in Section II.3. Then, in Section II.4, we focus on the HOR anode reaction, which involves the dissociation of hydrogen into protons and electrons. This discussion involves reactant transport into and out of the catalyst layer, poisoning of the catalyst by fuel impurities, and strategies to improve all of these issues. In Section II.5, we will focus on the even harder problem of oxygen reduction at the cathode. All the problems of reactant/product transport in and out of the catalyst layer that exist in the anode are there, but are compounded by corrosion problems associated with the high potential at this electrode, as well as competitive reactions that decrease ORR efficiency. We will review how advanced computational methods, particularly DFT methods, are producing predictions/calculations that are driving experimental discovery of new materials. Finally, in Section II.6, we present a synthesis, identifying a few key areas where surface physics can offer solutions to crucial catalyst design problems. To be perfectly frank, the inclusion of a review of the fuel cell technology within the pages of Solid State Physics may at first blush seem odd. After all, do not fuel cells sit in the realm of (physical) chemistry, and electrochemistry in particular? Certainly, from a historical perspective this is true, and is reflected in the older literature citations in this article. However, the confluence of three factors—(i) development of techniques to fabricate and control the morphology and microstructure of nanoscopic materials, (ii) the application of computational methods to the simulation of chemical reactions at atomic scale, and of course (iii) the critical importance of solving energy technology problems in order to alleviate climate change and geopolitical problems—has led to a change in perspective on the problem, and there are great opportunities for surface physics to contribute to these challenges. Professor Ehrenreich understood this, and this monograph is dedicated to his memory.
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II. Thermodynamics and Kinetics of the Hydrogen/Oxygen PEM Fuel Cell
Comprehensive discussions of the electrochemistry of fuel cells can be found in a number of sources, the review in Ref. [8] being particularly thorough. Here, we review the relevant thermodynamics and kinetics of PEM electrode reactions. We have tried to be microscopic physics oriented, aiming to highlight the assumptions and physics that dedicated electrochemists may take for granted. 1. T HE O PEN C IRCUIT P OTENTIAL OF THE H YDROGEN /O XYGEN PEM F UEL C ELL The electrochemistry of the fuel cell involves oxidation of hydrogen at an anode, and a reduction of oxygen to water at a cathode. The half reactions at each electrode are as follows: Anode : H2 ⇄ 2Hþ þ 2e ðhydrogen oxidation reaction; HORÞ
ð2:1Þ
Cathode: O2 þ 4Hþ þ 4e ⇄ 2H2 Oðoxygen reduction reaction; ORRÞ
ð2:2Þ
These reactions combine to form the overall reaction, Eq. (1.1). The basic hydrogen/oxygen fuel cell physically separates these reactions and allows the following spontaneous reactions to occur: a. Hydrogen dissociates at the catalytic anode, breaking into two electrons and two protons. This is an oxidation of hydrogen, functionally increasing the oxidation state of the H2 molecule by þ2, dissociating it into protons and electrons. Electrons leave this electrode (technically defining this electrode as an anode) and may flow through an external circuit to arrive at the cathode. b. At the cathode, electrons and protons from the anode are catalyzed to combine with oxygen being fed to the electrode to form water. Four electrons and four protons reduce two O2 molecules to two water molecules. This process is an example of an electrochemical reaction. When the reaction is spontaneous, and we can get current to flow and use it to do external work, the system is called a galvanic cell. More specifically, when one of the substances in the galvanic cell is consumed but we continuously replenish it, it is called a fuel cell. The anode here is the negative terminal, and the cathode is the positive one.
8
L. G. Austin, in Handbook of Fuel Cell Technology, ed. C. Berger, Prentice-Hill, New Jersey (1968), p. 5.
MATERIALS SCIENCE OF H/O FUEL CELL CATALYSIS
83
A spontaneous process at constant temperature T and pressure P results in a decrease in system Gibbs free energy. When the spontaneous process is run reversibly, this decrease in Gibbs free energy is equal to the maximum work that can be done by the system: dG ¼ dWmax ¼ dW rev :
ð2:3Þ
Here, both dG and dW rev will be negative for a spontaneous process. We conform to the convention that positive work represents work done on the system by the external environment. The work done by a system in making an electrical current flow is the product of the charge transported q (in Coulombs) and the potential difference Df (in Volts). In the fuel cell, if dn moles of protons are transported internally from the negative to the positive terminal, then dn moles of electrons, or Fdn moles of charge, are also transported along the external circuit from the negative terminal at potential fI to the positive terminal at potential fII ; here, F is Faraday’s constant, 9.65 104 C mol1. One way to make this process occur reversibly is shown in Figure II.1: let the ‘‘external load’’ be a battery with infinitesimally smaller potential difference than the cell inside the control surface shown. The spatially varying electrostatic potential f is sketched in the bottom half of the figure. The amount of work performed by the system in the reversible process under consideration is dW rev ¼ ðdnFÞðfII fI Þ
ð2:4Þ
or, defining the system’s electromotive force (EMF, in volts (J C1)) as Df ¼ fII fI , dW rev ¼ FDfdn:
ð2:5Þ
dG ¼ FDfdn:
ð2:6Þ
Therefore,
Now replace the external battery with a very large resistor (or make any other change so that the process is irreversible away from the system boundary of the fuel cell). As long as the process was reversible within and at the boundary of the system, then all system properties must change in the same way as above. This is because the system cannot tell whether reversible processes or irreversible processes are going on far from its borders. Therefore, the amount of work done on or by the system must be unchanged. For this reason, Eq. (2.6) still describes the drop in Gibbs free energy of the system for an externally irreversible process, so long as the process is reversible within and at the system boundary. In this case fðxÞ might vary in space in a complex way on the outside of the cell, but Df would have to be interpreted as the actual difference in
84
J. ERLEBACHER Electrochemical cell
“−”
2H+ + 2e−
H2
H+
External load
O2 + 4H+ + 4e−
“+”
Anode
i e−
2H2O
“+”
“−”
Cathode
i
e−
φΙΙ φΙ
Δφ Axial position, x
FIG. II.1. Electrochemical fuel cell (enclosed by dashed line representing a control surface; not to scale), opposing battery as external load, paths of electron and conventional current flow, and electrostatic potential versus position along the common axis of the cell and load. In a galvanic cell, the cathode (at which the reduction of oxygen in a PEM fuel cell occurs) has a more positive potential than the anode (at which the oxidation of hydrogen in the PEM fuel cell occurs).
electrostatic potential between the fuel cell terminals at the system boundary. In practice, internal reversibility is approached only when the current is small, and this occurs when the external load is large compared to the internal resistance of the cell. When one adds a large resistor (large enough to dominate all other resistances in the circuit) and an ammeter in series with the battery and the fuel cell, and continuously varies the electromotive force of the ‘‘external battery’’ (this could be a DC power supply instead, the output of which can be readily varied in this way; in fuel cell testing, one uses an ‘‘electronic load’’ for this purpose), the measured current density and power density of a typical hydrogen/oxygen PEM fuel cell is as shown in Figure II.2. The intersection of current/voltage curve with the vertical axis serves as our way to ‘‘measure’’ the electromotive force of the system. By analyzing the small current limit, we can predict the value of the open circuit potential. To do this, we correlate the free energy change upon transfer of dn electrons, Eq. (2.6), to the free energy change associated with all the reactant and products in the system. If each electrode is in local equilibrium, the chemical potentials of each species in each half reaction are related by þ 2manode mH2 ¼ 2manode e Hþ
ð2:7Þ
mO2 þ 4mcathode þ 4mcathode ¼ 2mH2 O : e Hþ
ð2:8Þ
85
Ideal cell voltage (~1.2 V) Activation losses
1.0
Resis
tance
0.8
0.7
(iR) lo
sses
Power density 0.5
Voltage 0.6
0.3
0.4 0.2
Cell power density (A/cm2)
Cell el (ope ectromo n cir t cuit ive force p o te n ti a l Cell voltage (V) )
MATERIALS SCIENCE OF H/O FUEL CELL CATALYSIS
0.1 0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Current density (A/cm2) FIG. II.2. Representative current density/voltage and current density/power density characteristics of a typical PEM fuel cell. Activation losses due to the high overpotential required to drive the cathodic oxygen reduction reaction dominate the overall cell losses, but are compounded by resistance losses at higher current densities.
Solving for the chemical potentials of the electrons at each electrode and taking their difference yields the free energy change to transfer dn moles of electrons between the electrodes; we find manode dn dG ¼ mcathode e e 2mH2 O mO2 4mcathode mH2 2manode Hþ Hþ dn: ð2:9Þ ¼ 4 2 Equating this expression to Eq. (2.8) gives us the desired potential difference: mH2 2manode 1 2mH2 O mO2 4mcathode Hþ Hþ : ð2:10Þ Df ¼ F 4 2 Most PEM fuel cells are run at temperatures and pressures greater than the thermodynamic standard (298.13 K, 1 bar). At a nonstandard pressure, we use the notation s for a pure component. Using this notation, we write the chemical potential of a generic component X in a mixture as mX ðT; PÞ ¼ ms X ðT Þ þ RTlnðaX Þ
ð2:11Þ
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J. ERLEBACHER
where it is understood that ms X ¼ mX ðT Þ þ RTlnðP=P Þ. Substituting into Eq. (2.10) for the chemical potential of each component and simplifying yields 1 1 1 1 s 1 s;cathode s;anode þ m mH2 O ms m m Df ¼ þ þ H F 2 4 O2 2 H2 F H " cathode 4 ## ð2:12Þ RT 2 aH þ þ ln aO2 ðaH2 Þ 4 4F aanode þ H
In an operating fuel cell at steady state (at any potential), protons will diffuse to the cathode, and away from the anode to form two charged double layers. Under open circuit conditions, the dimensions (width, charge) of these double layers are such that the chemical potential of protons (including the electrostatic energy contributions) is uniform across the cell and there is no flow of current. When current is drawn from a fuel cell, one generally still makes the assumption that each half-reaction is in quasi-equilibrium, just not sitting at the open circuit potential. Under this assumption, Eq. (2.12) still holds, except that the chemical potential profile of protons across the cell becomes more complex. One may note, however, that whenever current is flowing the supply of protons to the electrolyte from the anode must equal the rate at which protons are being consumed at the cathode. Therefore, we may approximate the concentration (activity) of protons as uniform across the cell, with their current being driven by a chemical potential gradient set up by a potential difference across the electrodes. We can thus analyze the proton current through the membrane by starting with the phenomenological law relating the flux J to the mobility Mð¼ D=RT Þ, the concentration C, and driving force rm, namely, JHþ ¼
DHþ CHþ rm: RT
ð2:13Þ
The flux of protons is proportional to the current of protons IHþ through the electrolyte cross-sectional area A2 JHþ ¼
IHþ A2
ð2:14Þ
and the gradient in chemical potential can be approximated by Dm=L, where L is the thickness of the electrolyte layer. Expressed in this way, we find 2 1 A DHþ CHþ s IHþ ¼ DmHþ ¼ ms ð2:15Þ Hþ ;cathode ðT Þ mHþ ;anode ðT Þ L RT
MATERIALS SCIENCE OF H/O FUEL CELL CATALYSIS
and this expression naturally leads to a proton resistivity 1 1 A2 DHþ CHþ rHþ ¼ : F L RT Substitute for DmHþ in Eq. (2.12) to find the overall result: i 1 1 1 1 s RT h 2 þ IH þ þ r mH2 O ms m ln P ð P Þ : Df ¼ O H H 2 2 F 2 4 O2 2 H2 4F
87
ð2:16Þ
ð2:17Þ
The second term of this expression is usually introduced as an ‘‘ohmic overpotential’’ ohmic ¼ rHþ IHþ associated with resistance losses. This is a proper interpretation, but the derivation here specifically addresses the thermodynamic origins of this quantity and indicates that this ohmic overpotential would exist even if proton transport across the electrolyte were reversible though not infinitely fast (granted, an unrealistic hypothetical). Overall, the term shows why increasing the proton diffusivity and/or decreasing the electrolyte thickness is advantageous (the latter must be done at the same time minimizing crossover of un-reacted species from one electrode to the other, which really is driven by concentration gradients). It also shows that drawing current intrinsically reduces the cell potential. In any analysis of fuel cell electrode kinetics, this term will be a linear offset. We turn now to the first term in Eq. (2.17). Water is a reaction product and is actively removed from the cell. Therefore, its activity is zero, and mH2 O is equal to the (temperature and pressure corrected) formation energy of water, DGf;H2 O . The variation of the molar free energies for water, oxygen, and hydrogen with temperature at constant pressure is found by applying dm ¼ Sm dT to each component, keeping in mind that the standard formation free energies of oxygen and hydrogen are zero. Integrating from the standard temperature with the reasonable assumption that the molar entropy is constant over experimental 1 1 1 1 temperature scales (S H2 O ¼ 6:6177 J mol K , SO2 ¼ 205:15 J mol K , 1 1 SH2 ¼ 130:68 J mol K ), we have that 1 1 1 1 1 1 m ms ms ¼ DGf;H2 O ðT T ÞS H2 O þ ðT T ÞSO2 2 H2 O 4 O2 2 H2 2 2 4 1 1 1 þ ðT T ÞS H2 116:6 J mol K ðT T Þ 2
ð2:18Þ
Of course, a better estimate will account for the variation in entropy and heat capacity with temperature, but this kind of estimate is often used. It is interesting to note that experimental data for the open circuit voltage (OCV), corrected for crossover, is best fit using a formation energy of liquid water, DGf;H2 O ¼237.18 kJ mol1, for which the corresponding cell potential is
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J. ERLEBACHER
1.229 V (i.e., multiplying by 1=2F), even for cells run at temperatures above 100 C. Especially for the higher temperature cells, one might expect that the formation of gaseous water is most relevant, for which the formation free energy is 228.59 kJ mol1(with corresponding cell potential 1.184 V). This observation indicates that the water formed in a fuel cell is always bound in a condensed phase, and thus must be actively transported through the various phases, that is, from the catalyst layer to the electrolyte and out through the exhaust. Taken together, Eq. (2.17) is written as i RT h ln PO2 ðPH2 Þ2 : Df ¼ 1:229 V 1:17 103 V K1 ðT T Þ rHþ IHþ þ 4F ð2:19Þ The prefactor to the second term varies in the literature, due to varying values for the entropies of oxygen and hydrogen, and uncorrected temperature dependencies for the entropy. The dependence of the OCV on temperature is a relatively minor effect anyway, lowering the theoretical value from 1.2 V at 25 C to 1.14 V at 120 C.9 2. K INETICS
OF
E LECTRODE R EACTIONS
In principle, if the reactions at each electrode were perfectly reversible, a fuel cell should be able to maintain a voltage near 1.2 V, or at least decrease linearly versus current with a slope equal to the cell’s internal proton resistance. In practice, the reactions at each electrode are not reversible and actual performances dramatically differ from ideal behavior. One central measure of the performance of a fuel cell is its polarization behavior, which relates the current passing through the cell to the voltage across it (an iV curve). Figure II.2 shows a schematic polarization curve and corresponding geometric power density (i.e., per area of electrode, not area of catalyst surface), for which the magnitude of the current density is approximately state-of-the-art at the time of writing (2008). Power densities near 1 W cm2 are now common. When extraordinary measures are made to optimize performance, power densities have been brought as high as 1.2 W cm2 (Figure II.3).10 A major source of loss in the PEM fuel cell is the ohmic resistance seen by protons crossing the polymer (ionomer) membrane. The most common material for the PEM fuel cell membrane is Nafion, whose composition is a hydrophobic
9
J. Zhang, Y. Tang, C. Song, J. Zhang, and H. Wang, J. Power Sources 163, 532 (2006). A. J.-J. Kadjo, P. Brault, A. Caillard, C. Coutanceau, J.-P. Garnier, and S. Martemianov, J. Power Sources 172, 613 (2007).
10
89
MATERIALS SCIENCE OF H/O FUEL CELL CATALYSIS
1.2
1.4
1.0
1.2
Ecell/V
0.8 0.6 0.6 0.4
0.4
0.2 0.0 0.0
P/W/cm−2
1.0
0.8
0.2 0.0 0.5
1.0
1.5 2.0 j/A cm−2
2.5
3.0
3.5
FIG. II.3. High performance current density/voltage and current density/power density characteristics of a H2/O2 PEM fuel cell. This figure has been adapted from Ref. [10], with other data removed from the figure using Photoshop. Cell operating conditions: anode Pt loading 0.35 mg cm2, cathode Pt loading 1 mg cm2, cell temperature ¼ 85 C, cell pressure ¼ 3 bar (anode and cathode gasses).
perflorinated hydrocarbon (Teflon-like) on which have been attached short side chains terminated with hydrophilic sulfonic acid (SO 3 ) head groups. Microstructurally, Nafion that has been soaked in water microphase separates into aqueous and nonaqueous domains. The exact details of the structure of hydrated Nafion continue to be debated, but there is growing consensus primarily due to analysis of small-angle X-ray and neutron diffraction studies that the water channels are arranged in tubular, inverted micelle cylinders with average diameter 2.4 nm that are mechanically bound by small regions of unhydrated Nafion crystallites that physically crosslink the micelles.11,12 As regards proton conductivity, protons move through Nafion via a hydronium ion complex (H3Oþ), hopping from sulfonic acid group to sulfonic acid group along the water/ionomer interface. Significant effort has been made to find a suitable material replacement for Nafion, but to date the proton conductivity through Nafion, 0.1 S cm1,13 remains higher than all others in the sub 100 C temperature range in which it is desired for PEM fuel cells to operate. The structural model for Nafion shows that the water channel diameter is of order or slightly smaller than the catalyst particles to and from which they direct protons, but the nature of the Nafion/catalyst junction is still largely unknown.
11
K. Schmidt-Rohr and Q. Chen, Nature 7, 75 (2008). O. Diat and G. Gebel, Nature 7, 13 (2008). 13 M. A. Hickner and B. S. Pivovar, Fuel Cells 5, 213 (2005). 12
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A recent experimental study has measured the interfacial resistance between the catalyst layer and a Nafion membrane (8–57 mO cm2, depending on the processing of the membrane), and this is an important but incomplete step toward clarifying the Nafion/catalyst junction.14 Another approach to this problem has been molecular dynamics modeling, and some of the first work in this area is appearing. Liu et al., for instance, studied Nafion adsorption onto simulated Pt(100) and found a strong surface/ionomer interaction, albeit using simple Lennard–Jones potentials in the absence of surface charging due to the cell potential.15 Another source of losses in the PEM fuel cell arises from reactant mass transport limitations at high current densities. Typically, the power generated at high current densities drops quickly (p2 A cm2). Amelioration of these kinetics losses are connected to design of the MEA housing, so we will not concern ourselves here with this issue. Rather, we will concentrate on the operating conditions near 0.7–0.9 V, where activation losses dominate, and the kinetic details of the catalyzed half-reactions are of primary concern. Figure II.4 shows an experimental polarization curve similar to the schematic in Figure II.2, but with the effect of the internal cell resistance subtracted out; this is the so-called resistance-corrected iV curve. The deviation from ideality is obvious even at low current densities, under which conditions the cell voltage drops from values typically near 1.0 to 0.9 V. This drop is primarily attributable to irreversibility of the electrode reactions (so-called activation losses). Under clean operating conditions, the majority of this irreversibility is attributable to an inefficient ORR at the cathode, as the HOR reaction over platinum catalysts is one of the most reversible catalytic reactions known. However, in the presence of impurities that strongly adsorb onto Pt, for example, residual CO in the fuel stream, the HOR reaction itself becomes more irreversible, and leads to further activation losses. Qualitatively, we can understand the decrease in cell voltage in the following way. First consider if the HOR is not efficient. In this case, a fixed current can only be drawn if the potential fI of the anode becomes more positive, that is, creating an environment in which it is easier to strip electrons (oxidize) from hydrogen (the absolute value of this potential shift is the anodic overpotential an ). Similarly, if ORR is not efficient, then a fixed current can only be drawn if the potential fII of the cathode becomes more negative, that is, creating an environment in which it is easier to add electrons (reduction) (the absolute value of this potential shift is the cathodic overpotential cath ; by convention, both an and cath are reported as positive quantities). In both cases, the cell
14 15
B. S. Pivovar and Y. S. Kim, J. Electrochem. Soc. 154, B739 (2007). J. Liu, M. Selvan, S. Cui, B. Edwards, D. Keffer, and W. Steele, J. Phys. Chem. 112, 1985 (2008).
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0.94 0.40/0.40 mgPt/Pt/cm2 0.92
0.40/0.24 mgPt/Pt/cm2 0.40/0.15 mgPt/Pt/cm2
0.90
EiR-free [V]
0.88 0.86 0.84 0.82 0.80 0.78 0.01
0.1
1
ieff [A/cm2]
FIG. II.4. Resistance-corrected iV plot from Ref. [21] showing exponential drop-off in current density at higher cell potentials for three cathode Pt loadings (0.15, 0.24, and 0.40 mg cm2) at the same anode Pt loading, 0.40 mg cm2. Experimental details are found in the reference. Note that in all cases, the Tafel slope is near 0.6 V per decade.
potential Df will decrease. The degree to which the potential is suppressed will depend on the particulars of the reaction; in PEM fuel cells, the loss is dominated by the ORR inefficiency. Activation losses are undesirable, obviously, because the power generated at any current density will be reduced, and the excess energy associated with the reaction inefficiency will be released as heat. This heat can be substantial in high power fuel cell stacks, and the extra complexity of drawing this heat away, or at least managing it efficiently, makes engineering of these stacks more complex. The reaction kinetics at each electrode interface are quite complex in a fuel cell, and strongly depend on current density, gas purity, operating voltage, the size, shape, composition and crystallography of the catalyst itself and the microscopic structural environment in which the catalyst sits. The last of these affects details of reaction kinetics via the reaction and mass transport mechanisms that are available for the various interface processes to occur. For example, in aqueous solution, protons can desorb anywhere on the surface via protonation of a nearby water molecule but, in contrast, although fuel cells work under humidified conditions, the catalyst does not sit in condensed water. Contact is made to the electrolyte through ‘‘three-phase points’’ (which in principle could be
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molecularly small) where Nafion ionomer is in proximity to the catalyst and reactant gases. Protons must either first diffuse to the three-phase point before protonating a water molecule or protonate a water molecule which itself then has to diffuse to the three-phase point. Such considerations exist even when mass transport limitations (i.e., at extremely high current densities) are not operative. The point is that reactant transport may not be rate limiting, but the mechanisms of reactant/product transport may differ in the fuel cell compared to aqueous systems. Comparisons between the two must be considered carefully. Despite the potential complexities at the microscopic level, the fuel cell is an electrochemical cell, and thus generalities associated with electrode kinetics must still apply. Electrode kinetics are described by charge transfer reactions occurring upon the electrode metal catalyst involving charged species transfer across electrical double layers. In the fuel cell schematic in Figure II.1, the electrical potential across the cell at open circuit and under load is shown. Two charged double layers are shown, which are due to the migration of charge through the electrolyte that builds up at each interface until a dynamic equilibrium (equal and opposite oxidation/reduction reactions) are established at each interface. Many models exist for the double layer in aqueous systems and are reviewed in any good electrochemistry text (e.g., Ref. [17]). They range in complexity from the simple Helmholtz model, which pictures each electrode as two charged planes establishing a parallel-plate capacitor, to the Gouy–Chapman model, which pictures a diffuse trail-off of charge into the electrolyte solution, to the models that combine elements of both, that is, a narrow charged double layer due to adsorption of ions onto the surface, along with a diffuse trail-off of charge into the solution. In a fuel cell, the nature of the double layer is debated. Central to the problem is the inability of the sulfonic acid groups on the ionomer to diffuse. At best, when deprotonated, these groups could be mechanically pulled to a cathode. Therefore, in porous catalyst layers with Nafion electrolytes, it is difficult to see how the double layer can be very diffuse as that would involve a gradient in charge along ionomeric chains that should be screened by nearby particles. This may be possible, but the likely scenario is that the electrical double layer is confined to molecular distances from the interface and mostly built up via adsorbed water, reactants, and intermediates. A natural question to ask at this point is how much potential is dropped at each interface? Even at open circuit conditions, this is a difficult question to answer, as it depends on the structure of the double layer. For analysis of electrode kinetics, it is generally unnecessary to know the specific voltage drop across a particular double layer. Rather, we focus on how changes in the potential difference across double layer affect current, using the OCV condition (zero net current across each double layer) as our baseline. Details of electrode kinetics in this context are associated with the Butler–Volmer (B–V) equation, which relates the current
MATERIALS SCIENCE OF H/O FUEL CELL CATALYSIS
93
density i across a double layer associated with a reaction Ox þ e ⇄ Rd to the a difference in actual electrode potential E and the equilibrium potential E0 , a quantity called the activation overpotential (the overpotential convention is to report it as positive, ¼ jE E0 j; at the anode E > E0 , and at the cathode E < E0 ). Here we choose to cite two forms of the generalized form of the B–V equation. The first allows for the activities of the reactants and products to be themselves functions of the overpotential, and the parameters are connected to microscopic physical processes. The second is a form used to measure an empirical parameter b, known as the Tafel slope, that can be compared to theories. For a cathodic overpotential ( ¼ E0 E), the expressions are 22 3 2 3 3 aOx aRd i ¼ i0 44 0 5expðnaF=RTÞ 4 0 5expð nð1 aÞF=RTÞ5 aOx aRd 2 0 1 0 1316 ð2:20Þ 2:3 2:3 A exp@ A5 : ¼ i0 4exp@ b b Derivations of the B–V equation may be found in a number of sources.17,18 Here, a is called the transfer coefficient (for net reactions limited by a step involving a single electron transfer a 0:5), n is the number of charges transferred in the rate-limiting reaction, and i0 is a materials-dependent function of the equilibrium oxidized/reduced species activities and the details of the interfacial reaction(s), but not the overpotential. The form of the this factor i0 can be expressed in terms of more fundamental kinetic and thermodynamic parameters by 1a 0 a i0 ¼ nFk3 a0Ox aRd : ð2:21Þ Here, k3 is the rate of oxidation or reduction in equilibrium, that is, k3 is an Arrhenius expression including an activation energy for the charge transfer reaction. In all of the above, the activities aOx and aRd are not individual activities, but are products of the activities of the particular oxidized and reduced species raised to the power of their stoichiometric coefficients. Because the oxidized or reduced species may be adsorbed, and because the adsorption conditions may themselves depend on the overpotential, aOx and aRd are usually dependent on the surface coverage and may also be functions of .
16
Tafel slopes are traditionally measured on a log base 10 scale; thus the factor of 2.3 in the exponentials. 17 J. O’M Bockris and A. Reddy, Modern Electrochemistry, Plenum Press, New York, (1970). 18 A. J. Bard and L. F. Faulkner, Electrochemical Methods: Fundamentals and Applications, Wiley, New York (1980).
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The experimental practice has been empirically to determine the dependence of i0 on external conditions such as reactant partial pressures, and compare these dependencies to theoretical models. First principles models and calculations should also be able to predict these parameters, and for such calculations the details of the double layer might play an important role. More specifically, a high electrical field always exists at the interface (if only 0.1 V is dropped across a double layer 1 nm thick, an electric field of 106 V cm1 will be established). This high field may affect the orientation of molecules at the surface possessing dipole moments (induced as well as intrinsic), and also kinetic parameters such as diffusion coefficients and mechanisms. At the very least, the influence of the field should be assessed so that it may be safely discounted or not.19,20 An estimated measure of the relative overpotential differences at each electrode of a fuel cell can be obtained using the B–V equation, and some crude approximations. Detailed discussion of each catalytic electrode reaction are provided below, but, following the estimates in Ref. [21], we can say that for 103 A cm2 and for the ORR, iORR 109 A cm2 . For a the HOR, iHOR 0 o catalyst layer with surface area 50 m2 g1 Pt and loadings between 0.04 and 0.4 mg cm2, pulling a current of 1 A cm2, a current density at the catalyst double layer of 4 105 A cm2 exists. At the anode for which the Tafel slope b 30 mV per decade, this corresponds to an overpotential of no more than an 0:05 V, whereas at the cathode, for which the Tafel slope b 60 mV per decade, cath 0:4 0:5 V. While only an approximation, the general conclusion is clear—most of the overpotential and activation losses are being used to drive the cathodic ORR. It is convenient (at least from the theoretical standpoint) that the anode potential is so low because it allows us to approximate the anode as a pseudo-reversible hydrogen electrode which corresponds to zero volts on the standard thermodynamic energy scale, and be able to correlate the potential of the materials at each electrode to their thermodynamic and chemical stability under the potential/pH conditions found in real fuel cells. Central to this discussion is the Pourbaix diagram, which is shown in Figure II.5 for Pt in water. The aqueous solution Pourbaix diagram (or, potential-pH diagram) is constructed by considering the possible oxidation/reduction reactions available to Pt in water and the Nernst potential associated with each. Some of these reactions involve protons, leading to a pH dependence, etc. Details of how to construct these diagrams are found in
19
P. J. Feibelman, Phys. Rev. B 64, 125403 (2001). J. K. Norskov, J. Rossmeisl, A. Logadottir, L. Lindqvist, J. R. Kitchin, T. Bligaard, and H. Jonsson, J. Phys. Chem. B, 108, 17886 (2004). 21 H. A. Gasteiger, J. E. Panels, and S. G. Yan, J. Power Sources 127, 162 (2004). 20
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−2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2.2 2.2 E(V) 2 2 3 PtO3.∴∞ H2O ? 1.8 1.8 Pt OH+++? PtO4−? 1.6 1.6 Pt OH++? 1.4 1.4 6 ? Pt O2.∞ H2O 1.2 6 1.2 Pt++? 1 5 4 1 2 1 ? 0.8 0.8 PtO2− 0.6 0.6 Pt (OH)2 0.4 0.4 ? 0.2 0.2 a 0 0 Pt −0.2 −0.2 −0.4 −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 −1.2 −1.2 −1.4 −1.4 −1.6 −1.6 −1.8 −1.8 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 pH
−2
0
2
4
1.2
6
8
10
12
14 16 2
2 E(V) 1.6
Passivation b
1.6
Corrosion?
1.2
0.8
? a
0
0.8 0.4
0.4
0
Immunity
0.4
0.4
0.8
0.8
1.2
1.2 1.6
1.6 −2
0
2
4
6
8
10
12
14
16 pH
FIG. II.5. (left) Pourbaix (potential-pH) diagram for Pt/water at 25 C. Regions of various stable oxides or hydroxide are indicated. (right) Schematic Pourbaix diagram showing the potential-pH regimes predicting whether Pt will corrode, form a passive oxide, or remain oxide-free. From Ref. [22].
Ref. [22]; one caveat is that the Pourbaix diagram does not account for catalyst curvature, which, as discussed below, will generally push oxidation potentials toward lower potentials. The potential-pH diagram for Pt as relevant for the fuel cell is read as follows: In the fuel cell, the concentration of protons is large, and the pH 0. For pH 0 and at 0 V, that is, the reversible potential for hydrogen, Pt is basically guaranteed to be found as an oxide-free metal and the HOR can be considered to occur on clean surfaces without completing surface oxides. As the overpotential for ORR ranges from near 0 (for low current densities) to 0.4 V (for high current densities), the relevant potential range that the cathode experiences varies from 1.1 down to 0.7 V. Referring to Figure II.4 and the Pourbaix diagram, we see that within this range, Pt may exist in a number of oxidized states. Some of these Pt oxidized states include soluble Pt ions, and others include hydroxyl and oxide terminated surfaces. There are three important implications to bring up at this point: first, when Pt is oxidized to a soluble state, it may corrode, and cathode corrosion is indeed a big problem, especially when the potential is cycled through the platinum corrosion region of the Pourbaix diagram during on/off cycles; second, during a variation in
22
M. Pourbaix, Atlas of Electrochemical Equilibria in Aqueous Solutions, Pergamon Press, Oxford, New York, (1966).
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J. ERLEBACHER
cathodic overpotential from the oxidizing regime to lower electrode potentials (high overpotential), there is a change in the nature of the surface chemistry. Not surprisingly, a change in ORR reaction mechanism at 0.8 V (resistance corrected) is observed for Pt. Details of this reaction are discussed below. Third, the presence of oxidized species at the cathode catalyst surface will strongly influence catalyst kinetics. 3. K INETICS
OF THE
H YDROGEN O XIDATION R EACTION
Because the hydrogen electrode is near its reversible potential, the kinetic description of reactions at this electrode must include the competing oxidation and reduction reactions of hydrogen, that is, both the forward and back reactions of Eq. (2.1) Three elementary reactions steps have been identified that are involved in hydrogen adsorption and oxidation on Pt surface sites (‘‘Pt’’ in the following), with overall reaction H2 ⇄ 2Hþ þ 2e :17 Tafel step: 2Pt þ H2 ⇄ 2ðPt Hads Þ. The adsorption of the hydrogen mole-
cule and dissociation into two adsorbed hydrogen atoms requires adsorption of the hydrogen molecule onto a region of surface with two unoccupied neighboring metal sites. Heyrovsky step: H2 þ Pt ⇄ Pt Hads þ Hþ þ e . Here, there is a simultaneous adsorption and dissociation. Volmer step: Pt Hads ⇄ Pt þ Hþ þ e . Electrons generated via hydrogen atom dissociation can travel in the external circuit. For a very long time, the Heyrovsky–Volmer sequence was believed not to commonly occur in hydrogen oxidation at low overpotential such as in fuel cells, instead perhaps playing an important role at high anodic overpotential and/or high hydrogen concentration; its reverse reaction is very important, however, in hydrogen evolution, in which the hydrogen electrode is driven toward negative potentials. For fuel cells, hydrogen oxidation was thought to occur via the Tafel– Volmer mechanism, for which the rate-limiting step is the Volmer step. Recently, however, new arguments have arisen that at low overpotential both reaction sequences (Tafel–Volmer and Heyrovsky–Volmer) are operative simultaneously; essentially, measurements of the exchange current density, and of the current as it varies at small overpotential are too large to be accounted for by one mechanism or the other. A dual-pathway mechanism has been invoked to explain these results.23
23
J. X. Wang, T. E. Springer, R. R. Adzic, J. Electrochem. Soc. 153, A1732 (2006).
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In practice, detailed analysis of the kinetics of HOR is quite difficult, and complicated by a number of factors, not the least being the crystallography of the catalyst, the nature of the porous matrix in which the particles are found, and the concentration of protons in the supporting electrolyte. Because the interface reaction is so facile, it is complex to deconvolute the overpotential associated with diffusion of hydrogen to reactive electrodes and the kinetic overpotential associated with the reaction, and careful analysis and experimental methodologies are required to eliminate these effects.24 In aqueous solution, rotating disk methods help mediate transport effects, but in fuel cells, analysis of HOR kinetics is even more difficult. In the following discussion we have attempted to present experimental results filtered from diffusion considerations, and concentrate only on electrode reaction kinetics. 4. P T C ATALYST E LECTRODE D YNAMICS A BSENCE OF H YDROGEN
IN
A CIDIC M EDIA
IN THE
The electrochemistry of Pt in proton-containing environments itself is quite rich. Figure II.6 shows cyclic voltammograms of Pt single crystals in acidic media. Early studies showed that in the low-potential range (50 to 300 mV) the charge passed could be associated with an adsorbed monolayer of hydrogen, so-called underpotential deposition (UPD) of hydrogen.25 There is a crystallographic surface orientation effect, but this general adsorption phenomenon has become the standard method employed to assess the surface area of a platinum catalyst. When performed in aqueous media, the measurement of Hupd involves proton adsorption from solution, that is, the reaction Hþ þ e ! Hupd :
ð2:22Þ
In fuel cells, the practice is to feed hydrogen to an anode, which acts as a quasireversible reference electrode, and an inert gas to the cathode. Protons dissociated at the anode diffuse across the membrane to the cathode, whose potential is controlled in order to get adsorption to occur or not. Integration of the charge associated with hydrogen UPD, and using an adsorption site density of adsorbed hydrogen equal to 210 mC cm2 , yields an assay of the surface area. The cyclic voltammograms measured in this way are generally symmetric in the Hupd region, which means that neither transport nor electrode kinetics are rate limiting. At more positive potentials, Pt oxides can form in aqueous solution. In the fuel cell,
24 25
P. M. Quaino, M. R. Gennero de Chialvo, and A. C. Chialvo, Phys. Chem. Chem. Phys. 6, 4450 (2004). F. G. Will, J. Electrochem. Soc. 112, 451 (1965).
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100
Pt(110)
50 0
Current density/mA cm−2
−50 50 Pt(100) 0
−50 40
Pt(111)
0 −40 0.0
0.2
0.4
0.6
0.8
1.0
E/V FIG. II.6. Cyclic voltammetry of Pt single crystals in acidic media (0.05 M H2SO4), showing the characteristic adsorption of a monolayer or partial monolayer of hydrogen (underpotential deposition, UPD) at potentials slightly positive of 0 V. From Ref. [26]. The Pt(111) voltammograms shows hydrogen UPD up to 0.4 V. The peak between 0.4 and 0.6 V is associated with bisulfate anion adsorption on highly oriented, clean surfaces with large terraces.
if care is taken to remove oxygen from the system, the oxide peaks can generally be suppressed during this kind of measurement. As the kinetics of hydrogen UPD are fast, these cyclic voltammograms essentially give us information about an adsorption isotherm, that is, the coverage of hydrogen as a function of potential. Markovic et al. have examined this isotherm in detail on low-index Pt single crystal surfaces.26 Figure II.7 shows the coverage
26
N. M. Markovic, B. N. Grgur, and P. N. Ross, J. Phys. Chem. B 101, 5404 (1997).
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MATERIALS SCIENCE OF H/O FUEL CELL CATALYSIS
250 Pt(111) Pt(100)
200
QHupd/mC cm−2
Pt(110) 150
100
50
0 0.1
0.2
0.3
0.4
0.5
E/V FIG. II.7. Adsorption isotherms for UPD hydrogen in 0.05 H2SO4, found by integrating the data in Figure II.5. The charge QHupd corresponds to one electron per hydrogen atom adsorbed and can be correlated to the surface atomic density of Pt atoms. From Ref. [26].
yHupd of this layer versus potential. One obvious observation is that the assumption of one H to one Pt atom is not valid for Pt(111) for which one monolayer (ML) corresponds to 240 mC cm2 ; rather, Pt(111) adsorbs 0.66 ML. Hydrogen UPD on Pt(100) is near the theoretical value of 208 mC cm2 , and for Pt(110)(1 2), the measured coverage is actually greater than the theoretical value of 147 mC cm2 , a discrepancy that can be accounted for by competing anion adsorption at the high potentials. In practice, an agreed-upon standard of comparison using 210 mC cm2 is used to average out these effects when comparing one catalyst to another.27 In all the examined cases, the equilibrium coverage of UPD hydrogen decreases with increasing potential, approaching yHupd 0 at 400 mV. This decrease is rationalized by a mathematical analysis of the adsorption isotherm,
27
T. Biegler, D. A. J. Rand, and R. Woods, J. Electroanal. Chem. 29, 269 (1971).
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but because analysis of isotherms is also relevant for ORR and HOR, we dwell for a moment on the common assumptions and analyses used in this context. In the simplest model for adsorption (Langmuir), an equilibrium is established between molecules in a vapor (gas), and an adsorbed monolayer (ads), in which the molecules do not interact with each other laterally, only with the substrate. Under these assumptions, the equilibrium constant for the general adsorption reaction of a component is given by ads m mgas kads 1 y ¼ exp ð2:23Þ K ¼ des ¼ k P1 y RT where P is the partial pressure of the adsorbing gas and y is the fraction of surface covered. Solving for the chemical potential of the adsorbate, we find y : ð2:24Þ mads ¼ mgas þ RT ln P ln 1y The last term in this expression is usually interpreted as TDSconf , where DSconf is configurational entropy change upon adsorption. We will only consider isobaric processes here, so ignoring the additive constant, y : ð2:25Þ DSconf ¼ R ln 1y More generally, we can also consider an enthalpy of adsorption which may include lateral interactions between adsorbed particles, so that mads ¼ mgas þ DH ads ðyÞ TDSconf :
ð2:26Þ
Now consider the specific case of hydrogen adsorption, and imagine a cell containing two electrodes in local equilibrium, the first (I) being a hydrogen reference electrode (Eqs. (2.1), (2.7)), and the second (II) the hydrogen UPD electrode (Eq. (2.22)). Following the same line of reasoning as in Section II.1, we can find the potential difference Df ¼ fII fI between the electrons at each electrode by finding the difference in chemical potentials of the electrons at each reaction: Df ¼
mH 1 II me mIe ¼ upd F F
ð2:27Þ
(the chemical potential of hydrogen is zero, by convention). This expression illustrates the general result—if there is adsorption at an electrode, the potential of that electrode must be different than the equilibrium potential. This difference is the overpotential , with ¼ E E0 ¼ Df:
ð2:28Þ
MATERIALS SCIENCE OF H/O FUEL CELL CATALYSIS
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With hydrogen UPD on Pt(111), it is found that the enthalpy of adsorption increases linearly with coverage, DH ads ¼ DHoads þ ry
ð2:29Þ
with r=RT 1618, at temperatures between 274 and 333 K. Combining these results gives us a functional relation between the overpotential and surface coverage: DG0upd ry RT y þ ln ð2:30Þ ¼ F F F 1y (we have written DG0upd , a free energy, instead of an enthalpy, to account for a nonzero adsorption entropy in Eq. (2.24); for reaction Eq. (2.22) and hydrogen, DG0upd 40 kJ mol1 ). This expression yields the approximately linear decrease in coverage from a full monolayer to near nothing between 50 and 300 mV. One common origin of the Frumkin (Temkin) isotherms as due to lateral interactions between the adsorbed species is dipole–dipole interactions between the molecules. The extra work associated with adding a dipole to an interface already populated with a coverage y of dipoles is simply due to the repulsive electrostatic field from the existing charges.28 For the case in which the charged double layer is solely due to the adsorbed charge, the potential difference across the interface is given by Df ¼ ðm=ei Þy
ð2:31Þ
where m is the dipole moment of the interface species, and e is the permittivity of the interface. The work to add a charge to this interface is simply qDV, where q is the elementary charge, and therefore the heat of adsorption changes as 0 þ qðm=eÞy: DHads ¼ DHads
ð2:32Þ
Temkin adsorption is thus not surprising for the adsorption of polar molecules such as CO, or oxygen reduction intermediates, but perhaps it is a bit surprising for hydrogen adsorption, for which calculated Pt–H dipoles are small in magnitude.29 Recent theoretical work has now successfully simulated the hydrogen UPD layer formation behavior for various crystalline facets of Pt without a priori assumptions. Karlberg et al. used DFT calculations for this purpose, including a minimal model incorporating a Pt substrate and a water bilayer.30 The details of
28
M. Eriksson, I. Lundstrom, and L.-G. Ekedahl, J. Appl. Phys. 82, 3143 (1997). X. Xu, D. Y. Bu, B. Ren, H. Xian, and Z.-Q. Tian, Chem. Phys. Lett. 311, 193 (1999). 30 G. S. Karlberg, T. F. Jaramillo, E. Skulason, J. Rossmeisl, T. Bligaard, and J. K. Norskov, Phys. Rev. Lett. 99, 126101 (2007). 29
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Pt(111) T = 274 K T = 303 K T = 333 K
50 0 −50
(c)
(d)
50
Monte carlo
i = dQ/dt (m A/cm2)
(b)
Analytic
i = dQ/dt (m A/cm2)
(a)
Pt(100)
0 −50
0.0
0.1
0.2 0.3 U vs. SHE (V)
0.4
0.0
0.1
0.2 0.3 U vs. SHE (V)
0.4
0.5
FIG. II.8. Predicted hydrogen UPD voltammograms using adsorption energies from density functional theory calculations (top) and Monte Carlo simulations (bottom) for Pt(111) and Pt(100) at different temperatures. From Ref. [30]
where the hydrogen atom sits is important for these calculations; for Pt(111), the threefold hollow site was found to be most stable (although this would yield a full monolayer of hydrogen, in contrast to the experiment, which yielded 0.66 ML). For Pt(100) the twofold bridge site is most stable. It was found that the system energy indeed rose linearly with hydrogen coverage. Interestingly, any effect of an applied electrostatic potential such as might be due to the presence of a charged double layer was found to be negligible. This means the calculations are also relevant for the fuel cell catalyst, for which the details of the near-surface electrostatic potential, the width of the charged double layer, and so on, are essentially unknown. Figure II.8 reproduces the results of these calculations, and they are clearly in semi-quantitative agreement with the experimental data in Figure II.6. 5. P T E LECTRODE C ATALYSIS IN A CIDIC M EDIA THE P RESENCE OF H YDROGEN In the presence of hydrogen, the fuel cell anode at even slight overpotentials will drive the HOR. It is actually quite difficult to measure what potential exactly the
MATERIALS SCIENCE OF H/O FUEL CELL CATALYSIS
103
electrode is sitting at, but measurements using microreference electrodes inserted into the Nafion electrolyte near the catalyst layer suggest that there is no more than a 50 mV overpotential at this electrode, and the exchange current density estimate above suggests that the overpotential may even be smaller. Within this potential regime, the interaction of underpotential hydrogen with the adsorbed hydrogen intermediate in the Volmer step of the HOR has been a subject of ongoing debate. Are they, in fact, the same? Markovic et al. suggested ‘‘no,’’ because HOR is observed to occur in aqueous solutions in the Hupd potential range even on Pt facets that have UPD coverages near 100% (i.e., Pt(100) and Pt (110)).26 They suggest the Hupd layer is ‘‘in’’ the surface, rather than ‘‘on’’ the surface. This discussion has since evolved into a clarification of ‘‘strongly bound’’ (UPD) hydrogen versus ‘‘weakly bound’’ hydrogen.29,31 For the purposes of clarifying the electrode kinetics of fuel cell anode catalysts, the general consensus seems to be that the strongly bound H-UPD layer is a spectator to weakly adsorbed hydrogen playing an actual role in hydrogen oxidation.23 Using the overpotential/coverage concepts developed above, we can predict an experimental current–overpotential relationship (this derivation loosely follows reasoning given by Vogel et al.32). First, we assume weak interactions in the hydrogen adlayer Hads , and approximate a Langmuir isotherm for the kinetics (r ¼ 0 in the equivalent of Eq. (2.30)). Now consider the current associated with the Volmer–Tafel adsorption steps. Let y equal the steady-state surface fraction covered by hydrogen atoms, let kct equal the desorption rate constant (the reverse reaction in the Tafel step), and let kat be the dissociation rate constant (the forward reaction in the Volmer step); that is, we have assumed Hads has reached a steady state and can thus consider the overall reaction 2Pt þ H2 ⇄
kct 2Pt kat
þ 2Hþ þ 2e :
ð2:33Þ
The number of electrons (or protons) created per unit time is the balance of this equation, that is, the steady-state current (corrected for the sign of the electron) is i ¼ 2Fkct y2 2Fkat CH2 ð1 yÞ2 :
ð2:34Þ
Note that the both terms are second order because desorption requires two hydrogen atoms in the same place at the same time, and dissociation requires a hydrogen molecule to land where there are two neighboring empty surface sites. For Langmuir adsorption kinetics, the overpotential/coverage relationship takes on the particularly simple form (with DG0 the free energy of adsorption)
31 M. E. Martins, C. F. Zinola, G. Andreasen, R. C. Salvarezza, and A. J. Arvia, J. Electroanal. Chem. 445, 135 (1998). 32 W. Vogel, J. Lundquist, P. Ross, and P. Stonehart, Electrochim. Acta 20, 79 (1975).
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DG0 RT y þ ln ¼ F F 1y and we can solve for y as a function of : y¼
1 1 þ C0 eF=RT
ð2:35Þ
where C0 ¼ eDG0 =RT . Note that in contrast to Eq. (2.30), the coverage of adsorbed hydrogen increases with in this case. Furthermore, C0 is expected to be rather large, so for small overpotentials we may use the approximation y
1 F=RT e : C0
ð2:36Þ
The current can also be also found using the B–V equation, Eq. (2.20). Because the current can be expressed in two ways, we can match terms to find 2Fkct y2 ¼ i0 e2:3=b :
ð2:37Þ
Analysis of the kinetics of the HOR at larger overpotentials and incorporating multiple parallel reaction sequences can be found in a number of sources.33,34 A particularly successful model has been proposed by Wang et al.,23who derive (using the same strategy as above), a model for the HOR kinetic current in the form h i F=2RT F=gRT F=2RT 1 e2F=gRT þ iHeyrovsky e e e ð2:38Þ i ¼ S iTafel 0 0 where S is a dimensionless scaling factor, the i0 s are exchange current densities associated with the Tafel and Heyrovsky steps, and g is a parameter that is dependent on the equilibrium coverage of adsorbed hydrogen and the kinetic rate constants of the three reaction steps (it is found that g 1:2). Equation (2.38) predicts that the effective exchange current density is high enough such that the anode overpotential required to maintain high current densities of 1 A cm2 will remain small even when the Pt loading at the anode is dropped to 0.05 mg cm2, as found experimentally by Gasteiger et al.21 The use of this model is already appearing in optimization studies of fuel cell loading, and predictions have been made that by judicious design of the catalyst layer and gas diffusion
33 34
M. R. Gennero de Chialvo and A. C. Chialvo, Electrochim. Acta 44, 841 (1998). P. M. Quaino, M. R. Gennero de Chialvo, and A. C. Chialvo, Electrochim. Acta 52, 7396 (2007).
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layer (GDL) (see Section III.8), the overall anode loading using pure platinum nanoparticles can be reduced to 0.018 mg cm2 without performance loss.35 A detailed application of any kinetic model must include surface crystallography. For instance, using (2.36) and (2.37), we expect that at low overpotential that the Tafel slope for the Volmer–Tafel sequence is given by 2:3RT=2F, or 30 mV per decade at room temperature. Experimentally, on single crystal surfaces, this slope is seen in aqueous solution for Pt(110), but not Pt(111) (for which b 2:3RT=F) nor for Pt(100) (for which two slopes are seen depending on current density).26 This effect has been interpreted as follows, namely, that on Pt(110), the Tafel–Volmer sequence predominates, and on Pt(100) the Heyrovsky–Volmer sequence predominates. On Pt(111), definitive conclusions about the reaction mechanism have not been made. In the fuel cell, in contrast, the elucidation of the reaction mechanism is complicated by diffusional limitations of gas to the electrode surface, and of protons away from the surface, and methods have been proposed to minimize this. Most commonly, an electrode employing a porous Nafion/Pt composite electrode is used in a sulfuric acid solution, the hope being that within the porous layer there is little aqueous solution, and measurements here tend to give slopes of 2:3RT=2F.36,37 Direct measurements in the fuel cell are rarer. One method uses a ‘‘hydrogen pump’’ in which two Pt electrodes in a fuel cell are made, one with a large Pt loading that acts as a reversible electrode, and the other with an extremely small Pt loading (<0.003 mg Pt cm2) for which the kinetics of HOR are slowed down significantly. Here, Tafel slopes of 2:3ð2RT Þ=F have been found.38 Unfortunately, deconvoluting the facet effect from the mechanism effect with catalyst particles is difficult, and no definitive conclusion can be made at this time. Atomistic simulations of HOR are beginning to clarify the details of the hydrogen/surface interaction. It has been recognized that the effects of the detailed structure of water on the Pt surface, as well as the effect of a biasing electric field, play a large role in the kinetics of HOR. How exactly one incorporates these effects into, for instance, DFT calculations is still evolving.39 In one method, termed ‘‘periodic density functional theory,’’ a periodic supercell is established with a thin metal layer biased by adding excess electrons to this slab, and a compensating charge is introduced uniformly into a thin water layer/ vacuum layer sitting over the slab. The voltage drop across the interface is 35
M. Secanell, K. Karan, A. Suleman, and N. Dijilali, J. Electrochem. Soc. 155, B125 (2008). J. Jiang and A. Kucernak, J. Electroanal. Chem. 567, 123 (2004). 37 R. M. Q. Mello and E. A. Ticianelli, Electrochim. Acta 42, 1031 (1997). 38 K. C. Neyerlin, W. Gu, J. Jorne, and H. A. Gasteiger, J. Electrochem. Soc. 154, B631 (2007). 39 C. D. Taylor and M. Neurock, Curr. Opin. Solid State Mater. Sci. 9, 49 (2005). 36
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found by referencing the potentials in each region to the vacuum. This method has found success in describing water adsorption over Pd40 (i.e., structural ordering of water due to the interface electric field), but neglects screening effects in the electrolyte which is of greater importance when the water contains charged, potentially reactive, species. A second method (‘‘effective screening medium’’) removes the periodicity in the direction normal to the surface, and uses a Green’s function methodology to add charge to the slab, which then dynamically induces the compensating charge distribution in the water layer over the slab. This method has been useful in clarifying the Volmer step in hydrogen evolution.41 In a third method, Skulason et al. have added excess protons to a water layer compensated with excess protons in the surface to study hydrogen evolution.42 Recall that although the HOR reaction on Pt is very facile and the total required platinum loading is quite low, the reaction surfers from irreversible CO adsorption. The effect of CO poisoning on the HOR on Pt is well described by a siteelimination model, which says that the mechanism of hydrogen adsorption and oxidation is unchanged except that the number of surface sites available for adsorption are reduced because of blocking by irreversibly adsorbed CO.43 The simplest formulation of this effect simply modifies the overpotential/coverage to DG0 RT y þ : ð2:39Þ ln ¼ F F 1 y yCO Following the same line of reasoning as before, we find an identical expression for the current/overpotential relationship, except that the exchange current density i0 in Eq. (2.37) is replaced by iCO 0 where 2 iCO 0 ¼ i0 ð1 yCO Þ :
ð2:40Þ
While known for a long time, it is nice to know this experimental relationship has also been seen experimentally inside fuel cells, rather than just in aqueous electrolytes.44,45 CO adsorption is nearly irreversible (adsorption/desorption rates are slow compared to hydrogen) under the anodic conditions and low overpotential of the hydrogen electrode, so over time, its effect will strongly
40
J. Filhol, M. Neurock, Ang. Chem. Int. Ed. 45, 403 (2006). M. Otani, I. Hamada, O. Sugino, Y. Morikawa, Y. Okamoto, and T. Ikeshoji, J. Phys. Soc. Jpn. 77, 024802 (2008). 42 E. Skulason, G. S. Karlberg, J. Rossmeisl, T. Bligaard, J. Greeley, H. Jonsson, and J. K. Norskov Phys. Chem. Chem. Phys. 9, 3421 (2007). 43 J. Zhang, H. Wang, D. P. Wilkinson, D. Song, J. Shen, and Z.-S. Liu, J. Power Sources 147, 58 (2005). 44 H. P. Dhar, L. G. Christner, A. K. Kush, and H. C. Maru, J. Electrochem. Soc. 133, 1574 (1986). 45 H. P. Dhar, L. G. Christner, and A. K. Kush, J. Electrochem. Soc. 134, 3021 (1987). 41
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degrade cell performance. This is the primary reason for anode failure in hydrogen/oxygen fuel cells, and the driver for methods to avoid this effect. 6. K INETICS
OF THE
O XYGEN R EDUCTION R EACTION
The four-electron ORR continues to be one of the most challenging reactions to elucidate within electrocatalysis generally, not just the specific context of fuel cells. The number of potential pathways, reaction intermediates, and so on, is quite large, and rate determining steps tend to dominate all experimental probes of the entire reaction sequence.46 A thorough review of the literature of oxygen reduction kinetics on a variety of materials in a variety of electrolytes (up to the early 1990s) is found in Ref. [47]. Here, our goal is to summarize the salient points and recent developments that will inform the later discussion of new materials. The overpotentials required to drive the ORR at reasonable rates are much larger than for the HOR, and thus the kinetics of the electrode reactions can be analyzed in the high overpotential regime. This allows us to neglect the anodic term in the Butler–Volmer Equation. Thus, on one hand, experiments to probe the ORR are usually straightforward to do, in the sense that extraneous effects such as diffusion limitations of reactants to and from the electrolyte do not play as major a role as they do for the HOR. On the other hand, the explanation of ORR measurements has been difficult to elucidate because the reaction pathways are complicated. Empirically, the ORR manifests the following properties: a. In aqueous half-cell experiments, the cathodic current/overpotential relationship takes the generic form 2:3 : ð2:41Þ i ¼ i0 exp b b. For potentials greater than 0.8 V (‘‘high-potential range’’), the absolute value of the Tafel slope is 60 mV per decade ( 2:3RT=F); for potentials less than 0.8 V (‘‘low-potential range’’), the absolute value of the Tafel slope is 120 mV per decade ð 2:3ð2RT Þ=FÞ. c. In the high-potential range, the functional dependence of the current density on proton activity (pH) and oxygen partial pressure is i ¼ j0 PO2 ðaHþ Þ3=2 eF=RT :
46 47
R. Adzic, in Electrocatalysis, eds. J. Lipowski, P. N. Ross, Wiley–VCH (1998), p. 197. K. Kinoshita, Electrochemical Oxygen Technology, Wiley, (1992).
ð2:42Þ
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d. In the low-potential range, the functional dependence of the current density on proton activity and oxygen partial pressure is i ¼ j0 PO2 aHþ eF=2RT :
ð2:43Þ
The hydrogen/oxygen fuel cell operates in the high-potential regime. Although the cell potentials are often reported at potentials less than 0.8 V, this is generally due to ohmic resistances, and one should not forget to correct for this. The nature of the transition at 0.8 V is highly correlated to the formation of surface oxides and hydroxides on the surface of Pt. These oxides are also related to the possibility of Pt corrosion. Obviously, elucidation of the surface oxide state on the surface of Pt nanoparticles in the fuel cell is of high interest. As discussed below, there has been progress in characterizing surfaces of single crystals, particularly in aqueous half-cell experiments, but again, transferring these ideas to the fuel catalyst layer is a challenge. One of the earliest models for (parts of) the ORR, due to Damjanovic, has been remarkably successful and is straightforward to describe.48–52 Recognizing that Pt within the high-potential regime is covered with oxygen containing intermediates, particularly hydroxide, Damjanovic and Brusic argued that the coverage with oxygen containing species is given by an expression of the form y ¼ K ½ð2:3RT=FÞpH þ þ f0 ¼ K
RT lnaHþ þ þ f0 F
ð2:44Þ
where f0 is the potential at which the coverage y goes to zero at pH ¼ 0; K is a constant. This expression is consistent with Temkin kinetics, as above (Eq. (2.30)), at intermediate coverages where lnðy=ð1 yÞÞ is slowly varying. The linear pH dependence of the coverage is purely empirical; presumably it comes from equilibration with solution protons but the detailed nature of this equilibration deserves further attention. The ORR begins with an adsorption of an oxygen molecule onto the metal: M þ O2 ⇄ M O2
48
ð2:45Þ
A. Damjanovic and V. Brusic, Electrochim. Acta 12, 615 (1967). A. Damjanovic and M. A. Genshaw, Electrochim. Acta 15, 1281 (1970). 50 D. B. Sepa, M. V. Vojnovic, and A. Damjanovic, Electrochim. Acta 26, 781 (1981). 51 D. B. Sepa, M. V. Vojnovic, L. M. Vracar, and A. Damjanovic, Electrochim. Acta 31, 91 (1986). 52 D. B. Sepa, M. V. Vojnovic, L. M. Vracar, and A. Damjanovic, Electrochim. Acta 32, 129 (1987). 49
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O O
O M
Griffiths
O
O M
O
M
Pauling (end-on)
M Bridge
FIG. II.9. O2 adsorption configurations. (left) Griffiths model, involving edge-on adsorption to a single substrate atom, (center) Pauling model, involving end-on adsorption forming one bond, and (right) a bridge configuration involving two bond to the substrate.
The nature of this adsorbed state has been of considerable debate.46 Three adsorption configurations are usually considered Figure II.9).53 The first involving only a single substrate atom (Griffiths model) can be formed via interactions between the p-orbitals of O2 and empty dz 2 orbitals of the substrate, is generally not considered likely. Instead, most consideration is given to the so-called Pauling configuration, an end-on adsorption in which a s orbital of O2 donates electron density to an empty dz 2 on the substrate, and a bridge configuration involving the formation of two bonds to the substrate. At this point, a rate-determining step consistent with Eq. (2.42) is the charge transfer reaction M O2 þ Hþ þ e ⇄ productðsÞ:
ð2:46Þ
At the time Damjanovic proposed this mechanism, the exact nature of the products and activated states was unknown. However, ab initio molecular dynamics simulations (see Section V.14) suggest a complicated picture in which the initial adsorption configuration is end-on, but upon protonation to form a M–OOH complex, the O–O bond is quickly extended and takes on a tophollow-bridge character before dissociating.54 With this picture, the current density should be first order in the oxygen partial pressure and the proton concentration, that is, using n ¼ 1, a ¼ 1=2 in Eq. (2.20),
i ¼ j0 aHþ PO2 eDG =RT eF=2RT
ð2:47Þ
where j0 is a constant. The parameter DG is the activation free energy for the reaction, which will depend on the surface coverage. According to simple activated state theory, if the energy difference DG between the initial or final
53 E. Yeager, M. Razaq, D. Gervasio, A. Razaq, and D. Tryk, Proc. of Workshop on Structural Effects in Electrocatalysis and Oxygen Electrochemistry, The Electrochemical Society, vol. 92–11, p. 440 (1993). 54 Y. Wang and P. B. Balbuena, J. Phys. Chem. B 108, 4376 (2004).
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state of a reaction changes by DðDGÞ, then the activation energy should change by DðDG Þ ð1=2ÞDðDGÞ (the factor of ½ here is called the symmetry factor and assumes the energy barrier sits centered on the reaction coordinate between the initial and final states). Making the further assumption that the entropy of adsorption of the oxygen containing species is small compared to the heat of adsorption, and that the heat of adsorption follows a Temkin isotherm, the factor DDG should be linear in y. With this in mind, Damjanovic wrote 1 RT ð2:48Þ F þ lnaHþ þ Ff0 DG ðyÞ ¼ DG ð ¼ 0Þ þ 2 F which is consistent with the heat of adsorption changing linearly with coverage as ðF=K Þy. Substitution of Eq. (2.48) into Eq. (2.47) yields the all the proper exponential functional dependencies in Eq. (2.42). The physical assumptions associated with the Damjanovic form for the ORR current have been critiqued and analyzed by a number of authors, especially by analysis of the multitude of rate equations associated with potential reaction mechanisms.55 It has been argued that Damjanovic’s model should be modified to include site blocking by adsorbed oxide species.56 In this case, Eq. (2.42) should be modified to i ¼ j0 PO2 ðaHþ Þ3=2 ð1 yad ÞeF=RT :
ð2:49Þ
The presence of the ð1 yad Þ term in Eq. (2.42) does not affect the Tafel slope. One concern in this picture has been the nature of the observed Pt–oxide species. Initially, it was thought that these were Pt–O, but there is now general consensus that the adsorbed species is Pt–OH. The adsorbed hydroxide layer not only is more thermodynamically stable, but also provides a mechanism for fast proton diffusion through the adsorbed layer. Such diffusion may occur by creation of a proton vacancy in a field of hydroxide via a hopping mechanism (we wonder if it may also occur by an interstitial mechanism in which an extra proton jammed into a field of hydroxide moves about, locally creating a bound water molecule wherever it currently resides). Details of such diffusion on Pt–OH have not been studied in detail, although they have been invoked in models of oxygen reduction on Pt for a long time (e.g., there is a model from the 1960s called the ‘‘pseudosplitting model’’ that invokes proton diffusion to active defect sites; the model relies on defects or step edges being the catalytic sites).57,58
55
V. S. Bagotsky and M. R. Tarasevich, J. Electroanal. Chem. 101, 1 (1979). N. M. Markovic, H. A. Gasteiger, B. N. Grgur, and P. N. Ross, J. Electroanal. Chem. 467, 157 (1999). 57 U. R. Evans, Nature 218, 602 (1968). 58 U. R. Evans, Electrochim. Acta 14, 197 (1969). 56
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Assuming this is also the case for the Pt–OH surface, an issue with ORR in fuel cells catalysts is nicely resolved, namely, in the PEM fuel cell ORR has been found to follow the same current/overpotential trends as in aqueous solution, that is, Eq. (2.42). But presumably ionomer is only touching distinct spots on a catalyst particle, or the outside of a catalyst cluster, otherwise we’d flood the catalyst, or block sites (such as when one uses to much Nafion as a binder); therefore, protons are being shuttled onto the catalyst only in these ionomer/ catalyst contact points. Fast diffusion of protons through a hydroxide layer thus allows the entire surface of catalyst particles to equilibrate with the electrolyte. The source of the hydroxide layer has been linked to dissociation of water, which will be plentiful at the cathode. Indeed, hydration of the oxygen gas is generally required for proper function. This hypothesis is supported by examination of oxygen reduction on surfaces that have been prepared by binding to them molecules that more strongly bind than hydroxide. Molecules that exhibit such specific binding include phosphate on Pt(111). Surfaces prepared with these molecules do not show oxidation until 1.2 V, and there is no change in Tafel slope behavior to the 60 mV per decade as potentials rise over 0.8 V. Occasionally, there is further discussion regarding the mechanism of oxygen reduction according to a second mechanism in which oxygen dissociation is the rate-limiting step, followed by fast charge transfer and protonation.46 This mechanism is due to Yeager et al. who observed that in phosphoric acid (a) an increase in current directly in proportion to the oxygen partial pressure and (b) no temperature dependence on the Tafel slope, which was fixed at 135 mV per decade over the 30–150 C temperature range.53,59,60 While interesting, these results are more relevant for the ‘‘low-potential regime,’’ where one generally sees 120 mV per decade Tafel slopes, because, as has already been noted, phosphate ions block oxidation of the Pt surface up to 1.2 V. As such, the results may be relevant for phosphoric acid fuel cells, which operate at 200 C, or for fuel cells using phosphoric acid-like electrolytes such a polybenzimidazole,61 but are likely not as relevant in the hydrogen/oxygen PEM fuel cell. For ORR specifically within the fuel cell, the historical progress of measurements of the ORR has been as follows. Initially, studies of ORR using trifluoromethane sulfonic acid electrolytes were invoked as analogues to the sulfonic acid head group environment of Nafion. Generally, the same kinetics (Tafel slope) were found for this electrolyte as for sulfuric acid based electrolytes.62,63 59
S. J. Clouser, J. C. Huang, and E. Yeager, J. Appl. Electrochem. 23, 597 (1993). M. M. Ghoneim, S. Clouser, and E. Yeager, J. Electrochem. Soc. 132, 1160 (1985). 61 Z. Liu, J. S. Wainright, M. H. Litt, and R. F. Savinell, Electrochim. Acta 51, 3914 (2006). 62 A. J. Appleby and B. S. Baker, J. Electrochem. Soc. 125, 404 (1978). 63 K.-L. Hsueh, H. H. Chang, D.-T. Chin, and S. Srinivasan, Electrochim. Acta 30, 1137 (1985). 60
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At more or less the same time, the kinetics of diffusion through porous electrodes were being established (and how to correct for them in any analysis).64 In the late 1980s and early 1990s, it was recognized that mixing Nafion ionomer with the catalyst layer, originally by seeping in a solution containing dissolved Nafion, and then directly mixing Nafion as a catalyst binder, significantly increased the current density of the PEM fuel cell by allowing protonic contact to the catalyst (see below for further discussion). These observations also led to a number of new studies of ORR, with an eye toward elucidation of ORR within the fuel cell environment. Initial studies employed planar electrodes coated with a film of ‘‘re-cast’’ Nafion, and the serendipitous result was found that, although the Tafel slope kinetics of the ORR remained the same as for aqueous electrolytes, the exchange current density rose, the idea being that Nafion shuttles reactants to the surface only at a few contact points, and since the sulfonic acid groups are bound to the perflourinated backbone of the ionomer, they do not specifically adsorb on the surface and block active sites.65 Played out on the pages of the Journal of the Electrochemical Society, experimental improvements removed all aqueous acids from the catalyst electrode,66,67 and the same generic observation was made—ORR kinetics appeared to follow the ‘‘high potential’’ kinetics associated with hydroxide covered Pt, but the exchange current density was higher due to the lack of anion adsorption.68,69 Inevitably, careful experimental methods have been devised to probe ORR kinetics within the actual PEM fuel cell environment, being careful to correct for diffusional considerations, ohmic losses due to transport through the membrane, active surface area of catalyst nanoparticles, and so on. Exchange current densities, Tafel slopes, and reaction orders with respect to O2 pressure and proton activity have been elucidated.70 Essentially, the original dependencies (Eq. (2.42)) that describe ORR in the high-potential regime in aqueous electrolyte are found to also describe ORR in fuel cell nanoparticle-based catalysts.
64
P. Stonehart and P. N. Ross, Catal. Rev. Sci. Eng. 12, 1 (1975). S. Gottesfeld, I. D. Raistrick, and S. Srinivasan, J. Electrochem. Soc. 134, 1455 (1987). 66 A. Parthasarathy, C. R. Martin, and S. Srinivasan, J. Electrochem. Soc. 138, 916 (1991). 67 F. A. Uribe, T. E. Springer, and S. Gottesfeld, J. Electrochem. Soc. 139, 765 (1992). 68 A. Parthasarathy, B. Dave, S. Srinivasan, and A. J. Appleby, J. Electrochem. Soc. 139, 1634 (1992). 69 A. Parthasarathy, S. Srinivasan, and A. J. Appleby, J. Electrochem. Soc. 139, 2530 (1992). 70 O. Antoine, Y. Bultel, and R. Durand, J. Electroanal. Chem. 499, 85 (2001). 65
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III. The Catalyst Layer in PEM Fuel Cells
7. G ENERAL D ESCRIPTION The heart of the PEM fuel cell is a layered electrode sandwich called the MEA comprising a proton conducting membrane between the two catalytic electrodes (Figure I.1). The MEA is housed between two current collectors, monolithic highconductivity plates in which have been machined flow channels through which pass the reactant gases and product water. The challenges in this area include identification of materials for the housing and current collectors (‘‘bipolar plates’’) that are (a) inexpensive, (b) easy to machine, (c) non-corrosive under acidic operating conditions, (d) nonporous, and (e) lightweight. Thin metal plates, coated surfaces, and composites have been examined, but by far the most common material has been high-density graphite.71,72 The science and engineering of the gas flow channels has also been of interest. These flow channels are usually machined into a serpentine pattern, so that the gases are following a one-dimensional path across the surface of the MEA. This is done for two reasons. First, and most importantly, if the gas pressure or reactant concentration were nonuniform across a region of MEA, lateral concentration gradients would be established leading to dissipative parasitic currents. Second, suspension of a large piece of MEA is mechanically challenging and would eliminate the sometimes useful possibility of having different pressures at the cathode and anode. Also, as discussed shortly, the MEA is highly water sensitive and can change its volume dramatically when hydrated. In the science of flow channel design, the primary challenge has been water management, which also plays a large role in the catalyst layer design. Water plays multiple roles in the PEM fuel cell. On the anode side, water must be fed with the hydrogen gas to keep the Nafion membrane hydrated. Because Nafion conducts protons as hydronium ions, if one did not hydrate the hydrogen, then the membrane would dry out due to osmotic drag. Water is created on the cathode side of the membrane and needs to be removed as fast as it is created. With these observations in mind, it is perhaps odd to note that the cathode oxygen inlet stream must also be humidified to maintain high performance. From the catalyst standpoint, when the cathode is not sufficiently humidified, the shift in Tafel slope for the ORR from 60 to 120 mV per decade shifts to potentials near 0.9 V. This is clearly undesirable, as it leads to a faster decline in power at higher current densities. It has been speculated that the change in Tafel slope within the
71 72
A. Hermann, T. Chaudhuri, and P. Spagnol, Int. J. Hydrogen Energy 30, 1297 (2005). Y. Wang, D. O. Northwood, Electrochim. Acta 52, 6793 (2007).
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fuel cell context is due to increases in the ohmic and mass transport losses, but given the discussion of ORR above, we are free to speculate when the hydration level at the cathode is too low, then the surface of the Pt particles are not sufficiently hydroxilated to allow the 60 mV per decade reaction mechanism to operate. Perhaps water fed to the cathode plays an important role in keeping the surface of the catalyst covered in hydroxide and allows facile transport of protons to and from reaction sites. When the correct level of hydration is not maintained, one can either dry out the cell, or flood the electrode(s) with liquid water, in the two extremes. In the latter case, submersal of the surfaces of the catalyst layer is thought to kill their activity by simply blocking reactant transport to the surface. Recent innovations in the area of water management have been primarily in the area of diagnostics. See, for instance, the recent work on neutron scattering imaging performed at NIST, which used the higher scattering cross-section of neutrons from hydrogen to directly image the formation of water inside an operational fuel cell,73 or the optical visualization of water condensation at fuel cell anodes.74 8. T HE C ATALYST L AYER
IN THE
M EMBRANE E LECTRODE A SSEMBLY
An important aspect of water management is to keep liquid water from entering the MEA and to help water formed at the cathode to be drawn away. For these reasons, MEAs generally possess porous carbon paper or woven carbon cloth outer layers that have been treated with a Teflon emulsion to render them hydrophobic. The catalyst layer is most often applied to one side of these layers, which are called GDLs for obvious reasons. Figure III.1 shows a cartoon of the five-layer sandwich structure of an MEA, including two GDLs, two electrodes and the proton conducting membrane. The proton conducting membrane most often used is Nafion, whose microstructure was discussed above. The structure of the catalyst layer has seen a remarkable historical development. Early Pt-based fuel cells employed a fine mesh of platinum as the electrodes pressed up against the membrane.75 Obviously, this is an expensive strategy, but it did work. As these meshes were of order at least tens of microns thick, very little Pt was in contact with the ionomer. That is, the ‘‘three-phase region’’ in which reactant gasses, electrolyte, and catalyst are all in contact was very small. Interestingly, however, the active surface area of the mesh could be measured in
73
D. J. Ludlow, C. M. Calabrese, S. H. Yu, C. S. Dannehy, D. L. Jacobson, D. S. Hussey, M. Arif, M. K. Jensen, and G. A. Eisman, J. Power Sources 162, 271 (2006). 74 S. Ge and C.-Y. Wang, J. Electrochem. Soc. 154, B998 (2007). 75 J. McBreen, J. Electrochem. Soc. 132, 1112 (1985).
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~0.5 mm
115
~10 mils ~0.5 mm ~10 microns ~10 microns
Hydrophobic mesh gas diffusion layer (GDL)
Nafion membrane
Catalyst layer: Carbon support (50 nm diam, Vulcan XC-72R) +Pt particles (3 nm diam) + ionomeric binder (CF backbone, SO3− terminated sidechains)
FIG. III.1. Cartoon of a typical MEA: (left) basic five-layer structure including hydrophobic meshes sandwiching two porous, catalytic electrodes that in turn are sandwiching the proton conducting membrane; (right) the catalyst layer is comprised of catalyst nanoparticles supported by larger carbon particles that are bound by proton conducting membrane ionomer.
situ via hydrogen adsorption and yielded a surface area equal to the geometric surface area of the mesh. Recall that this experiment feeds hydrogen to a quasireversible reference electrode anode, and an inert gas to the cathode. By adjusting the cathode voltage, one can drive hydrogen UPD on this electrode. The mesh experiments are most easily interpreted with a picture in which protons can access the entire surface of the Pt via surface diffusion. We believe surface diffusion is also responsible for proton transport in our own experiments on Pt-coated mesoporous metal electrodes, which are essentially nanoscale mesh electrodes.76,77 In an effort to reduce costs, the catalyst structure shifted from the mesh to Pt nanoparticles supported on carbon. Here, Pt particles typically with diameter less than 5 nm are physisorbed onto the surface of larger carbon nanoparticles to create a high dispersion. Typical carbon particles are 40–50 nm; a common brand name used is Vulcan XC72 (Cabot Corp.). Creating a good dispersion of catalyst on the carbon usually involves drying a solution of Pt salts onto a sample of carbon, and then reducing the Pt salts either with a reducing agent or in a flowing
76 77
Y. Ding, M. Chen, and J. Erlebacher, J. Am. Chem. Soc. 126, 6876 (2004). R. Zeis, A. Mathur, G. Fritz, J. Lee, and J. Erlebacher, J. Power Sources 165, 65 (2007).
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hydrogen environment at elevated temperature. Once made, the catalyst-carbon particles are combined with a polymeric binder (often containing some fraction of Teflon to keep hydrophobic channels open) to form a catalyst ‘‘paint.’’ This paint can be applied directly onto the carbon cloth or paper, or tape cast into sheets that, once dried, are applied to the carbon cloth/Nafion (the latter method generally gives more uniform catalyst layer thickness and is not quite as subject to the ‘‘art’’ of laying down that catalyst layer). Typical catalyst layers are 10 mm thick. Early Pt nanoparticle based catalyst did not perform very well relative to today’s expected performance. Large increases in performance were made when a thin layer of solublized Nafion was painted onto the catalyst layer,78,79 and really huge increases (that really precipitated the current interest) were made in the early 1990s when solublized Nafion ionomer was mixed directly into the binder of the catalyst layer.6,7 Ionomer in the catalyst layer greatly improved proton transport in and out of the electrode and led to performances that greatly eclipsed the mesh electrodes. Getting the exact right amount of Nafion correct, so that it does not block too many surface sites on one hand, yet yields a high enough proton conductivity on the other, is a matter of art and depends on the carbon used, the method of catalyst dispersion, and so on. One can easily imagine a number of ways that the supported catalyst layer might underperform, particularly by thinking about how the Pt nanoparticles must be in electrical contact with the external circuit. If they become disconnected from the external circuit in any way, they are rendered useless. Indeed, estimates for Pt utility are not 100%, instead levels near 80% are more common. Typical specific Pt mass per electrode geometric area (the ‘‘loading’’) is 0.4 mg cm2 for the cathode, and an order of magnitude less at the anode. Catalyst particles must also be in protonic contact with the membrane. However, the carbon particles are often on which they are supported are observed to cluster within the catalyst layer, surrounded by binder. Thus, the interior of these clusters is not in intimate contact with the binder. Estimates, however, for the active surface area generally cannot discount the ‘‘interior’’ area of Pt in these clusters, that is, reactants are getting to them. To explain this behavior, it can be hypothesized that transport of protons to and from the catalysts layers occurs via sites where Nafion microchannels are in contact with carbon/catalyst clusters. Within these clusters, protons diffuse quickly via surface diffusion both over the
78 E. A. Ticianelli, C. R. Derouin, A. Redondo, and S. Srinivasan, J. Electrochem. Soc. 135, 2209 (1988). 79 S. Srinivasan, D. J. Manko, H. Koch, M. A. Enayetullah, and A. J. Appleby, J. Power Sources 29, 367 (1990).
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carbon and catalyst particles. As these clusters are small (characteristic length scales 100 nm) this local transport is fast enough to sustain high current densities (for a recent simulation model, see Ref. [80]). Without Nafion in the binder, protons need to diffuse from the junction between the catalyst layer and the primary membrane along a tortuous path from particle to particle to distances of microns away—a much slower process. 9. S YNTHESIS M ETHODS
OF
C ATALYSTS
FOR
PEM F UEL C ELLS
The number of ways to synthesize platinum nanoparticles for PEM fuel cells is quite large. Most methods use reduction of Pt salts, particularly the hexachloroplatinate salts. To form supported particles on carbon, a slurry of metal salts (typically chlorides) and carbon powder will be heated to 600–900 C in hydrogen. The hydrogen reduces the salts to pure metal particles on the carbon. If a mixture of salts is used, alloy nanoparticles can be formed. Depending on conditions, particle size distributions centered at diameters as small as 2 nm can be made. The shape, facet orientation, and defect structure of Pt nanoparticles has been studied by a number of groups.81 Theoretically, like all fcc metals, the shape of these particles should be cube-octahedra exposing {111} and {100} facets; given the surface energy difference between the two facets, there should be a slight preponderance of {111} facets. Transmission electron microscopy has been the primary tool to study the shape of these particles, and micrographs of Pt nanoparticles supported on carbon are shown in Figure III.2. Analysis of the facet orientation distribution is consistent with the surface energy prediction of many {111} facets. More difficult has been elucidation of the shape of Pt nanoparticles supported on carbon. One might expect that the shape is unchanged, but recent work suggests that there is some sort of shape change leading to particles becoming more disk-like, with aspect ratios of 1–3.81 Some of these observations might be attributable to the microscopy technique, as TEMs that are not aberration corrected sometimes lead to optical illusions, but one cannot discount the possibility that the interfacial energy between the particle and the carbon support is high enough to drive surface diffusion in such a way that the particle/support area is maximized.
80
P. B. Balbuena, E. J. Lamas, and Y. Wang, Electrochim. Acta 50, 3788 (2005). Y. Shao-Horn, W. C. Sheng, S. Chen, P. J. Ferreira, E. F. Holby, and D. Morgan, Top. Catal. 46, 285 (2007).
81
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FIG. III.2. Transmission electron microscope (TEM) micrographs of Pt nanoparticles synthesized by reduction of Pt salts on high surface area carbon, at increasing magnification from (a)–(d). Pt nanoparticles often show pronounced {111} facets, but also appear to possess a nonuniform aspect ratio disk-like morphology. From Ref. [81].
The synthesis of next-generation catalysts for PEM fuel cells centers around the formation of multi-component nanoparticles, and nanoparticles with controlled core-shell composition. The theoretical reasons for this are discussed below. Multi-component nanoparticles are generally synthesized again through reduction of metal salts, but with multiple reducible salts in the solution. Obviously the chemistry here becomes more complex, and careful elucidation of the final nanoparticle composition to check if it is uniform is of tantamount importance. Many successful systems have been synthesized (see Section II.5).
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To make core-shell nanoparticles, three techniques have been receiving significant attention. The first, galvanic displacement, involves submersion of a particle with a surface composed of a less-noble species (e.g., Cu) into a solution of a salt of a more-noble species (e.g., Pt). As the reduction potential of Cu is less than that of Pt, there is a driving force for Pt from the solution to exchange with Cu from the surface.82 This leads to the exquisite formation of a monolayer of Pt on the particle surface. While this has been shown to be very well controllable on model single-crystal surfaces, the formation of galvanically displaced layers on actual nanoparticles has been more difficult to realize experimentally. By displacing the surface layer in a solution containing multiple noble metal salts, one can form multi-component shell compositions. Actually, the example used is a bit spurious purposely. In practice, it is difficult to make Cu nanoparticles because they oxidize so easily. It is much easier to make more noble nanoparticles such as Au or Pd (or Pt). To use galvanic displacement here, one takes advantage of the fact that under certain potential regimes, Cu will form a UPD layer on these metals. In the second method to make core-shell nanoparticles, this UPD layer is subsequently displaced. A third method to make a core-shell nanoparticle is chemical dealloying, where the less noble component of a solution of two metals is selective oxidized and dissolved away. When dealloying is performed on bulk samples, surface diffusion of the remaining, noble component, leads to the formation of extended, three dimensionally, open porous metals, typically with pore sizes 10–20 nm. When nanoparticles are dealloyed, the chemical environment is also such that surface diffusion of the more noble component is fast, but because the dimensions are so small, the particle surface simply enriches in the more noble component (the diameter shrinks), and one is left with a shell comprised of the more noble component, and a core comprised of the original alloy composition. 10. D EGRADATION
OF
C ATALYST L AYERS
The deterioration of catalysts within the PEM fuel cell matrix, and degradation of the matrix itself, has received significant attention because it impacts lifetime requirements. The DOE benchmark required for automotive applications is >5000 h. Here, we will not dwell on the degradation of the Nafion membrane, which can deteriorate by chemical interaction from catalyst reaction
82
S. R. Brankovic, J. X. Wang, and R. R. Adzic, Surf. Sci. 474, L173 (2001).
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intermediates,83–85 or from mechanical cycling due to humidity variations within the membrane. Within the catalyst layers, the anode tends to deteriorate because of CO poisoning from the fuel stream, which can in principle be avoided (see Section IV.11). At the catalyst level, cathode degradation is the primary source of failure. As reviewed by Shao-Horn,81 there are multiple mechanisms for cathode degradation: Dissolution of the carbon support from trace hydrogen peroxide. An alternate
pathway for oxygen reduction at the cathode is a two-electron reduction of O2 to H2O2 (peroxide), which reacts with carbon to form CO2. Although Pt does not generally support the two-electron reduction of oxygen, it is thought that trace amounts might form overtime.86 Carbon can also catalyze this reaction itself slightly. Sintering and agglomeration of catalyst nanoparticles. Here, whether because the carbon support is being dissolved away from underneath, or because there is significant diffusion of the particles themselves, the catalyst particles will agglomerate and sinter. This obviously leads to an undesired reduction in surface area. Dissolution and re-precipitation of the catalyst material. At the potentials and pH experienced by the cathode, Pourbaix suggests that dissolution of the Pt is possible. Indeed, it has been found that overtime, particles of Pt have been found within the Nafion membrane near the catalyst layer, and the likely explanation is that Pt ions have dissolved and precipitated (see Figure III.3).87 An Ostwald ripening process (curvature-driven evaporation from smaller particles, and condensation onto larger particles) has also been invoked in this context, as the particle size distribution shifts to an average larger size overtime.88 Curvature effects on dissolution. The potential for the onset of oxidation of Pt may be strongly affected by curvature (ala a Gibbs–Thomson effect),89 as
83
C. Iojoiu, E. Guilminot, F. Maillard, M. Chatenet, J.-Y. Sanchez, E. Claude, and E. Rossinot, J. Electrochem. Soc. 154, B1115 (2007). 84 V. O. Mittal, H. Russell Kunz, and J. M. Fenton, J. Electrochem. Soc. 154, B652 (2007). 85 J. Xie, D. L. Wood III, D. M. Wayne, T. A. Zawodzinski, P. Atanassov, and R. L. Borup, J. Electrochem. Soc. 152, A104 (2005). 86 V. A. Sethuraman, J. W. Weidner, A. T. Haug, S. Motupally, and L. V. Protsailo, J. Electrochem. Soc. 155, B50 (2008). 87 E. Guilminot, A. Corcella, F. Charlot, F. Maillard, and M. Chatanet, J. Electrochem. Soc. 154, B96 (2007). 88 P. J. Ferreira, G. J. la O’, Y. Shao-Horn, D. Morgan, R. Makharia, S. Kocha, and H. A. Gasteiger, J. Electrochem. Soc. 152, A2256 (2005). 89 K. Sieradzki, J. Electrochem. Soc. 140, 2868 (1993).
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(a)
2 mm
(b)
20 nm
Mag = 3.00 k x EHT = 20.00 kV WD = 10 mm
Signal A = STEM Signal B = InLens
Mixing = Off Signal = 1.000
Rate : 26 Apr 2006 Time : 10:38:34
(c)
(d)
(e)
20 nm
20 nm
20 nm
INPG CMTC
FIG. III.3. Cross-section electron microscopy of a used PEM fuel cell membrane. The anode is to the left, and the cathode is to the right. Particles (both individual nanoparticles and sintered agglomerates) of Pt are found within the membrane, separated from the cathode. From Ref. [87].
argued by Shao-Horn et al.81 who predicts that the onset of Pt oxidation may be shifted anodically by as much as a couple hundred millivolts, and experiments to probe the dissolution of platinum nanoparticles are beginning to appear.90 In the context of degradation, this may lead to enhanced Pt dissolution, particularly at potentials where one might think the Pt is unpassivated (the ‘‘high overpotential regime,’’ Section II.6). In the context of catalysis, it further bolsters the suggestion that ORR occurs only on Pt particles passivated with an oxide or hydroxide.
90
Q. Xu, E. Kreidler, D. O. Wipf, and T. He, J. Electrochem. Soc. 155, B228 (2008).
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A final practical issue associated with Pt corrosion is the effect of cycling the cell potential. In on/off cycling, the potential of the cathode can drop to a regime where Pt is no longer passivated with hydroxide, and thus residual oxide can desorb overtime. Cycling back again takes the Pt through the corrosion region of the Pourbaix diagram, where Pt metal can be oxidized directly to Pt2þ and dissolved into the electrolyte, and it is thus not surprising that cells subjected to such cycling deteriorate significantly faster overtime compared to cells held at fixed potential.88 IV. Improving the Hydrogen Oxidation Reaction
11. G ENERAL C ONSIDERATIONS The HOR reaction at the fuel cell anode works quite well, as long as the hydrogen fuel stream is sufficiently pure of irreversibly adsorbing species such as CO. This is due to the large exchange current density, which allows high current densities at only small overpotential. The total amount of Pt catalyst in the anode is thus relatively low, and anode loadings of <0.05 mg cm2 are routinely shown to be effective in laboratory cells. From the Materials Science standpoint, the primary challenge is improvement of the adsorbate tolerance of the anode catalyst. One way to do this would be to place a preprocessor in the fuel line before the anode. Within the preprocessor one could use a catalyst for the water–gas shift reaction (WGS), CO þ H2 O ⇄ CO2 þ H2 :
ð4:1Þ
This reaction is favored because the fuel stream is already humidified, as required for fuel cell operation, and produces only more hydrogen and carbon dioxide, the latter being inert. The problem with WGS as a solution is not only that it adds extra engineering complexity to the fuel cell system, but also that there currently is not a catalyst for this reaction that works near the operating temperature of the fuel cell. Currently used catalysts for WGS are copper based, for example, RaneyTM copper made by selective etching of Al from CuAl alloys in hot basic solution, or copper nanoparticles supported on oxide supports.91,92 Unfortunately, both these catalysts work with appreciable activity at temperatures greater than 400 C. There is significant attention being paid to WGS, as it is also employed in methanol
91
A. J. Smith and D. L. Trimm, Annu. Rev. Mater. Res. 35, 127 (2005). C. V. Ovesen, B. S. Clausen, B. S. Hammershoi, G. Steffensen, T. Askgaard, I. Chorkendorff, J. K. Norskov, P. B. Rasmussen, P. Stoltze, and P. Taylor, J. Catal. 158, 170 (1996).
92
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synthesis, coal gasification and other technologies. Recent theoretical advances have concentrated on elucidation of the WGS reaction. Using periodic, selfconsistent DFT, Gokhale et al. have shown that the rate-limiting step in this reaction on Cu(111) is hydrogen abstraction from adsorbed water, that carboxyl intermediates are playing an important role,93 and that such advances may lead to the identification of better, lower temperature catalysts for this reaction.94 One interesting variation of the attack to this problem has been to catalytically drive the water–gas shift as a half reaction at an anode, where the reaction products are carbon dioxide, protons, and electrons; the reaction energy is captured by shuttling the protons and electrons to a conventional fuel cell cathode.95 A second interesting variation on the strategy to avoid CO poisoning has been to feed a small amount of oxygen to the anode, mixed with the fuel stream. Pt nanoparticles seem to drive catalytic oxidation of CO, and nearly eliminate poisoning. While addition of oxygen to the fuel stream does raise safety and engineering questions, a more fundamental question is raised when it is noted that CO oxidation over Pt does not occur with bulk Pt until temperatures greater than 100 C, that is, higher than cell operating temperatures. One wonders if the effect is similar to the extraordinary catalytic ability of oxide-supported gold nanoparticles and oxide-free nanoporous gold to catalyze this reaction, even at sub-zero temperatures96,97, an effect linked to crystalline step edges as the active sites.98 The more common solution to CO adsorption has been the use of Pt alloy particles, where the binding affinity of CO to the surface is reduced due to the presence of a secondary species. How this secondary species interacts with CO depends on the nature of the adsorbed CO layer. While CO does act to block reactive sites, the details of this blockage are only now being clarified by combinations of spectroscopy, X-ray scattering, and scanning probe microscopy.99 Analogously to hydrogen UPD, CO is found to adsorb on Pt(111) in two adsorbed states.100 For CO adsorption at low potentials, such as in a fuel cell, there is simultaneous adsorption to a weakly bound state that is oxidized at lower potentials than a more strongly bound state. The particular potentials at which
93
A. Gokhale, J. A. Dumsic, and M. Mavrikakis, J. Am. Chem. Soc. 130, 1402 (2008). J. A. Rodriguez, S. Ma, P. Liu, J. Hrbek, J. Evans, and M. Perez, Science 318, 1757 (2007). 95 W. B. Kim, T. Voitl, G. J. Rodriguez-Rivera, and J. A. Dumesic, Science 305, 1280 (2004). 96 M. Haruta, T. Kobayashi, H. Sano, and N. Yamada, Chem. Lett. 4, 405 (1987). 97 C. Xu, J. Su, X. Xu, P. Liu, H. Zhao, F. Tian, and Y. Ding, J. Am. Chem. Soc. 129, 42 (2007). 98 N. Lopez, T. V. W. Janssens, B. S. Clausen, Y. Xu, M. Mavrikakis, T. Bligaard, and J. K. Nrskov, J. Catal. 223, 232 (2004). 99 B. Beden, C. Lamy, N. R. deTacconi, and A. J. Arvia, Electrochim. Acta 35, 691 (1990). 100 N. M. Markovic, B. N. Grgur, C. A. Luca, and P. N. Ross, J. Phys. Chem. 103, 487 (1999). 94
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oxidation can be driven depends on Pt surface orientation, but usually sits at potentials greater than 0.5 V. The presence of the CO adsorbates inhibits both hydrogen oxidation and hydrogen UPD. When CO is allowed to adsorb onto Pt(111) at potentials near 0 V, it is found to form an ordered p(2 2)-3CO structure with 3/4 CO per Pt atom;101 within each unit cell there are two CO in threefold hollow sites, and one atop CO. This adstructure leads to a large surface expansion, of order 4%.100 Not surprisingly, this expansion is also correlated to large changes in surface stress that can be measured using cantilever bending experiments.102 At higher potentials (0.25 V), results of IR spectroscopy are consistent with replacement of the threefold site with bridge sites.103 Of the two types of adsorption sites seen at fuel cell operating potentials, calculations using the density functional theory with the generalized gradient approximation (DFT-GGA) confirm that the atop adsorption site is the more strongly bound site.104,105 This system is really quite a nice, and well studied, example of how DFT has been assisting in the elucidation of the electronic aspects of catalysis. As reviewed by Hammer and Norskov,106 the CO-metal bond is due to hybridization of the CO 5s and 2p states with empty d states of the metal. This type of interaction is quite general for molecular adsorbates with transition metals, and there are often correlations between the center of the d-band in the metal and the binding affinity of molecules. Of course, the details of the binding will depend on particular atoms, their strain state, composition, the binding site, and so on, but it has become a useful tool to consider correlations in terms of the average d-band center. Here, for instance, it has been found that the adsorption energy of CO diminishes in magnitude as the d-band center of the substrate shifts more negative from the value of pure Pt (2.45 eV vs. vacuum).107 12. M ATERIALS
FOR
CO-R ESISTANT HOR C ATALYSTS
With these structural, mechanical, and electronic aspects of CO binding in mind, various materials strategies can be envisioned to help ameliorate CO binding and 101
I. Villega and M. J. Weaver, J. Chem. Phys. 101, 1648 (1994). L. Mickelson, Th. Heaton, and C. Friesen, J. Phys. Chem. 112, 1060 (2008). 103 F. Kitamura, M. Takahashi, and M. Ito, Surf. Sci. 223, 493 (1989). 104 G. Kresse, A. Gil, and P. Sautet, Phys. Rev. B 68, 073401 (2003). 105 S. E. Mason, I. Grinberg, and A. M. Rappe, Phys. Rev. B 69, 161401(R) (2004). 106 B. Hammer and J. K. Norskov, Adv. Catal. 45, 71 (2000). 107 Y. Gauthier, M. Schmid, S. Padovani, E. Lundgren, V. Bus, G. Kresse, J. Redinger, and P. Varga, Phys. Rev. Lett. 87, 036103 (2001). 102
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adsorption: (a) in order to avoid adsorption onto threefold hollow sites, alloy surfaces can be considered in which the Pt atoms are not densely percolating, (b) in order to strain surface Pt atoms, core-shell alloys can be considered that mechanically change their surface lattice spacing, and (c) alloys can be designed that modify the binding affinity by changing the electronic structure of the Pt d-band. Of course, for each change, one must also consider the effect on hydrogen adsorption and the kinetics of the HOR. It has long been known that PtRu alloys are significantly more tolerant toward CO poisoning than Pt alone.108 It is believed that this property is not due to an enhanced ability of PtRu to perform HOR in the presence of CO, primarily because the apparent activation energy for HOR on Pt and PtRu is the same. Instead, it is thought that Ru surface sites act to oxidize adsorbed CO, keeping more Pt surface sites open and available for HOR.109 As such, there is an obvious trade-off in composition—the more Pt sites, the higher the overall reaction rate but the lower the CO tolerance, and vice versa. Not surprisingly, in environments with 1% CO, a 50% Ru alloy works best.110 In this sense, PtRu alloys do not ‘‘enhance’’ HOR. They facilitate the HOR in CO containing environments. The observations of HOR on PtRu logically lead to a hypothesis that the surface composition is the most important aspect of CO tolerant catalysts. Both to test this hypothesis and also to reduce the precious metals content in the anode catalyst, a materials architecture has been developed for which the idea is to start with pure Ru nanoparticles, and then deposit upon them a sub-monolayer of Pt.111,112 This is accomplished via a spontaneous deposition effect seen when oxide-free Ru is immersed in an H2PtCl6 solution. Instead of oxidizing in solution, it is hypothesized that the metallic Ru reduces the Pt salt leaving small Pt clusters on the Ru particle surface (see Figure IV.1). Using this method, particles with as small an average Pt composition as PtRu20 (at a loading of 1 mg cm2) have been shown to exhibit the same electrochemical characteristics (Tafel slope and exchange current density) as pure Pt. In addition, the materials show the high CO tolerance. A somewhat different strategy to reduce Pt content and yet retain CO poisoning tolerance has been to use intermetallic alloys of Pt, which forms catalytically active ordered compounds with a number of elements including Mn, Pb, Sb, and 108
P. N. Ross, K. Kinoshita, A. J. Scarpellino, and P. Stonehart, J. Electroanal. Chem. 63, 97 (1975). W. F. Lin, M. S. Zei, M. Eiswirth, G. Ertl, T. Iwasita, and W. Vielstich, J. Phys. Chem. B 103, 6968 (1999). 110 H. A. Gasteiger, N. M. Markovic, and P. N. Ross, J. Phys. Chem. 99, 8290 (1995). 111 S. R. Brankovic, J. X. Wang, and R. R. Adzic, Electrochem. Solid State Lett. 4, A217 (2001). 112 J. X. Wang, S. R. Brankovic, Y. Zhu, J. C. Hanson, and R. R. Adzic, J. Electrochem. Soc. 150, A1108 (2003). 109
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FIG. IV.1. Transmission electron microscopy of PtRu20 nanoparticles: (left) (a) 2.5 nm diameter particles supported on 50 nm carbon supports, (b)–(e) high resolution images and selected area diffraction patterns showing the particles possess the Ru hcp crystal structure; (right) proposed structural model of the PtRu20 catalyst particle, illustrating the notion that small Pt clusters decorate Ru nanoparticle cores. From Ref. [111].
Sn.113,114 These materials offer a strategy for HOR catalysts more in-line with the strategies suggested by first principles calculations in the following senses: (i) Pt atoms are never found in surface clusters, and thus CO adsorption into threefold hollow sites is stymied, and (ii) the electronic structure of the material should be different from Pt, and perhaps tunable to a certain extent by the
113
A. F. Innocente and A. C. D. Angelo, J. Power Sources 162, 151 (2006). L. M. C. Pinto, E. R. Silva, R. Caram, G. Tremiliosi-Filho, and A. C. D. Angelo, Intermetallics 16, 246 (2008).
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alloying component. That the electronic structure of the alloy affects HOR kinetics is dramatically demonstrated by the electrochemistry of HOR on these intermetallics.113 The HOR on Pt, it will be recalled, exhibits a Tafel slope of 30 mV per decade and an exchange current density of 1 mA cm2. In contrast, PtSb and PtSn ordered intermetallics exhibit Tafel slopes of 60 mV per decade and larger exchange current densities, of 2.35 and 1.53 mA cm2, respectively. Not only are the exchange current densities larger than Pt alone, but the mechanism of HOR must be affected in an, as yet, undetermined way. V. Improving the Oxygen Reduction Reaction
13. N EW I NSIGHTS INTO O XYGEN R EDUCTION D ENSITY F UNCTIONAL T HEORY
FROM
The development of new catalysts materials for the ORR has been illustrative of the more global trend of using quantum mechanical calculations, especially DFT, as drivers in the identification of new materials systems. Most of the new materials discussed below were ‘‘virtually’’ assessed for ORR in the computer prior to any physical manifestation.115 Here, we give a very brief overview of the computational techniques being brought to bear on catalysis. To briefly summarize DFT, we note first that a computational framework suitable for surfaces must be one in which a sufficient number of atoms can be included in the calculation; at least 1000 electrons has been sufficient for a variety of problems.106 One also needs the capability of not just finding equilibrium ground states, but also to dynamically track the motion and reaction of adsorbates with surfaces. Such calculations are beyond the computational ability of wave-function based ab initio methods, but sufficient accuracy at these spatial temporal scales can be accomplished using DFT. Instead of solving a many-body Schrodinger equation, DFT begins with the assertion (proved by Hohenberg and Kohn116) that under an external potential vðrÞ there exists a universal functional G½r, dependent only on the electron density variation in space rðrÞ, such that the ground-state energy E of an interacting inhomogeneous electron gas can be written in the form 0 ðð ð rðrÞr r 3 3 0 1 d rd r þ G½r: ð5:1Þ E ¼ vðrÞrðrÞd3 r þ 2 jr r0 j
115 116
Z. Shi, J. Zhang, Z.-S. Liu, H. Wang, and D. P. Wilkinson, Electrochim. Acta 51, 1905 (2006). P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
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A variational approach to minimizing the total energy leads to a series of one-dimensional Schrodinger equations and the wavefunction solutions of these equations can be used to construct r. The difficulty, however, is that the functional G must include exchange-correlation effects. Kohn and Sham showed that this can be included in an approximate way,117 leading to the following set of equations: ( ) ð 0 r r 1 2 0 d3 r þ mxc ðrðrÞÞ ci ðrÞ ¼ ei ci ðrÞ r þ vðrÞ þ ð5:2Þ 2 jr r0 j where mxc is an appropriate approximating function for the exchange-correlation potential. For adsorption at surfaces, the electron density changes relatively rapidly, and the local density approximation (LDA) for the exchange and correlation energy of electrons usually yields incorrect energies.106 In these circumstances, so-called generalized gradient approximations (GGAs) are used instead. These come in many flavors—see, for instance, references in Refs. [115, 118]—with the unifying theme that they employ semiempirical approximations for the gradient-corrected exchange energy functional. DFT-GGA is most often employed to calculate adsorption energies or other thermodynamic quantities that are difference energies between two states that are assumed to be in equilibrium. The extension of DFT methods into the realm of dynamics began with Car and Parinello in 1985,119 who recognized that after an initial condition solution of the Kohn–Sham equations, the system of wavefunctions and ion positions can be allowed to follow a dynamical trajectory determined by Lagrangian dynamics. In this process, a Lagrangian is constructed from the kinetic energy of all ions, electrons, external constraints such as strain and so on, from which are derived equations of motion for the wavefunctions and ion cores. The method, sometimes called ‘‘ab initio molecular dynamics’’ is a good approximation as long as the Born–Oppenheimer assumption that the electrons are always equilibrated relative to the motion of the ion cores holds (which is usually true even during surface chemical reactions).120 Molecular dynamics simulations are good for examination of the approach of molecular entities to each other or surfaces, but the calculation of activation barrier energies is often hampered by statistical meandering of the system. To overcome this limitation, so-called nudged elastic band (NEB) methods have
117
W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992). 119 R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985). 120 J. S. Tse, Annu. Rev. Phys. Chem. 2002, 249 (2002). 118
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been developed in the last decade.121 These work by creating a virtual tether in configuration space between an initial and final state of a chemical reaction, and then dynamically adjusting the nodes of the tether (via a molecular dynamics-like procedure) until it follows the minimum energy path (MEP). According to classical transition state theory,122,123 the activation barrier for the reaction can be associated with a saddle point on the MEP, as long as it possesses one. It is only very recently that NEB methods have been applied to fuel cell catalysis reactions, but appear in a number of the calculations discussed below.42,124,125 14. C ALCULATIONS P ERTAINING
TO THE
O XYGEN R EDUCTION R EACTION
The development of simulation tools is mirrored by its application to the problem of oxygen reduction.126 In the 1990s DFT methods were first used to examine adsorption of oxygen molecules on Pt surfaces, but naturally focused on smaller scale systems including only oxygen/Pt interactions in the absence of adsorbed hydroxide.127–129 The proper insertion of a potential drop across the metal/ electrolyte interface was first probed by charging a small cluster of Pt atoms with a thin solvation shell.130 Further thermodynamic studies using DFT, including biasing the energy of electrons in a slab substrate and adsorbed water, suggested some generic trends that could be applied to the identification of good ORR catalysts. Norskov et al.20 calculated the relative electrochemical potential associated with a few potential reaction steps in the ORR, specifically,
121
1 Oxygen adsorption and dissociation : O2 þ M ⇄ O M 2
ð5:3Þ
First protonation : O M þ Hþ þ e ⇄ HO M
ð5:4Þ
Second protonation : HO M þ Hþ þ e ⇄ H2 O þ M:
ð5:5Þ
G. Henkelman, B. P. Uberuaga, and H. Jonsson, J. Chem. Phys. 113, 9901 (2000). G. H. Vineyard, J. Phys. Chem. Solids 3, 121 (1957). 123 G. Mills, H. Jonsson, and G. K. Schenter, Surf. Sci. 324, 305 (1995). 124 J. Zhang, M. B. Vukmirovic, Y. Xu, M. Mavrikakis, and R. R. Adzic, Angew. Chem. Int. Ed. 44, 2132 (2005). 125 S. Meng, E. G. Wang, and S. Gao, Phys. Rev. B 69, 195404 (2004). 126 T. V. Albu and S. E. Mikel, Electrochim. Acta 52, 3149 (2007). 127 A. Eichler and J. Hafner, Phys. Rev. Lett. 79, 4481 (1997) 128 A. B. Anderson and T. V. Albu, J. Am. Chem. Soc. 121, 11855 (1999). 129 A. B. Anderson and T. V. Alvu, J. Electrochem. Soc. 147, 4229 (2000). 130 T. Li and P. B. Balbuena, Chem. Phys. Lett. 367, 439 (2003). 122
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They found that at 1.2 V the energy differences between the protonation steps are approximately the same (0.45 eV), and both are uphill. They therefore conclude that in order to reach an equilibrated situation where the forward and reverse reaction rates for all three reactions are equal, an overpotential of 0.4–0.5 V must be applied, and this roughly agrees with experiment. The calculation suggests the general insight that good ORR activity on transition metals results partly from a synergy between O and OH binding affinity and proton transfer facility. For instance, Ni binds O and OH so strongly that the potential differences between the reactions (5.5) and (5.6) are each 1 eV, and thus protonation of these adsorbed species is slow. Pt, in contrast, binds O and OH more weakly, allowing facile proton transfer reactions to occur. This trend can be overshot, too. Au, for example binds O and OH very weakly, so one would expect easy protonation, but in fact O and OH are bound so weakly this never has a chance to occur. As a result, for elemental species, there is a peak in the trend of oxygen reduction activity versus O and OH binding affinity, as illustrated in Figure V.1. Materials with moderate oxygen binding affinity are predicted to be the best ORR catalysts, and amongst the elements, Pt is the best.
0.0 Pt −0.5
Pd
Activity
Ir −1.0
Rh Ru
−1.5
Cu
Ni
Co
−2.0
Au
Mo Fe
W −2.5 −3
Ag
−2
−1
1 0 ΔEO(eV)
2
3
4
FIG. V.1. Trend in oxygen reduction activity versus oxygen binding affinity in pure elements. Elements that too-strongly bind and dissociate oxygen (toward more negative values) are ineffective for ORR, and so are elements that too-weakly bind oxygen, such as Au. Pt, the most active pure elemental catalyst, sits near the top of this ‘‘volcano’’ plot. From Ref. [20].
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Norskov’s simulations do not consider the adsorbed reaction steps of the Damjanovic model, namely: Oxygen adsorption : M þ O2 ⇄ OO M
ð5:6Þ
First protonation : OO M þ Hþ þ e ⇄ M OOH:
ð5:7Þ
That this is the reaction sequence, and not Eqs. (5.3)–(5.5) has been dramatically suggested in a series of ab initio molecular dynamics simulations by Wang and Balbuena that for the first time have tracked an oxygen molecule through the full four electron reduction to water.54,131 Unlike Norskov’s work, these simulations support the Damjanovic idea that the rate-limiting step is the first electron transfer to an adsorbed oxygen molecule, Eq. (5.7). However, what happens next is more complex, and the simulations show a number of possible routes. One route is a second reduction step, leading to a one-end adsorbed hydrogen peroxide intermediate that readily dissociates into two co-adsorbed hydroxyls. A second route is that the OOH complex can be reduced to an atop water and an oxygen atom that is readily protonated by a nearby water molecule, a conclusion also supported by other DFT calculations.132 In either case, the adsorbed hydroxyls are further reduced to adsorbed water. This adsorbed water is relatively strongly bound, but can ultimately desorb, completing the reaction. Given these somewhat different simulation approaches, it is fair we think to point out a certain tension in the status of the quantum mechanical description of the ORR. On one hand, Norskov’s simulations suggest a good material for ORR should be able to easily dissociate O2, but not bind OH very strongly. On the other hand, Wang and Balbuena suggest a good ORR material should (i) easily form M–OOH intermediates and (ii) easily reduce adsorbed OH.133 From an electronic standpoint, these two criteria are never simultaneously optimal for any single element. Those elements with vacant valence d-orbitals tend to form M–OOH intermediates easily, but those elements with fully occupied valence d-orbitals tend to favor reductions of M–OH. Of single elements, Pt has the best trade-off. Figure V.2 shows a illustration of the free energy of these reactions (from DFT) relative to Pt. It strikes us that Norskov’s model is more applicable to ORR in the ‘‘high overpotential’’ regime where there is little adsorbed hydroxide species, and Wang and Balbuena’s model is more applicable to conditions where the surface is more or less covered with adsorbed hydroxide, that is, operational fuel cells. The role of proton diffusion within the hydroxide layer needs to be clarified
131
Y. Wang and P. B. Balbuena, J. Phys. Chem. B 109, 14896 (2005). M. P. Hyman and J. Will Medlin, J. Phys. Chem. B 110, 15338 (2006). 133 Y. Wang and P. B. Balbuena, J. Phys. Chem .B 109, 18902 (2005). 132
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5
ΔΔG4/3
V 4 Mn 3
Cr
2
Ir
Ru
Re
(a) Co
Ni
Os
1 Rh
ΔΔG1
Pt
0 −2
−1.5
−1
−0.5
−1
0
0.5
1
Cu 1.5
Pd −2 Au −3
Zn Ag
Cd
−4 (c)
−5
(b)
FIG. V.2. Trends in oxygen reduction activity in pure elements from ab initio molecular dynamics. x-axis: free energy of protonation relative to Pt; y-axis: free energy of M–OOH dissociation relative to Pt. In region (a), metals have vacant valence d-orbitals, promoting the formation of M–OOH; in region (b), the metals possess fully occupied valence d-orbitals, and promote reductions of M–O and M–OH. From Ref. [133].
for this system, as well as the inclusion of a realistic hydration layer in larger system sizes. The ab initio molecular dynamics simulation methodology used by Desai and Neurockto study water dissociation over Pt–Ru alloys may be useful for this purpose.134 The experimental status of oxygen reduction under fuel cell conditions is still unclear. The M–OOH intermediate has not been observed (to our knowledge) but 135 and this has led to the superoxide anion (O 2 ) has been seen in alkaline media, a tendency to discount the OOH intermediate in rate equation reaction modeling.136 But the ab initio molecular dynamics simulations support the notion that OOH is very unstable, and in highly acidic media will tend to very quickly decay into two hydroxyls.
134
S. K. Desai and M. Neurock, Phys. Rev. B 68, 075420 (2003). M.-H. Shao, P. Liu, and R. R. Adzic, J. Am. Chem. Soc. 128, 7408 (2006). 136 J. X. Wang, J. Zhang, and R. R. Adzic, J. Phys. Chem. A 111, 12702 (2007). 135
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Regardless of the details, the two pictures more-or-less give the same strategy for new materials development because elements that promote O2 dissociation also promote the formation of M–OOH intermediates (the vacancies in the d-band principle). The idea is to make mixtures of elements, one of which promotes O2 dissociation/M–OOH formation, and the other of which promotes reduction of OH. Such mixing can be simple bulk alloying, but more sophisticated ideas are to make core-shell structures, where mixing can occur laterally by surface alloying in the outermost surface layers and also to influence the ‘‘mixing’’ of adsorbate interaction modes from the subsurface core composition. 15. E XPERIMENTAL D EVELOPMENT OF N EW M ETALLIC C ATALYSTS FOR ORR We discuss here three approaches to the fabrication of core shell, near surface alloys for ORR: using binary alloy nanoparticle catalysts, fabricating Pt-skins over nanoparticles by galvanic displacement of adsorbed overlayers, and selectively dissolving one component from a multi-component alloy. No technique in particular is very easy—in fact, some dramatic demonstrations such as oxygen reduction over Pt3Ni have not moved past single crystal half-cell experiments— and challenges will have to be surmounted to scale up production. However, with each technique, materials have been found that exhibit a higher oxygen reduction activity than pure platinum. 16. B INARY A LLOY E LECTROCATALYSTS In fact, it has long been known that certain binary alloys of Pt, particularly alloys with the base transition metals V, Cr, Ti enhance oxygen reduction kinetics in phosphoric acid fuel cells. As mentioned, however, phosphate is strongly adsorbed onto Pt, in contrast to the PEM fuel cell electrolyte, and it was not clear that alloying would help in this case. However, alloys with Ni, Co, and Cr were found to significantly enhance ORR activity.137 Enhancements manifested themselves primarily by increases of 2–5 times in the exchange current density, the prefactor in Eq. (2.20); the Tafel slope under operating potentials sits in the 60 mV per decade range. XRD studies of the structure of the particles used indicated they primarily possessed the LI2 structure (facecentered cubic with the majority species on the cube faces, and the minority species at the cell corners), which is consistent with Pt3X (X ¼ Ni, Co, Cr), and
137
S. Mukerjee and S. Srinivasan, J. Electroanal. Chem. 357, 201 (1993).
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some PtX material with the L10 structure. These observations are in-line with the bulk phase diagram; Pt–Cr, for instance, has a miscibility gap opening up at a bit higher than 850 C, which separates precursor material in the 55–75 at.% range into these two compositions. The origin of the enhanced activity of these alloys was originally considered to be a compression of the Pt lattice parameter by these smaller atomic radii elements. This correlates to the XRD results, and the idea is that a smaller Pt lattice parameter supports O2 adsorption in a bridge configuration. However, note that the original materials used a mix of 50/50 and 75/30 Pt–X alloys. Further electrochemical characterization of alloy nanoparticles with better controlled stoichiometry showed that although PtCo worked better than Pt and Pt3Co, the trend was not generalized to Ni–Pt alloys, that is, NiPt has a lower activity than Pt3Ni.138 The above-referenced further electrochemical characterization of Pt/Ni and Pt/Co alloys sheds additional light on this issue. The linear dependence of the current on competing adsorbates such as hydroxide is given by Eq. (2.42), and this relationship was confirmed on these alloys. However, the onset of hydroxide formation was shifted positive by ca. 50 mV, suggesting a somewhat weaker binding affinity for hydroxide. And, as predicted by DFT calculations, a weaker binding affinity for hydroxide while keeping the affinity for oxygen dissociation should lead to better catalysts. Another effect, however, that may be operative here is the so-called ‘‘common-ion’’ effect whereby OH ligands on surface Ni and Co (assuming they exist at the surface) discourages the lateral formation of OH species on nearby Pt atoms. Work on alloy catalyst such as Pt/Ni, Pt/Co, or Pt/Cr is complicated by the problem that neither Ni, Co nor Cr is stable at the cathode potentials and pH of the PEM fuel cell, and they should corrode.22 Indeed, corrosion of Co from Pt/Co nanoparticles in operating fuel cells has been observed.139 The problem of selective dissolution of one component from a multi-component alloy is known as dealloying and is discussed further below. At this point it suffices to say that selective dissolution from alloy nanoparticles should enrich their surfaces with Pt. This likely obviates the common-ion effect in earlier studies where catalyst surface composition was not of key interest, but the idea is now being engineered into the so-called Pt-monolayer electrocatalysts (Section V.17) with more stable species, such as Ir. Well-prepared model single crystal studies of Pt3Ni have shown that the reduction of OH binding affinity can lead to quite dramatic effects. In single crystal studies of Pt3Ni, a 90-fold increase in ORR activity over Pt(111) was
138
U. A. Paulus, A. Wokaun, G. G. Scherer, T. J. Schmidt, V. Stamenkovic, V. Radmilovic, N. M. Markovic, and P. N. Ross, J. Phys. Chem. B 106, 4181 (2002). 139 S. Koh, J. Leisch, M. F. Toney, and P. Strasser, J. Phys. Chem. C 111, 3744 (2007).
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reported on a particular facet, namely Pt3Ni(111).140 In this study, samples were annealed in ultra-high vacuum (UHV) and formed a Pt-skin structure due to surface segregation of Pt. Surface X-ray scattering indicates that while the bulk of the material is a random solution of Pt and Ni, the topmost three layers show interesting segregation behavior, namely, the topmost layer is Pt, the second layer is Ni-rich (52%), and the third layer is Pt-rich (87%). Background corrected ultraviolet photoemission spectroscopy (UPS) was used to measure the d-band density of states for Pt3Ni(111), as well as the (001) and (110) orientations. For the (111) orientation, the shift in the d-band center from 3.10 to 2:76 eV was significantly larger than for the other orientations. We have already commented that DFT calculations have been used to correlate a shift in the d-band center to lower hydroxide binding affinity, and new DFT calculations on Pt3Ni(111) bolstered this idea. 17. P T -S KIN E LECTROCATALYSTS Translation of UHV annealed single crystals to nanoparticle catalysts integrated into a fuel cell is difficult, to say the least, and as of writing no one has reported Pt3Ni nanoparticles with well-defined (111) facets. The difficulty here is that the good performance of Pt3Ni is due to the Pt-skin structure, which was formed via surface segregation during a high temperature anneal. Nanoparticles subjected to the sample treatment will sinter. A different strategy has been used to overcome this obstacle. The concept is to use a single-component particle of a less-expensive metal (such as Pd or Au) as a substrate, then plate a monolayer or partial monolayer of Cu using UPD, and then galvanically displace the Cu with Pt, leaving a core-shell structure. For a review of this approach, see Ref. [141], and references therein. Apart from potential advantages in Pt-based ORR kinetics, this approach has the potential to significantly reduce the overall Pt use in fuel cells. Initially, work in this area also concentrated on single crystal studies, where the ORR activity of Pt monolayers on a variety of substrates (Au(111), Ir(111), Rh(111), Pd(111), Ru(0001)) was experimentally examined and correlated to DFT calculated binding affinities for OH adsorption and O2 dissociation (i.e., following the strategy of Norskov et al.20).142 Figure V.3 shows their primary
140 V. R. Stamenkovic, B. Fowler, B. S. Mun, G. Wang, P. N. Ross, C. A. Lucas, and N. M. Markovic, Science 315, 493 (2007). 141 R. R. Adzic, J. Zhang, K. Sasaki, M. B. Vukmirovic, M. Shao, J. X. Wang, A. U. Nilekar, M. Mavrikakis, J. A. Valerio, and F. Uribe, Top. Catal. 46, 249 (2007). 142 J. Zhang, M. B. Vukmirovic, Y. Xu, M. Mavrikakis, and R. R. Adzic, Angew. Chem. Int. Ed. 44, 2132 (2005).
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6 18
1.3 2
15
1.1 5
9
4
0.9 3
5
0.7 6
6
2
Ea/eV
−jk / mA cm−2
12
0.5
3 1 0 −4.5
−4.0 BEO/eV
−3.5
0.3 −3.0
FIG. V.3. Trends in Pt monolayer catalysts versus calculated binding energy of oxygen. (gray squares) Experimental kinetic currents (akin to the exchange current density) at 0.8 V for ORR in 0.1 M HClO4 solution, (filled circles) calculated activation energy for O2 dissociation, and (open circles) calculated activation energy for OH formation. 1. PtML/Ru(0001), 2. PtML/Ir(111), 3. PtML/Rh(111), 4. PtML/Au(111), 5. Pt(111), 6. PtML/Pd(111). Adapted from Ref. [133].
result: PtML/Pd(111) shows a significantly higher activity toward oxygen reduction than Pt alone, by a factor of about two. This particular combination is better than the other combinations because it demonstrates the best trade-off between OH adsorption and O2 dissociation. Having established that the Pt monolayer skin concept could be used to create oxygen reduction catalyst, work in the area has progressed to the fabrication of Pt-skin nanoparticles of a number of varieties. PtML/Pd nanoparticles were fabricated by galvanic displacement of UPD deposited Cu by Pt.143 In the fuel cell, performance (power) increases by a factor of 2 on a mass specific basis. These are comparable increases to the binary alloy catalysts, but with much less Pt. Of course, using Pd only reduces the costs of the catalysts moderately, and the logical next step has been to fabricate Pt-skins on non-noble metal nanoparticle cores. One method has been studied by Zhang et al. who used alloy nanoparticles containing Ni or Co alloyed with a more noble element on which UPD could be driven (e.g., Au, Pd, Pt). Exposure to elevated temperatures prior to mixing with Nafion to form a catalyst ink tends to segregate the noble species to the surface, and the Cu UPD and galvanic displacement can be performed. In this way,
143
J. Zhang, Y. Mo, M. B. Vukmirovic, R. Klie, K. Sasaki, and R. R. Adzic, J. Phys. Chem. B 108, 10955 (2004).
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Pt/Au/Ni, Pt/Pd/Co and Pt/Pt/Co (shell/surface/core) catalysts have been formed. While their overall catalyst performance, measured on a catalyst area basis, was not significantly higher than Pt-based devices, on a per mass basis, all three outperformed Pt by an order of magnitude. A final variation on this idea has been to use mixed monolayers (Pt plus one other element). Here, the idea is to use a second element during galvanic displacement that oxidizes much more easily (adsorbs oxygen or hydroxyls) than Pt by itself. In this case, the common-ion effect whereby the nearby secondary species weakens hydroxide adsorption on Pt should occur. Zhang et al. experimentally and theoretically assessed this construction first on singlecrystal surfaces, in which context they found a threefold increase in the kinetic current over pure Pt(111) by using (Pt80Ir20)ML/Pt(111), and then found a remarkable 21-fold increase in the kinetic current by using carbon-supported Pd nanoparticles as the substrate, that is, (Pt80Ir20)ML/Pd/C.144 Other examples of these constructed catalysts can be found in Ref. [141]. The excitement surrounding these new core-shell catalysts for ORR is well justified by the increase in performance, especially as most of the experimental work has been complemented by DFT simulations that show the alloy/adsorbate interactions are following the expected rules of thumb—d-band center shifts promoting oxygen dissociation, reduction of hydroxide adsorption affinity, and so on. 18. D EALLOYED C ATALYSTS The problem of alloy activity is strongly coupled with the problem of alloy stability. Elements that tend to easily oxidize under acidic conditions also tend to be soluble in acidic electrolytes. As mentioned, even platinum is soluble to a detrimental extent, such that small particles tend to dissolve away. One solution is to use somewhat larger particles, and create core-shell material as discussed above. Another solution is to take advantage of the corrosion of less noble alloy components to create self-assembled core-shell catalysts. The corrosion of the less noble component of an alloy is known as dealloying. Dealloying has a long history, especially within the corrosion community. First observed in the civil war era in the context of selective dissolution of Zn or Sn from brasses,145 the study of dealloying has progressed from a corrosion phenomenon to be avoided into a powerful tool to generate nanostructured materials.
144 J. Zhang, M. B. Vukmirovic, K. Sasaki, A. U. Nilekar, M. Mavrikakis, and R. R. Adzic, J. Am. Chem. Soc. 127, 12480 (2005). 145 C. Calvert and R. Johnson, J. Chem. Soc. 19, 434 (1866).
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Dealloying can be used to create nanoporous materials. For instance, the dissolution of silver from silver/gold alloys will generate a mesoporous metal called nanoporous gold, which possess pore sizes tunable from 5 up to 15 nm, depending on the applied electrochemical potential used to drive silver dissolution. Porosity forms in this system in silver-rich precursor alloys. These alloys are nearly ideal solid solutions and are not formed by the excavation of a buried bicontinuous microstructure. Rather, our current understanding is that porosity evolves due to a pattern forming instability associated with a complex choreography between three kinetic factors:146 (i) the dissolution rate of silver, (ii) a short time scale interfacial segregation of gold to step edges that avoids supersaturating the interface with low-coordination gold adatoms, and (iii) a long time scale capillarity-driven smoothening of the gold atoms that accumulated in the surface layer. Intrinsic length scales of the porous material (e.g., the ligament diameter) are set by the relative rate of dissolution and surface diffusion, but experimentally the ligament size has often coarsened considerably by the time samples are removed from the electrolyte (an effect that can be ameliorated by the judicious addition of a minority ternary component in the precursor alloy, such as Pt in Ag/Au alloys). Dealloyed mesoporous metals have been used as scaffolds for epitaxial layers of Pt, and these Pt shell/porous metal core composites work very effectively as low-Pt content anodes and cathodes in PEM fuel cells.147 In these studies Pt was electrolessly deposited on to the surface of the porous metal,76 and by using ultra-thin foils of porous gold only 100 nm thick (pore/ligament diameters 15 nm),148 fuel cells were made with precious metal loading of less than 0.05 mg cm2 Pt and less than 0.12 mg cm2 Au were made. The power generated on a per gram Pt basis was near 5 kW g1 Pt, although the power per unit area was less than nanoparticle-based catalysts. These cells are interesting for the following reasons: (i) they demonstrate that the catalyst layer does not need to be nanoparticulate, nor be infiltrated with Nafion ionomer. Rather, in these cells, proton transport must occur by surface diffusion within the Pt/vapor interface and (ii) no carbon supports are used, and thus the material is not susceptible to the same carbon corrosion dissolution mechanisms. Looking at dealloying more generally, if an alloy nanoparticle is subjected to selective dissolution conditions, then porosity evolution should not occur. Instead, capillarity-driven surface diffusion should always keep the particle smooth, but as more and more of the major component is dissolved away, the
146
J. Erlebacher, M. J. Aziz, A. Karma, N. Dimitrov, and K. Sieradzki, Nature 410, 450 (2001). R. Zeis, A. Mathur, G. Fritz, J. Lee, and J. Erlebacher, J. Power Sources 165, 65 (2007) 148 Y. Ding, Y. J. Kim, and J. Erlebacher, Adv. Mater. 16, 1897 (2004). 147
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surface will enrich in the more noble component left behind. This effect has been used to advantage by Mani et al. who started with PtCu particles and then dealloyed them chemically in situ within a fuel cell.149 Characterization of the dealloyed nanoparticles confirmed their alloy core-Pt shell structure, and the ORR activity of this material rivaled core-shell materials made by thermally driven surface segregation. First principles calculations have begun to appear regarding the dissolution of cathode catalysts, again using in these early stages thermodynamic models to assess the relative energy of atoms solvated in solution compared to sitting in terraces on the surface of alloys, rather than any dynamics.150,151 Trends matching the expectations of electrochemical measurements (especially Pourbaix diagrams) have been found, namely, that metals which alloy with Pt but form stable oxides (such as Ir or Rh) tend to stabilize the alloy toward dealloying, but less noble metal components (such as Ni or Co) should tend to dissolve from the alloy. VI. Synthesis and Outlook
The development of the PEM fuel cell has seen remarkable progress over the last 40 years and is a good example of a ‘‘nanotechnology’’ that was mature even before the term was coined. Power densities and durability are at a point where significant technological impact can be envisioned, and it is probably a good time to reflect on how much better fuel cells can become. Here, we have been focused only on the catalysts, and within this context it is fair to conclude that poisonresistant, high performance anodes for the HOR already exist, and the level of Pt use can be brought to extremely low levels. The ORR, on the other hand, has a way to go. Although increases in activity over Pt by an order of magnitude or more have been demonstrated, the ORR remains orders of magnitude more sluggish than the HOR. Even with advances in core-shell synthesis of multi-component alloys, significantly more Pt will have to be used in the cathode compared to the anode. That said, we are on the cusp of viability. Consider the following simplistic analysis: currently in a gasoline powered car, the catalytic converter contains at least 1 g Pt. Catalytic converters are not necessary in fuel cell vehicles, so we can speculate how much power can be generated with the 1 g of Pt used in a fuel cell instead. If, for a power density of 1 W cm2, the anode loading can be brought to 1 mg cm2 using PtRux catalysts, and if the cathode loading can be brought to 0.01 mg cm2 at this same power
149
P. Mani, R. Srivastava, and P. Strasser, J. Phys. Chem. C 112, 2770 (2008). Z. Gu and P. B. Balbuena, J. Phys. Chem. A 110, 9783 (2006). 151 J. Greeley and J. K. Norskov, Electrochim. Acta 52, 5829 (2007). 150
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density (representing a bit more than an order of magnitude increase in current ORR activity), then the cell in principle can generate 105 W (130 hp), enough to power the vehicle. There are many combinations of elements and materials structures still left to explore for the ORR. The Pt3Ni(111) system, although still confined to single crystal half-cell experiments, shows that in principle nearly two orders of magnitude higher activities can be found using bimetallic catalysts, and the surface alloy strategy shows this could be increased a number of times further. Given the fast recent developments, marked by an usually effective synergy between experimental and simulation approaches, there is reason to be optimistic that an effective transition metal catalyst for ORR will be discovered. There are other strategies for the development of ORR catalysts that do not utilize any precious metals at all. For instance, metal-organic porphyrin rings complexing Co or Cu ions are used by biological organisms to enzymatically catalyze ORR at very little overpotential. Other examples of good ORR catalysts are found in enzymes. For instance, the active site of laccase, another good ORR catalyst, is comprised of Cu ions complexed by imidazole groups (structurally similar to the porphyrins), and fuel cells have even been made that bind these enzymes to a substrate.152,153 The power generated by this strategy is orders of magnitude lower than Pt, but the approach is interesting. Perhaps synthetic metalorganic systems inspired by these enzymatic structures will be viable. A significant step in this direction was taken recently by Bashyam and Zelenay, who used a modified conducting polymer intercalated with metal ions to achieve power densities of 0.15 W cm2.154 There is still significant work, however, at even beginning to elucidate the mechanism of fuel cell catalysis in these metal-organic systems, and integration into robust, durable fuel cells capable of supplying high power is further away still. In the meantime, the development of new transition metal catalysts will have the dual function of generating new technologies and perhaps more importantly combined simulation/experimental methodologies that can be generalized to other areas of catalysis.
152 M. R. Tarasevich, A. I. Yaropolov, V. A. Bogdanovskaya, and S. D. Varfolomeev, J. Electoanal. Chem. 104, 393 (1979). 153 S. C. Barton, H.-H. Kim, G. Binyamin, Y. Zhang, and A. Heller, J. Am. Chem. Soc. 123, 5802 (2001). 154 R. Bashyam and P. Zelenay, Nature 443, 63 (2006).
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Acknowledgments
I am grateful to the Department of Energy, Basic Energy Sciences for the Hydrogen Fuel Initiative for support during the preparation of this manuscript. In addition, I would like to thank M. J. Aziz, K. Sieradzki, J. Snyder, and C. B. Arnold for helpful discussions and critiques.
SOLID STATE PHYSICS, VOL. 61
Effect of Dislocations on Electrical and Optical Properties in GaAs and GaN J. H. Y OU Department of Mechanical Science & Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801
H. T. J OHNSON Department of Mechanical Science & Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Review of Dislocation Types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Edge Dislocations as Electron Acceptors . . . . . . . . . . . . . . . . . . . . . . . . 4. Review of Dislocation Effects on Electrical and Optical Properties in Wurtzite GaN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Review of Dislocation Effects on Optical Properties in Zinc-Blende GaAs . . . . . 6. Article Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Modeling of Dislocations in Semiconductor Materials . . . . . . . . . . . . . . . . . . . 7. Electrostatics of Edge Dislocations in Semiconductor Materials . . . . . . . . . . . 8. Structure of Edge Dislocations in GaAs and GaN . . . . . . . . . . . . . . . . . . . 9. Screw Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Electron Scattering due to Threading Edge Dislocations . . . . . . . . . . . . . . . . . . 10. Modification of Read Model for 2q Edge Dislocations in GaN . . . . . . . . . . . 11. Classical Scattering Model for Electron Mobility . . . . . . . . . . . . . . . . . . . 12. Comparison with Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Single Electron Model for Calculating Dislocation Effects on Optical Properties . . . . 14. The Real Space kp Hamiltonian Approach . . . . . . . . . . . . . . . . . . . . . . . 15. Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16. Spontaneous Emission Spectrum Calculation . . . . . . . . . . . . . . . . . . . . . . 17. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Effect of Edge Dislocations on Optical Properties of GaN and GaAs . . . . . . . . . . . 18. Edge Dislocation Strain Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. Inhomogeneous Band Edge Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. Changes of Densities of States and Wave Functions due to an Edge Dislocation in GaN and in GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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21. Calculated Spontaneous Emission Spectra for GaN and GaAs . . . . . . 22. Reduction in Band Edge Peak as a Function of Dislocation Density . . . 23. Peaks Below Band Gap Energy in GaN . . . . . . . . . . . . . . . . . . . 24. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effect of Open-Core Screw Dislocations on Optical Properties in GaN . . . . 25. Strain Field Associated with Screw Dislocations . . . . . . . . . . . . . . 26. Inhomogeneous Band Edge Shifts due to Screw Dislocation . . . . . . . 27. Changes of Densities of States for Electrons and Holes due to Screw Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28. Calculated Spontaneous Emission Spectra and Comparison with Experimental Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions: Understanding Dislocation Effects Across the III–V Materials . 30. Effect of Dislocations in Other Materials . . . . . . . . . . . . . . . . . . 31. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction
Group III–V compound semiconductor materials, particularly alloys and heterostructures of GaAs and GaN, are of significant interest for the development of electronic and optical devices. Lattice mismatch between these materials and available substrates has long been a challenge, and it leads to device layers with extremely high dislocation densities. In the present work, the effects of dislocations on electrical and optical properties in III–V semiconductor materials are reviewed. The materials GaAs and GaN are considered in particular, because despite their chemical similarities, GaAs is known to be more sensitive to dislocations than GaN. Before discussing the details of dislocations in GaN and GaAs and their effects on electrical and optical properties in Sections II–VI, some general background about these materials and about dislocations in semiconductors is given in Section I. 1. M OTIVATION Crystalline perfection is almost impossible to achieve in most materials. There are inevitably defects that affect many material properties, including mechanical, electrical, and optical. Defects in materials can be classified into four categories: point, line, surface, and volume defects. A point defect is an imperfection associated with one or two atomic positions, such as a vacancy or an impurity. If a crystalline imperfection extends along a line, it is called a linear defect.
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A dislocation is a linear defect often classified as one of two distinct types: edge or screw, depending on how the material is locally distorted. Polycrystalline materials contain a large number of misoriented grains, and each grain boundary is an example of a surface defect. Twin boundaries and stacking faults are also surface defects. Precipitates and voids are examples of volume defects. For over a century, dislocations in materials have been studied intensively. In 1899, the presence of slip bands was observed during permanent unrecoverable (plastic) deformation of metals,1 which was later found to be attributed to dislocation motion. Until the 1930s, the microscopic view of plastic deformation had not been clearly explained. The first attempt to explain plastic deformation with respect to dislocation motion was made by Taylor in 1934.2 It was motivated by a large discrepancy between the theoretical shear strength to slide one plane of atoms over the adjacent plane and the experimentally observed strength. The theoretical strength is greater than observed values by several orders of magnitude.3 This discrepancy is explained by the presence of dislocation motion on an atom by atom level instead of the sliding of entire planes of atoms. One of most important material properties in mechanics is mechanical strength, which is determined by plastic deformations. Therefore, the presence of dislocations in materials plays an important role in characterizing mechanical properties of materials. Most semiconductor devices today are fabricated by growth of a device layer on a substrate which is typically a different material. During growth and processing, dislocations are generated in the growing material to relieve misfit strain energy due to material mismatch in lattice constants or thermal expansion coefficients between the film and substrate. Dislocations in semiconductors have been studied for over 60 years4–6 and are known to be detrimental to device performance. In particular, most efforts have been in studying electrical and optical properties of dislocations due to their significant influence on electron mobility and light emission efficiency.7–9 However, the details of fundamental processes of electron scattering and luminescence intensity reductions due to dislocations are still not well understood.
1
J. A. Ewing and W. Rosenhain, Phil. Trans. Roy. Soc. A 193, 353 (1899). G. I. Taylor, Proc. R. Soc. Lond. A 145, 362 (1934). 3 D. Hull and D. J. Bacon, Introduction to Dislocations, International Series on Materials Science and Technology, Vol. 37, Pergamon Press, Oxford (1984). 4 D. L. Dexter and F. Seitz, Phys. Rev. 86, 964 (1952). 5 W. T. Read Jr., Philos. Mag. 46, 111 (1955). 6 H. J. Queisser and E. E. Haller, Science 281, 945 (1998). 7 W. Heinke and H. J. Queisser, Phys. Rev. Lett. 33, 1082 (1974). 8 H. J. Queisser, Appl. Phys. 10, 275 (1976). 9 E. M. Kuznetsova, Fiz. Tverd. Tela 3, 1987 (1961). 2
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2. R EVIEW
OF
D ISLOCATION T YPES
A dislocation is a line defect along which the lattice is locally distorted by misalignments of atoms. There are two basic types of dislocations: edge and screw. Examples of edge and screw dislocations in a simple cubic structure are shown in Figure I.1. The edge dislocation is located at the termination of an additional half plane of atoms. In Figure I.1(a), the additional half plane is located in the upper part of the crystal, so the upper part is in compression while the lower part is in tension. A screw dislocation is formed by the application of shear stress, and the distorted lattice is shown in Figure I.1(b). Lattice distortions in edge and screw dislocations have different directions and magnitudes. The magnitude and direction of the lattice distortion is described by the Burgers vector. For an edge dislocation, the Burgers vector is perpendicular to the dislocation line while it is parallel to the dislocation line for a screw dislocation. In real materials, combinations of edge and screw type dislocations, or mixed dislocations, are also observed. Mixed dislocations have components and properties of both edge and screw type dislocations.10 In epitaxially grown materials, dislocation formation is induced by mismatch of lattice constants or thermal expansion coefficients between the growing layer and the substrate. When the thickness of an epitaxial layer is larger than a certain critical thickness, stored strain energy in the system (a)
Additional half plane Burgers vector b
Edge dislocation line
(b) Screw dislocation line
Burgers vector b
FIG. I.1. (a) An edge dislocation and (b) a screw dislocation in a simple cubic structure.
10 W. D. Callister Jr., Material Science and Engineering, An introduction, Wiley, New York (2003), 6th ed.
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Threading dislocation
Layer Substrate
Misfit dislocation
FIG. I.2. Misfit dislocations lie on the interface between the substrate and growing film material. Threading dislocations are the portions of the dislocations propagating into the growing materials.
will be reduced by formation of dislocations threading from the free surface to the bimaterial interface (threading dislocations) and along the bimaterial interface (misfit dislocations), as shown in Figure I.2. The crystallographic direction(s) of the dislocation lines and the crystallographic plane(s) on which the dislocations lie are different in different crystalline structures. Dislocations move in specific directions (slip directions) on specific planes (slip planes), depending on the crystalline structure. The slip plane is typically the plane with the highest planar density of atoms, and the slip direction is typically the direction with the highest linear density of atoms on the slip plane. In the face-centered cubic (fcc) structure, which is the basis of the zinc-blende (ZB) structure, slip planes are of type {111} and slip directions are of type h110i. In the hexagonal close-packed (hcp) structure, which is the basis of the wurtzite structure, there are three different slip planes: {0001}, f1010g, and f1011g, with h11 20i as the slip direction. 3. E DGE D ISLOCATIONS
AS
E LECTRON A CCEPTORS
An edge dislocation line contains dangling bonds which generate energy states for electrons, called dislocation states (ET). If dislocation states are below the Fermi level within the band gap, free electrons in the conduction band will lower their energy to occupy the dislocation states. Figure I.3 shows schematically how edge dislocations in n-type material trap free electrons. This transition for free electrons to be trapped at the dislocation states results in a reduction in free carrier concentration. However, the electron acceptors along dislocations are not always fully filled by electrons. The probability of occupancy at the dislocation states is a function of many parameters, such as Fermi level, dislocation state levels, temperature, and so on.11 Read11 has derived the filling fraction of
11
W. T. Read Jr., Philos. Mag. 45, 775 (1954).
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(a)
(b) EC EF
ET Ev FIG. I.3. (a) A free electron traveling near an edge dislocation tends to lower its energy by becoming trapped at the dangling bonds at the dislocation core. (b) The energy level of the dislocation state should be below the Fermi level within the band gap.
electron acceptors for edge dislocations in germanium by minimizing the total increase in free energy to move a free electron into a dislocation state at a particular filling fraction. Details of the filling fraction calculation by Read are reviewed in Section II.7. Once an edge dislocation traps free electrons, the edge dislocation becomes locally negatively charged and takes on an electrostatic potential. The electrostatic potential associated with negatively charged edge dislocations plays an important role in modifying the electrical and optical properties of dislocations. This potential scatters free traveling electrons resulting in a reduction of electron mobility, as is shown in Section III. It also repels free electrons, but attracts holes, resulting in a spatial separation of electrons and holes, reducing the optical emission intensity, as is shown in Section V. 4. R EVIEW OF D ISLOCATION E FFECTS ON E LECTRICAL AND O PTICAL P ROPERTIES IN W URTZITE G A N In recent years, wurtzite GaN and other III–V nitride semiconductors have received much attention for their application in blue light-emitting diodes,12,13 blue lasers,14 and high-power transistors15 because of their wide band gap and
12
H. Sakai, T. Koide, H. Suzuki, H. Yamaguchi, M. Yamasaki, S. Koike, M. Amano, and I. Akasaki, Jpn. J. Appl. Phys. Part 2 34, L1429 (1995). 13 S. Nakamura, M. Senoh, N. Iwasa, and S. Nagahama, Appl. Phys. Lett. 67, 1868 (1995). 14 S. Nakamura, M. Senoh, S. Nagahama, N. Iwasa, T. Yamada, T. Matsushita, H. Kiyoku, and Y. Sugimoto, Jpn. J. Appl. Phys. Part 2 35, L74 (1996). 15 R. J. Trew, M. W. Shin, and V. Gatto, Solid-State Electron. 41, 1561 (1997).
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high temperature stability. Despite these advantages, the large lattice mismatch between epilayer and available substrates (e.g., 14% with sapphire) induces a high density of dislocations. All types of threading dislocations, including edge, screw, and mixed, have been observed in epitaxially grown wurtzite GaN with h0001i orientation.16,17 The typical areal density is on the order of 107–1011 cm2 for edge dislocations18,19 and 105–107 cm2 for screw dislocations.16 Edge dislocations are found to act as electron acceptors, becoming negatively charged. Using a combination of atomic force microscopy and scanning capacitance microscopy, Hansen et al.20 report the presence of negative charge in the vicinity of edge dislocations. Selective photoenhanced wet etching has been used to reveal dislocation microstructures of n-type GaN films, and GaN whisker formation indicates negative charging of dislocations.21 Electron holography studies measuring charge on dislocations confirm that edge dislocations along h0001i in n-type GaN are negatively charged at a level of two electrons per structural unit length.22,23 Piezoelectric effects due to strain fields around edge dislocations are found to be insignificant compared to dislocation charging effects.23 Effects of edge dislocations on electron mobility in n-type wurtzite GaN have been studied experimentally by Weimann et al.24 In their work, it is found that a high density of charged threading dislocation lines (above 109 cm2) in GaN affect mobility for low carrier concentrations of less than 1018 cm3. Look and Sizelove25 have solved for the Hall mobility as a function of temperature within the framework of the Boltzmann transport equation and matched with experimental data very well. They assume that the edge dislocation contains the one
16 T. Hino, S. Tomiya, T. Miyajima, K. Yanashima, S. Hashimoto, and M. Ikeda, Appl. Phys. Lett. 76, 3421 (2000). 17 Y. Xin, S. J. Pennycook, N. D. Browning, P. D. Nellist, S. Sivananthan, F. Omne`s, B. Beaumont, J. P. Faurie, and P. Gibart, Appl. Phys. Lett. 72, 2680 (1998). 18 P. Hacke, A. Kuramata, K. Domen, K. Horino, and T. Tanahashi, Phys. Stat. Sol. (b) 216, 639 (1999). 19 M. Iwaya, S. Terao, T. Sano, S. Takanami, T. Ukai, R. Nakamura, S. Kamiyama, H. Amano, and I. Akasaki, Phys. Stat. Sol. (a) 188, 117 (2001). 20 P. J. Hansen, Y. E. Strausser, A. N. Erickson, E. J. Tarsa, P. Kozodoy, E. G. Brazel, J. P. Ibbetson, U. Mishra, V. Narayanamurti, S. P. DenBaars, et al., Appl. Phys. Lett. 72, 2247 (1998). 21 C. Youtsey, L. T. Romano, and I. Adesida, Appl. Phys. Lett. 73, 797 (1998). 22 D. Cherns and C. G. Jiao, Phys. Rev. Lett. 87, 205504 (2001). 23 E. Mu¨ller, D. Gerthsen, P. Bru¨ckner, F. Scholz, Th. Gruber, and A. Waag, Phys. Rev. B 73, 245316 (2006). 24 N. G. Weimann, L. F. Eastman, D. Doppalapudi, H. M. Ng, and T. D. Moustakas, J. Appl. Phys. 83, 3656 (1998). 25 D. C. Look and J. R. Sizelove, Phys. Rev. Lett. 82, 1237 (1999).
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negative charge per structural unit length along the dislocation line and conclude that edge dislocations are negatively charged in GaN. Unlike electrical properties of dislocations, effects of dislocations on optical properties are studied mainly experimentally. Ponce et al.26 report a spatial dependence of the band edge intensity using cathodoluminescence in n-type GaN. Weak band edge luminescence is observed in low-angle grain boundaries where the dislocation density is high. Hacke et al.18 and Iwaya et al.19 find that the intensity of band edge luminescence decreases as dislocation density increases for dislocation density higher than 108 cm2. Photoluminescence measurements in InGaN/GaN multiple quantum wells by Zhang et al.27 reveals that the integrated band edge intensity is reduced and the band edge peak broadens as dislocation density increases. They suggest that broadening of the band edge peaks is attributed to the strain field around dislocations. For screw dislocations in GaN, Xin et al.17 directly observe the atomic structure of full edge dislocations and also report the presence of full-core screw dislocations using atomic-resolution Z-contrast imaging. Open-core screw dislocations were first observed by Qian et al.28 The open-cores have hexagonal ˚ and are so called nanopipes. shape with radii in the range between 35 and 500 A Arslan and Browning29 experimentally observe three different kinds of threading screw dislocations: open, filled, and full-core screw dislocations. Open-core screw dislocations are essentially hexagonal voids and form nanopipes in the material. Filled-core screw dislocations are open-core nanopipes that are filled at a later stage of growth, while full-core screw dislocations are never open. Using electron-loss spectroscopy, it has been found that as the probe is closer to the edge of the open-core, the nitrogen signal decreases while the oxygen signal increases. For a full-core screw dislocation, no oxygen signal is observed at the core. Hawkridge and Cherns30 also experimentally confirm that open-core screw dislocations form because the oxygen atoms migrating on the inner surface block the crystal growth laterally. They find that about 1.7 monolayers of nitrogen from the inner core surface are replaced by oxygen. Screw dislocations are found to be electrically neutral.31,34 Hino et al.16 report on the effects of screw
26
F. A. Ponce, D. P. Bour, W. Go¨tz, and P. J. Wright, Appl. Phys. Lett. 68, 57 (1996). J. C. Zhang, D. S. Jiang, Q. Sun, J. F. Wang, Y. T. Wang, J. P. Liu, J. Chen, R. Q. Jin, T. Dai, and Q. J. Jia, Appl. Phys. Lett. 87, 071908 (2005). 28 W. Qian, G. S. Rohrer, M. Skowronski, K. Doverspike, L. B. Rowland, and D. K. Gaskill, Appl. Phys. Lett. 67, 2284 (1995). 29 I. Arslan and N. D. Browning, Phys. Rev. Lett. 91, 165501 (2003). 30 M. E. Hawkridge and D. Cherns, Appl. Phys. Lett. 87, 221903 (2005). 31 Y. Shreter and Y. Rebane, Proceedings of the International Conference on Extended Defects in Semiconductors-96, Gien, France, 7–12 September (1996). 27
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dislocations on photoluminescence in Si-doped GaN epitaxial layers on sapphire and find degradation of photoluminescence band edge intensity as screw and mixed dislocation density increases. Shi et al.32 find that screw dislocations and edge dislocations both decrease photoluminescence intensity. In GaN, besides the band edge peak, photoluminescence studies show other peaks below the band edge transition energy. Yellow luminescence (YL) with a broad Gaussian shape is observed at 2.16–2.4 eV with a width of 0.8–1.0 eV.26,32–35 Other peaks are observed at 45–500 meV below the band edge transition energy. There is some speculation that these peaks are from donor–acceptor pairs (DA), but the precise identity of the donor and acceptor states is unidentified to date.26,33–37 Spatial dependence of the YL, found by cathodoluminescence studies, suggests that dislocations at low-angle grain boundaries or point defects nucleating dislocations might be the origin of YL.26 A strong correlation between dislocation density (ndis), YL intensity (IYL) and DA intensity (IDA) has been observed experimentally; relative ratios of IYL and IDA over IBE increase as dislocation density increases.32,34,35 Experimental observations of green luminescence (GL) and blue luminescence (BL) are also reported. Reshchikov et al.38 find a transition from YL to GL centered at 2.5 eV when increasing the excitation power density and conclude that both YL and GL are from the same defects having multicharge acceptors, such as VGa–ON. Mickevicˇius et al.39 find that BL centered at 2.8 eV is also from the same structural defects as YL. In summary, both edge and screw types of dislocations are found in wurtzite GaN with a h0001i growth direction. For edge dislocations, the dangling bonds along edge dislocations act as electron acceptors resulting in the reduction of free carrier concentration. Edge dislocations become negatively charged at a level of two electrons per structural unit length. The negatively charged edge dislocations diminish the electron mobility due to electron scattering. Open-, filled-, and fullcore screw dislocations have been observed in GaN and are found to be electrically neutral. Formation of hexagonal open-core screw dislocations is attributed to oxygen atoms migrating on the inner surface preventing the crystal from 32
J. Y. Shi, L. P. Yu, Y. Z. Wang, G. Y. Zhang, and H. Zhang, Appl. Phys. Lett. 80, 2293 (2002). E. Calleja, F. J. Sa´nchez, D. Basak, M. A. Sa´nchez-Garcı´a, E. Mun˜oz, I. Izpura, F. Calle, J. M. G. Tijero, J. L. Sa´nchez-Rojas, B. Beaumont, et al., Phys. Rev. B 55, 4689 (1997). 34 X. G. Qiu, Y. Segawa, Q. K. Xue, Q. Z. Xue, and T. Sakurai, Appl. Phys. Lett. 77, 1316 (2000). 35 G. Li, S. J. Chua, S. J. Xu, W. Wang, P. Li, B. Beaumont, and P. Gibart, Appl. Phys. Lett. 74, 2821 (1999). 36 W. Go¨tz, N. M. Johnson, R. A. Street, H. Amano, and I. Akasaki, Appl. Phys. Lett. 66, 1340 (1995). 37 G.-C. Yi and B. W. Wessels, Appl. Phys. Lett. 68, 3769 (1996). 38 M. A. Reshchikov, H. Morkoc¸, S. S. Park, and K. Y. Lee, Appl. Phys. Lett. 81, 4970 (2002). 39 J. Mickevicius, R. Aleksiejunas, M. S. Shur, S. Sakalauskas, G. Tamulaitis, Q. Fareed, and R. Gaska, Appl. Phys. Lett. 86, 041910 (2005). 33
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growing laterally. With respect to optical properties, both edge and screw dislocations reduce the intensity of band edge luminescence. Besides the band edge peak, YL, GL, and BL have been observed and are found to be from the same kind of structural defects. Strong correlations have been observed between YL and dislocation density. Therefore, a better understanding of the characteristics of each type of dislocation is needed to predict electrical and optical properties. 5. R EVIEW OF D ISLOCATION E FFECTS IN Z INC -B LENDE G A A S
ON
O PTICAL P ROPERTIES
Zinc-blende GaAs is widely used for red light-emitting devices; it has a direct band gap of 1.519 eV at 0 K and 1.424 eV at 300 K.122 Zinc-blende GaAs is typically grown along the h001i direction; dislocation densities are reported over a wide range between 102–108 cm2.40,41 Recently, high quality GaAs has been successfully grown along h111i on Ge/Si1-xGex on top of a nanostructured Si substrate with dislocation density on the order of 105 cm2.42 Calculation of viscoplastic constitutive equation coupling microscopic dislocation density with macroscopic plastic deformation predicts the dislocation density reduction for h111i growth by an order of magnitude compared to growth along the h001i direction.43 It is widely known that emission efficiency diminishes as dislocation density increases in GaAs.7,44–48 Reduced emission intensity near dislocations is observed using spatially resolved photoluminescence measurements.49 A broad photoluminescence emission band with a peak at 0.95 eV, which is commonly observed in Si-doped GaAs, increases in intensity near the core region of dislocations while the band edge emission intensity decreases.50,51 Recent ab initio density function theory calculations reveal that edge dislocations induce dislocation states within the band gap.52 Photoluminescence studies indicate 40
A. H. Herzog, D. L. Keune, and M. G. Craford, J. Appl. Phys. 43, 600 (1972). H. C. Webber and S. E. R. Hiscocks, J. Mater. Sci. 8, 295 (1973). 42 G. Vanamu, A. K. Datye, R. Dawson, and S. H. Zaidi, Appl. Phys. Lett. 88, 251909 (2006). 43 X. A. Zhu and C. T. Tsai, J. Appl. Phys. 88, 2295 (2000). 44 P. Petroff and R. L. Hartman, Appl. Phys. Lett. 23, 469 (1973). 45 W. D. Johnston Jr., Appl. Phys. Lett. 24, 494 (1974). 46 R. Ito, H. Nakashima, and O. Nakasa, Jpn. J. Appl. Phys. 13, 1321 (1974). 47 A. L. Esquivel, S. Sen, and W. N. Lin, J. Appl. Phys. 47, 2588 (1976). 48 Y. D. Woo, H. I. Lee, T. W. Kang, T. W. Kim, and K. L. Wang, J. Mater. Sci. Lett. 14, 1340 (1995). 49 K. Bo¨hm and B. Fischer, J. Appl. Phys. 50, 5453 (1979). 50 R. Toba, M. Warahina, and M. Tajima, Mater. Sci. Forum 196–201, 1785 (1995). 51 Y. Chriqui, L. Largeau, G. Patriarche, G. Saint-Girons, S. Bouchoule, D. Bensahel, Y. Campidelli, O. Kermarrec, and I. Sagnes, Mater. Res. Soc. Symp. Proc. 809, B2.4.1 (2004). 52 S. P. Beckman and D. C. Chrzan, Physica B 340–342, 1001 (2003). 41
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another broad peak at 1.22 eV with width of 1.0 eV that is believed to be from a vacancy–donor complex.53 Finally, it is well known that photoluminescence intensity in GaAs is more sensitive to dislocation density than in GaN;54 an edge dislocation density of less than 106 cm2 is required for light-emitting devices based on GaAs, while GaN devices with edge dislocation densities of 1010 cm2 are acceptable. Such differences have not been explained, and more work is needed to reveal the underlying mechanisms. 6. A RTICLE O UTLINE This chapter presents both a review of electrical and optical effects of dislocations in semiconductors, and original results—first presented in a series of recent journal articles by the authors on the effects of dislocations in GaN and GaAs, where electrical and optical properties are calculated in the presence of dislocations.55–58 Section II covers electrostatic and structural modeling of dislocations including reviews of atomistic structure studies and experimental observations. For edge dislocations, charge distributions are calculated to identify the positions of electron acceptors. Based on charge distributions from Section II, the so-called filling fraction is calculated for an edge dislocation in wurtzite GaN, and electron mobilities are calculated and compared with experimental results in Section III. To review dislocation effects on optical properties, a multisubband Schro¨dinger equation method is formulated in real space and solved using the finite element method (FEM). Details of the finite element formulation and emission spectrum calculations are described in Section IV. In Sections V and VI, effects of dislocations on optical properties in GaN and GaAs are presented, including changes of densities of states and emission spectra, and a detailed comparison is made between the effects of edge dislocations in GaN and GaAs. In Section VII, implications of this approach for other III–V and III–Nitride materials are discussed.
53
H. J. Queisser and C. S. Fuller, J. Appl. Phys. 37, 4895 (1966). S. D. Lester, F. A. Ponce, M. G. Craford, and D. A. Steigerwald, Appl. Phys. Lett. 66, 1249 (1995). 55 J. H. You, J.-Q. Lu, and H. T. Johnson, J. Appl. Phys. 99, 033706 (2006). 56 J. H. You and H. T. Johnson, J. Appl. Phys. 101, 023516 (2007). 57 J. H. You and H. T. Johnson, Phys. Rev. B 76, 115336 (2007). 58 J. H. You, J.-Q. Lu, and H. T. Johnson, Math. Mech. Solids 13, 267 (2008). 54
154
J. H. YOU AND H. T. JOHNSON
II. Modeling of Dislocations in Semiconductor Materials
Basic models of both edge and screw dislocations in semiconductors are presented in the following sections. For the case of edge dislocations, in particular, the Read model11 for electrostatic features of dislocations in n-type Ge is summarized in Section 7. The basic crystal structure of edge dislocations in GaAs is then presented. For GaN, various dislocation structures are considered based on a literature survey of atomistic studies and experimental observations. GaN edge dislocations are shown to be negatively charged at a level of two electrons per structural unit length and are considered as a possible source of YL in n-type GaN. Also in GaN, open-, filled-, and full-core screw dislocations have been observed and found to be neutral. An overview of the mechanical origins of these unique hexagonal structures is presented in Section II.9.
7. E LECTROSTATICS OF E DGE D ISLOCATIONS IN S EMICONDUCTOR M ATERIALS
a. Dangling Bond Models of Semiconductor Dislocations The effects of edge dislocations in semiconductors have been studied over a half century, from the time when a reduction of carrier concentration was observed in plastically deformed n-type germanium.59–61 Read11 explains this observation by modeling edge dislocations as electron acceptors, whereby each dangling bond could accept one electron. Conversely, donor-like behavior of edge dislocations is observed under some conditions in p-type Ge62,63 and Si.64 Labusch and Schro¨ter65 explain these two different observations by a simple linear combination of atomic orbitals (LCAO) model. On each dangling bond along a dislocation there is only one electron, due to the missing single neighbor atom. Each dangling bond can then accept one electron from a chemical donor in n-type materials, so that the acceptor-like dislocation line becomes negatively charged.
59
G. L. Pearson, W. T. Read, and F. J. Morin, Phys. Rev. 93, 666 (1953). R. A. Logan, G. L. Pearson, and D. A. Kleinman, J. Appl. Phys. 30, 885 (1959). 61 R. M. Broudy, Adv. Phys. 12, 135 (1963). 62 L. Bliek and W. Schro¨ter, Phys. Stat. Sol. 14, K 55 (1966). 63 W. Schro¨ter, Phys. Stat. Sol. 21, 211 (1967). 64 H. Weber, W. Schro¨ter, and P. Haasen, Helv Phys. Acta 41, 1255 (1968). 65 R. Labusch and W. Schroter, in Dislocations in Solids, Other Effects of Dislocations. Disclinations 5, Chap. 20, ed. F R N Nabarro, Amsterdam, North-Holland (1980), pp. 127–192. 60
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
155
Alternatively, the single electron can be removed from the dislocation core, making the donor-like dislocations positively charged in p-type materials. Thus, two different models of electron filling fraction along an edge dislocation are available: one by Read11 and one by Labusch and Schro¨ter.66 Read treats the dislocation electronic state as single acceptor state, so the filling fraction is zero in a neutral material and can increase up to one in n-type materials. Labusch and Schro¨ter account for a splitting of dislocation states due to overlapping dangling bonds, leaving one state above the Fermi level and one state below. In their model, even in a neutral material the lower dislocation states are filled by electrons so that the dislocation band is half filled, and the occupation ratio of dislocation states is rescaled to accommodate both n-type and p-type materials. However, it is found that applying these different physical assumptions lead to only small differences in computing resulting dislocation properties.65 In Section II.7.b, Read’s model for n-type materials is summarized in more detail. The Labusch and Schro¨ter model is explained in detail in Refs. [65,66].
b. Read Model for Electrostatic Filling of Dislocations in n-Type Ge In the model developed by Read, dangling bonds along an edge dislocation act as electron acceptors and the edge dislocation takes on a negative line charge. However, the electron acceptor sites along an edge dislocation are not always fully filled by electrons. The extent of electron trapping, called the filling fraction, is a function of many physical parameters, such as the Fermi level, the energy levels of the dislocation state, the temperature, and the lattice parameter. Read11 calculates the filling fraction for an edge dislocation lying on the (111) plane in n-type germanium. As shown in Figure II.1(a) the edge dislocation in germanium contains a straight-line array of single traps that form a –1q line charge per structural unit as the fully charged state, where q is the magnitude of the electron charge. In the Read model, trapped electrons come from the ionized donors near the edge dislocation, so that the edge dislocation is surrounded by a positively charged cylinder with a depletion region of radius R, as shown in Figure II.1(b). The filling fraction is then calculated by minimizing the total increase of free energy per site along the dislocation. The filling fraction, f is defined as f ¼
66
b a
R. Labusch and W. Schroter, Phys. Stat. Sol. 36, 359 (1969).
ð2:1Þ
156
J. H. YOU AND H. T. JOHNSON
(a)
[111] (b) −
b −
−
Calculation disk
0
−
a −
R FIG. II.1. (a) Edge dislocation on a (111) plane in germanium at the fully charged state. The edge dislocation contains a straight-line array of single traps that form a – 1q line charge per structural unit as the fully charged state. (b) An example of filling fraction equal to 1/3. a is the spacing between trapped electrons, or occupied sites, and b is the spacing between dangling bonds. The edge dislocation is surrounded by a positively charged cylinder with a depletion region of radius R.
where a is the spacing between trapped electrons, or occupied sites, and b is the spacing between dangling bonds. The depletion radius, R can be found from the neutrality condition, qpR2 (ND–NA) ¼ q/a where q is the magnitude of the electron charge. Trapping one electron can change the electrostatic energy within the calculation disk per electron as ES ¼ Ee þ Ec þ Eec þ Ece
ð2:2Þ
where Ee is the electron–electron interaction energy along the dislocation, Ec is the positive space charge energy, Eec is the interaction energy of electrons with the positive space charge, and Ece is the interaction energy of the positive space charge with electrons. Assuming the dislocation line is infinite, ES can be solved within the middle calculation disk, as shown in Figure II.1(b), because ES is periodic along the dislocation.
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
157
For the electron–electron interaction energy Ee, the potential ce0 at the electron 0 in the calculation disk, due to other electrons is 2q X 1 4pea n ¼ 1 n N=2
ce0 ¼
ð2:3Þ
where e is dielectric constant. In the limit of the sum as N ! 1, lim
N=2 X 1
N!1
n¼1
¼ ln
n
N þ 0:577 2
ð2:4Þ
where 0.577 is known as Euler’s constant. Then the electron–electron interaction energy Ee is 1 q2 N ln þ 0:577 : ð2:5Þ Ee ¼ qce0 ¼ 4pea 2 2 The positive space charge energy, Ec, can be written as ðR a Ec ¼ s rcc ðrÞ2pr dr 2
ð2:6Þ
0
and cc (r) can be found by integrating Poisson’s equation as cc ðrÞ ¼
pr 2 ðR r 2 Þ; 4pe 1
prR2 cc ðrÞ ¼ 4pe
Na=2 ð
Na=2
dz pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r þ z2
2prR2 Na ln ; 4pe r
ð2:7Þ
for r < R
:
ð2:8Þ
for r > R
Equating the two solutions from Eqs. (2.7) and (2.8) at r ¼ R gives R21 ¼ R2 ½1 þ 2 lnðNa=RÞ: Then, cc ðrÞ ¼
q Na r 2 1 þ 2 ln 2 R 4pea R
ð2:9Þ
ð2:10Þ
158
J. H. YOU AND H. T. JOHNSON
and from Eq. (2.6) Ec becomes Ec ¼
q2 Na 1 ln þ : 4pea R 4
ð2:11Þ
The interaction energy of electrons with the positive space charge, Eec and the interaction energy of the positive space charge with the electrons, Ece, are equal to each other. The interaction energy of electrons with the positive space charge can be written as Eec ¼ 12 qcc ðr ¼ 0Þ. Therefore, from Eq. (2.10), q2 Na 1 þ 2 ln ð2:12Þ ¼ Ece : Eec ¼ 8pea R The total electrostatic energy per electron can be found as q2 R ln 0:866 : ES ¼ Ee þ Ec þ 2Eec ¼ 4pea a
ð2:13Þ
The total increase of free energy per site is then fFð f Þ ¼ f ðET EF Þ þ f ES ð f Þ f TSð f Þ
ð2:14Þ
where ET is the localized energy level of the dislocation within the band gap, EF is BT ln W represents the effect of configurational the Fermi level, and f TSð f Þ ¼ kM entropy, where there are W ways of arranging N electrons in M sites. kB is Boltzmann’s constant and T is temperature. The filling fraction, f at given ET and EF, should be one that minimizes the total increase of free energy from Eq. (2.14). This approach from Read to calculate the filling fraction is good for a straightline array of single traps, such as an edge dislocation in germanium as shown in Figure II.1(a). However, edge dislocations in wurtzite GaN do not contain a straight-line array of single traps, but they have a double-line array of traps. This is explained in Section II.7.c based on ab initio electronic structure calculations. Modification of the Read model for a double-line array of traps is then calculated in Section III.10.
c. Electrostatic Potential due to Edge Dislocations Read11 also derived the electrostatic potential for a negatively charged edge dislocation as a continuous line charge along the axis of a cylinder (radius of R) of uniformly distributed positive charge. The boundary condition imposed is that the potential vanishes at the boundary of cylinder. The electrostatic potential derived by Read is
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
" # r 2 q2 R 2 ln VRead ðr RÞ ¼ f 1þ 4pea r R
159
ð2:15Þ
where e is dielectric constant, q is the magnitude of the electron charge, a is the spacing between electron acceptors along a dislocation, r is radial distance measured from the core of the dislocation, and the multiplication factor, f is the filling fraction of electron acceptors along the dislocation. Therefore, the strength of the electrostatic potential is determined by the filling fraction, which ranges from 0 to 1. The potential becomes zero outside the cylinder implying that all of the electrons within the cylinder with a sharp boundary will be trapped at the dislocation, so the dislocation zone is distinct from the neutral zone of the crystal. Bonch-Bruevich and Glasko (B–G)67 independently solve for the electrostatic potential within the framework of quantum mechanics, treating the dislocation as a structural defect that can capture electrons. The Schro¨dinger equation is solved with the Poisson equation for a screened negatively charged infinitely long dislocation line and the electrostatic potential is obtained as VBG ðrÞ ¼
2q2 K0 ðr=lD Þ 4pea
ð2:16Þ
where lD is Debye screening length and K0(r/lD) is the modified Bessel function of the second kind of order 0 evaluated at r/lD. The main characteristic in Eq. (2.16) is that the dislocation zone overlaps with neutral zones of the crystal, but is not constrained by a sharp boundary of a depletion region. Figure II.2 shows the Read potential and the B–G potential with normalized radial distance from the core of the edge dislocation. The Read potential terminates at r ¼ R while the B–G potential extends infinitely. 8. S TRUCTURE
OF
E DGE D ISLOCATIONS
IN
GAAS
AND
GAN
a. Crystallographic Structure of Edge Dislocations in Zinc-Blende GaAs Gallium arsenide is based on a zinc-blende structure, which contains two facecentered cubic (fcc) lattices, each with different species of atoms; one fcc structure is shifted by [a/4a/4a/4] from the other. The slip plane in an fcc structure is of type {111} and the slip direction is of type h 110i. As an example, in an fcc crystal grown along a [001] direction, a misfit dislocation lies on the interface
67
V. L. Bonch-Bruevich and V. B. Glasko, Fiz. Tverd. Tela 3, 36 (1960).
160
J. H. YOU AND H. T. JOHNSON
(a)
4
Electrostatic potential (eV)
3.5 3 2.5 2 1.5 1 0.5 0
(b)
0
0.5
1
1.5 r/R
2
2.5
3
4
Electrostatic potential (eV)
3.5 3 2.5 2 1.5 1 0.5 0 0
0.5
1
1.5
2
2.5
3
r/lD FIG. II.2. Electrostatic potential profiles from (a) Read model and (b) B–G model. The potential from the Read model has a sharp boundary where the potential terminates. The potential from the B–G model extends infinitely.
with the substrate along a ½110 direction on the (111) plane as shown in Figure II.3(a). Figure II.3(b) shows that the (111) plane contains two possible threading dislocation directions: ½01 1 and ½10 1 on the (111) slip plane. In this
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a)
161
(b)
[001]
[101]
[011]
A D
A [110] C
[010]
B
C
[110]
B [100] FIG. II.3. (a) A misfit dislocation lies on the interface with a substrate along the ½110 direction on a (111) slip plane. (b) Two possible directions of threading dislocations: ½001 and ½101 on the (111) slip plane.
example, an additional half plane of atoms lies on the (111) plane and terminates at the ½ 110 misfit dislocation and either the ½01 1 or the ½101 threading dislocation. The Burgers vector, which is a measure of the direction and magnitude of misalignment of the atoms as a result of the dislocation, is b ¼ a/2h110i. The angle between the misfit and threading dislocations is 60 , and thus the threading dislocation is a 60 dislocation. Another possible slip plane, on which the ½110 misfit dislocation lies, is the ð 1 11Þ plane passing through the atoms B, C, and D in Figure II.3(a). On the ð1 11Þ slip plane, another set of two possible directions for threading dislocations exist: [011] and [101]. Therefore, with a misfit dislocation along the ½ 110 direction, a threading dislocation can be aligned with either ½01 1, ½10 1, [011], or [101] with an angle of 60 from the misfit dislocation as shown in Figure II.4(a). For GaAs, the slip plane and direction are the same as in the fcc structure because adding an fcc lattice of As (or Ga) to the fcc lattice of Ga (or As) with an offset of [a/4a/4a/4] does not change the highest planar density plane and linear density direction. In ZB GaAs with a h001i growth direction, two types of threading edge dislocations have been found experimentally; one has an angle of 60 with the misfit dislocation and the other is parallel to the h001i direction, creating an angle of 90 with the misfit dislocation. The 60 dislocation is on a (111) plane as shown in Figure II.4(a), while the 90 dislocation is generated by segments of two 60 dislocations.68 The Burgers vector is b ¼ a/2h110i, where a is the lattice constant of GaAs.
68
Y. Yang, H. Chen, Y. Q. Zhou, X. B. Mei, Q. Huang, J. M. Zhou, and F. H. Li, J. Mater. Sci. 31, 829 (1996).
162
J. H. YOU AND H. T. JOHNSON
(a)
[001]
(b)
[001]
[110] [010]
[100]
[110] [010]
[100]
FIG. II.4. (a) With a misfit dislocation along ½110 direction, there are four possible directions for threading dislocation: ½001, ½101, [011], and [101]. All four possible directions have angle of 60 with the misfit dislocation. (b) Model of threading edge dislocation along [001] direction from averaging angles of ½00 1, ½101, [011], and [101].
To model threading edge dislocations in GaAs grown along the h001i direction, a representative edge dislocation is taken to be parallel to the growth direction as shown in Figure II.4(b). This is based on averaging the four possible 60 dislocations, and it also represents the 90 dislocation. In Section V effects of threading edge dislocations on optical properties are calculated in a bulk like layer of GaAs. Since the electrostatic potential and strain fields vary only on plane perpendicular to the dislocation, but are uniform along the dislocation as shown in Eqs. (2.15) and (2.16), threading edge dislocations may be modeled as two-dimensional features.
b. Atomistic Structure of GaN Edge Dislocations The crystal structure of GaN edge dislocations is simpler in some ways than the crystal structure of GaAs edge dislocations; the core is simply aligned along the h0001i direction. However, the atomistic details of the core reconstruction are much more complex. Many atomistic structural models have been studied and suggested to explain the characteristics of negatively charged edge dislocations and to find possible sources of YL in GaN. In reviewing various atomistic structures for edge dislocations, the main properties of interest are (i) structural stability, or lowest formation energy, (ii) the electrostatic charge distribution at a level of two electrons per structural unit length, (iii) the electrical activity, or the presence of dislocation electron states below the Fermi level within the band gap in n-type material, and (iv) the possibility that the dislocation is a source of YL.
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a)
163
(b)
A
[0001]
FIG. II.5. (a) A full-core edge dislocation contains Ga and N atoms at the dislocation core (point A) and (b) an open-core edge dislocation has vacancies of both Ga and N atoms at the core. Images are reproduced with permission from Ref. [71]. Gallium atoms are shown as large white circles and nitrogen atoms are shown as small black circles. A gallium (or nitrogen) vacancy edge dislocation can be constructed by removing only a gallium (or nitrogen) atom from the core. A VGa–ON defect complex edge dislocation has the gallium vacancy and an oxygen atom replacing the nitrogen atom at the core.
Before discussing details of the atomistic models, it is useful to review the atomistic structures of the basic dislocations. Figure II.5 shows the atomistic structures for full-core and open-core edge dislocations. Point A in Figure II.5(a) indicates the position of the dislocation core; the dislocation line is along with h0001i direction. The full-core edge dislocation contains Ga and N atoms along the dislocation core, while the open-core has vacancies of both Ga and N atoms, as shown in Figure II.5(b). A gallium (or nitrogen) vacancy edge dislocation can be constructed by removing only a gallium (or nitrogen) atom from the core. A VGa– ON defect complex edge dislocation has the gallium vacancy and an oxygen atom replacing the nitrogen atom at the core. Elsner et al.69 use the ab initio local-density functional (LDF) cluster method, AIMPRO, and the density functional method based on tight binding (DF-TB) to calculate the structural and electronic properties of threading edge and screw dislocations. An edge dislocation is found to be energetically favorable with a full-core but electrically inactive, with a band gap free from deep dislocation state levels. Leung et al.70 use the Vienna ab initio simulation package (VASP), a DFT program, to study the effects of doping and growth condition on edge dislocations. ¨ berg, J. Elsner, R. Jones, P. K. Sitch, V. D. Porezag, M. Elstner, Th. Frauenheim, M. I. Heggie, S. O and P. R. Briddon, Phys. Rev. Lett. 79, 3672 (1997). 70 K. Leung, A. F. Wright, and E. B. Stechel, Appl. Phys. Lett. 74, 2495 (1999). 69
164
J. H. YOU AND H. T. JOHNSON
Edge dislocations with a gallium vacancy (VGa) at their core are predicted to be stable and to act as electron acceptors under nitrogen-rich growth conditions, while edge dislocations with a nitrogen vacancy (VN) at their core are predicted to be stable and to act as hole acceptors under gallium-rich growth conditions. Lee et al.71 investigate full-core, open-core, VGa, and VN edge dislocations using selfconsistent-charge density-functional tight-binding approaches, and find the opencore edge dislocation to be stable under gallium-rich growth conditions and to be inactive. They also find the VGa edge dislocation to be stable under nitrogen-rich growth conditions and to be inactive. Neugebauer and Van de Walle72 first suggested that the VGa–ON defect complexes may be trapped at extended defects, such as dislocations or lowangle grain boundaries, and that they might be the source of YL, which is often found in photoluminescence studies of GaN. The VGa–ON defect complex acts as an acceptor, which is consistent with the experimentally observed behavior of YL; the YL is suppressed in p-type materials. Later, Elsner et al.73 considered the behavior of the oxygen defects, gallium vacancies, and related defect complexes, such as [VGa–(ON)i](3i) (i ¼ 1, 2, and 3), at various positions, such as at bulk sites, near dislocations, and at the edge dislocation cores. Their results show that VGa–(ON), VGa–(ON)2, and VGa–(ON)3 have considerably lower negative formation energy at the core of edge dislocations, 2.3, 2.5, and 3.0 eV, respectively, while formation energies at bulk positions are 1.1, 0.7, and 0.8 eV. The [VGa–(ON)i](3-i) defect complexes at the dislocation cores have lower formation energy than VGa and ON by more than 2 eV. In particular, the VGa–ON complex at the edge dislocation core has a deep state at 1 eV above the valence band maximum, suggesting that VGa–ON complex defects trapped at dislocations are the origin of YL. Also, the VGa–ON complex defect edge dislocation is electrically active with two electrons trapped per structural unit, which is consistent with experimental observations.22,23 Various types of edge dislocations are summarized in Table II.1. Based on the formation energies, the electrical activity with a charge state of two electrons, and the possibility as a YL source, the VGa–ON complex defect edge dislocation is selected for further consideration and is used for further calculations to study the effects of edge dislocations in wurtzite GaN in the present work.
71
S. M. Lee, M. A. Belkhir, X.Y. Zhu, Y. H. Lee, Y. G. Hwang, and T. Frauenheim, Phys. Rev. B 61, 16033 (2000). 72 J. Neugebauer and C. G. Van de Walle, Appl. Phys. Lett. 69, 503 (1996). 73 J. Elsner, R. Jones, M. I. Heggie, P. K. Sitch, M. Haugk, Th. Frauenheim, S. o¨berg, and P. R. Briddon, Phys. Rev. B 58, 12571 (1998).
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES TABLE II.1. VARIOUS TYPES
OF
EDGE DISLOCATIONS
IN
165
WURTZITE GAN
Types
Stability
Electrical activity
References
Full-core Open-core
Stable Stable for n-type Ga-rich condition Stable under N-rich condition Stable under Ga-rich condition Stable (lowest formation energy)
Inactive Inactive
Elsner et al.69 Lee et al.71
Active
Leung et al.70
Active
Leung et al.70
Active (YL source)
Elsner et al.73
VGa VN VGa–ON
c. Ab initio Calculation of Charge Distribution Along Edge Dislocations in GaN As shown in the previous section, Read calculates the filling fraction for a dislocation in germanium, which contains straight-line arrays of single traps, by minimizing the total increase of free energy per site of the dislocation. To use the Read model for wurtzite GaN, it is necessary to identify the amount of charge per structural unit and the positions of electron acceptors for the VGa–ON edge dislocation model under consideration. For this purpose, a charge distribution is calculated using the VASP.74–77 For a test case, a calculation for bulk wurtzite GaN is performed using ultra soft pseudopotentials (US-PP) and the projectoraugmented wave method (PAW). Table II.2 shows the comparison of the calculated lattice constants with experimental results78,79 for bulk wurtzite GaN. Lattice constants from both US-PP and PAW are very close to experimental measurements, but the US-PP result is slightly closer to the experimental data, so US-PP is used for the charge distribution calculation for a VGa–ON edge dislocation. First, a full-core edge dislocation structure is calculated in a 142-atom supercell containing a dislocation dipole and calculated with US-PP and the
74
G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). G. Kresse and J. Hafner, Phys. Rev. B 49, 14251 (1994). 76 G. Kresse and J. Hafner, Phys. Rev. B 54, 11169 (1996). 77 G. Kresse and J. Furthmu¨ller, Comput. Mater. Sci. 6, 15 (1996). 78 H. Schulz and K. H. Thiemann, Solid State Commun. 23, 815 (1977). 79 J. H. Edger (Ed.), Properties of Group III nitrides, Electronic Materials Information Service (EMIS) Datareviews Series, Institution of Electrical Engineers, London (1994). 75
166
J. H. YOU AND H. T. JOHNSON TABLE II.2. COMPARISON
OF
LATTICE CONSTANTS
FOR
BULK WURTZITE GAN
˚) Lattice constants (A
Experiment78 Experiment79 US-PP PAW
a ¼ 3.189, c ¼ 5.185 a ¼ 3.190, c ¼ 5.189 a ¼ 3.1813, c ¼ 5.1755 a ¼ 3.2247, c ¼ 5.2441
gradient-generalized approximation (GGA) in the exchange-correlation functional. The full-core edge dislocation contains Ga and N atoms at the dislocation core as shown in Figure II.6(a). The relaxed structure is shown in Figure II.6(a) and the charge distribution is shown in Figure II.6(b). Gallium atoms are shown as large white circles and nitrogen atoms are shown as small black circles. For the VGa–ON complex edge dislocation, the N atom at the dislocation core is replaced by an O atom and the Ga atom is removed. The relaxed structure for the VGa–ON edge dislocation without doping is shown in Figure II.6(c). For n-type materials, some Ga atoms are replaced by Si atoms. Figure II.6(e) shows the relaxed structure for the n-type case, where six Ga atoms are replaced with Si atoms in symmetric positions. Oxygen atoms are red and silicon atoms are purple only in the n-type case. Charge distributions on the plane perpendicular to h0001i and cutting the center of two N atoms near the Ga atom (or vacancy) at the core, are shown in Figure II.6(b) for the full core, in Figure II.6(d) for VGa–ON complex in the undoped case, and in Figure II.6(e) for the VGa–ON complex in the n-type case. Comparing the charge distributions between the full-edge and undoped VGa–ON complex, the two N atoms near the Ga vacancy in the undoped VGa–ON complex dislocation contain less charge than in the full-core case, and the distance between the two N atoms in the undoped VGa–ON complex dislocation is smaller due to the Ga vacancy at the core. Comparing the charge distribution between the undoped and n-type materials for the VGa–ON complex dislocation, the charge density around the two N atoms near the Ga vacancy along the core becomes larger for the n-type case. In comparing Figure II.6(c) and (e), the larger separation between the two N atoms in the n-type case is consistent with greater repulsion due to the presence of additional negative charge on each N atom. To see any symmetry effects of Si positions, six Si atoms are located in asymmetric positions as shown in Figure II.7(a) for the relaxed structure and Figure II.7(b) for the charge distribution. Compared with the symmetrically doped case (Figure II.6(e) and (f)), the asymmetric dopant position does not change the results significantly. Figure II.7(c) and (d) shows the relaxed structure and charge distribution for the n-type case with four Si atoms with symmetric positions.
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a)
167
(b)
0.004 0.0036 0.0032 0.0027 0.0023 0.0019 0.0015 0.001 0.0006 0.0002
Dislocation core (c)
(d)
(e)
(f)
N atom near VGa
FIG. II.6. Relaxed edge dislocation structures containing a dislocation dipole: (a) Full-core edge dislocation with a 142-atom supercell, (c) VGa–ON complex edge dislocation, undoped, and (e) VGa–ON complex edge dislocation, n-doped with six Si atoms. Gallium atoms are shown as large white circles and nitrogen atoms are shown as small black circles. Oxygen atoms (in (c) and (e)) are red and silicon atoms (in (e) only) are purple. Charge distribution on a plane perpendicular to [0001] crossing the two N atoms near the Ga vacancy at the lower layer: (b) Full-core, (d) undoped, and (f ) n-doped with six Si atoms. The two N atoms near the Ga vacancy act as electron acceptors, taking on nearly one extra charge per atom and repelling each other significantly.
168 (a)
J. H. YOU AND H. T. JOHNSON
(b) 0.004 0.0036 0.0032 0.0027 0.0023 0.0019 0.0015 0.001 0.0006 0.0002
N atom near VGa (c)
(d)
FIG. II.7. Relaxed structure of a VGa–ON threading edge dislocation with a 142-atom supercell containing a dislocation dipole: (a) n-doped with six Si atoms at asymmetric positions and (c) n-doped with four Si atoms at symmetric positions. Gallium atoms are shown as large white circles and nitrogen atoms are shown as small black circles. Oxygen atoms are red and silicon atoms are purple. Charge distribution on a plane perpendicular to [0001] crossing two N atoms near the Ga vacancy at the lower layer: (b) n-doped with six Si atoms at asymmetric positions and (d) n-doped with four Si atoms at symmetric positions.
Charge around the N atoms near the Ga vacancy is calculated by integrating ˚ radius of each N atom.80 For the charge density over volume within a 1.32 A comparison, the charge around a N atom in bulk WZ GaN is found to be 7.1407q. The total charge around a N atom near a Ga atom at the core site in the full-core edge dislocation is found to be nearly neutral – about 0.02q less than around a
80
Ionic radius for nitrogen atom in 4-coordinate tetrahedral structure. The value is taken from http://www.webelements.com
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES TABLE II.3. SPACE BETWEEN THE TWO CORE N ATOMS EACH CORE N ATOM
AND
CHARGE AROUND
N atom near VGa at core of dislocation
Full-core edge dislocation VGa–ON edge Undoped dislocation Four Si n-doped (symmetric) Six Si n-doped (symmetric) Six Si n-doped (asymmetric)
169
N atom at bulk
Distance between ˚) two N atoms (A
Charge (q)
Charge (q)
2.9585 1.5180 2.8864
7.1201 6.3891 7.1419
7.1412 7.1421 7.1484
3.0044
7.1513
7.1586
2.9899
7.1521
7.1641
bulk N atom. This result is consistent with other reported works.69,70 For the undoped VGa–ON edge dislocation shown in Figure II.6(c) and (d), the charge of each of the two N atoms near the Ga vacancy is found to be 0.753q less than the charge around a N atom at a bulk site; the distance between the two N atoms is ˚ . After n-doping, the charge at the N atom near the Ga vacancy is almost 1.5180 A the same as around a N atom at a bulk site and the separation distance between the two N atoms becomes larger. These results are summarized in Table II.3. Therefore, it is clear that the two N atoms near the Ga vacancy act as electron acceptors: each N atom accepts nearly one electron per structural unit of the VGa–ON edge dislocation. Thus, for the purpose of recourse to an analytical model, it is assumed that each N atom near the Ga vacancy may trap up to one electron, so the dislocation unit cell takes on 2q as its maximum charge state. Now, filling fraction calculations from the Read model can be modified for the VGa–ON edge dislocation which has two acceptor sites per structural unit length. Details of these calculations are presented in Section III.10. 9. S CREW D ISLOCATIONS
a. Frank Model for Open-Core Screw Dislocation Formation It has been observed that screw dislocations in some wurtzite materials form hollow cores to reduce their total energy. Frank81 explains the formation
81
F. C. Frank, Acta Cryst. 4, 497 (1951).
170
J. H. YOU AND H. T. JOHNSON
of open-core screw dislocations by equating the increased surface energy with the reduced strain energy associated with forming the open-core structure. For a screw dislocation core with an empty cylinder of radius r, the shear strain at the surface of the cylinder is b/2pr, where b is the magnitude of the Burgers vector of the screw dislocation. Then, the strain energy density is Ee ¼ Gb2/8p2r2. The increase of total energy per unit length of screw dislocations by increasing dr is dF ¼ 2pg dr ðGb2 =8p2 r 2 Þ2pr dr
ð2:17Þ
where g is the surface energy on the inner surface of the cylinder. There is an equilibrium length when dF/dr ¼ 0, or r ¼ Gb2 =8p2 g:
ð2:18Þ
The magnitude of the Burgers vector for screw dislocations takes on discrete values of b ¼ nc, where n is an integer number and c is the lattice parameter along the screw dislocation. It implies that the radius should also be discrete as r ¼ n2 c2 G=8p2 g;
with n ¼ 1; 2; 3 . . . :
ð2:19Þ
Frank theory indicates that a screw dislocation with Burgers vector exceeding a critical n with a large shear modulus has a hollow core with the equilibrium radius as given by Eq. (2.19). For wurtzite SiC, screw dislocations with both full-cores and hexagonal opencores are observed.82,83 Typical densities of screw dislocations are on the order of 103 cm2 for full-core and 10 cm2 for open-core screw dislocations. The critical magnitude of the Burgers vector to create the open-core hollow screw dislocations has been reported as 2c.84,85 Observed radii of hollow open-core screw dislocations on the order of micrometers (called micropipes) are discrete and match well with the predictions from the Frank model.
b. Hexagonal Open-Core Screw Dislocations in GaN In wurtzite GaN, typical screw dislocation densities are between 106 and 108cm2, and full-, filled-, and open-core screw dislocations have been
82
J. Heindl, W. Dorsch, H. P. Strunk, St. G. Mu¨ller, R. Eckstein, D. Hofmann, and A. Winnacker, Phys. Rev. Lett. 80, 740 (1998). 83 J. Heindl, W. Dorsch, R. Eckstein, D. Hofmann, T. Marek, St. G. Mu¨ller, H. P. Strunk, and A. Winnacker, J. Cryst. Growth 179, 510 (1997). 84 M. Dudley and X. Huang, Mater. Sci. Forum 338–342, 431 (2000). 85 T. A. Kuhr, E. K. Sanchez, M. Skowronski, W. M. Vetter, and M. Dudley, J. Appl. Phys. 89, 4625 (2001).
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
171
observed.28,29,86,87 Unlike in SiC, it is found that the Frank model does not hold for screw dislocations in wurtzite GaN.28,86,88,89 From the Frank model, radii of holes should have a discrete distribution proportional to n2c2, however, the radii observed have a random distribution of sizes within the range of 3.5–75 nm (called nanotubes), while the most commonly observed magnitude of the Burgers vector, b, is 1c. The origin of open-core screw dislocations with small Burgers vector is an open question. There are several proposed theoretical and experimental models to explain the existence of open-core screw dislocations. In reviewing various atomistic structures for screw dislocations, the main properties of interest are (i) the structural stability, (ii) the observations by experiments, and (iii) the electrical inactivity (neutrality). Figure II.8 shows the various GaN screw dislocation structures considered here: (a) open-core, (b) Ga-filled closed-core, (c) H-passivated open-core, and (d) VGa–(ON)3 complex defect open-core. A full-core screw dislocation can be constructed by filling the hollow core in Figure II.8(a). In Figure II.8(a)–(c), the structures are viewed along the h0001i direction, while Figure II.8(d) shows the inner surface of opencore where a gallium atom has been removed and three nitrogen atoms at 1, 2, and 3 have been replaced by oxygen. Ab initio LDF cluster method calculations and the density functional calculations based on tight-binding by Elsner et al.69 reveal that a full-core screw dislocation is not energetically favorable due to the elastic energy stored at the highly distorted core, and that an open-core screw dislocation with b ¼ 1c is ˚ . In terms of the electrical activity, energetically stable with the diameter of 7.2 A the full-core screw dislocation has deep states at 0.9–1.6 eV above the valence band maximum and shallow states at 0.2 eV below the conduction band minimum. Those dislocation defect states move away from the band gap as the radius of the core increases, and finally no states within the band gap are observed for ˚ . However, the energy of screw dislocations with diameters larger than 20 A ˚ increases due to increased surface energy. Cherns et al.86 diameter d > 7.2 A suggest that the formation of large open-core screw dislocations with small Burgers vectors is from the coalescence of misaligned interfacial growth islands at the early stages of growth that are affected by surface steps in the substrate.
86 D. Cherns, W. T. Young, J. W. Steeds, F. A. Ponce, and S. Nakamura, J. Cryst. Growth 178, 201 (1997). 87 W. Qian, M. Skowronski, K. Doverspike, L. B. Rowland, and D. K. Gaskill, J. Cryst. Growth 151, 396 (1995). 88 Z. Liliental-Weber, Y. Chen, S. Ruvimov, and J. Washburn, Phys. Rev. Lett. 79, 2835 (1997). 89 J. E. Northrup, R. Di Felice, and J. Neugebauer, Phys. Rev. B. 56, R4325 (1997).
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J. H. YOU AND H. T. JOHNSON
(a)
(b)
[0001]
(d)
(c)
[0001] 1 [1210]
2 3
FIG. II.8. (a) Open-core screw dislocation in GaN, (b) Ga-filled closed-core, (c) H-passivated opencore, and (d) VGa–(ON)3 complex defect open-core screw dislocation. A full-core screw dislocation, not shown here, can be constructed by filling the hollow core in (a). (d) shows the inner surface of open-core where a gallium atom has been removed and three nitrogen atoms at 1, 2, and 3 have been replaced by oxygen. Gallium atoms are shown as large white circles and nitrogen atoms are shown as small black circles. (a) is from Ref. [69], (b) and (c) from Ref. [90], and (d) from Ref. [92], all reproduced with permission.
Northrup90,91 proposes two theoretical models; Ga-filled closed-core and H-passivated open-core screw dislocations. The Ga-filled closed-core screw dislocation is based on the observation that Ga is much more compliant than GaN, so that replacing the highly strained GaN at the core with Ga makes the system energetically more stable. The H-passivated open-core is based on the idea that migration of H atoms into the inner surface blocks the lateral growth of GaN and forms the hollow core.
90 91
J. E. Northrup, Phys. Rev. B. 66, 045204 (2002). J. E. Northrup, Appl. Phys. Lett. 78, 2288 (2001).
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
173
Liliental-Weber et al.88 experimentally showed that pinholes seed the opencore screw dislocations, and that the density of pinholes increases as the concentration of oxygen increases. Elsner et al.92 showed that VGa–(ON)3 complex defects are stable at the inner surface of open-core screw dislocations and electrically inactive. The VGa–(ON)3 complex defect screw dislocation model is consistent with experimental observations;29–31 at close to the edge of the opencore the nitrogen signal decreases while the oxygen signal increases, and the oxygen atoms migrating on the inner surface block the crystal growth laterally leading to formation of neutral open-cores. Various types of screw dislocations studied experimentally and computationally are tabulated in Table II.4. Based on experimental observations28–31,86,87 and theoretical calculations,92 open-core screw dislocations are modeled here as electrically neutral hexagonal voids with radius varying between 3.5 and 75 nm and Burgers vector b ¼ 1c. In summary, the VGa–ON complex defect edge dislocation has been selected as an edge dislocation model of interest based on the formation energies, the electrical activity with a charge state of two electrons, and the possibility as a YL source. Charge distributions calculated by density functional theory calculations reveal that the two N atoms near the gallium vacancy at the core act as electron acceptors resulting in two straight lines of electron acceptors. This configuration of electron acceptors is different from the germanium dislocation charge distribution studied by Read in Section III.7.b. Therefore, filling fraction calculations from Read should be modified for the VGa–ON complex defect edge dislocation; this is presented in Section III.10. Two electrostatic potentials of negatively charged edge dislocations have been presented: Read and Bonch-Bruevich and TABLE II.4. VARIOUS TYPES
OF
SCREW DISLOCATIONS
IN
WURTZITE GAN
Stability Types
Experiment
Calculation
Electrical activity
Full-core Filled-core Open-core
Observed Observed Observed
Unstable69
Inactive Inactive for ˚ 69 d > 20 A
Ga-filled closed-core H-passivated open-core VGa–(ON)3 open-core
Observed29
Stable for ˚ 69 d ¼ 7.2 A 91 Stable Stable90 Stable92
Inactive92
¨ berg, Appl. J. Elsner, R. Jones, M. Haugk, R. Gutierrez, Th. Frauenheim, M. I. Heggie, and S. O Phys. Lett. 73, 3530 (1998).
92
174
J. H. YOU AND H. T. JOHNSON
Glasko (B–G) in Section II.7.c. For screw dislocations, the open-core type is neutral and has a hexagonal void with radius varying between 3.5 and 75 nm, and Burgers vector b ¼ 1c. In Section III, effects of the VGa–ON complex defect edge dislocations on electron mobility are presented using the two electrostatic potentials by Read and B–G. III. Electron Scattering due to Threading Edge Dislocations
A simple model is developed to predict the effect of VGa–ON complex edge dislocations on electron mobility as a function of free carrier concentration and dislocation density. As shown in Section II.8.c, the VGa–ON complex edge dislocation contains two N atoms serving as electron acceptors per structural unit along the dislocation so that the line takes on 2q per structural unit as its fully charged state. Thus, it is necessary to modify the Read model for the filling fraction calculation. Modification of the Read model for the VGa–ON complex edge dislocation is shown in Section III.10. Then, dislocation scattering is combined with other scattering mechanisms (ionized impurity, acoustic deformation and piezoelectric phonon, optical polar scattering) in a classical scattering model to calculate total electron mobility in Section III.11. The calculated total drift mobility is compared with experimental results in Section III.12. The effect of dislocation density on total drift and Hall mobilities is also predicted for various free carrier concentrations. Results are summarized in Section III.13. This work includes (i) the first filling fraction calculations for two-column arrays of dislocation acceptors and (ii) the only closed-form solutions available for drift and Hall mobility by dislocation scattering using the potential by Bonch-Bruevich and Glasko.67 The work presented here is reported in detail in You et al.55,58 10. M ODIFICATION D ISLOCATIONS
OF IN
R EAD M ODEL GAN
FOR
2 Q E DGE
As shown in Section II.8.c, the edge dislocations in wurtzite GaN act as electron acceptors and become 2q charged per structural unit length along the dislocation line. The complicated dislocation structure in wurtzite GaN contains two equivalent columns of available sites with separation distance between the columns given by the basal lattice constant a as shown in Figure III.1. In this case it is helpful to define a new variable called strength factor (s), given by, s¼
actual line charge density of dislocation l ¼ maximum line charge density of dislocation 2q=c
ð3:1Þ
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
175
(b)
(a) Ga
Ga ON VGa
N−
−
−
N−
Ga
Ga ON VGa N−
−
−
Ga
N−
Ga
[0001]
C
C
ON VGa N−
−
−
a
N− a
FIG. III.1. The core of the VGa–ON edge dislocation. (a) Replacing N with O and removing Ga opens one possible acceptor site at each N atom near the Ga vacancy. (b) Array of electron acceptors along dislocation. Dashed circles represent available sites for electron trapping and the dashed vertical line represents the center line of edge dislocation core. Acceptor sites form two columns along the core; the inter-column distance is a and the inter-site vertical distance is c within each column.
where c is the lattice spacing in [0001] direction, and l is the average line charge density on the dislocation line. If s ¼ 0, no sites are filled; exactly half of the available sites are filled for s ¼ 0.5. If s ¼ 1, each acceptor site contains one electron and the dislocation takes on the largest possible scattering potential of –2q per unit c along the dislocation. A classical electrostatic dislocation filling fraction calculation in the spirit of the Read model can then be constructed. Electrons along the dislocation line are mutually repelled to reduce the overall electrostatic energy of the dislocation. Thus, a zigzag filling configuration gives the minimum energy as shown in
176
J. H. YOU AND H. T. JOHNSON
Figure III.2. Read assumed the existence of a positive spatially charged cylinder pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in n-type material with the effective radius as R ¼ 2s=pcðND NA Þ, by charge neutrality, according to qpR2 ðND NA Þ ¼ 2sq=c
ð3:2Þ
where ND and NA are the concentration of chemical donors and acceptors. The strength factor, s, can be calculated by minimizing the total increase of free energy per site within a calculation disk representing a unit cell along the dislocation line, separately for the cases of 0 < s 0.5 and 0.5 < s 1 for analytical convenience only. First, for 0 < s 0.5, each disk with thickness d contains only one electron so that each electron per disk is separated by a vertical distance d and a horizontal distance a as shown in Figure III.2(a). Substituting l ¼ q/d into Eq. (3.1) gives (a)
(b)
d
C
a
(c)
d⬘
C
a
FIG. III.2. Arrangement of acceptors along the GaN edge dislocation. (a) 0 < s < 0.5, (b) s ¼ 0.5, and (c) 0.5 < s 1. The case of s ¼ 0.5 is a special case of 0 < s < 0.5 when d ¼ c. The marks and O represent the trapped electrons and empty sites, respectively. The lattice spacing along h0001i is denoted by C, a is basal lattice constant, and d and d0 are the sizes of disk representing a unit cell defined in each case. Electrons tend to form a zigzag filling configuration to reduce the electrostatic formation energy.
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
177
s ¼ c/(2d). Then, from Eq. (3.2), the neutrality condition for 0 < s 0.5 is given by qpR2 ðND NA Þ ¼ q=d:
ð3:3Þ
The total length of the dislocation is (N þ 1)d as shown in Figure III.3a. It is assumed that the dislocation is infinitely long so that Es can be derived for the middle disk without consideration of end effects. The electron–electron interaction energy per electron for the middle disk is Ee ¼ 12 ðqÞVe0 where Ve0 is the potential at electron PN=20 due to all other electrons. From the symmetry of the system, Ve0 ¼ 2 i ¼ 1 Vi0 , where Vi0 is the potential at electron 0 due to electron i and
(a)
(b) (b/2)⬘
N/2
3u + 1
N/2-1 Nd/2
2u + 2 d⬘ (2u + 1)⬘ 2u
3 2
u 2
1 (N + 1)d
d
(b/2)
bc
0 -1
d⬘
2u + 1 2u - 1 u+1 uc
0
1 0⬘
-2
1
-u
-2 -3
2u + 3
-2u
d⬘ -(2u + 1)⬘ -2u - 2
-N/2 + 1
c uc
-u - 1 -2u + 1 -2u- 1 -2u- 3 -3u- 1
-N/2
-(b /2)⬘
-(b /2)
FIG. III.3. Filling configuration of trapped electrons along an edge dislocation in GaN. (a) 0 < s 0.5 (b) 0.5 < s 1. For 0 < s 0.5, the unit cell or calculation ‘‘disk’’ contains one electron at the center row and these electrons form a zigzag filling configuration. For 0.5 < s 1, the calculation disk contains (2u þ 1) electrons from a half filling and one additional electron at the center site of the disk. The additional electron in each disk is also assumed to form in a zigzag configuration with respect to the additional electrons in neighboring disks. The additional electrons are numbered as 00 , (2u þ 1)0 , 2(2u þ 1)0 , and so on.
178
J. H. YOU AND H. T. JOHNSON
8 q 1 N > qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; for i ¼ 1; 3; 5; . . . ; > > > 4pe 2 2 > < a2 þ ðidÞ 0 1 Vi0 ¼ > > q 1 N > > for i ¼ 2; 4; 6; . . . ; @ 1A > : 4pe id ; 2
ð3:4Þ
where e is the dielectric constant. Assigning N/2 as odd or even has no effect since the dislocation is considered to be very long. After adding all Vi0 terms and manipulating, Ee becomes 2 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 2 c 41 1 N2 1 Nd 1 @ d d A5 g þ ln 1þ Ee ¼ Eo þ g2 ðsÞ þ ln þ ln d 2 1 2 4 2 a 2 a a ð3:5Þ
where
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ða=dÞ2 þ 1 ðNd=aÞ N=2 þ ða=dÞ2 þ ðN=2Þ2 1 þ ðd=aÞ2 d=a is
used in the limit of N ! 1, and Eo ¼ q2/(4pec), g1 is Euler’s constant and g2(s) is a modified Euler’s constant which is actually a function of s because d ¼ c/(2s), defined as: 1 0 ðN ð2Þ=4 ðNX 2Þ=4 1 1 C B ð3:6Þ lim dnA g1 ¼ @ n n ðN 2Þ=4 ! 1 n¼1 1
and 0 g2 ðsÞ ¼
lim
ðN þ 2Þ=4 ! 1
B @
ðNX þ 2Þ=4 n¼1
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða=dÞ þ ð2n 1Þ2
ðN þð2Þ=4
1
1 1 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dnA: 2 2 ða=dÞ þ ð2n 1Þ
ð3:7Þ The modified Euler’s constant g2(s) is function of s, while Euler’s constant g1 is independent of s. The convergence of g1 and g2 (s ¼ 0.2) is shown in Figure III.4(a) and (b). g1 and g2(s) as a function of s are shown in Figure III.4(b) and (d). By solving Poisson’s equation, the potential for the positive space charge, Vc, is given, for r < R, as q Nd 2 1 ðr=RÞ þ 2 ln : ð3:8Þ Vc ðr < RÞ ¼ 4ped R
179
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a)
(b) 0.65
0.65 0.6
0.6 g1
0.5772
0.55
g1
0.55
0.5772
0.5
0.5
0.45
0.45
0.4
(c)
0
1000
2000
3000 4000 (N − 2)/4
5000
0.4
6000
(d)
0.65
0
0.1
0.2
0.3
s
0.4
0.5
0.65
0.6121 0.6 g2 (σ)
0.6
g2
0.55 0.5
0.55 0.5
0.45
0.45
s = 0.2
0.4 0
1000
2000
3000 4000 (N + 2)/4
5000
6000
0.4
0
0.1
0.2
s
0.3
0.4
0.5
FIG. III.4. (a) Convergence of Euler’s constant, (b) Euler’s constant is independent of s as g1 ¼ 0.5772, (c) Convergence of modified Euler’s constant at s ¼ 0.2, and (d) g2 (s) as a function of s.
The energy of the positive space charge per electron, therefore, is given by ðR 1 c 1 Nd þ ln 2pdrVc ðr < RÞr dr ¼ Eo Ec ¼ 2 d 4 R
ð3:9Þ
0
where r is the volume density of positive space charge, r ¼ q(NDNA), and the neutrality condition for 0 < s 0.5, rpR2d ¼ q, is used. The interaction energy between electrons within the disk and the positive space charge is readily obtained by evaluating Vc at r ¼ a/2. Both interaction energies per electron are equal and are given by: a 2 Eo c Nd 1 : ð3:10Þ þ 2 ln Eec ¼ Ece ¼ 2 d 2R R
180
J. H. YOU AND H. T. JOHNSON
After adding all four terms from Eqs. (3.5), (3.9), and (3.10), Es becomes independent of N and is given by Es ðsÞ ¼ 2sEo
8 <1 :2
3 a cpðND NA Þ 1 g1 þ g2 ðsÞ þ þ ln 4 8s 2 2
s2 ½
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 1 þ ðc=2asÞ2 c=2as= pac2 ðND NA Þ
;
ð3:11Þ where charge neutrality and d ¼ c/(2s) are used to eliminate R and d, and the electrostatic energy per site can be obtained as sEs(s). The effect of configurational entropy is taken into account by adding a term of the form sTS(s) to the total increase of free energy per site. The total increase of free energy per site, sF(s), is then sFðsÞ ¼ sðET EF Þ þ sEs ðsÞ sTSðsÞ;
ð3:12Þ
where ET is the localized energy level of the dislocation within the band gap which is BT 1 eV above the valence band maximum,33,73 EF is the Fermi level, sTSðsÞ ¼ kM In W, where there are W ways of arranging N electrons in M sites, kB is Boltzmann’s constant, and T is temperature. Minimizing sF(s) with respect to s gives the following nonlinear equation which can be solved numerically for the strength factor:
d d sEs ðsÞ ¼ EF ET þ sTSðsÞ : ð3:13Þ ds ds For 0.5 < s 1, each disk with dimension d0 contains one additional electron at the center row in addition to the ‘‘zigzag’’ arranged electrons from the case of s ¼ 0.5 as shown in Figure III.2(c). From Eq. (3.1), the strength factor is s ¼ (c þ d0 )/(2d0 ) after substituting l ¼ -q(1 þ d0 /c)/d0 . Another variable, u, is introduced, and defined as d0 ¼ (2u þ 1)c as shown in Figure III.3(b). One disk, in this case, contains (2u þ 2) electrons and s can be expressed as s¼(u þ 1)/(2u þ 1). The neutrality condition is then given by, qpR2 ðND NA Þ ¼
ð2u þ 2Þq : d0
ð3:14Þ
The electron–electron energy per electron in the disk is Ee ¼
m X q Vðri Þ 2ð2m þ 2Þ i ¼ u;00
ð3:15Þ
where V(ri) is the potential at ri due to all other electrons, and ri is the position of electron i within the disk only. It can be separated as u X i ¼ u;0
Vðri Þ ¼ 0
u X i ¼ u;0
V ZIG ðri Þ þ 0
u X i ¼ u;0
V ADD ðri Þ 0
ð3:16Þ
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
181
where VZIG (ri) in the first term represents the potential at ri due to all zigzagarranged electrons from the case s ¼ 0.5, while VADD (ri) in the second term is the potential due to all electrons in excess of s > 0.5. A detailed derivation for Ee is provided in Appendix A; an expression for Ee is given in Eq. (8.9). The energy of the positive space charge per electron is given by c 1 Nd 0 ð3:17Þ Ec ¼ Eo ð2u þ 2Þ 0 þ ln R d 4 and the energy, Eec ¼ Ece is found to be Eec ¼ Ece ¼
a 2 Eo ð2u þ 2Þ c Nd 0 : 1 þ 2 ln R 2 d0 2R
ð3:18Þ
Equations (3.17) and (3.18) are in the same form as Eqs. (3.9) and (3.10), respectively, but for s > 0.5, where d is replaced with d 0 , q with (2u þ 2)q, and the entire expressions are divided by the number of electrons per disk. Now, Es(s) is the sum of Ee, Ec, Ece, and Eec. It becomes independent of N, and is given by 0 1 9 8 > > > > 1 2s 1 c > > @ þ C2 ðsÞ þ C3 ðsÞ þ C4 ðsÞA > > C1 þ > > > > 2s 2s a > > > > > > > > = < 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ð2s 1Þ pa cðN N Þ D A 2 2 Es ðsÞ ¼ Eo þ ln ð2s 1Þ þ ðc=aÞ c=a þ > > 2s 4 > > > > > > > > > > 2 > > 2s 3s 4s þ 1 > > > > þs ln ln 2 > > ; : pac2 ðND NA Þ 2 4s ð3:19Þ where u ¼ (1 s)/(2s 1) is used to get s dependency, and the neutrality condition and d0 ¼ (2u þ 1)c are used to eliminate R and d0 . C1, C2(s), C3(s), and C4(s) are defined in Appendix A. As before in Eq. (3.12), minimizing the total increase in free energy per site with respect to s gives a nonlinear equation that can be solved numerically. It can be readily shown that Eqs. (3.11) and (3.19) become identical at s ¼ 0.5. The proof is shown in Appendix B. Figure III.5 shows the strength factor at various densities of dislocations as a function of free carrier concentration. Compared to previous work,24 it generally indicates lower filling fractions, but it saturates at 1 at a higher carrier concentration because there are more sites to be filled. The strength factor changes significantly for ndis > 108cm2, but there is only a very small difference in the strength factor for ndis < 108cm2. If ionized donor density is smaller than the
182
J. H. YOU AND H. T. JOHNSON
1 T = 300 K 0.9
Strength factor
0.8 0.7 0.6
1011 cm−2
0.5
ndis = 107 cm−2 ndis = 108 cm−2 ndis = 109 cm−2 ndis = 101 0 cm−2 ndis = 101 1 cm−2
1010 cm−2 0.4
109 cm−2
0.3
107 cm−2
1015
1016
1017 1018 1019 1020 −3 Free carrier concentration (cm )
1021
FIG. III.5. Strength factor s for VGa–ON edge threading dislocations. Strength factor increases as free carrier concentration and dislocation density increase.
concentration of electron trapping at dislocations, all carriers are trapped. Therefore, the dislocation density sets the lower limit of accessible doping range. The relation between carrier concentration and doping concentration is n ¼ ðNDþ NA Þ 2sndis =c, which implies that some electrons from ionized doping are trapped in edge dislocations. Figure III.6 shows the relation between free carrier concentration and doping density. To have the same free carrier concentration, a system with high dislocation density should have more ionized impurities especially in the low carrier concentrations because more carriers are trapped with high dislocation densities. 11. C LASSICAL S CATTERING M ODEL
FOR
E LECTRON M OBILITY
To evaluate electron mobility due to edge dislocations, two models of scattering potentials are compared for the charged dislocations: one by Read11 and one by Bonch-Bruevich and Glasko (B–G)67 as described in Section II.7.c. To compare these two models, the drift mobility is calculated and compared with the existing experimental results by Weimann et al.24 The Read potential from Eq. (2.15) is modified for 2q the charged dislocation (as the maximum) using VRead ðr < RÞ ¼ 2sEo ½lnðR=rÞ2 1 þ ðr=RÞ2 :
ð3:20Þ
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
Free carrier concentration [cm−3]
1021
183
ndis = 107 cm−2 ndis = 108 cm−2 ndis = 109 cm−2 ndis = 101 0 cm−2 ndis = 101 1 cm−2
1020 1019 1018 1017
1016 107 cm−2 108 cm−2
1015 1015
1010 cm−2
109 cm−2
1016
1011 cm−2
1017 1018 1019 Doping density [cm−3]
1020
1021
FIG. III.6. Free carrier concentration vs. doping concentration.
The matrix element including Thomas–Fermi screening is readily calculated to be M*kscr0 *k ðd? Þ ¼ ? ?
4p d2? TF S
2sEo ½1 J0 ðd? RÞ J2 ðd? RÞ *
ð3:21Þ *
*
where d? is the magnitude of the scattering wave vector defined as d? ¼ k ?0 k? , * * where k ?0 and k? are the final and initial wave vectors perpendicular to the dislocation line, and S is the area. The function J0/2(d?R) is the Bessel function of the first kind of order 0/2 evaluated at d?R and TF is the Thomas–Fermi dielectric function definedffi as TF ¼ 1 þ (lD d?)2 with Debye screening pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi length lD ¼ ekB T=q2 n0 , where the effective screening concentration is n0 ¼ n þ ðn þ NA Þ ½1 ðn þ NA Þ=NDþ , which involves both free and bound carriers as a function of the ionized donor and acceptor concentrations. Using the * Fermi–Golden rule and integrating over all d? s, the inverse of scattering time, or scattering rate, is obtained as ndis m
¼ tRead ðkÞ 2ph3 k3 1
2k ð
0
d2? ~ ? Þ j2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dd? j Mðd 1 ðd? =2kÞ2
ð3:22Þ
~ ? Þ ¼ SM*scr* ðd? Þ, ndis is density of dislocations, m* is the electron where Mðd 0 k? k?
*
*
effective mass, h is the reduced Planck’s constant, and k ¼ j k? j ¼ j k 0? j for
184
J. H. YOU AND H. T. JOHNSON
elastic scattering. The dislocation scattering contribution to drift mobility is then obtained as mdrift Read
qhtRead ðEÞi 4q ¼ ¼ ∗ m 3p1=2 m∗ ðkB TÞ5=2
1 ð
tRead E3=2 expðE=kB TÞdE: ð3:23Þ 0
Similarly, Hall mobility can be calculated using mHall Read ¼
qht2 i : m hti
ð3:24Þ
Equations (3.23) and (3.24) cannot be reduced in closed form, so they are numerically evaluated. The B–G scattering potential from Eq. (2.16) is VBG ðrÞ ¼ 4sEo K0 ðr=lD Þ
ð3:25Þ
which is taken to be twice as strong as the original potential, given the double acceptor dislocation instead of the single acceptor dislocation. The scattering time, drift mobility, and Hall mobility due to dislocation scattering are then given explicitly as tBG ðkÞ ¼
mdrift BG
mHall BG
h3 ðecÞ2 ð4k2 l2D þ 1Þ3=2 4s2 ndis q4 m∗ l4D
ðh2 ecÞ2 exp z=ð8kB TÞ qhtBG ðEÞi ¼ ¼ K TÞ z=ð8k ffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 B m
8 2pkB T ndis q3 s2 ðm∗ Þ5=2 l5D
ð3:26Þ
ð3:27Þ
pffiffiffi 2 2 pðh ecÞ ðkB TÞ7=2 ½ðz=kB TÞ3 þ 30ðz=kB TÞ2 þ 420ðz=kB TÞ þ 2520
¼ pffiffiffi 2 2ndis q3 s2 ðm∗ Þ5=2 l5D z4 exp z=ð8kB TÞ K2 z=ð8kB TÞ
ð3:28Þ =ð2m l2D Þ
where z ¼ h and K2(z/(8kBT )) is the modified Bessel function of second kind of order 2 evaluated at z/8kBT. Equation (3.26) is due to Po¨do¨r,93 and the closed form expressions for mobility in Eqs. (3.27) and (3.28) are results of the work presented in reference 55. 2
93
B. Po¨do¨r, Phys. Stat. Sol. 16, K167 (1966).
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
185
To calculate the total mobility, scattering due to ionized impurities (ti i.), acoustic deformation (tad.) and piezoelectric (tap.) phonons, and optical polar phonons (top.) are included as 1 1 1 1 1 1 ¼ þ þ þ þ : ttotal tdis: ti i: tad: tap: top:
ð3:29Þ
The scattering relaxation time for ionized impurities (ti i.) is given by Conwell and Weisskopf 94 as pffiffiffiffiffiffiffiffi 16pe2 2m 1 ti i: ðEÞ ¼ ð3:30Þ E3=2 GðEÞ q4 NI where NI is the ionized impurity concentration and G(E) is defined as
2 2 1=3 E2 : GðEÞ ¼ ln 1 þ 4pe=q NI
ð3:31Þ
Acoustic-mode lattice vibrations induce lattice spacing changes, which also change the band gap from point to point. This deformation potential induced from the acoustic-mode lattice vibrations scatters the traveling electrons. The scattering relaxation time for acoustic deformation phonon scattering (tad.) is given as95 ph4 rs2 tad: ðEÞ ¼ pffiffiffi E1=2 2a2dp ðm Þ3=2 ðkB TÞ
ð3:32Þ
where r is the crystal density, s is the average velocity of sound, and adp is the deformation potential. All material properties related to scattering mechanisms are listed in Table X.1 in Appendix C. The atomic displacements produced by the acoustic-mode lattice vibrations introduce a piezoelectric effect if the atoms are partially ionized. The scattering relaxation time for acoustic piezoelectric phonon scattering (tap.) is95 given as pffiffiffi 2 2ph2 rs2 tap: ðEÞ ¼ ð3:33Þ E1=2 2 1=2
ðqhpz =eÞ ðm Þ ðkB TÞ where hpz is the piezoelectric constant. The dipole moments formed by the ionic charges on the atoms with the optical-mode lattice vibrations also scatter the traveling electrons. The scattering relaxation time for optical polar phonon scattering (top.) is95 given as
94 95
E. M. Conwell and V. F. Weisskopf, Phys. Rev. 77, 388 (1950). D. C. Look, Electrical Characterization of GaAs Materials and Devices, Wiley, New York (1989).
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J. H. YOU AND H. T. JOHNSON
top: ðEÞ ¼
pffiffiffi 2 2ph2 ½expðTpo =TÞ 1wðTpo =TÞ 1 q2 ðm Þ1=2 ðkB Tpo Þðe1 1 e Þ
E1=2
ð3:34Þ
where Tpo is the polar phonon Debye temperature and w(Tpo/T) is a slow varying function of T given as95 wðTpo =TÞ ¼ 1 0:5841ðTpo =TÞ þ 0:2920ðTpo =TÞ2 0:037164ðTpo =TÞ3 þ 0:0012016ðTpo =TÞ4 : ð3:35Þ Examples of total drift mobility and mobilities due to each scattering mechanism at 300 K as a function of free carrier concentration at dislocation densities of ndis ¼ 108 and 1010 cm2 are shown in Figure III.7(a) and (b), respectively. In these examples, the B–G potential is used for the dislocation scattering. As shown in Eqs. (3.32)–(3.34) the electron mobilities due to phonon scattering mechanisms are functions of temperature, but independent of dislocation density. The mobility due to ionized impurity scattering is affected by dislocation density through n ¼ ðNDþ NA Þ 2sndis =c as shown in Figure III.6. Because the high dislocation density case requires more ionized dopants to have the same free carrier concentration, the mobility due to ionized impurities is expected to be lower in the high dislocation density case. Total mobility is less than any mobility contribution due to other scattering mechanisms so that the scattering mechanism producing the lowest mobility at a given free carrier concentration acts as the upper limit for total mobility. For dislocation density ndis ¼ 108 cm2 (Figure III.7(a)), optical polar phonon scattering is dominant for free carrier concentrations less than about 2 1017 cm3, while the ionized impurity scattering is dominant for high free carrier concentrations. For dislocation density ndis ¼ 1010 cm2 (Figure III.7(b)), dislocation scattering is dominant over optical polar phonon scattering in the low free carrier concentration regime. Therefore, dislocation scattering lowers the total electron mobility in the low free carrier concentration regime, while the ionized impurity scattering is still dominant for high carrier concentrations.
12. C OMPARISON
WITH
E XPERIMENTAL D ATA
Total drift mobility is compared with experimental results by Weimann et al.24 for two samples (one with ndis ¼ 8 109 cm2 and another with ndis ¼ 2 1010 cm2), in Figure III.8. The B–G potential predicts reduced mobility as free carrier concentration decreases at low carrier concentrations, while the mobility based on the Read potential under these conditions is nearly constant. This deceasing mobility at low
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a) 107
Drift mobility (cm2/V s)
106
187
Dislocation (B-G) Ionized imp. Acoustic deformation Acoustic piezoelectric Optic polar Total drift mobility
105 104 103 102 101 15 10
(b) 107
Drift mobility (cm2/V s)
106
1016
1017 1018 1019 1020 Free carrier concentration (cm−3)
1021
Dislocation (B-G) Ionized imp. Acoustic deformation Acoustic piezoelectric Optic polar Total drift mobility
105
104
103
102
101 15 10
1016
1017 1018 1019 1020 −3 Free carrier concentration (cm )
1021
FIG. III.7. Total mobility and mobilities due to each scattering mechanism (a) at ndis ¼ 108 cm2 and (b) ndis ¼ 1010 cm2. Phonon scattering is temperature dependent, but independent of dislocation density. For ndis ¼ 108 cm2, optical polar phonon scattering dominates the total mobility in the low carrier concentration regime, while ionized impurity scattering dominates in the high carrier concentration regime. For ndis ¼ 1010 cm2, dislocation scattering lowers the total electron mobility for low free carrier concentrations, while the ionized impurity scattering is still dominant for high carrier concentrations.
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J. H. YOU AND H. T. JOHNSON
B-G, ndis = 8 ⫻ 109 cm−2
500
B-G, ndis = 2 ⫻ 101 0 cm−2
Drift mobility (cm−2/V s)
Read, ndis = 8 ⫻ 109 cm−2 Read, ndis = 2 ⫻ 101 0 cm−2
200
ndis = 8 ⫻ 109 cm-2 ndis = 2 ⫻ 101 0 cm-2
100
50
20 1015
1016
1017 1018 1019 Free carrier concentration (cm−3)
1020
1021
FIG. III.8. Comparison of drift mobility for two different potentials. The results based on two potentials diverge because the radius of the Read potential becomes nearly constant at low carrier concentrations. Experimental data are taken from Ref. [24].
carrier concentrations is known to be due to the dislocation scattering effect,24,96 which is consistent with the B–G potential dislocation scattering model. Therefore, the B–G potential is the basis for further studies in this work. The behavior predicted by the Read potential at low carrier concentrations can be understood based on the radius of the positively charged cylinder. The radius of cylinder in Read potential is shown in Figure III.9(a). As shown in Eq. (3.2) the radius of the cylinder, R, is a function of ionized dopant density, which is not equal to free carrier concentration, especially for low carrier concentrations. For low donor concentrations, most electrons are trapped at dislocations, resulting in low free carrier concentration. For high donor concentrations exceeding the volumetric density of electrons trapped at dislocations, free carrier concentration becomes linearly proportional to donor concentration as shown in Figure III.9(b). Therefore, R becomes nearly constant at low carrier concentrations resulting in the constant mobility predicted by Read potential. The predicted drift and Hall mobilities using the B–G potential for various dislocation densities are given in Figure III.10. The dislocation
96
H. M. Ng, D. Doppalapudi, T. D. Moustakas, N. G. Weimann, and L. F. Eastman, Appl. Phys. Lett. 73, 821 (1998)
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a)
189
70 ndis = 8 ⫻ 109 cm−2 ndis = 2 ⫻ 1010 cm−2
Radius of read potential (nm)
60 50 40 30 20 10 0 1015
(b)
1016
1017 1018 1019 Free carrier concentration (cm−3)
1020
1021
Free carrier concentration (cm−3)
1021
1020
1019
1018
1017
1016 ndis = 8 ⫻ 109 cm−2 ndis = 2 ⫻ 1010 cm−2
1015 1017
1018
1019 1020 −3 Doping concentration (cm )
1021
FIG. III.9. (a) Radius of Read potential as a function of free carrier concentration and (b) free carrier concentration as a function of donor concentration. For low donor concentration, most electrons are trapped at dislocations, resulting in low free carrier concentration. For high donor concentration exceeding the volumetric density of electrons trapped at dislocations, free carrier concentration becomes linearly proportional to donor concentration. From the charge neutrality condition, the Read potential radius is inversely related to donor concentration, so it varies slowly at low free carrier concentrations.
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J. H. YOU AND H. T. JOHNSON
(a)
ndis = 107 cm−2
107 cm−2
Drift mobility (cm2/V s)
103
ndis = 108 cm−2
108 cm−2
ndis = 109 cm−2 ndis = 1010 cm−2 ndis = 1011 cm−2
109 cm−2
T = 300 K 1010 cm−2 102
1011 cm−2
101 1015
(b) 103
1016
1017 1018 1019 Free carrier concentration (cm−3)
107 cm−2
ndis = 107 cm−2
108 cm−2
ndis = 108 cm−2
1021
ndis = 109 cm−2
109 cm−2 Hall mobility (cm2/V s)
1020
ndis = 1010 cm−2 ndis = 1011 cm−2 T = 300 K
1010 cm−2 102 1011 cm−2
101 1015
1016
1017
1018
1019
Free carrier concentration
1020
1021
(cm−3)
FIG. III.10. (a) Predicted drift mobility and (b) Hall mobility including dislocation scattering, ionized impurity scattering, acoustic deformation and piezoelectric phonon scattering, and optical polar phonon scattering for different dislocation densities. The B–G scattering potential is used for both cases. Dislocation scattering reduces the mobility at low carrier concentrations.
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
191
scattering effect dominates for ndis > 108 cm2 for low carrier concentrations. The range of carrier concentrations dominated by dislocation scattering widens as ndis increases. For example, dislocation scattering dominates for n < 1019 cm3 with ndis ¼ 1011 cm2 and n < 1018 cm3 with ndis ¼ 1010 cm2. It should also be noted that the mobility is insensitive to dislocation density when ndis < 108cm2. As shown in Figure III.8, it is because the phonon scattering becomes more dominant than dislocation scattering for low dislocation densities, which implies that unless the mobility due to lattice scattering is exceptionally high, total mobility will not be improved by reducing dislocation density less than 108 cm2. To show how much the total mobility can be improved by reducing dislocation density, the ratio of mobility with ndis ¼ 108 cm2 to mobility with ndis ¼ 107 cm2 as a function of m op. is shown in Figure III.11 for different carrier concentrations. It indicates that for typical values of lattice mobility ( mop. 103 cm2 V s1 ) and carrier concentration (n ¼ 1016 cm3 or higher), there is little benefit to be had by reducing dislocation densities below ndis ¼ 108 cm2.
1
m(ndis = 108 cm−2)/m(ndis = 107 cm−2)
1018 cm−3 0.95 1017 cm−3 0.9 0.85
1016 cm−3
0.8 0.75 0.7 0.65 101
1015 cm−3
n = 1015 cm−3 n = 1016 cm−3 n = 1017 cm−3 n = 1018 cm−3 102
103 mlat.
(cm2/V
104
s)
FIG. III.11. Sensitivity of mobility to reduction of dislocation density from ndis ¼ 108 to 107 cm2. For typical values of lattice mobility on the order of mlat. ¼ 103 cm2 V1 s1 or lower, there is little advantage in reducing the dislocation density below ndis ¼ 108 cm2. The mobility is insensitive to dislocation density for all but the lowest values of carrier concentration.
192
J. H. YOU AND H. T. JOHNSON
13. S UMMARY A closed-form analytical model is derived for the effect of electrically active VGa–ON edge dislocations on drift and Hall mobilities in GaN. The two N atoms near the Ga vacancy at the dislocation core are found to be electron acceptors by a charge distribution study of the dislocation core structure. An accurate formula for the dislocation scattering strength factor is then derived for the double acceptor VGa–ON dislocation, with up to 2q charge per structural unit. Two electrostatic dislocation scattering potentials are compared as bases for predicting drift and Hall mobility. At low carrier concentrations, the B–G potential correctly predicts reduced mobility, while the Read potential predicts constant mobility. Closedform solutions for the dislocation contribution to drift and Hall mobility are derived for the B–G potential. Simulated drift mobility results match with the experimental data reasonably well. Both drift and Hall mobilities are strongly affected by dislocation densities higher than 108 cm2 in devices with carrier concentrations in the range of 1015–1021 cm3. Maximum mobility occurs at higher carrier concentrations as dislocation density increases. For n > 1016 cm3, reducing the dislocation density below ndis ¼ 108 cm2 has a negligible effect on total mobility if the mobility due to lattice scattering is about 103 cm2 V1 s1. IV. Single Electron Model for Calculating Dislocation Effects on Optical Properties
A method to calculate effects of dislocation density on optical properties is presented in this section. To calculate the optical emission spectrum, the Schro¨dinger equation is solved within the kp Hamiltonian single electron approximation, which includes interactions between subbands. In prior studies, the kp Hamiltonian method has been used mainly to analyze band structure in reciprocal space.97,98 However, for spatially inhomogeneous systems, it is advantageous to transform the kp Hamiltonian into real space because spatially varying potentials must be considered. For wurtzite GaN with a wide band gap, interactions between conduction and valence bands are negligible, so the valence band energy levels are solved separately from the conduction band. For zinc-blende GaAs, the narrower gap requires a coupled 8 8 kp Hamiltonian formulation. These Hamiltonians are described in Section IV.14 and details of the finite element formulation are explained in Section IV.15. The calculation of the spontaneous emission coefficient, which is a measure of optical emission, is presented in Section IV.16. 97 98
S. L. Chuang and C. S. Chang, Phys. Rev. Lett. 54, 2491 (1996). T. B. Bahder, Phys. Rev. B 41, 11992 (1990).
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
14. T HE R EAL S PACE
KP
193
H AMILTONIAN A PPROACH
a. Wurtzite Crystal Structure (GaN) The kp Hamiltonian is a well-known method to calculate the band structure of semiconductors, and thus is commonly used in k-space. This method provides an approximate solution to the single electron steady state Schro¨dinger equation given by Hkp j ci ¼ E j ci
ð4:1Þ
where c is the wave function and E is the energy of an electron state. Hkp is the appropriate kp Hamiltonian that models the energy interactions between subbands. Since wurtzite GaN has a band gap as wide as 3.44 eV it is reasonable to neglect the interactions between conduction band and valence band, so the conduction and valence bands are calculated separately. Chuang and Chang97 have developed a 6 6 kp Hamiltonian for the valence bands in strained wurtzite structure given by, 2 3 0 0 0 F K þ H þ 6 K G H 0 0 D 7 6 7 þ 6 l 0 D 0 7 H H VB 6 7 Hkp ðkÞ ¼ 6 ð4:2Þ 0 0 F K H 7 6 0 7 4 0 G H þ 5 0 D K þ 0 D 0 Hþ H l where F ¼ D1 þ D2 þ l þ y; h2 l¼ ½A1 kz2 þ A2 ðkx2 þ ky2 Þ þ le ; 2m0 y¼
h2 ½A3 kz2 þ A4 ðkx2 þ ky2 Þ þ ye ; 2m0
h2 A5 ðkx þ iky Þ2 þ D5 eþ ; 2m0 pffiffiffi D ¼ 2D3 ; e ¼ exx 2iexy eyy ; K¼
G ¼ D1 D2 þ l þ y le ¼ D1 ezz þ D2 ðexx þ eyy Þ ye ¼ D3 ezz þ D4 ðexx þ eyy Þ H¼
ð4:3Þ
h2 A6 kz ðkx þ iky Þ þ D6 ezþ 2m0
ez ¼ ezx ieyz
kx, ky, kz are components of the wavevector, and constants A1–6, D1–6, and D1–3 are material parameters. The superscript þ means Hermitian conjugate. The deformation potential due to strain is modeled via le, ye, eþ in K, and ezþ in H.
194
J. H. YOU AND H. T. JOHNSON
The Hamiltonian for the conduction band has a hydrostatic energy shift97 due to strain, and is given by CB ðkÞ ¼ EC þ Hkp
h 2 k2 þ aC1 eZZ þ aC2 ðeXX þ eYY Þ 2me
ð4:4Þ
where EC is the conduction band edge in the absence of strain, aC1 and aC2 are deformation potentials, and eij are the strain components. Unlike the valence band, the conduction band kinetic energy term ( k2) is isotropic in k-space. Chuang and Chang have derived the Hamiltonian in Eq. (4.2) without spin–orbit interaction and set the split-off (SO) energy level (E3) as a reference level without spin–orbit interaction as shown in Figure IV.1, and then separately include the spin–orbit interaction effect. The spin–orbit interaction shifts the energy levels by some amount for all subbands. The energy levels including the spin–orbit interaction are 20 meV for heavy holes (HH), 14.2 meV for light holes (LH), 2.2 meV for crystal-field split-off holes (CH), and 3460 meV for conduction band (EC ¼ Egap þ 20 meV) corresponding to a band gap of 3.44 eV. For uniform systems, such as uniformly strained materials, all parameters including strain components in Eq. (4.3) are independent of position. In such a (a)
Ec = Eg + Δ1+ Δ2
Ec = Eg + Δ1 Eg Eg
E10 = Δ1+ Δ2
E10 = E20 = Δ1 E30 = 0
Δ1−Δ2
E30 =
Δ1−Δ2
2 2
+ −
Δ1−Δ2
2
+2Δ32
2 Δ1−Δ2 2
2
+2Δ32
With spin-orbit interaction (Δ2 = Δ3 = 0)
Without spin-orbit interaction (Δ2 = Δ3 = 0) (b)
E20 =
50 Energy (meV)
HH 0
Unstrained
LH CH
−50
−100 0.12 0.08 0.04 kz (1/Å)
0
0.04 0.08 0.12 kx (1/Å)
FIG. IV.1. (a) Band edge energy shifts with spin–orbit interaction and (b) band structure for unstrained wurtzite GaN. From Ref. [97].
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
195
case, the Schro¨dinger equation in Eq. (4.1) can be solved in k-space resulting in the band structure for uniformly strained materials. However, if there is a spatially inhomogeneous potential in the material, solving the Schro¨dinger equation (4.1) in k-space is not possible unless the Fourier transform of the potential is known. Also, the wave functions should then be obtained by inverse Fourier transform, which is difficult in general. Spatially inhomogeneous potentials can be induced by defects (e.g., dislocations), localized strain, or heterojunctions (e.g., based on quantum dots). For spatially inhomogeneous systems, it is advantageous to solve the Schro¨dinger equation (4.1) in real space because the spatial variations of potentials can be easily implemented in real-space form. Especially for dislocations, the electrostatic potential due to the electron acceptor nature of edge type dislocations, and the strain field associated with both edge and screw type dislocations, are localized and nonuniform. Therefore, it is advantageous to transform the kp Hamiltonian into real space to consider systems with dislocations. Transformation of the kp Hamiltonian in Eq. (4.2) into real space can be accomplished via the substitution k ¼ ir, resulting in second-order partial differential equations in real space. After transformation into real space the terms in Eq. (4.3) can be written as zÞ lðx; y; zÞ ¼ lk ðx;2y; zÞ þ le ðx; y; 0
13 2 2 h2 4 @2 @ @ lk ðx; y; zÞ ¼ A1 2 A2 @ 2 þ 2 A5 2m0 @z @x @y le ðx; y; zÞ ¼ D1 ezz ðx; y; zÞ þ D2 ½exx ðx; y; zÞ þ eyy ðx; y; zÞ zÞ yðx; y; zÞ ¼ yk ðx;2y; zÞ þ ye ðx; y; 0 13 yk ðx; y; zÞ ¼
h2 4 @2 @2 @2 A3 2 A4 @ 2 þ 2 A5 2m0 @z @x @y
ye ðx; y; zÞ ¼ D3 ezz ðx; y; zÞ þ D4 ½exx ðx; y; zÞ þ eyy ðx; y; zÞ : Kðx; y; zÞ ¼ Kk ðx; y; zÞ þ D5 eþ ðx; y; zÞ 0 12 h2 @ @ A5 @i þ A Kk ðx; y; zÞ ¼ 2m0 @x @y Hðx; y; zÞ ¼ Hk ðx; y; zÞ þ D 06 ezþ ðx; y; zÞ 1 Hk ðx; y; zÞ ¼ i
h2 @ @ @ A6 @i þ A 2m0 @z @x @y
e ðx; y; zÞ ¼ exx ðx; y; zÞ 2iexy ðx; y; zÞ eyy ðx; y; zÞ ez ðx; y; zÞ ¼ ezx ðx; y; zÞ ieyz ðx; y; zÞ
ð4:5Þ
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J. H. YOU AND H. T. JOHNSON
b. Zinc-Blende Crystal Structure (GaAs) To calculate the optical properties in GaAs, a 8 8 kp Hamiltonian is solved to include interactions between the conduction and valence bands due to its relatively narrow band gap of 1.519 eV. The 8 8 kp Hamiltonian has been derived by Bahder98 and is given in the form Hkp ðkÞ ¼ H k ðkÞ þ H e ðkÞ:
ð4:6Þ
The Hamiltonian Hk (k) contains kinetic energy terms and He (k) contains strain effects with deformation potentials as pffiffiffi pffiffiffi pffiffiffi 3 A 0 ðT þ VÞþ 0 3ðT VÞ 2ðW UÞ W U 2ðT þ VÞþ pffiffiffi pffiffiffi pffiffiffi 6 þ þ 7 0 A 2ðW UÞ 3ðT þ VÞ 0 T V 2ðT VÞ ðW þ UÞ 7 6 pffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 7 6 6 T þV 2ðW UÞþ P þ Q Sþ R 0 3=2S 2Q 7 6 pffiffiffi pffiffiffi 7 pffiffiffi 7 6 6 0 3ðT þ VÞ S P Q 0 R 2R S= 2 7 pffiffiffi pffiffiffi þ 7 H k ðkÞ ¼ 6 7 6 pffiffi3ffiðT VÞþ þ þ þ 0 R 0 P Q S S = 2 2R 6 pffiffiffi pffiffiffiffiffiffiffiffi þ 7 pffiffiffi 7 6 þ þ 7 6 2ðW UÞþ ðT VÞ 0 R S P þ Q 2 Q 3=2 S 7 6 pffiffiffiffiffiffiffiffi þ pffiffiffi pffiffiffi þ pffiffiffi pffiffiffi 7 6 þ þ 2ðT VÞ 2R S= 2 2Q Z 0 3=2S 5 4 ðW UÞ p ffiffiffiffiffiffiffi ffi pffiffiffi pffiffiffi p ffiffi ffi p ffiffi ffi 2ðT þ VÞ W þU 2Q Sþ = 2 2R 3=2S 0 Z 2
ð4:7Þ and pffiffiffi pffiffiffi pffiffiffi 3 ac e 0 ðt vÞþ 0 3ðt þ vÞ 2ðw þ uÞ wþu 2ðt vÞþ pffiffiffi pffiffiffi pffiffiffi 6 þ þ 7 0 ac e 2ðw þ uÞ 3ðt vÞ 0 tþv 2ðt þ vÞ ðw uÞ 7 6 pffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 7 6 þ þ 6 t v 2 ðw þ uÞ p þ q s r 0 3=2s 2q 7 6 p ffiffi ffi pffiffiffi 7 p ffiffi ffi 7 6 6 0 3ðt vÞ s p q 0 r 2r s= 2 7 p ffiffi ffi p ffiffi ffi þ 7 p ffiffi ffi H e ðkÞ ¼ 6 7 6 3ðt þ vÞþ þ þ þ 0 r 0 p q s s = 2 2 r 6 pffiffiffi pffiffiffiffiffiffiffiffi þ 7 pffiffiffi 7 6 6 2ðw þ uÞþ ðt þ vÞþ 0 rþ s p þ q 2q 3=2s 7 7 6 pffiffiffiffiffiffiffiffi þ pffiffiffi pffiffiffi þ pffiffiffi pffiffiffi 7 6 þ þ 2ðt þ vÞ 2r s= 2 2q av e 0 3=2s 5 4 ðw þ uÞ p ffiffiffiffiffiffiffi ffi pffiffiffi pffiffiffi p ffiffi ffi p ffiffi ffi þ 2ðt vÞ wu 2q s = 2 2r 3=2s 0 av e 2
ð4:8Þ where
2
3 2 h 5ðk2 þ k2 þ k2 Þ A ¼ EC þ 4A0 þ x y z 2m0 P ¼ EV þ g1
h2 2 ðk þ ky2 þ kz2 Þ 2m0 x
h2 2 ðk þ ky2 2kz2 Þ Q ¼ g2 2m0 x 1 U ¼ pffiffiffi P0 kz 3
ð4:9Þ
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
197
1 V ¼ pffiffiffi P0 ðkx iky Þ 6 pffiffiffi h2 R¼ 3 ½g ðk2 ky2 Þ 2ig3 kx ky 2m0 2 x 1 W ¼ i pffiffiffi Bkx ky 3 1 T ¼ pffiffiffi Bkz ðkx þ iky Þ 6 pffiffiffi h2 S ¼ 2 3g3 kz ðkx iky Þ 2m0 Z ¼ Ev D g 1
h2 2 ðk þ ky2 þ kz2 Þ 2m0 x
and p ¼ av ðexx þ eyy þ ezz Þ; 1 w ¼ i pffiffiffi b0 exy ; 3 X 1 ezj kj ; u ¼ pffiffiffi P0 3 j s ¼ dðexz ieyz Þ;
q ¼ b½ezz ðexx þ eyy Þ=2 1 t ¼ pffiffiffi b0 ðexz þ ieyz Þ 6 : X 1 v ¼ pffiffiffi P0 ðexj ieyj Þkj 6 j e ¼ exx þ eyy þ ezz
ð4:10Þ
The modified Luttinger parameters, g1, g2, and g3 are related to the parameters used by Luttinger,99 gL1 , gL2 , and, gL3 by g1 ¼ gL1
EP ; 3Egap þ D
g2 ¼ gL2
1 EP ; 2 3Egap þ D
g3 ¼ gL3
1 EP 2 3Egap þ D ð4:11Þ
where the band gap is Egap ¼ EC EV and the conduction-valence band mixing parameter, EP, is related with the conduction-valence band momentum matrix element, P0 by EP ¼
2m0 2 P0 : h2
ð4:12Þ
All parameters for GaAs are tabulated in Table X.3 in Appendix C. The transformed Schro¨dinger equation can be solved as a real space eigenproblem using the FEM, and thus the energy level spectrum and wave functions for
99
J. M. Luttinger, Phys. Rev. 102, 1030 (1956).
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conduction and valence states are determined in real space. This approach is desirable for a spatially large inhomogeneous system in which some knowledge of the spatial wave function is needed, such as in computing the optical response. Details of the finite element kp Hamiltonian method are described for the 2D case in Ref. [100] and details of the FEM formulation for the 3D real space kp Hamiltonian are shown in Section IV.15. 15. F INITE E LEMENT F ORMULATION In this section, details of the FEM formulation of the 3D real space kp Hamiltonian method for wurtzite structures are shown; the formulation for the zinc-blende Hamiltonian is similar and is not shown in detail here. As shown in Eq. (4.5), the kp Hamiltonian in Eq. (4.1) contains second-order differential equations in real space and can be solved by FEM. The basic idea of FEM is to seek an approximate solution satisfying a governing equation over the domain of interest in an integral or average sense. From Eq. (4.1), the residual function R is ~ Ec ~ R ¼ Hkp c ~ is the approximate solution. The weak form of Eq. (4.13) is where c ð ~ EcÞ ~ dO ¼ 0 P ¼ wðHkp c
ð4:13Þ
ð4:14Þ
O
where O is the volume of the domain and w is the weight function. To apply Eq. (4.14) to a kp Hamiltonian, it is advantageous to rewrite the Hamiltonian into the sum of two Hamiltonians based on the order of the derivative term, as Hkp ðx; y; zÞ ¼ H2 ðx; y; zÞ þ H1 ðx; y; zÞ þ H0 ðx; y; zÞ
ð4:15Þ
where H2, H1, and H0 are the Hamiltonians containing only second-order derivatives, first-order derivatives, and constant terms, respectively. Because the 6 6 kp Hamiltonian in Eq. (4.14) does not have first derivative terms, H1 is zero. The Hamiltonians H2 and H0 can be found as 2 3 l k þ yk Kkþ Hkþ 0 0 0 6 Kk lk þ yk Hk 0 0 0 7 6 7 þ 6 Hk Hk lk 0 0 0 7 6 7 H2 ¼ 6 ð4:16Þ Kk Hk 7 0 0 lk þ y k 6 0 7 þ þ5 4 0 0 0 Kk lk þ yk Hk 0 0 0 Hkþ Hk lk
100
H. T. Johnson, L. B. Freund, C. D. Akyu¨z, and A. Zaslavsky, J. Appl. Phys. 84, 3714 (1998).
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
and
199
2
3 Dþ þ le þ ye Keþ Heþ 0 0 0 6 D þ le þ ye He 0 0 D 7 Ke 6 7 þ 6 H l 0 D 0 7 H e e e 6 7 H0 ¼ 6 Ke He 7 0 0 0 D þ þ le þ ye 6 7 4 0 0 D Keþ Dþ le þ ye Heþ 5 0 D 0 Heþ He le ð4:17Þ where Dþ ¼ D1 þ D2 and D ¼ D1 D2. The weak form in Eq. (4.14) can be rewritten as ð ð ~ ~ ~ dO: wH2 c þ wH0 c dO ¼ wEc ð4:18Þ O
O
~ contains six subbands as From a 6 6 Hamiltonian, the total wave function c ~ ¼ hc ~ 1c ~ 2c ~ 3c ~ 4c ~ 5c ~ 6 iT : c
ð4:19Þ
In FEM, any function can be expressed as a sum of products of shape functions and nodal values at a given node. For an m node system, the approximate wave function in subband j can be written as ~j ¼ c
m X i¼1
Ni Dij ¼ N1 D1j þ N2 D2j þ . . . þ Nm Dmj ;
j ¼ 1; 2; . . . ; 6:
ð4:20Þ
~ can be written as Then, the total wave function c ~ ¼ ½N fDg c |{z} |{z} |{z} 6 1
ð4:21Þ
6 6m 6m 1
where [N] is the shape function matrix, or 2 N1 0 0 0 0 0 Nm 6 0 N1 0 0 0 0 0 6 6 0 0 0 0 0 N1 0 ½N ¼ 6 0 0 0 0 N1 0 |{z} 6 6 0 4 0 6 6m 0 0 0 N1 0 0 0 0 0 0 0 N1 0
0 Nm 0 0 0 0
0 0 Nm 0 0 0
0 0 0 Nm 0 0
0 0 0 0 Nm 0
3 0 0 7 7 0 7 7 0 7 7 0 5 Nm ð4:22Þ
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J. H. YOU AND H. T. JOHNSON
and {D} is the nodal value vector, or fDg ¼ hD11 D21 D31 D41 D51 D61 D1m D2m D3m D4m D5m D6m iT |{z}
ð4:23Þ
6m 1
where the subscript index indicates the node number from 1 to m and the superscript index refers to the six subbands. The first term including [H2] from Eq. (4.18) can be rewritten by introducing the gradient matrix [r] as ½H2 ¼ ½rT ½L ½r |ffl{zffl} |{z} |{z}
ð4:24Þ
6 18 18 18 18 6
where the gradient matrix is defined as 2 @x 0 6 @y 0 6 6 @z 0 6 6 0 @x 6 6 0 @y 6 6 ½r ¼ 6 0 @z |{z} 6 6 18 6 6 6 6 60 0 6 40 0 0 0
0 0 0 0 0 0 .. . 0 0 0
0 0 0 0 0 0 .. 0 0 0
0 0 0 0 0 0 . 0 0 0
0 0 0 0 0 0
3
7 7 7 7 7 7 7 7 7 7: 7 7 7 7 7 @x 7 7 @y 5 @z
ð4:25Þ
Now the matrix [L] can be defined. All elements in [H2] can be expressed in matrix form through 2 0 13 2 2 2 2 h 4 @ @ @ A1 2 A2 @ 2 þ 2 A5 lk ðx; y; zÞ ¼ 2m0 @z @x @y 2 38 9 A2 0 0 < @x = 2 ð4:26Þ h 4 ¼ h @x @y @z i 0 5 @y 0 A2 : ; 2m0 @z 0 0 A1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
½Dl
2 0 13 2 2 h2 4 @2 @ @ yk ðx; y; zÞ ¼ A3 2 A4 @ 2 þ 2 A5 2m0 @z @x @y 2 38 9 A4 0 0 < @x = 2 h 4 ¼ h @x @y @z i 0 5 @y 0 A4 : ; 2m0 @z 0 0 A3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
½Dy
ð4:27Þ
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
0 12 h2 @ @ Kk ðx; y; zÞ ¼ A5 @i þ A 2m0 @x @y 2 38 9 A5 iA5 0 < @x = h2 4 ¼ h @x @y @z i A5 0 5 @y iA5 : ; 2m0 @z 0 0 A3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
201
ð4:28Þ
½DK
0 1 h2 @ @ @ @A Hk ðx; y; zÞ ¼ i A6 i þ 2m0 @z @x @y 2 38 9 0 0 A6 =2 < @x = ð4:29Þ h2 4 0 0 iA6 =2 5 @y : ¼ h @x @y @z i : ; 2m0 0 @z A6 =2 iA6 =2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
½DH
Then, the (18 18) matrix [L] can be assembled from (3 3) matrices [Dl], [Dy], [DK], and [DH] as 2 3 ½0 ½0 ½0 ½Dl þ Dy ½DKþ ½DHþ 6 ½DK ½Dl þ Dy ½DH ½0 ½0 ½0 7 6 7 þ 6 ½DH 7 ½Dl ½0 ½0 ½0 ½DH 6 7: ½L ¼ 7 |{z} 6 ½0 ½0 ½0 ½Dl þ Dy ½DK ½DH 6 7 4 18 18 ½0 ½0 ½0 ½DKþ ½Dl þ Dy ½DHþ 5 ½0
½0
½0
½DHþ
½DH
½Dl ð4:30Þ
After replacing the weight functions with the shape functions and applying integration by parts to the second-order derivative term, Eq. (4.18) can be rewritten as 0 1 0 1 ð ð @ ½rN T ½L ½rN þ ½ N T ½H0 ½ N d OA f Dg ¼ E @ ½ N T ½ N d OA f Dg |fflffl{zfflffl} |{z} |ffl{zffl} |{z} |{z} |{z} |{z} |{z} |{z} |{z} O 6m 8 18 8 18 6m 6m 6 6 6 6 6m O 6m 6 6 6m 6m 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 6m 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 6m 6m
6m 6
ð4:31Þ
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J. H. YOU AND H. T. JOHNSON
Equation (4.31) is in the form of a generalized eigenvalue problem, given by ½AfDg ¼ E½BfDg
ð4:32Þ
with E being the energies of electrons (eigenvalues) and {D} containing the nodal values of wave functions (eigenvectors). In FEM the matrices [A] and [B] are evaluated within each element and are used to assemble the global matrix. For each element, the matrix components are ð ð4:33Þ aij ¼ ½rNT ½L ½rN þ½NT ½H0 ½N dOe |fflffl{zfflffl} |ffl{zffl} |ffl{zffl} |{z} |{z} |{z} Oe 48 18 18 18 18 48
48 6 6 6 6 48
ð bij ¼ e
½NT ½N dOe |{z} |{z}
ð4:34Þ
O 48 6 6 48
where is the volume of each element and eight node brick elements (m ¼ 8) are assumed. Oe
16. S PONTANEOUS E MISSION S PECTRUM C ALCULATION After the generalized eigenvalue problem in Eq. (4.32) is solved, the energy spectrum and corresponding wave functions in real space are obtained. Once the real space wave functions are known, the spontaneous emission spectrum can be calculated by evaluating the spontaneous emission coefficient. The spontaneous emission coefficient for the wurtzite case, in units of (1 cm1), is expressed as101,102 gðoÞ ¼
C0 X * * j e p CV j2 j hC j Vi j2 fC ðEC Þ 1 fV ðEV Þ d½EC EV ho o C;V ð4:35Þ
where C0 ¼ 2pq The constant C0 containing the volume of system O, electron charge q, mass m0, permittivity e0, refractive index nr, and speed of light c has no effect on relative magnitude along the emission spectrum at a 2
101
=ðe0 cnr m20 OÞ.
J. H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge University Press, Cambridge (1998). 102 S. L. Chuang, Physics of Optoelectronic Devices, Wiley–Interscience, New York (1995).
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
203
* 2 particular frequency o of interest. The first term j * e p cv j in the summation over conduction and valence states jCi and jVi contains the interaction between polarization of light and the Bloch wave function for the conduction and valence bands, and determines the polarization of emitted light. The second squared term, jhCjVij2, represents the square of spatial overlap between the electron envelope wave function and the hole envelope wave function, which is related to the relative intensity of light emitted or absorbed through a transition between a particular pair jCi and jVi. The occupation functions represent the probability of carrier occupation in each subband defined from the quasi-Fermi level. The final Dirac delta function equates the energy difference between jCi and jVi with the emitted or absorbed energy ho. This part determines the transition energy in the spectrum. To compare with experimental PL intensity data, g(o)/C0 is calculated to obtain the emission spectrum. After obtaining the g(o)/C0 spectrum as a function of transition energy, the emission spectrum is broadened with a Gaussian distribution to obtain a continuous spectrum that can be compared with experimental data. The width of the Gaussian expansion is related to the thermal energy, kBT, where kB is Boltzmann’s constant and T is temperature. It is worth noting here that defect studies in semiconductors often involve transient response measurements such as time resolved and time integrated photoluminescence measurements.103,104 From these transient responses, radiative and nonradiative decay times can be evaluated to classify the defects as radiative or nonradiative recombination centers. However, the approach presented in this section is based on steady state responses and the assumption that carriers are continuously provided by steady excitation sources. Equation (4.35) gives the steady state response of light intensities in the presence of defects. More discussion on this point is provided in Section V where the effects of edge dislocations on optical properties are discussed.
17. S UMMARY A method to calculate the optical emission spectrum in a semiconductor layer with a nonuniform effective potential is introduced. Since dislocations generate spatial inhomogeneity in systems it is advantageous to transform the k-space Hamiltonian to real space. Then, using FEM, the Schro¨dinger equation is solved in real space as an eigenvalue problem to obtain the energy levels and
103
M. S. Minsky, S. Watanabe, and N. Yamada, J. Appl. Phys. 91, 5176 (2002). C. L. Andre, J. J. Boeckl, D. M. Wilt, A. J. Pitera, M. L. Lee, E. A. Fitzgerald, B. M. Keyes, and S. A. Ringel, Appl. Phys. Lett. 84, 3447 (2004). 104
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J. H. YOU AND H. T. JOHNSON
corresponding spatial wave functions of electron states. From spatial overlap between the electron and hole wave functions, spontaneous emission coefficients are obtained as a function of transition energies, and Gaussian broadening is applied to generate a continuous optical emission spectrum that can be compared with experimental observations. Using this approach, the effects of edge dislocations on the optical emission spectra in GaN and GaAs are presented in Section V, and the effect of screw dislocations on the optical spectra in GaN are presented in Section VI. V. Effect of Edge Dislocations on Optical Properties of GaN and GaAs
Using the charged dislocation model described in Section II, the effects of negatively charged edge dislocations on optical properties in GaAs and GaN are now considered by solving the kp Hamiltonian in real space using FEM as described in Section IV. The electrostatic potential due to the electron acceptor nature of edge type dislocations and the strain fields associated with edge dislocations are considered in these calculations. As described previously, this fully electrostatic approach requires no a priori assumptions about the radiative or nonradiative nature of individual transitions, which gives an effectively steady state picture of the optical spectra. Finite element meshes and inhomogeneous strain fields for edge dislocations in GaAs and GaN are presented in Section V.18. The electrostatic potential and deformation potential that induce band edge shifts for each subband are presented in Section V.19 in the form of band edge shift plots for the conduction and valence bands. The inhomogeneous band edge shifts change the densities of states and induce different spatial distributions for electrons and holes, which are described in Section V.20. Then, spontaneous emission spectra with various dislocation densities and carrier concentrations are calculated in Section V.21. Comparisons of calculated band edge peak intensities for GaN and GaAs as a function of dislocation density are presented in Section V.22. Other peaks observed with energies below the band gap are discussed in Section V.23, particularly for the case of radiative recombination as found in the analysis of edge dislocations in GaN. In this section emission spectra are presented for cases in which the samples contain the effects of only edge dislocations. These spectra can be directly compared with photoluminescence measurements from experiments. The results indicate the existence of other below-band-edge peaks associated with edge dislocations and illustrate their behavior with varying dislocation density and carrier concentration.
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
205
18. E DGE D ISLOCATION S TRAIN F IELDS
a. Isotropic Dislocation Fields in Wurtzite GaN To study the effects of edge dislocations in GaN, a periodic computational domain is prepared in three-dimensional real space as shown in Figure V.1. The computational domain contains a dipole of edge dislocations with opposite Burgers vector directions. The thickness of the sample along the z-direction is ˚ and the crystal orientation h0001i is aligned with the assumed as 1000 A z-direction. This structural unit cell is based on the same symmetry considerations as the unit cell commonly used for atomic scale calculations of edge dislocations as shown in Figure II.6. Periodic boundary conditions have been applied in the lateral boundaries along the x–y plane while a termination boundary condition (or zero-valued Dirichlet condition) has been applied on the top and bottom boundaries along the z-direction. Therefore, the system acts as a wide quantum well along the z-direction and represents an infinitely broad domain in the x–y plane with the periodic perturbation of evenly distributed edge dislocations. To study the dislocation density effect, the domain area A is changed according to the desired dislocation density using A ¼ 2/ndis, where ndis is the areal dislocation density and the factor of 2 is from the existence of the dislocation dipole in the unit cell. Since the edge dislocation line, shown in Figure V.1, is aligned with the h0001i axis, the nonzero strain components induced by the dislocation are on the basal plane (x–y plane in the current coordinates), which is well known to be elastically isotropic in a hexagonal close-packed crystalline material.105 The stress components around an edge dislocation in an isotropic material are given by105 sxx ¼
Gb yð3x2 þ y2 Þ ; 2pð1 nÞ ðx2 þ y2 Þ2
Gb xðx2 y2 Þ sxy ¼ ; 2pð1 nÞ ðx2 þ y2 Þ2
syy ¼
Gb yðx2 y2 Þ 2pð1 nÞ ðx2 þ y2 Þ2
ð5:1Þ
szz ¼ nðsxx þ syy Þ; sxz ¼ syz ¼ 0
¼ 3:189 Å, the Poisson where the magnitude of Burgers vector is b ¼ 13 h1120i ratio is n ¼ C12/(C12þC11), and shear modulus is G ¼ C66 ¼ (C11–C12)/2. All material properties are listed in Table X.2 in Appendix C. Assuming that the ˚ ) is large enough compared to the displacements of the edge thickness (1000 A
105
J. P. Hirth and J. Lothe, Theory of Dislocations, 2nd ed., Krieger Publishing Company, Malabar, FL (1992).
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J. H. YOU AND H. T. JOHNSON
(a)
z (Å) 1000 0 3000 2000 1000
3000 2000 1000 0 y (Å) −1000 −2000 −3000
0 −1000 x (Å) −2000 −3000
(b) 3000 2000
0
y (Å)
1000
−1000 −2000 −3000 −3000 −2000−1000
0 1000 2000 3000 x (Å)
FIG. V.1. Example of the finite element computational domain, showing a finite element mesh with ndis ¼ 109 cm2. The domain contains a dipole of edge dislocations with opposite Burgers vector directions. (a) Perspective view and (b) Plan view. The thickness of the sample along the z-direction is ˚ and the crystal orientation h0001i is aligned with the z-direction. 1000 A
dislocation (which is linked to the magnitude of the Burgers vector), allows use of a plane strain approximation. The strain field for the case of plane strain is exx ¼
C11 sxx C12 syy C11 syy C12 sxx ; eyy ¼ ; 2 2 C11 C12 C211 C212
exy ¼ eyx ¼
sxy 2C66
ð5:2Þ
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a)
exx
207
3000 2000 1000 0
Tension
0.001 0.0008 0.0006 0.0004 0.0002 0 −0.0002 −0.0004 −0.0006 −0.0008 −0.001
y(Å)
Compression
−1000 −2000 −3000 −3000 −2000 −1000
1000
0
2000
3000
x(Å) 3000
exy
3000
2000
2000
1000
1000
0
0
−1000
−1000
−2000
−2000
−3000 −3000 −2000 −1000
0
x(Å)
1000
2000
3000
y(Å)
(c)
eyy
y(Å)
(b)
−3000 −3000 −2000 −1000
0
1000 2000 3000
x(Å)
FIG. V.2. Strain components around edge dislocations with ndis ¼ 109 cm2. (a) exx, (b) eyy, and (c) exy ¼ eyx. Other strain components are zero. Positive strain indicates tension while negative strain indicates compression. The strain field is localized near edge dislocations resulting in spatially inhomogeneous deformation potential.
while all other strain components are zero. The strain components around an edge dislocation are shown in Figure V.2. The strain exx in Figure V.2(a) is compressive above the dislocation at the center of domain and tensile below. Strain components are localized near an edge dislocation and their magnitudes decay with distance away from the dislocation. These inhomogeneous strain fields produce spatially inhomogeneous deformation potentials. The spatially inhomogeneous deformation potentials around an edge dislocation plus the electrostatic potential at the core induce spatially inhomogeneous band edge shifts as will be shown in Section V.19.
b. Anisotropic Dislocation Fields in Zinc-Blende GaAs Isotropic strain fields around edge dislocations are on planes perpendicular to the edge dislocations as shown in Eq. (5.2). For edge dislocations along the [001] direction, the strain fields are on the (001) plane where elastic properties
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J. H. YOU AND H. T. JOHNSON
are anisotropic. To include the effects of these dislocation strain fields, calculations are carried out on the (001) plane and transformed to crystallographic coordinates for which the kp Hamiltonian in Eq. (4.6) has been derived. Thus, two coordinate systems are defined. The coordinates x and y align with the crystallographic system [100] and [010], and x 0 and y 0 align with the dislocation coordinates as shown in Figure V.3(a). z and z0 are aligned with the dislocation line which is along the [001] direction. A periodic computational domain is prepared in two-dimensional real space as shown in Figure V.3(b). Like the GaN calculations, the computational domain contains a dipole of edge dislocations with opposite Burgers vector directions, and periodic boundary conditions are applied in the lateral boundaries on the x–y plane. The system has an infinite thickness along the z-direction, representing infinitely long dislocation lines. Therefore, the system represents an infinitely broad domain in the x–y plane with the periodic perturbation of evenly distributed edge dislocations. The domain area A is changed according to the desired dislocation density using A ¼ 2/ndis. Strain fields around edge dislocations on the anisotropic plane are described in Ref. [105] and summarized here. For zinc-blende GaAs, three elastic constants are needed to describe the elastic behavior: C11, C12, and C44. The relationship between components of stress and strain is
(a)
(b)
y⬘, [110] x⬘, [110]
2000 1000
y(Å)
y, [010]
0 −1000
x, [100] z // z⬘, [001]
−2000 −2000 −1000
0
1000 2000
x(Å)
FIG. V.3. (a) x and y are aligned with the crystallographic directions [100] and [010], and x 0 and y 0 are aligned with the dislocation coordinates, [110] and ½110. The Burgers vector is along the [110] direction. (b) Example of the finite element computational domain in two-dimensional real space, showing the finite element mesh with ndis ¼ 109 cm2. The domain contains a dipole of edge dislocations with opposite Burgers vector directions. The sample has an infinite thickness along the z-direction.
209
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
9 2 8 C11 s11 > > > > > > 6 > > s C > 22 > > 6 12 > > > 6 C12 > s33 > > > > > > 6 > > = 6 < s23 > 6 0 s31 ¼ 6 6 0 > > > > 6 0 s > > 12 > > 6 > > > > 6 0 > > s 32 > > 6 > > > > 4 0 > > s > > 13 ; : 0 s21
C12 C11 C12 0 0 0 0 0 0
C12 C12 C11 0 0 0 0 0 0
0 0 0 C44 0 0 C44 0 0
0 0 0 0 C44 0 0 C44 0
0 0 0 0 0 C44 0 0 C44
0 0 0 C44 0 0 C44 0 0
0 0 0 0 C44 0 0 C44 0
38 9 e11 > 0 > > > > > > 0 7 e22 > > 7> > > > > > > 0 7 e > 33 > > 7> > > > > 0 7 e < 7 23 = 0 7 e 7> 31 > > e12 > C44 7 > > > 7> > > > > > 0 7 e 32 > 7> > > > > 5 > 0 > e > > ; : 13 > C44 e21 ð5:3Þ
where sij and eij are the stress and strain components. All material properties for GaAs are tabulated in Appendix C. The 9 9 elastic constant matrix [C] containing Cij is based on the crystallographic coordinates (x, y, z). To obtain the anisotropic strain fields around the edge dislocations on (x 0 , y0 , z0 ), the elastic constant matrix [C] should be transformed to [C0 ], where C0 ij is on the (x0 , y0 , z0 ) coordinates. The relationship between (x, y, z) and (x0 , y0 , z0 ), from Figure V.3(a), is 1 1 1 1 x0 ¼ pffiffiffi ½110 ¼ pffiffiffi ðx þ yÞ; y0 ¼ pffiffiffi ½ 110 ¼ pffiffiffi ðx þ yÞ; z0 ¼ z ¼ ½001: 2 2 2 2 ð5:4Þ Thus, the coordinate transformation matrix, [Tij], to transform (x, y, z) to (x0 , y0 , z0 ) can be built as 8 09 2 38 9 1 1 0 <x= <x = 1 y0 ¼ pffiffiffi 4 1 1 p0ffiffiffi 5 y : ð5:5Þ : 0; : ; 2 0 0 z 2 z Then, the transformation of the 9 9 elastic constant matrix can be carried out by Qmnkl ¼ Tkm Tln
ð5:6Þ
where the matrix element Qij ¼ Q(i, j) can be found by the following index mapping:
i or j mn or kl
1 11
2 22
3 33
4 23
5 31
6 12
7 32
8 13
9 21
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J. H. YOU AND H. T. JOHNSON
For an example,
pffiffiffi pffiffiffi Qð9; 6Þ ¼ Q96 ¼ Q2112 ¼ T12 T21 ¼ Tð1; 2ÞTð2; 1Þ ¼ ð1= 2Þð1= 2Þ ¼ 1=2:
Using the coordinate transformation matrix, [Tij], from Eq. (5.5), the transformation matrix, [Q], can be found as 2
1 61 6 60 6 60 16 ½Q ¼ 6 0 26 61 6 60 6 40 1
1 1 0 0 0 1 0 0 1
0 0 2 0 0 0 0 0 0
0 0 1 0 0 0 1 0 0 0 0 0 pffiffiffi 2 p0ffiffiffi 0 0 ffiffiffi p 0 2 0 2 0 1 p0ffiffiffi p0ffiffiffi 0 2 0 2 pffiffiffi 2 0 0 0 0 0 1 0
3 0 1 0 17 7 07 p0ffiffiffi 7 2 07 7 0 07 7 0 1 7 7 07 7 p0ffiffiffi 2 05 0 1
ð5:7Þ
and the transformed elastic constant matrix [C0 ] on (x0 , y0 , z0 ) coordinates can be calculated by ½C0 ¼ ½QT ½C½Q:
ð5:8Þ
Stress fields of edge dislocations in anisotropic materials, with Burgers vector b ¼ (bx0 ,by0 ,0), are given as105 ( ) 0 0 2 C66 Mbx0 C 0 0 0 2 3 0 0 0 0 0 0 0 11 sxx ¼ ½ðC11 C12 ÞðC11 þC12 þ2C66 ÞC11 C66 x y þ 0 y 2pr4 C022 C22 0 0 Mby0 C66 C11 0 0 2 0 3 ð5:9Þ x y x 2pr4 C022 0
0
1 0 Mbx0 C66 @ 0 2 0 C11 0 3 A 0 syy ¼ x y þ 0 y 2pr4 C22 0
þ
Mby0 0 0 0 0 0 03 0 0 0 0 0 0 2 0 f½ðC11 C12 ÞðC11 þ C12 þ 2C66 Þ C11 C66 x y þ C22 C66 x g 4 2pr C11 ð5:10Þ
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
0 sxy ¼
0 0 0 Mby0 C066 Mbx0 C66 C11 0 0 2 C11 0 3 03 02 0 x þ x y x y þ y 2pr4 C022 2pr4 C022
where
"
0 C11 C012 M ¼ ðC11 þ C12 Þ 0 0 0 C22 C66 ðC11 þ C012 þ 2C066 Þ 0
r ¼ 4
2 0 0 0 ðC11 þ C012 ÞðC11 C012 2C066 Þ 0 2 0 2 C11 0 2 x þ 0 y þ x y C22 C022 C066 02
0 C11 ¼
ð5:11Þ
#1=2
0
and
211
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C011 C022 :
ð5:12Þ
ð5:13Þ
ð5:14Þ
Stresses in Eqs. (5.9)–(5.11) are defined in the dislocation coordinates (x 0 , y0 , z0 ). For the edge dislocation shown in Figure V.3(b), the Burgers vector component 0 along the y0 axis, by0 , is zero, pffiffiffi and the x component, bx0 , is equal to the magnitude of the Burgers vector, a= 2. After evaluating the stresses, strain components can be found by fs0 g ¼ ½C0 fe0 g
ð5:15Þ
which leads to the expressions for strain fields, for the case of plane strain, of 0 exx ¼
0 0 0 0 0 C022 sxx C012 syy C011 syy C012 sxx sxy 0 0 0 ; e ¼ ; e ¼ e ¼ : 0 yy xy yx 2 2 0 2C066 ðC11 Þ ðC012 Þ ðC11 Þ2 ðC012 Þ2
ð5:16Þ
Using material properties from Table X.3 in Appendix C, 0 0 0 ¼ C0 22 ¼ 1493:5 GPa, C12 ¼ 293:5 GPa, C66 ¼ 327:5 GPa, M ¼ 1.79, and C11 r4 ¼ (x 0 2þy0 2)2þ2x0 2 y0 2. Then, the stresses from Eqs. (5.9)–(5.11) become 0 ¼ sxx
ð1:79Þbð327:5Þ y0 ð5x0 2 þ y0 2 Þ 2p r4
ð5:17Þ
0 syy ¼
ð1:79Þbð327:5Þ y0 ðx0 2 y0 2 Þ 2p r4
ð5:18Þ
0 sxy ¼
ð1:79Þbð327:5Þ x0 ðx0 2 y0 2 Þ : 2p r4
ð5:19Þ
Compared to the stress fields in isotropic materials shown in Eq. (5.1), the functional forms are similar, but the coefficients are different. Calculated anisotropic strain fields for edge dislocations in GaAs are shown in Figure V.4; these
212
J. H. YOU AND H. T. JOHNSON
are comparable to the strain fields shown in Figure V.2, the strain magnitudes are different. The strains are calculated in the dislocation coordinates (x0 , y0 z0 ), so it is necessary to express them in crystallographic coordinates (x, y, z) before including them in the Hamiltonian, He(k), in Eq. (4.6). Strains {e0 } in Eq. (5.16) can easily be converted to {e} by the transform {e}[Q]{e0 }, and are shown in Figure V.5. Figures V.4 and V.5 show the same strain states but in different reference coordinates. Just as in GaN, the inhomogeneous strain fields and the electrostatic potential here induce inhomogeneous band edge shifts for each carrier. The inhomogeneous band edge shift maps are shown in Section V.19
(a)
e⬘xx
2000
y
y(Å)
1000
y⬘
0.001 0.0008 0.0006 0.0004 0.0002 0 −0.0002 −0.0004 −0.0006 −0.0008 −0.001
0
−1000 −2000 −2000 −1000
0
1000
x⬘ x
2000
x(Å)
(b)
e⬘yy
(c) e⬘xy
2000
1000
y(Å)
1000
y(Å)
2000
0
0
−1000
−1000
−2000
−2000 −2000 −1000
0
x(Å)
1000
2000
−2000 −1000
0
1000
2000
x(Å)
FIG. V.4. Strain components around edge dislocations with ndis ¼ 109 cm2. (a) e 0 xx, (b) e 0 yy, and (c) e0 xy ¼ e 0 yx based on dislocation coordinates, (x 0 , y 0 , z 0 ). Other strain components are zero. Positive strain indicates tension while negative strain indicates compression. Axes x and y are along the crystallographic coordinates [100] and [010]. Axes x 0 and y0 are along the dislocation coordinates [110] and ½ 110.
213
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a)
exx
2000
y y(Å)
x⬘
y⬘
1000 0.001 0.0008 0.0006 0.0004 0.0002 0 −0.0002 −0.0004 −0.0006 −0.0008 −0.001
0 −1000 −2000 −2000 −1000
0
1000
x
2000
x(Å)
(c)
(b)
eyy
exy
2000
1000
y(Å)
1000
y(Å)
2000
0
0
−1000
−1000
−2000
−2000 −2000 −1000
0
1000
−2000 −1000
2000
x(Å)
0
1000
2000
x(Å)
FIG. V.5. Strain components around edge dislocations with ndis ¼ 109 cm2. (a) exx, (b) eyy, and (c) exy ¼ eyx based on dislocation coordinates, (x, y, z). Other strain components are zero. Axes x and y are along the crystallographic coordinates [100] and [010]. Axes x 0 and y 0 are along the dislocation coordinates [110] and ½110. Note that these strains are in the same states as those in Figure V.4, but expressed in different reference coordinates.
19. I NHOMOGENEOUS B AND E DGE S HIFTS
a. Wurtzite GaN For a negatively charged edge dislocation, the B–G electrostatic potential from Eq. (2.16) and the strain fields from Eq. (5.2) produce spatially inhomogeneous band bending for the conduction and valence bands. The B–G electrostatic potential from Eq. (2.16) is given again here as VðrÞ ¼
2q2 K0 ðr=lD Þ: 4pea
ð2:16Þ
214
J. H. YOU AND H. T. JOHNSON
To include the electrostatic potential in the calculations, an additional Hamiltonian needs to be added to Eq. (4.1), so that ðHkp þ Helect Þ j ci ¼ E j ci
ð5:20Þ
where the Hkp term is the kp Hamiltonian from Eqs. (4.2) and (4.4) including the kinetic energy contributions and the deformation potential contributions, and Helect is the Hamiltonian that includes band bending due to the electrostatic potential for a negatively charged edge dislocation. The electrostatic potential CB ¼ VðrÞ for the conduction band and Hamiltonian Helect can be written as Helect VB Helect ¼ VðrÞ I6 6 for the valence bands, where I6 6 is the 6 6 identity matrix. The combined effect of the spatially inhomogeneous deformation potential and the electrostatic potential at the dislocation core can be viewed as a spatially inhomogeneous band edge shift. The inhomogeneous band edge shift maps for each subband are shown in Figure V.6. In each valence subband, band edge levels in unstrained bulk WZ GaN are 20 meV for heavy holes (HH), 14.2 meV for light holes (LH), and 2.2 meV for crystal-field split-off holes (CH). The conduction band edge level in unstrained bulk WZ GaN is 3460 meV as shown in Figure IV.1. In each plot, the reference band edge level is green. Lower energy indicates that the conduction band edge shifts downward so that the electrons tend to be localized; higher energy acts as a repulsive potential for electrons. As shown in Figure V.6(a), electrons are localized in the tensile strained area while the electrostatic potential and compressive strain repel the electrons. Therefore, there are electron states with energy within the band gap, with corresponding wave functions localized in the tensile strained region. For valence subbands, higher energy indicates that the band edge shifts upward so that holes tend to be localized. Lower energy represents a repulsive potential for holes. The electrostatic potential and strain field near edge dislocations localizes HH carriers, so there are HH states with energy inside the band gap, with corresponding wave functions localized at the core and near the dislocation. For LH carriers, strain acts as a repulsive potential while the electrostatic potential is attractive, localizing the LH wave functions at the dislocation core. The electrostatic potential and tensile strain both localize CH carriers while compressive strain repels CH carriers. These different inhomogeneous band edge shifts for different carriers perturb the densities of states and result in spatial separations between electrons and holes. The spatial separations reduce the spatial overlap between the conduction and valence band states resulting in an overall reduction of emission intensity as shown in Eq. (4.35).
215
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(b) 3000 2000
Electron
3000 2000
HH 24 23.2 22.4 21.6 20.8 20 19.2 18.4 17.6 16.8 16
y(Å)
1000
3464 3463.2 3462.4 3461.6 3460.8 3460 3459.2 3458.4 3457.6 3456.8 3456
0
−1000 −2000 −3000 −3000 −2000 −1000
0
1000 0 −1000 −2000 −3000 −3000−2000 −1000
1000 2000 3000
x (Å)
y(Å)
(a)
0
1000 2000 3000
x (Å)
(d) 3000 2000
LH
0
y(Å)
1000
18.2 17.2 16.6 15.8 15 14.2 13.4 12.6 11.8 11 10.2
−1000 −2000 −3000 −3000 −2000 −1000
0
x (Å)
1000 2000 3000
3000 2000
CH
1000
2.2 1.36 0.52 −0.32 −1.16 −2 −2.84 −3.68 −4.52 −5.36 −6.2
0
y(Å)
(c)
−1000 −2000 −3000 −3000−2000−1000
0
1000 2000 3000
x(Å)
FIG. V.6. Spatially inhomogeneous band edge shift maps for (a) electron, (b) Heavy hole (HH), (c) Light hole (LH), and (d) Crystal-field split-off hole (CH) in units of meV due to electrostatic and deformation potential fields for edge dislocations for an example case with ndis ¼ 109 cm2 and n ¼ 2 1018 cm3. Green indicates the band edge levels for the unstrained bulk case for each plot. Valence band edge levels for unstrained bulk GaN are 20 meV for HH, 14.2 meV for LH, and 2.2 meV for CH. The conduction band edge level in unstrained bulk GaN is 3460 meV (giving a band gap of 3.44 eV relative to the HH band edge) as shown in Ref. [97]. The high energy at the dislocation core in each plot is due to electrostatic potential from the negative charge at the dislocation core. For the conduction band, blue indicates that the band edge shifts downward, so the electrons tend to be localized, while red indicates a repulsive potential. For valence subbands, red indicates that the band edge shifts upward so that holes tend to be localized, while blue indicates a repulsive potential.
b. Zinc-Blende GaAs Band edge shift maps for each subband, in the presence of edge dislocations, are calculated including the electrostatic potential and strain fields from Figure V.5. For the electrostatic potential, the B–G potential from Eq. (2.16) is pffiffiffi used, replacing a with a= 2 which is the spacing between electron acceptors for edge dislocations in GaAs as shown in Figure II.3. Band edge shift maps for
216
J. H. YOU AND H. T. JOHNSON
GaAs are shown in Figure V.7. Compared to the GaN edge dislocation case (Figure V.6), the general trends are similar except for the SO subband, where the polarity is shifted. The SO holes are attracted in the compressive region, and repelled in the tensile region, even though the magnitude of the electrostatic potential is much larger than the deformation potential for the SO subband. An important observation can be made based on a comparison of the band edge shift magnitudes with the GaN case. Figures V.6 and V.7 are plotted with same range of band edge shifts; each plot has its legend within 4 meV from the band
(a)
(b) 2000
2000
HH
1000
1000
0
0 4 3.2 2.4 1.6 −1000 0.8 0 −0.8 −1.6 −2000 −2.4 −3.2 −4
y(Å)
y(Å)
Electron
1523 1522.2 1521.4 −1000 1520.6 1519.8 1519 1518.2 −2000 1517.4 1516.6 −2000 1515.8 1515
−1000
0
1000
2000
x (Å)
(c)
−2000
−1000
0
1000
2000
1000
2000
x(Å)
(d) 2000
2000
LH
SO 1000
0 4 3.2 2.4 1.6 −1000 0.8 0 −0.8 −2000 −1.6 −2.4 −3.2 −4
0 −326 −326.8 −327.6 −328.4 −1000 −329.2 −330 −330.8 −2000 −331.6 −332.4 −333.2 −334
y(Å)
y(Å)
1000
−2000
−1000
0
x (Å)
1000
2000
−2000
−1000
0
x(Å)
FIG. V.7. Spatially inhomogeneous band edge shift maps in GaAs for (a) electron, (b) HH, (c) LH, and (d) SO in units of meV due to electrostatic and deformation potentials associated with edge dislocations for the case of ndis ¼ 109 cm2 and n ¼ 2 1018 cm3. Green indicates the band edge levels for the unstrained bulk case for each plot. Valence band edge levels for unstrained bulk GaAs are 0 meV for the HH and LH bands, and 330 meV for the SO band. The conduction band edge level in unstrained bulk GaAs is 1519 meV. Compared to band edge shifts in GaN shown Figure V.6 plotted using the same color scale, effects of strain fields through the deformation potential in GaAs are more significant than in GaN.
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
217
edge level in the unstrained case. In this view, it can be seen that effects of strain through the deformation potential induce larger shifts in GaAs than in GaN. 20. C HANGES OF D ENSITIES OF S TATES AND W AVE F UNCTIONS AN E DGE D ISLOCATION IN G A N AND IN G A A S
DUE TO
An example of the computed densities of states for the GaN case with dislocation density ndis ¼ 109 cm2 and ncarr ¼ 2 1018 cm3 is shown in Figure V.8. In general, each energy level is of mixed subband character; the sum of hcijcii over all subbands i equals one. Band edge levels for the unstrained bulk case for each subband are marked with an arrow. For electrons, there are localized states below the conduction band edge with the corresponding wave functions localized in the tensile strained region. An example of the probability density, jcij2 localized in the tensile strained region is shown in Figure V.9(a). For the valence subbands, there are more hole states within the band gap. The hole states with energy between 1.01 and 1.06 eV in Figure V.8 are the localized states at the dislocation core due to the electrostatic potential; the corresponding wave functions have monopole symmetry on the x–y plane as shown in Figure V.2(b). The states between 0.11 and 0.20 eV are also localized at the dislocation core due to the electrostatic potential, but have dipole symmetry as shown in Figure V.9(c). The states between the valence band edge maximum up to 0.049 eV consist of quadrupole symmetry states and strain-localized states. Optical emission is based on the spatial overlap between carriers. For an example, transitions from the localized electron state with energy of 3456 meV (Figure V.9(a)) to the localized hole state with energy of 1058 meV (Figure V.9(b)) produce only weak emission with transition energy of about 2.4 eV because of the small spatial overlap between wave functions. The electron is localized in the tensile strained region excluding the core of the dislocation while the hole with monopole symmetry is localized at the core. However, spatial overlap between the localized electron state (Figure V.9(a)) and the hole state (Figure V.9(d)) is relatively large, compared with the transition to the hole state with monopole symmetry. This transition produces a relatively strong emission, with transition energy at about 3.436 eV, which contributes to band edge emission. To determine the overall optical response, the spontaneous emission spectrum in Eq. (4.35) is calculated in Section V.21. All calculations for GaAs are carried out in a 2D approximation. As noted in Section II.8, the assumed orientation of the threading dislocation segments is along the z (or growth) direction. Since the electrostatic potential and the strain fields for a dislocation are uniform in this orientation, the edge dislocation can be modeled in the x–y plane. Then, wave functions along the z-direction are traveling waves and the wave vector, kz, can be any real value. For the strictly
218
J. H. YOU AND H. T. JOHNSON
1 Electron
(a)
0 3440 3450 3460 3470 3480 3490 3500 3510 3520 3530 3540 3550 1 HH
(b)
0
0
100
200
300
400
500
600
700
800
900 1000 1100
0
100
200
300
400
500
600
700
800
900 1000 1100
0
100
200
300
400
500
600
700
800
900 1000 1100
1 LH
(c)
0
1 CH
(d)
0
Energy (meV) FIG. V.8. Densities of states for GaN (a) electron and (b, c, d) hole (HH, LH, and CH) states for the system with ndis ¼ 109 cm2 and n ¼ 2 1018 cm3. Band edge levels for the unstrained bulk case for each subband are marked with an arrow. The electron states below the conduction band edge are due to localization in the tensile strained region. For the valence band, the electrostatic potential and strain field localize hole states within the band gap into three different groups: states between 1.01 and 1.06 eV are localized at the core due to the electrostatic potential and the corresponding wave functions have monopole symmetry in x–y plane; states between 0.11 and 0.20 eV are also localized at the core due to the electrostatic potential, but have dipole symmetry; states between the valence band edge and 0.049 eV consist of quadrupole symmetry states and strain-localized states. For each energy state in the valence band, the sum of (b), (c), and (d) is equal to one.
2D case, kz in the kp Hamiltonian in Eq. (4.6) becomes an input value to determine the energy along z-direction, and the spontaneous emission coefficient in Eq. (4.35) becomes
3000
(a)
2000
y(Å)
1000 0 −1000
ya
−2000
2
−3000 −3000 −2000 −1000
1000
0
2000
3000
3000
(b)
2000
y(Å)
1000 0 −1000
yb
−2000
2
−3000 −3000 −2000 −1000
1000
0
2000
3000
|⬍ya|yb ⬎|2 =1.312 × 10−9 3000
(c)
2000
y(Å)
1000 0 −1000
yc |⬍ya |yc
−2000
2
−3000 −3000 −2000 −1000
⬎|2 =1.010 × 10−5
1000
0
2000
3000
3000
(d)
2000
0
y(Å)
1000
−1000
yd |⬍ya |yd
⬎|2 = 1.081 × 10−4
−2000
2
−3000 −3000 −2000 −1000
0
1000
2000
3000
x(Å)
FIG. V.9. Examples of the GaN probability density, jcj2 for (a) the lowest electron state localized in tensile strained region with E ¼ 3456 meV, (b) the highest hole state, strongly localized at the dislocation core by the electrostatic potential, with monopole symmetry and E ¼ 1058 meV, (c) one of the hole states localized at the dislocation core with dipole symmetry with E ¼ 203 meV, and (d) one of the hole states localized between dislocations with E ¼ 20.2 meV. The squared value of spatial overlap, jhCjVij2, between the electron state (a) and hole states (b), (c), and (d) are shown. Strong localization of the hole states in (b) and (c) results in relatively small overlap with the electron state in (a).
220
J. H. YOU AND H. T. JOHNSON
gðoÞ ¼
2 C0 X X * * e p C? V? jhC? j V? ij2 fC? ðEC? Þ o k C ;V z ? ?
1 fV? ðEV? Þ d½EC? EV? ho
ð5:21Þ
where jC?i and jV?i are the conduction and valence band states on the x–y plane, perpendicular to the edge dislocation. Figure V.10 shows the GaAs densities of states for electron and hole (heavy, light, and split-off) states for dislocation density of ndis ¼ 106 cm2 and carrier concentration of n ¼ 2 1018 cm3, with the lowest band in the z-direction (kz ¼ 0). Similar to the GaN case, there are localized electron states within the band gap corresponding to wave function localization in the tensile strained area as shown in Figure V.11(a). The electrostatic potential due to negatively charged edge dislocations also leads to hole states within the band gap with monopole, dipole, and quadrupole symmetry at the dislocation cores. States between 1.1 and 1.25 eV are localized at the core due to the electrostatic potential, and the corresponding wave functions have monopole symmetry in the x–y plane as shown in Figure V.11(b). States between 0.14 and 0.24 eV are also localized at the core due to the electrostatic potential, but have dipole symmetry, and states between the valence band edge and 0.048 eV consist of quadrupole symmetry states and strain-localized states. 21. C ALCULATED S PONTANEOUS E MISSION S PECTRA FOR G A N AND G A A S
a. GaN Spectra To determine the overall optical response, the spontaneous emission spectrum in Eq. (4.35) is calculated. After obtaining the g(o)/C0 spectrum as a function of transition energy, a continuous spectrum is obtained using a narrow 6 K Gaussian thermal broadening. To compare with experimental PL intensity data,18,19 which are reported in arbitrary units, the Gaussian-broadened g(o)/C0 are normalized by the experimental data points at the lowest dislocation density. Examples of normalized g(o)/C0 for GaN are shown in Figure V.12 with different dislocation densities and carrier concentrations. The inset in each figure shows the band edge peak only. Besides band edge peaks (IBand Edge), there are three distinct peaks observed with below band edge transition energy. They correspond to the transitions to dislocation-localized hole states with (i) monopole symmetry (Im), (ii) dipole symmetry (Id), and (iii) quadrupole symmetry (Iq) due to the electrostatic potential of dislocations as shown in Figures V.8 and V.9. Transition energies to strain-confined states are so close to the band edge
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
1
Electron
(a)
221
0 1490
1500
1510
1520
1530
1540
1550
HH
(b) 1
0
0
200
400
600
800
1000
1200
0
200
400
600
800
1000
1200
LH
(c) 1
0 1
SO
(d)
0
−200
0
200
600 400 Energy (meV)
800
1000
1200
FIG. V.10. Densities of states for GaAs electron and hole (HH, LH, and SO) states for the system with ndis ¼ 106 cm2 and n ¼ 2 1018 cm3, with ground state symmetry in the z-direction (kz ¼ 0). Band edge levels for the unstrained bulk case for each subband are marked with an arrow. The electron states below the conduction band edge are due to localization in the tensile strained region. For the valence band, the electrostatic potential and strain field localize hole states within the band gap into three different groups: states between 1.1 and 1.25 eV are localized at the core due to the electrostatic potential and the corresponding wave functions have monopole symmetry in the x–y plane; states between 0.14 and 0.24 eV are also localized at the core due to the electrostatic potential, but have dipole symmetry; states between the valence band edge and 0.048 eV consist of quadrupole symmetry states and strain-localized states. For each energy state in the valence band, the sum of 20.3 (a), 20.3 (b), 20.3 (c), and 20.3 (d) is equal to one.
transition that they are not distinguishable. The magnitudes and energy levels of these below band gap transition peaks vary with free carrier concentrations and dislocation densities.
222 (a)
J. H. YOU AND H. T. JOHNSON
(b)
2000
2000 1000 y(Å)
y(Å)
1000 0
0
−1000
−1000
−2000
−2000
−2000 −1000
0 x(Å)
1000
2000
−2000 −1000
0 x(Å)
1000
2000
FIG. V.11. Examples of the GaAs probability density, jcj2 for ndis ¼ 106 cm2 and n ¼ 2 1018 cm3. (a) The lowest electron state localized in tensile strained region with E ¼ 1510 meV and (b) the highest hole state, strongly localized at dislocation core by the electrostatic potential, with monopole symmetry and E ¼ 1202 meV. Figures are zoomed in to show the localized probability clearly. Actual domain size for ndis ¼ 106 cm2 is about a thousand times larger than the domain shown here.
The emission spectra with different carrier concentrations at the same dislocation density (ndis ¼ 109 cm2) are shown in Figure V.12(a)–(c). As carrier concentration increases both IBand Edge and Im increase. However, Im increases more than IBand Edge, so the ratio (Im/IBand Edge) increases with increasing carrier concentration. Energy levels of Im, Id, and Iq also vary with carrier concentration; they blue shift as carrier concentration increases, while the energy of the band edge peak does not vary significantly. For high carrier concentrations, such as n ¼ 5 1017 cm3 (Figure V.12(b)) and 2 1018 cm3 (Figure V.12(c)), the Iq peak becomes undistinguishable. The transition to the localized monopole symmetry hole state is more sensitive to carrier concentration than transitions to dipole and quadrupole symmetry states; Im increases more and its corresponding energy blue shifts more with increasing carrier concentration. Figure V.12(d) shows the normalized g(o)/C0 for ndis ¼ 1011 cm2 with n ¼ 5 1017 cm3. Compared to the case of ndis ¼ 109 cm2 as shown in Figure V.12(b), the magnitudes of IBand Edge, Im, and Id decrease with increasing dislocation density. However, IBand Edge decreases much more significantly than Im and Id, so the ratios Im/IBand Edge and Id/IBand Edge increase with the increase of dislocation density. The energy level of the Im peak blue shifts about 0.2 eV with increasing dislocation density from ndis ¼ 109 to 1011 cm2 while the energy levels of the IBand Edge and Id peaks do not change much. As dislocation density increases, the IBand Edge peak becomes broader which is consistent with experimental evidence.27
223
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a) 0.045
Normalized g(w )/C0
0.035 0.03
0.04 0.03 0.02 0.01 0 3400
IBand Edge
IBand Edge
ndis = 109 cm −2
3500
Id ⫻5
0.01
Im ⫻ 500
Iq ⫻5
0.005
0.16
0.1
0.14
0.05
0.12
0.08
0.1
Im ⫻ 20
0.06
Id ⫻1
0.02 1800 2000 2200 2400 2600 2800 3000 3200 3400 Transition energy (meV)
IBand Edge
ncarr= 2 ⫻ 10 cm 18
0.4
IBand Edge
ndis = 109 cm −2
0.016 3
0.014
−3
0.012
0.2 0 3400
(d)
3500
Normalized g(w)/C0
Normalized g(w)/C0
0.6
3600
0.3 0.2
3600
0
(c)
0.4
3500
ncarr= 5 ⫻ 1017 cm −3
0.1
1800 2000 2200 2400 2600 2800 3000 3200 3400 Transition energy (meV)
0.5
0 3400
IBand Edge
ndis = 109 cm −2
IBand Edge
0.04
0
0.6
0.2 0.15
3600
0.02
0.2 0.18
ncarr= 1 ⫻ 1017 cm −3
0.025
0.015
(b)
Normalized g(w )/C0
0.04
Im ⫻2
0.01
⫻2
0 1800 2000 2200 2400 2600 2800 3000 3200 3400 Transition energy (meV)
IBand Edge
ncarr= 5 ⫻ 1017 cm −3
1 0 3400
ndis = 1011 cm −2
3500
Id ⫻1
3600
0.008 0.006 0.004
Id
⫻ 10−3
2
0.002
Im ⫻ 20
IBand Edge
0 1800 2000 2200 2400 2600 2800 3000 3200 3400 Transition energy (meV)
FIG. V.12. Calculated spontaneous emission spectra for different dislocation densities and free carrier concentrations. The inset in each figure shows the band edge peak only. As carrier concentration increases (a–c) the band edge peak intensity (IBand Edge) and the transition intensity to monopole symmetry hole states (Im) increase significantly. The energy levels of Im and Id increase significantly while the energy of IBand Edge does not vary. For higher carrier concentrations, Iq disappears. Note that the scales are different for each plot. As dislocation density increases (b, d) IBand Edge, Im, and Id decrease. However, IBand Edge decreases much more significantly than Im and Id, so the ratios Im/IBand Edge and Id/IBand Edge increase with increasing dislocation density. Also the band edge peak broadens as the dislocation density increases.
Within the range of dislocation densities and carrier concentrations considered in this work (ndis ¼ 106 – 1011 cm2 and n ¼ 1016 – 1019 cm3), the energy levels of Im vary from 1.4 up to 3 eV. When energy levels of Im are below 1.9 eV, the ratio of Im / IBand Edge is less than 105, which represents essentially negligible Im intensity. Therefore, the observable energy levels of Im correspond to yellow (YL), green (GL), and blue (BL) luminescence, which is consistent with the experimental observations concluding that YL, GL, and BL are from the same defects.38,39 The blue shift of the Im energy with increasing carrier concentration is also consistent with the experimentally observed transition to GL with increasing excitation power.38 The present calculation suggests
224
J. H. YOU AND H. T. JOHNSON
that this defect is the negatively charged VGa–ON edge dislocation. Specifically, YL, GL, and BL are from the transition to the dislocation-localized hole state with monopole symmetry. Also, if one associates the computed Im peak with the experimentally observed YL, the behavior of Im/IBand Edge with varying dislocation density and carrier concentration is consistent with experimental observations; the intensity ratio IYL /IBand Edge increases as dislocation density increases32,34,35,106 and increasing carrier concentration enhances relative YL intensity.106 The energy levels of Id and Iq are from 3.0 up to 3.4 eV. This energy range is consistent with the energy levels of experimentally observed states previously reported to be from donor–acceptor pairs (DA), and the enhancement of Id/IBand Edge with increasing dislocation density is consistent with experimental observations; IDA/IBand Edge increases as dislocation density increases.34 This implies that the states observed at 45–500 meV below the band edge transition energy might be attributed to the charged edge dislocation. Considering the inhomogeneous distribution of dislocations and dislocation densities, it seems plausible that Im contributes to the broad range of emission energies including YL, GL, and BL at the same time, and that experimental findings26 of spatial inhomogeneity of the band edge intensity and the YL intensity can be explained by this model. Detailed behaviors of Im, Id, and Iq as a function of dislocation density and carrier concentration are discussed in Section V.23.
b. GaAs Spectra GaAs spontaneous emission spectra calculated based on Eq. (5.21), for different dislocation densities at carrier concentration of n ¼ 2 1018 cm3 are shown in Figure V.13. Each figure shows the band edge peak; the inset shows entire spectrum over a wider energy range. Similar to the GaN cases, there are peaks from transitions to the monopole, dipole, and quadrupole symmetry localized hole states with energies below the band gap energy. As a final note on the results presented in the preceding sections, the model adopted in this chapter for both GaAs and GaN assumes that edge dislocations take on a negative charge. Thus, ground state holes are locally attracted to the dislocation cores and ground state electrons are electrostatically repelled. This simplification is consistent with the view that dislocation cores act as nonradiative recombination centers, since holes localized hear the dislocation cores are assumed to not recombine with trapped electrons along the dangling bonds.
106 D. G. Zhao, D. S. Jiang, H. Yang, J. J. Zhu, Z. S. Liu, S. M. Zhang, J. W. Liang, X. Li, X. Y. Li, and H. M. Gong, Appl. Phys. Lett. 88, 241917 (2006).
225
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a)
1 0.9
(b)
0.8
0.7 0.8
0.6
0.6
0.6
0.4
0.5
0.2
0.4
0
300
600
900
1200
Normalized g(w)/C0
Normalized g(w)/C0
0.8 0.7
0.8
1
1500
0.3
0.5
0.7 0.6 0.5 0.4 0.3
0.4
0.2 0.1
0.3
0
300
600
900
1200
1500
0.2
0.2 0.1
0.1
Normalized g(w)/C0
0.45
(d)
0.5
0.4
0.4
0.35
0.3
0.3
0.2
0.14
0.25 0.1
0.2 0.15
0
0 1500 1510 1520 1530 1540 1550 1560 1570 1580 Transition energy (meV) 0.18 0.16
300
600
900
1200
1500
Normalized g(w)/C0
(c)
0 1500 1510 1520 1530 1540 1550 1560 1570 1580 Transition energy (meV) 0.5
0.12 0.1
0.2 0.15 0.1
0.08
0.05
0.06
0
0.1
0.04
0.05
0.02
0 1500 1510 1520 1530 1540 1550 1560 1570
0.25
300
600
900
1200
1500
0 1500 1510 1520 1530 1540 1550 1560 1570
1580
Transition energy (meV)
1580
Transition energy (meV)
FIG. V.13. Calculated spontaneous emission spectra for the band edge peaks with different dislocation densities at carrier concentration of n ¼ 2 1018 cm3. (a) ndis ¼ 102 cm2, (b) ndis ¼ 104 cm2, (c) ndis ¼ 106 cm2, and (d) ndis ¼ 108 cm2. The inset in each figure shows the spectrum over a wider energy range. As dislocation density increases, the band edge peak decreases and broadens.
22. R EDUCTION IN B AND E DGE P EAK OF D ISLOCATION D ENSITY
AS A
F UNCTION
a. Comparison Between GaAs and GaN In both GaAs and GaN, as the dislocation density increases, the band edge peak decreases and broadens. To make a direct comparison, GaN is also solved in the two-dimensional approximation, as done for GaAs in Section V.21.b. Comparisons between GaAs and GaN, and comparisons with GaAs experimental data, are shown in Figure V.14 as a function of dislocation density with carrier concentration n ¼ 2 1018 cm3. As shown in Figure V.14, the band edge peak intensity in GaAs decreases significantly for dislocation density higher than ndis ¼ 103 cm2, while the GaN peak decreases significantly only for dislocation density higher than ndis ¼ 107 cm2. For example, to maintain at least 50% of the
226
J. H. YOU AND H. T. JOHNSON
PL intensity (arb. units)
100
10−1
GaN 2D:ncarr=2⫻1018 cm−3 GaAs 2D:ncarr=2⫻1018 cm−3
10−2 102
GaAs Experiment 1 (300 K) GaAs Experiment 2 (300 K) GaAs Experiment 3 (194 K) GaAs Experiment 4 (77 K) 104
106
108
1010
Dislocation density (cm−2) FIG. V.14. Band edge peak intensity comparison between GaAs and GaN for carrier concentration of n ¼ 2 1018 cm3. Experiment 1 is from Ref. [40] and experiments 2, 3, and 4 are from Ref. [41]. Band edge peak intensity in GaAs decreases as dislocation density increases for dislocation density higher than ndis ¼ 103 cm2 while intensity in GaN decreases for dislocation density higher than ndis ¼ 107 cm2.
ideal intensity, dislocation density in GaAs should be less than 106 cm2 while dislocation density in GaN can be up to ndis ¼ 108 cm2. This confirms that GaAs is more sensitive to dislocations than GaN in terms of the band edge peak emission intensity. The different sensitivity of GaAs can be explained by two reasons: (i) there is larger deformation around edge dislocations in GaAs than in GaN and (ii) effective masses are smaller in GaAs than in GaN. Edge dislocation strain fields in GaAs (Figure V.4) have larger magnitudes than those in GaN (Figure V.2). These large strain values reduce the densities of states near the band edge, resulting in a reduction of band edge emission intensity. Systems with smaller effective mass are perturbed more significantly by the presence of dislocations. An edge dislocation can be thought of as an infinite barrier for electrons due to the electrostatic potential. For simplicity, a one-dimensional array of dislocations can be considered as a one-dimensional periodic array of infinite potentials with equal spacing between potentials. Between two infinite potentials, the energy
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
227
levels and wave functions of electrons follow the well-known ‘‘particle in a box’’ behavior, where the quantized energies are inversely proportional to the effective mass and the square of the spacing between potentials. Therefore, as dislocation density increases, the spacing between dislocations decreases and the electron energy separation increases resulting in a reduction of the densities of states, and consequently, a reduction of the optical emission intensity. Small effective mass also increases the energy spacing, causing a reduction in the density of states. The electron effective mass for GaAs is 0.067 and 0.2 for GaN; thus, for a given dislocation density, the density of states near the band edge in GaAs is lower than in GaN.
b. GaN Band Edge Peak Comparison with Experimental Data The comparison of calculated GaN band edge peaks (IBand Edge) with experimental data,18,19 for different dislocation densities and carrier concentrations, is shown in Figure V.15. The calculated band edge peak intensities are normalized with the case for carrier concentration of n ¼ 5 1017 cm3. The IBand Edge peak decreases significantly for dislocation density higher than 107 cm2. As free carrier concentration increases, IBand Edge increases. The IBand Edge attenuation rates with dislocation density are very similar for carrier concentrations higher than 5 1017cm3, but IBand Edge attenuates more significantly for high dislocation density with carrier concentrations lower than 5 1017 cm3. The effect of carrier concentration on the IBand Edge attenuation for different dislocation densities is shown in Figure V.16. Data points are normalized at the carrier concentration of 1019 cm3. This result shows that even GaN, which is more dislocation tolerant than other III–V materials, becomes dislocation-sensitive if insufficient carriers are present. 23. P EAKS B ELOW B AND G AP E NERGY
IN
GAN
As briefly discussed in Section V.21, besides the band edge peaks, there are peaks observed with transition energies below the band gap energy. These peaks arise in GaN due to transitions to monopole (Im), dipole (Id), and quadrupole (Iq) symmetry hole states that are confined at the dislocation core due to the electrostatic potential associated with dangling bonds. The energy levels of the Im peak correspond to the observed levels of YL, GL, and BL and the behavior of Im/IBand Edge with varying dislocation density and carrier concentration is consistent with experimental observations.32,34,35,38,39,106 The energy levels of Id and Iq are consistent with the energy levels of experimentally observed peaks believed to be from donor–acceptor pairs and the enhancement of Id/IBand Edge
228
J. H. YOU AND H. T. JOHNSON
101
PL intensity (arb. units)
100
10−1
10−2
10−3
10−4
10−5
10−6 106
Hacke et. al. 1999 Iwaya et. al. 2001 n=1⫻1019 cm−3 n=2⫻1018 cm−3 n=5⫻1017 cm−3 n=1⫻1017 cm−3 n=1⫻1016 cm−3 107
108 109 Dislocation density (cm−2)
1010
1011
FIG. V.15. Comparison of calculated band edge peak intensities with experimental data for different dislocation densities and different carrier concentrations. The experimental data are taken from Refs. [18,19]. The IBand Edge decreases significantly as dislocation density increases for ndis > 107cm2. As free carrier concentration increases, IBand Edge increases. The IBand Edge peak attenuates most severely for high dislocation density and low carrier concentration.
with increasing dislocation density is consistent with experimental observations of IDA/IBand Edge.34 In this section, the dependence of Im, Id, and Iq on dislocation density and carrier concentration is discussed in detail. Figure V.17 shows examples of electron and hole energy levels as a function of dislocation density with carrier concentration of 2 1018 cm3. The figure shows 6408 states for the conduction band and 38,448 states for the valence band. Within the valence band, the calculation finds 144 monopole states, 384 dipole states, and 294 quadrupole states and strain localized states within the band gap. The other valence states are found below the valence band maximum. For dislocation density less than 109 cm2, the states in each subband are energetically very close together. As the dislocation density increases, the energy of states diverges widely due to the perturbations of the dislocations. For lower carrier concentrations, there is a larger perturbation by the dislocations because the electrostatic potential is larger due to reduced screening.
229
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
100
PL intenstiy (arb. units)
10−1
10−2
10−3 ndis = 106 cm−2 ndis = 107 cm−2 ndis = 108 cm−2 ndis = 109 cm−2 ndis = 1010 cm−2 ndis = 1011cm−2
10−4
10−5 1016
1017 1018 Free carrier concentration (cm−3)
1019
FIG. V.16. Effect of carrier concentration on the IBand Edge attenuation at different dislocation densities. Each case with different dislocation densities is normalized with the case of n ¼ 1019 cm3. For low carrier concentrations, the band edge peak intensity becomes most sensitive to dislocation density.
The valence band monopole state energies are distinguishable from the other symmetry states for the full range of dislocation densities, from 106 to 1011 cm2. The dipole states remain separated from other symmetry states for dislocation densities less than or equal to 109 cm2. However, for dislocation densities higher than 109 cm2, the dipole states become energetically indistinguishable due to mixing with the quadrupole states and strain-localized states. Energy levels of the quadrupole states and strain-localized states are close enough to the valence band maximum as to be indistinguishable from the band edge peak, as shown in Figure V.12. Figure V.18 shows magnitudes and energy levels of the emission intensity associated with monopole states (Im) as a function of dislocation density at various carrier concentrations. As dislocation density increases, Im and Im/IBand Edge both increase for the carrier concentration of n ¼ 1019 cm3. However, for carrier concentrations less than n ¼ 1019 cm3, there is a transition point where both Im and Im/IBand Edge decrease. For example, at the carrier
230
J. H. YOU AND H. T. JOHNSON
5 4.5 4
Conduction band
3.5
Energy (eV)
3 2.5 Band gap 2 1.5
Monopole
1 0.5
Dipole
0 −0.5 106
Quadrupole and valence band 107
108 109 Dislocation density (cm−2)
1010
1011
FIG. V.17. Example of energy levels of states as a function of dislocation density with carrier concentration of 2 1018 cm3. The number of states is 6320 states for the conduction band and 38,448 states for the valence band. Within the valence band, there are 144 monopole states, 384 dipole states, 294 quadrupole states within the band gap. The rest of valence states are below the valence band maximum. As dislocation density increases, the densities of states decrease. For dislocation density higher than 109 cm2, the dipole states can not be distinguished from quadrupole states.
concentrations of n ¼ 2 1018 and 5 1017 cm3, both Im and Im/IBand Edge decrease at ndis ¼ 1010 cm2 and increase again at ndis ¼ 1011 cm2. This transition dislocation density decreases as the carrier concentration decreases. At the carrier concentrations of n ¼ 1017 and 1016 cm3, Im decreases at ndis ¼ 109 cm2. This trend can be explained by comparing the overlap of the electrostatic and deformation potentials between adjacent dislocations and the region where electrons are localized. Figure V.19 shows the band edge shifts in the tensile strained region between dislocations (between points A and B in Figure V.19(a)) for different dislocation densities with n ¼ 5 1017 cm3. The dashed line in Figure V.19(b)–(f) represents the conduction band edge minimum in the unstrained case as a reference level, which is 3460 meV. A dipole of dislocations is located at points A and B in each case. For dislocation density of ndis ¼ 107 cm2 (Figure V.19(b)), the distance between dislocations is large enough that the band edge at the midpoint between dislocations is approximately
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
Im (arb. units)
100
(b) 104
Edge n=1⫻1019 cm−3 Edge n=2⫻1018 cm−3 Edge n=5⫻1017 cm−3 Edge n=1⫻1017 cm−3 Edge n=1⫻1016 cm−3
102 Im/IBand Edge (arb. units)
(a) 105
100
231
Edge n=1⫻1019 cm−3 Edge n=2⫻1018 cm−3 Edge n=5⫻1017 cm−3 Edge n=1⫻1017 cm−3 Edge n=1⫻1016 cm−3
10−2
10−5
10−4 10−6
10−10 106
Transition energy Im of (meV)
(c) 105
100
107
1010 108 109 Dislocation density (cm−2)
1011
10−8 106
107
1010 108 109 Dislocation density (cm−2)
1011
Edge n=1⫻1019 cm−3 Edge n=2⫻1018 cm−3 Edge n=5⫻1017 cm−3 Edge n=5⫻1017 cm−3 Edge n=1⫻1016 cm−3
10−5
10−10 106
107
1010 108 109 Dislocation density (cm−2)
1011
FIG. V.18. (a) Im, (b) Im/IBand Edge, and (c) transition energy of Im as a function of dislocation density at different carrier concentrations. There is a transition point of dislocation density where both Im and Im/IBand Edge decrease. This transition dislocation density point decreases as the carrier concentration decreases. While the Im and Im/IBand Edge highly depend on dislocation density and carrier concentration, the energy levels of Im depends more on carrier concentrations than dislocation densities.
3460 meV and the electrons are localized in valleys near the dislocation cores. As dislocation density increases, the spacing between dislocations decreases and the deformation potential, which is relatively long-ranged compared to the electrostatic potential, starts to overlap between dislocations resulting in the band edge shifting below the conduction band minimum. This causes more electron states to be localized within the tensile region. For dislocation densities up to ndis ¼ 109 cm2, the tensile strain fields generate locally sharp valleys due to the deformation potentials near the cores and electrons are localized in the valleys. Then, as dislocation density increases up to ndis ¼ 109 cm2, the reduced band edge increases the number of electron states below the conduction band minimum resulting in an increase of Im. However, at dislocation density ndis ¼ 1010 cm2, the spacing between dislocations is comparable to the effective distance of the strain fields so that the potential valleys near the core smooth out and the potential
232
J. H. YOU AND H. T. JOHNSON
(b) 3468
(a)
3000 3466 2000
y(Å)
A
0
−1000
Energy (meV)
3464 1000
3462 3460 3458
−2000 3456
B −3000 −2000 −1000
0
−3000 3454
1000 2000 3000
x(Å)
(d)
3468
3464
Energy (meV)
3466
3464 3462 3460
3458 3456 −9000 −8000 −7000 −6000 −5000 −4000 −3000 −2000 −1000
Energy (meV)
(e)
3454
0
A
3468
(f)
3466 3464
3458
−1000 −900 −800 −700 −600 −500 −400 −300 −200 −100
B
0
x104 A
−3000
−2500
−2000
−1500
−1000
−500
0
A
3462 3460 3458
3456 3454
−0.5
3468
3464
3460
−1
B
3466
3462
−1.5
3460
3456
B
−2
3462
3458
3454
−2.5
3468
3466
Energy (meV)
Energy (meV)
(c)
−3
B
3456
0
A
3454
−300
B
−250
−200
−150
−100
−50
0
A
FIG. V.19. Band edge shifts in the tensile strained region between dislocations (between points A and B) with n ¼ 5 1017 cm3 for different dislocation densities (b) ndis ¼ 107 cm2, (c) ndis ¼ 108 cm2, (d) ndis ¼ 109 cm2, (e) ndis ¼ 1010 cm2, and (f) ndis ¼ 1011 cm2. The dashed line in (b), (c), (d), (e), and (f), represents the conduction band edge minimum in the unstrained case as a reference level, which is 3460 meV. Dislocations are located at points A and B in each case.
approximates a square well. In this case, electrons are not localized near the core, but rather in the region between dislocations, resulting in decreasing spatial overlap with localized hole states at the core and, thus, reduced Im. For dislocation density
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
⫻10−3
(b) 1.8
1.4
1.6
1.2
1.4
1 0.8 0.6 0.4 0.2 0 3000 3100 3200 3300 3400 3500 3600 3700 3800 Transition energy (meV)
Normalized g(w)/C0
Normalized g(w)/C0
(a) 1.6
233
⫻10−4
1.2 1 0.8 0.6 0.4 0.2 0 3000 3100 3200 3300 3400 3500 3600 3700 3800 Transition energy (meV)
FIG. V.20. Emission spectra for (a) ndis ¼ 109 cm2 and (b) ndis ¼ 1010 cm2 with n ¼ 1016 cm3. Im, Id, and Iq are indistinguishable from the band edge peak.
of ndis ¼ 1011 cm2, electrons are also localized between dislocations. However, the confining region is highly limited by the spacing between electrostatic potentials. Therefore, transition intensities involving monopole symmetry hole states increase again. The dislocation density of ndis ¼ 1010 cm2 represents a transition case from strain-localization near the core to localization between core electrostatic potentials. For smaller carrier concentrations, the electrostatic potential is larger due to reduced screening so that the transition occurs at a lower dislocation density. Energy levels of Im are shown in Figure V.18(c). While Im and the ratio Im/IBand Edge depend on both dislocation density and carrier concentration, the energy levels of Im depend more on carrier concentration than dislocation density. For the cases with n ¼ 1016 cm3 and dislocation densities higher than 108 cm2, Im becomes indistinguishable from Id, Iq, or IBand Edge so that separate data points are not presented for Im. Examples of these cases are shown in Figure V.20. Magnitudes and energy levels of dipole states are shown in Figure V.21. Generally, the ratio Id/IBand Edge increases with increasing dislocation density. Effects of carrier concentration on Id/IBand Edge are clear only for dislocation densities less than ndis ¼ 108 cm2; as carrier concentration increases, the ratio Id/IBand Edge decreases. This trend can be compared with the experimentally observed decrease of DA peaks at 3.28 eV for Mg-doped GaN.37 Undoped GaN shows n-type behavior with carrier concentration about 1017 cm3. In the Mg-doped case, the carrier concentration decreases to 1016 cm3 and the ratio IDA/IBand Edge increases. This is consistent with the prediction in Figure V.21(b); as carrier concentration decreases, the ratio Id/IBand Edge increases. However, detailed trends for Id and the transition energy associated with Id are not clear,
234
J. H. YOU AND H. T. JOHNSON
101 10
Id (arb.units)
10
0
−1
(b)
Edge n = 1 ⫻ 1019 cm−3 Edge n = 2 ⫻ 1018 cm−3 Edge n = 5 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1016 cm−3
10
Id /IBand Edge (arb.units)
(a)
10−2
10 10
−4
0
10
10
10−3
Edge n = 1 ⫻ 1019 cm−3 Edge n = 2 ⫻ 1018 cm−3 Edge n = 5 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1016 cm−3
2
10
−2
−4
−5
10
10−6 10−7 6 10
107
108
109
Dislocation density (cm−2)
1010
1011
Transition energy of Id (meV)
(c) 3500
10
−6
−8
10
6
7
10
8
10
10
9
Dislocation density (cm−2)
10
10
11
10
3400
3300
3200
3100 Edge n = 1 ⫻ 1019 cm−3 Edge n = 2 ⫻ 1018 cm−3 Edge n = 5 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1016 cm−3
3000
2900 6 10
10
7
10
8
10
9
10
10
10
11
Dislocation density (cm−2)
FIG. V.21. (a) Id, (b) Id/IBand Edge, and (c) transition energy of Id as a function of dislocation density at different carrier concentrations. Generally, the ratio Id/IBand Edge increases with increasing dislocation density. However, detailed trends for Id are not obvious. Effects of carrier concentration on Id/IBand Edge are clear only for dislocation densities less than ndis ¼ 109cm2; as carrier concentration increases the ratio Id/IBand Edge decreases.
because for high dislocation densities the density of states is severely perturbed, resulting in overlapping with Iq as illustrated in Figure V.17. Magnitudes and energy levels of quadrupole states are shown in Figure V.22. For dislocation densities higher than ndis ¼ 109 cm2, Iq becomes indistinguishable as shown in Figure V.12. Only the cases where Iq is energetically distinguishable from other peaks are presented. Generally, Iq and Iq/IBand Edge increase with increasing dislocation density and carrier concentration. The transition energy associated with Iq increases as carrier concentration increases, but it is not significantly affected by dislocation density. 24. S UMMARY A multiband kp Hamiltonian is solved using the finite element method in a real space formulation to study the effect of threading edge dislocation density on the
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a)
(b)
10−1 10−2
n = 1 ⫻ 1019 n = 2 ⫻ 1018
10−2
cm−3 cm−3
Edge Edge Edge n = 5 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1016 cm−3
Iq /IBand Edge (arb.units)
100
Iq (arb.units)
235
10−3 10−4 10−5 10−6
10−3
Edge n = 1 ⫻ 1019 cm−3 Edge n = 2 ⫻ 1018 cm−3 Edge n = 5 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1016 cm−3
10−4
10−6
−7
10
10−8 6 10
(c) Transition energy of Iq (me.V)
3460 3440
107 Dislocation density (cm−2)
10
8
108 106
107 Dislocation density (cm−2)
108
Edge n = 1 ⫻ 10−3 cm−3 Edge n = 2 ⫻ 1018 cm−3 Edge n = 5 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1017 cm−3 Edge n = 1 ⫻ 1016 cm−3
3420 3400 3380 3360 3340 106
107 Dislocation density (cm−2)
108
FIG. V.22. (a) Iq, (b) Iq/IBand Edge, and (c) the transition energy associated with Iq as a function of dislocation density at different carrier concentrations. Iq and Iq/IBand Edge increase with increasing dislocation density and carrier concentration. The transition energy associated with Iq increases as carrier concentration increases, while dislocation density does not affect the transition energy significantly.
optical properties in wurtzite GaN and zinc-blende GaAs. The model is used to interpret numerous recent experimental findings. The electrostatic potential at dislocation cores localizes holes generating states within the band gap. Transition energy levels and intensities vary with dislocation density and free carrier concentration. Transition energy levels to the primary hole states (Im) in the band gap correspond to YL, GL, and BL which are experimentally found to be from the same extended defects;38,39 the behavior and functional dependence of the associated emission intensity ratio, Im/IBand Edge, is also consistent with experimental findings for the IYL/IBand Edge ratio.32,34,35,106 Transition energy levels to other hole states (Id and Iq) in the band gap are in the same range as donor–acceptor pairs, or ‘‘DA’’ states, and the behavior of this emission intensity is consistent with the IDA observed experimentally.34 As dislocation density increases, the band edge peak intensity decreases for dislocation densities higher
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than ndis ¼ 107 cm2 and the band edge peak broadens to an extent consistent with experimental observations.18,19,27 As carrier concentration increases, the band edge peak intensity increases. The band edge peak attenuates severely with high dislocation densities and low carrier concentrations. The model implies that GaN becomes dislocation-sensitive if insufficient carriers are present. There is a transition dislocation density above which Im decreases. This transition point is associated with changing of the electron-localization mechanism from strain-localization near the core to localization between electrostatic potentials associated with core dangling bonds. For lower carrier concentrations, the electrostatic potential is larger due to reduced screening so that the transition occurs at lower dislocation density. Generally, the ratios Id/IBand Edge and Iq/IBand Edge increase as dislocation density increases. However, due to overlap with other peaks, detailed trends are not clearly observable. Energy levels of the Im, Id, and Iq transitions depend more on carrier concentration than on dislocation density. VI. Effect of Open-Core Screw Dislocations on Optical Properties in GaN
In this section, effects of open-core screw dislocations on optical properties are calculated for the specific case of GaN, where these defects are widely observed experimentally. The calculations are made using the kp Hamiltonian method presented in Section IV. The inhomogeneous shear strain fields and the hexagonal hollow core of the screw dislocation induce inhomogeneous band edge shifts in a different way than in the edge dislocation case. The inhomogeneous strain fields and band edge shifts are shown in Sections VI.25 and VI.26. Changes in densities of states and examples of wave functions are shown in Section VI.27. Spontaneous emission spectra are calculated and compared with experimental results in Section VI.28. 25. S TRAIN F IELD A SSOCIATED
WITH
S CREW D ISLOCATIONS
The real space three-dimensional finite element computational domain for the open-core screw dislocation case is shown in Figure VI.1. Each domain contains a screw dislocation dipole, one at the center of the domain and the other at the corners, with opposite Burgers vector direction. At the top and bottom boundaries along the z-direction, a termination boundary condition (or zero-valued Dirichlet condition) is applied so that the system acts as a wide quantum well in the z-direction. Periodic boundary conditions are applied in the lateral boundaries along the x–y plane. The crystal orientation h0001i is aligned with the z-direction. The termination boundary condition is also applied to the inner surfaces of the
237
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
(a)
(b) 2000 1000
z (Å) 1000
0
y (Å)
0 −1000
2000
2000 1000
y (Å)
0 −1000 −2000
1000 0 −1000 x (Å) −2000
−2000 −2000 −1000
0
1000 2000
x (Å)
FIG. VI.1. Example of the 3D finite element computational domain, showing the finite element ˚ . The domain mesh for the open-core screw dislocation case with ndis ¼ 109 cm2 and a radius of 300 A contains a dipole of open-core screw dislocations with opposite Burgers vector direction. (a) Perspec˚ and the crystal tive view and (b) plan view. The thickness of the sample along the z-direction is 1000 A orientation h0001i is aligned with the z-direction.
hollow open-core. Hexagonal core radius, R, is defined by equating the hexagonal area with a circular area of radius R. To study the dislocation density effect, the domain area A is changed according to the desired dislocation density using A ¼ 2/ndis, where ndis is the areal dislocation density and the factor of 2 accounts for the presence of the dislocation dipole in the unit cell. Strain components for the hexagonal open-core screw dislocation have been ˚ along h0001i as shown in calculated using a Burgers vector of c ¼ 5.185 A Figure VI.2. For open-core screw dislocations, only exz ¼ ezx and eyz ¼ ezy are non-zero but all other strain components are zero. Like edge dislocations in the previous section, the strain fields are spatially inhomogeneous, resulting in inhomogeneous band edge shifts. Band edge shifts are discussed in Section VI.26. 26. I NHOMOGENEOUS B AND E DGE S HIFTS TO S CREW D ISLOCATION
DUE
Band edge shift maps for each subband are calculated and shown in Figure VI.3. Again, the valence band edge levels in unstrained bulk WZ GaN are 20 meV for heavy holes (HH), 14.2 meV for light holes (LH), and 2.2 meV for crystal-field split-off holes (CH). The conduction band edge energy level in unstrained bulk WZ GaN is 3460 meV (EC ¼ Egap þ 20 meV), where the bulk band gap is 3.44 eV. Since the conduction band has deformation potential only from hydrostatic strain according to Eq. (4.4), shear strain fields near the screw dislocations
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(a)
(b)
1000
1000
0
−2000 −1000
0
1000 2000
0
y (Å)
2000
y (Å)
2000
−1000
−1000
−2000
−2000
−2000 −1000
x (Å)
0
0.0021 0.00168 0.00126 0.00084 0.00042 0 −0.00042 −0.00084 −0.00126 −0.00168 −0.0021
1000 2000
x (Å)
FIG. VI.2. Nonzero strain components around the hexagonal open-core screw dislocations with ˚ . (a) exz ¼ ezx and (b) eyz ¼ ezy. All other strain components are zero. ndis ¼ 109 cm2 and radius of 300 A
do not affect electron states in the conduction band. For the valence subbands, strain fields near the screw dislocations lead to localized states for the HH band, so there are some strain-confined HH states within the band gap with wave functions locally confined near the screw dislocations. The same localizing tendency is present for the LH band. However, the crystal-field split-off, or CH band is repelled by the deformation potential near the screw dislocation core so that no strain-confined states exist there. 27. C HANGES OF D ENSITIES OF S TATES DUE TO S CREW D ISLOCATIONS
FOR
E LECTRONS
AND
H OLES
An example of the densities of states for the electron, HH, LH, and CH bands in the presence of screw dislocations is shown in Figure VI.4. This example is for an open-core screw dislocation density of ndis ¼109 cm2 with radius ˚ . Each computed energy level is, in general, of mixed subband r ¼ 300 A character; the relative nature of each state is given by hci j cii so that the sum over subbands i is equal to one, where ci is the wave function for each subband. The band edge for unstrained WZ GaN is marked for reference. There are no strain-confined electron states, but the absence of material at the core increases the energy levels. For the HH and LH subbands, there are confined states due to the strain field around the screw dislocations. For the CH subband, the repulsive deformation potential and the absence of material reduces the energy levels. Thus, the absence of material in the open-cores and
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
239
(b)
(a)
2000
2000
−1000 3464 3460 3456
−2000 −2000 −1000
0
1000
24 23.2 22.4 21.6 20.8 20 19.2 18.4 17.6 16.8 16
0 −1000 −2000 −2000 −1000
1000 2000
y (Å)
0
y (Å)
1000
0
1000 2000
x (Å)
x (Å) (c)
(d) 2000
2000
−1000 −2000 −2000 −1000
0
x (Å)
1000 2000
1.8 1 0.2 −0.6 −1.4 −2.2 −3 −3.8 −4.6 −5.4 −6.2
1000 0
y (Å)
0
y (Å)
1000
16.2 15.8 15.4 15 14.6 14.2 13.8 13.4 13 12.6 12.2
−1000 −2000 −2000 −1000
0
1000 2000
x (Å)
FIG. VI.3. Spatially inhomogeneous band edge shifts for (a) electron, (b) HH, (c) LH, and (d) CH in ˚ . Green units of meV due to open-core screw dislocations with ndis ¼ 109 cm2 and radius of 300 A indicates the band edge for the unstrained bulk case for each plot. Valence band edge levels for unstrained bulk GaN are 20 meV for HH, 14.2 meV for LH, and 2.2 meV for CH. The conduction band edge level is 3460 meV, corresponding to a band gap of 3.44 eV. For valence subbands, red indicates band edge shifts toward the band gap, so that holes tend to be locally confined, while blue indicates a repulsive potential. For the conduction band, shear strain near the screw dislocation does not affect the conduction band edge.
the dislocation strain field both change the density of states for each subband. The small peaks in the LH and CH subbands above the band edge are due to band mixing caused by the dislocations and diminish with decreasing dislocation density. The probability densities, jcj2, for the first electron state and the first HH state are shown in Figure VI.5(a) and (b), respectively. The first electron wave function is not localized by the deformation potential, but the absence of material forces the wave function away from the dislocation core. By contrast, the first HH wave function is localized near the dislocation core by the local strain field, as suggested in Figure VI.3. This inhomogeneity in
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(a) Electron
1
0 3459
(b)
3460
3462
3461
3463
3464
3465
HH
1
0 −10
(c)
−5
0
5
10
15
20
−5
0
5
10
15
20
25
−5
0
5
10
15
20
25
25
LH
1
0 −10
(d)
CH
1
0 −10
Energy (meV)
FIG. VI.4. (a) Electron (EL) and (b, c, d) hole (HH, LH, CH) states for the system with ndis ¼ 109 ˚ . HH and LH carriers are confined due to the strain field around the cm2 and radius of 300 A dislocations while EL and CH carriers are not confined. Band edge levels for the unstrained bulk case for each subband are marked with an arrow. The band gap is taken as 3.44 eV. For each valence band energy state, the sum of (b), (c), and (d) is equal to one. The small peaks in the LH and CH subbands above the band edge are due to band mixing caused by the dislocations, and diminish with decreasing dislocation density.
the spatial distribution of carriers implies that the open-core screw dislocations reduce the optical response, as suggested in Ref. [107]. The spatial overlap between the first electron state (Figure VI.5(a)) and the first HH states (Figure VI.5(b)), which partially determines the transition from the first electron state to the first heavy-hole state, is almost zero. Therefore, the transition between these states does not contribute to the optical response.
107
S. J. Rosner, E. C. Carr, M. J. Ludowise, G. Girolami, and H. I. Erikson, Appl. Phys. Lett. 70, 420 (1997).
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
241
(a)
z(Å) 1000 0 2000
2000 1000
1000 y(Å)
0 −1000 −2000
0 −1000 −2000
x(Å)
E = 3460.2 meV
(b)
z(Å) 1000 0 2000
2000 1000 y(Å)
1000 0 −1000 −2000
0 −1000 −2000
x(Å)
E = 22.6 meV FIG. VI.5. (a) Probability density, jcj2 for the first electron state and (b) probability density for the first HH state. The HH state is localized due to the strain field near the dislocation cores while the electron state is not. The transition between the first electron and first HH levels does not contribute to the optical response of the sample.
28. C ALCULATED S PONTANEOUS E MISSION S PECTRA WITH E XPERIMENTAL D ATA
AND
C OMPARISON
To determine the overall optical response, the spontaneous emission spectrum, given by Eq. (4.35), is calculated for different dislocation densities and different radii of open-core screw dislocations. After obtaining the g(o)/C0 spectrum as a function of transition energy, the spectrum is broadened with a narrow Gaussian
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(a)
(b)
1000 ndis = 106 cm−2
Normalized g(w)/C0
800
8
−2
ndis = 10 cm
ndis = 109 cm−2
600
400
103 PL intensity (arb. units)
ndis = 107 cm−2
102
Hino et. al. (2000) Screw r = 50A (Calculation) Screw r = 300A (Calculation) Screw r = 500A (Calculation)
200
0 3435
3440
3445
3450
3455
Transition energy (meV)
3460
3465
101 106
107
108
109
Dislocation density (cm−2)
˚ at FIG. VI.6. (a) Examples of normalized g(o)/C0 for systems with screw dislocation radius of 300 A different dislocation densities and (b) comparison with experimental data from Ref. [16]. The PL intensity values in (b) are obtained from the peak values in (a). The discrepancy at high dislocation densities may be due to the fact that the experimental data includes both screw and mixed dislocations while only screw dislocations are considered in this calculation. Secondary peaks at higher energies in g(o)/C0 shown in (a) are due a quantization effect in the growth direction (z-direction in Figure VI.1(a)) of the epilayer and the assumed perfectly terminating boundary conditions (ci ¼ 0) at the top and bottom surfaces.
distribution corresponding to a 6 K thermal broadening. The electron free carrier concentration is assumed to be 2 1018 cm3. Figure VI.6(a) shows the 6 K Gaussian broadened g(o)/C0 spectrum for screw ˚ at four different dislocation densities as an dislocations with radii of 300 A example. The peak point for each case is marked with a circle. As dislocation density increases, the peaks decrease and the peak transition energy blue-shifts. The strain and absence of material in the core alter the densities of states, resulting in diminished PL intensity and broadened PL spectrum, which is similar to the effect imposed by edge dislocations. The secondary peaks at higher energies are due to a quantization effect in the growth direction (z-direction in Figure VI.1(a)) of the epilayer caused by the assumed perfectly terminating boundary conditions (ci ¼ 0) at the top and bottom surfaces; the thickness is ˚ which is relatively small compared to experimental cases. assumed to be 1000 A Figure VI.6b shows the calculated band edge peak intensity compared with experimental data16 where the measured dislocation density includes both screw and mixed dislocations. Calculated results and experimental data both show significant reductions in optical response for the cases with dislocation densities higher than 107 cm2. The discrepancy at high dislocation densities may be due to the fact that the experimental data include both screw and mixed dislocations while only screw dislocations are considered in this work. The mixed dislocations contain edge type components that are partially negatively charged.
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TABLE VI.1. NORMALIZED VALUES OF g(o)/C0 AT DIFFERENT DISLOCATION DENSITIES AND RADII OF OPEN-CORE SCREW DISLOCATIONS ndis ¼ 106cm2
ndis ¼ 107cm2
ndis ¼ 108cm2
ndis ¼ 109cm2
850.00 888.35 890.17
754.96 779.49 787.79
436.18 456.60 468.11
122.22 114.85 98.99
˚ r ¼ 50 A ˚ r ¼ 300 A ˚ r ¼ 500 A
The dominant mechanisms for reducing the optical response change for different radii and different dislocation densities as tabulated in Table VI.1. For ndis 108 cm2, the optical response is stronger in the presence of larger radius dislocations, while for ndis > 108 cm2, the optical response is stronger in the presence of smaller radius dislocations. The transition point in PL reduction mechanisms, occurring around ndis 108 cm2, can be explained by two competing factors: the strain field around the core and the absence of material inside the core. Both disturb the densities of states from the bulk perfect crystal case, so that the PL spectrum decreases and broadens. Smaller radius screw dislocations have strain of larger magnitude, but more material inside the core, while the larger radius screw dislocations have reduced strain magnitude but less material inside the core. For dislocation densities below the transition point, the effect of the material absence is relatively small compared to strain field effect. However, for dislocation densities above the transition point, the effect of missing material becomes the dominant mechanism for reducing PL intensity.
29. S UMMARY A real-space finite element kp Hamiltonian method is used to study the effect of screw dislocation density on the optical response in wurtzite GaN. It is found that the shear strain field around screw dislocations localizes the heavy- and light-hole states. By comparison, electron and split-off holes are repelled from the screw dislocation cores. This inhomogeneity in the spatial distribution of carriers reduces the recombination efficiency of the sample, and thus the PL intensity, and broadens the PL spectrum. Two competing factors, the absence of material at the core and the strain field associated with the dislocation, dominate at different dislocation densities with a transition density around ndis 108 cm2. The calculated optical response matches with experimental PL intensity measurements and both indicate a significant decrease for screw dislocation densities of ndis >107 cm2.
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VII. Conclusions: Understanding Dislocation Effects Across the III–V Materials
In this section, effects of dislocations in other III–V materials are briefly reviewed and dislocation sensitivity is predicted based on two key factors identified for GaAs and GaN: the effective mass and the magnitude of the Burgers vector, as shown in Section V.22. 30. E FFECT
OF
D ISLOCATIONS
IN
O THER M ATERIALS
There has been considerable experimental work leading to insight on dislocation effects in other III–V and III–Nitrides materials. For example, the light-emitting efficiency in GaP is also known to be more sensitive to dislocation density than that of GaN, but less so than that of GaAs.54,108,109 Other wurtzite binary and tertiary III–Nitrides are found to have high internal quantum efficiency. InGaN has very high dislocation densities, like GaN and its efficiency is high.110,111 There have been several suggested explanations for the unusually high optical efficiency of wurtzite III–Nitride materials. Wurtzite III–Nitrides have polar structure and internal electric fields are generated from this polarization. Bernardini et al.112 show that wurtzite III–Nitrides, such as AlN, GaN, and InN, have larger piezoelectric constants in the polar direction by an order of magnitude than zincblende III–V materials. Mendez et al.113 report an increase of PL intensity with increasing external field applied perpendicular to GaAs quantum well layers. However, Gallart et al.114 conclude that the polar-axis electric field is not enough to explain the high efficiency, but that a reduction of dislocation density and the in-plane carrier mobility does improve the light-emission efficiency. Many of the explanations of dislocation effects in these materials are based on nonuniform potential effects in the material. Chichibu et al.111 argue that the high efficiency in InGaN is because the indium-rich regions capture holes due to
108 W. A. Brantley, O. G. Lorimor, P. D. Dapkus, S. E. Haszko, and R. H. Saul, J. Appl. Phys. 46, 2629 (1975). 109 T. Suzuki and Y. Masumoto, Appl. Phys. Lett. 26, 431 (1975). 110 J. Abell and T. D. Moustakas, Appl. Phys. Lett. 92, 091901 (2008). 111 S. F. Chichibu, A. Uedono, T. Onuma, B. A. Haskell, A. Vhakraborty, T. Koyama, P. T. Fini, S. Keller, S. P. Denbaars, J. S. Speck, et al., Nat. Mater. 5, 810 (2006). 112 F. Bernardini, V. Fiorentini, and D. Vanderbilt, Phys. Rev. B 56, R10024 (1997). 113 E. E. Mendez, G. Bastard, L. L. Chang, L. Esaki, H. Morkoc, and R. Fischer, Phys. Rev. B 26, 7101 (1982). 114 M. Gallart, P. Lefebvre, A. Morel, T. Taliercio, B. Gil, J. Alle`gre, H. Mathieu, B. Damilano, N. Grandjean, and J. Massies, Phys. Stat. Sol. (a) 183, 61 (2001).
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES TABLE VII.1. OTHER III–V
Decreasing dislocation sensitivity
InSb (ZB) InAs (ZB) GaSb (ZB) GaAs (ZB) Ga0.95AI0.05As InP (ZB) GaP (ZB) InN (WZ) GaN (WZ) BN (WZ) A1N (WZ)
245
SEMICONDUCTORS
Type
Effective mass
Direct Direct Direct Direct Direct Direct Indirect Direct Direct Direct Direct
0.0135 0.026 0.039 0.0670 0.06715 0.0795 0.13 0.12 0.2 0.24 0.28
Lattice ˚) constant (A
6.4794 6.0583 6.0959 5.653 5.6537 5.8697 5.4505 3.545 3.189 2.55 3.112
localized valence states and that the holes generate localized excitons to emit light. On the contrary, Cherns and Jiao22 and Cai and Ponce115 suggest that negatively charged dislocations prevent electrons from migrating to the dislocations. Another model has been suggested by Hangleiter et al.116 whereby the high emission efficiency is attributed to self-screening of defects rather than random localization due to chemical inhomogeneity, and that the potential barrier around defects prevents carriers from nonradiatively recombining. In short, physical explanations for high light-emitting efficiency in III–nitrides are still widely discussed and debated today. The simple model presented here based on the electrostatic and deformation potential of dislocations shows that the effective mass and the magnitude of the Burgers vector can be among the key factors that determine the PL sensitivity to dislocations. Based on these factors the PL sensitivity to dislocation density can be qualitatively estimated for other III–V binary semiconductors. Table VII.1 shows values for other III–V materials, where the materials are listed in order of expected increasing dislocation sensitivity; PL in InSb is expected to be most sensitive to dislocation density, while PL in AlN is expected to be most robust to deleterious effects of dislocation density. Typically materials with zinc-blende structure have
115
J. Cai and F. A. Ponce, Phys. Status Silidi A 192, 407 (2002). A. Hangleiter, F. Hitzel, C. Netzel, D. Fuhrmann, U. Rossow, G. Ade, and P. Hinze, Phys. Rev. Lett. 95, 127402 (2005).
116
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effective masses and lattice constants similar to those of GaAs. Indeed, Roedel et al.117 experimentally find that PL in Ga0.95Al0.05As is slightly more robust than in GaAs to dislocation density. This can be explained by a slightly heavier effective mass and a larger lattice constant. It can be expected, then, that InAs, InP, GaSb, and InSb should have similar PL sensitivity to dislocation density as GaAs. For wurtzite structures, AlN would be expected to be more robust than GaN because it has a heavier effective mass than that of GaN, while InN and BN should be more sensitive than GaN based on this simple picture. GaP has a lighter effective mass than GaN but a larger lattice constant. While the model presented here may allow for some qualitative material comparisons, in real systems, piezoelectric effects, carrier dynamics, and chemical interactions also contribute to light emission, so the prospects for highly accurate predictions quantitative comparisons based on only electrostatic and deformation potential effects are limited. Nevertheless, the results presented in the previous sections show that this approach does yield useful information about these complex, spatially nonuniform systems. 31. S UMMARY
AND
C ONCLUSIONS
The main results and observations in this chapter can be summarized in three categories: effects of dislocations on (i) electrical properties in GaN, (ii) optical properties in GaN, and (iii) optical properties in GaAs relative to GaN.
a. Effects of Dislocations on Electrical properties in GaN To study the effects of edge dislocations on electrical properties in GaN, electron drift and Hall mobilities due to dislocation scattering are calculated using the Fermi–Golden rule. Total electron mobilities including the ionized impurity scattering, acoustic deformation and piezoelectric phonon scattering, optical polar phonon scattering, and dislocation scattering are calculated and compared with the available experimental data. The dislocation scattering is computed by first obtaining charge distributions along the VGa–ON complex defect edge dislocation in wurtzite GaN using density functional theory calculations. The calculations reveal that the two N atoms near the gallium vacancy at the core act as electron acceptors resulting in two straight lines of electron acceptors along the dislocation line. A filling fraction calculation
117
R. J. Roedel, A. R. VonNeida, R. Caruso, and L. R. Dawson, J. Electrochem. Soc. 126, 637 (1979).
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for the Ge dislocation by Read is then modified for the VGa–ON complex defect edge dislocation in GaN. To calculate electron mobility due to dislocations, two electrostatic potentials for negatively charged edge dislocations are compared: one by Read and one by Bonch-Bruevich and Glasko (B–G). Closed-form solutions for the electron drift and Hall mobility due to the B–G electrostatic dislocation potential are derived. At low carrier concentrations, the B–G potential correctly predicts reduced mobility, while the Read potential predicts constant mobility. Calculated drift mobility results match with the experimental data reasonably well. Both drift and Hall mobilities are strongly affected by dislocation densities higher than 108 cm2 in devices with carrier concentrations in the range of 1015–1021 cm3. Maximum mobility occurs at higher carrier concentrations as dislocation density increases. For n > 1016 cm3, reducing the dislocation density below ndis ¼ 108 cm2 has a negligible effect on total mobility if the mobility due to lattice scattering is above 103 cm2 V1 s1.
b. Effects of Dislocations on Optical Properties in GaN To study the effects of dislocations on optical properties, the 6 6 kp Hamiltonian is solved for the valence band using the finite element method in a real space formulation in wurtzite GaN. The conduction band is solved separately, taking advantage of wide band gap of 3.44 eV. After obtaining the energy levels and wave functions for electrons and holes, the spontaneous emission spectra are calculated and broadened with a Gaussian distribution to compare to the available experimental data. For an edge dislocation, the electrostatic potential due to its electron acceptor nature localizes holes at the core and generates hole states within the band gap. Transition energy levels and intensities vary with dislocation density and free carrier concentration. Transition energy levels to the monopole localized hole states (Im) in the band gap correspond to YL, GL, and BL which are experimentally found to be from the same extended defects;38,39 the behavior and functional dependence of the associated emission intensity ratio, Im/IBand Edge, is also consistent with experimental findings for the IYL/IBand Edge ratio.32,34,35,106 Transition energy levels to dipole and quadrupole localized hole states (Id and Iq) in the band gap are in the same range as donor–acceptor pairs, ‘‘DA’’ states, and the behavior of this emission intensity is consistent with the IDA observed experimentally.34 As the dislocation density increases, the densities of states at the band edge decrease so that the band edge peak intensity decreases for dislocation densities higher than ndis ¼ 107 cm2 and the band edge peak broadens to an extent
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consistent with experimental observations.18,19,27 As the carrier concentration increases, the band edge peak intensity increases. The band edge peak attenuates severely with high dislocation densities and low carrier concentrations implying that GaN becomes dislocation-sensitive if insufficient carriers are present. There is a transition dislocation density above which Im decreases. At this transition point there is a change of electron-localization mechanisms from strainlocalization near the core to localization due to the electrostatic potential between dislocations. For lower carrier concentrations, the electrostatic potential is larger due to reduced screening so that the transition occurs at lower dislocation density. Generally, the ratios Id/IBand Edge and Iq/IBand Edge increase as dislocation density increases. Energy levels of Im, Id, and Iq depend more on carrier concentration than on dislocation density. For an open-core screw dislocation, the shear strain field around the core localizes heavy- and light-hole states. Electrons and split-off holes are repelled from the screw dislocation core. Therefore, shear deformation near the dislocations generates spatial separation between electrons and holes, resulting in the reduction of optical emission efficiency of the sample. As dislocation density increases the band edge peak intensity decreases and broadens. Both calculated band edge peak intensity and experimental data indicate a significant reduction for screw dislocation densities of ndis > 107 cm2. For screw dislocation density less than ndis 108 cm2, a sample with larger core screw dislocations has stronger emission intensity while for ndis > 108 cm2, a sample with smaller core screw dislocations has stronger intensity. Two competing factors, the absence of material at the core and the strain field associated with the dislocation, dominate at different dislocation densities with a transition density around ndis 108 cm2.
c. Effects of Dislocations on Optical Properties in GaAs Relative to GaN The effects of edge dislocations on optical properties in zinc-blende GaAs are calculated and compared with those in GaN. A 8 8 kp Hamiltonian is solved using the finite element method in a two-dimensional real space formulation to include interactions between the conduction and valence bands. Anisotropic strain fields associated with edge dislocation on the (001) plane are calculated. Compared to GaN, the deformation fields around edge dislocations in GaAs have larger magnitudes. Band edge peak intensity in GaAs decreases as dislocation density increases for dislocation density higher than 103 cm2, while it decreases for dislocation density higher than 107 cm2 in GaN. Larger deformation around GaAs edge dislocations and smaller GaAs
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249
electron effective mass makes the GaAs system more sensitive to dislocations than the GaN system.
d. Future Work In this chapter, the real space kp Hamiltonian method is applied to study dislocation effects. The method can be used for many optical applications where inhomogeneous potentials and material discontinuities exist. Useful applications are effects of defects (ionized impurities and grain boundaries), low-dimensional systems (quantum dots, wires, and wells), heterostructures, and their combinations. Ionized impurities are known to have electrostatic and deformation potentials localized near the impurities resulting in inhomogeneous potential fields; these effects can be analyzed, under the present framework. Low-angle grain boundaries can be modeled as arrays of dislocations and the effects of grain size can be studied for applications related to dislocation effects. Also, quantization effects in low-dimensional systems (quantum dots, wires, wells) with inhomogeneous strain fields and material property discontinuities can also be studied, with a view toward understanding the mechanics of defect and disorder effects on optoelectronic properties of materials. Acknowledgements
The original work carried out by the authors in preparing this review was supported by the National Science Foundation grant #CMS-02-96102. Portions of this review contain copyrighted text excerpts and figures reproduced, with permission, from the authors’ prior work in references 55–58. VIII. Appendix A
In Appendix A, the electron–electron energy for 0.5 < s 1 is derived. The Pu ZIG ðri Þ, can be written as a sum of two first term in Eq. (3.16), 0V i ¼ u;0 Pu P u ZIG ZIG ZIG parts, ðr Þ ¼ V ðr ðr00 Þ. Then, the electron– 0V i iÞ þ V i ¼ u;0 i ¼ u electron energy per electron is " # u u 1 qX q ZIG q X ZIG ADD V ðri Þ V ðr00 Þ V ðri Þ : ð8:1Þ Ee ¼ ð2u þ 2Þ 2 i ¼ u 2 2 i ¼ u;00 The first term in Eq. (8.1) can be approximated as q2 ð2u þ 1ÞV ZIG ðr0 Þ for infinitely long dislocation lines; q2 V ZIG ðr0 Þ is the same form as Ee in
250
J. H. YOU AND H. T. JOHNSON
Eq. (3.5) replacing d with c and N with N(2uþ1). Then, the first term in Eq. (8.1) becomes u qX V ZIG ðri Þ 2 i ¼ u qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g1 1 b2 1 bc 2 0 þ ln þ g 2 þ ln 1 þ ðc=aÞ ðc=aÞ ¼ ð2u þ 1ÞEo 2 2 4 2 a
ð8:2Þ where bc is the total length of dislocation line as b ¼ N(2u þ 1), g1 is Euler’s constant and g0 2 is a modified Euler’s constant defined as: 0 g0 2 ¼
B @
lim
ðbX þ 2Þ=4
ðb þ 2Þ=4 ! 1
n¼1
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða=cÞ2 þ ð2n 1Þ2
ðb þð2Þ=4
1
1 1 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dnA: ða=cÞ2 þ ð2n 1Þ2
ð8:3Þ The second term in Eq. (8.1) can be expanded as q V ZIG ðr00 Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c 1 b 1 ðb 2Þc þ g3 þ ln þ g4 þ ln 1 þ ð2c=aÞ2 2c=a ¼ Eo 2a 2 2 2 a ð8:4Þ where
0 g3 ¼
lim
ðb þ 2Þ=4 ! 1
B @
ðbX þ 2Þ=4 n¼1
1 2n 1
0 g4 ¼
lim
B @
ðb 2Þ=4 ! 1
ðbX 2Þ=4 n¼1
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða=cÞ þ ð2nÞ2
ðb þð2Þ=4
1 1 C dnA 2n 1
ð8:5Þ
1 ðb ð2Þ=4
1
1 1 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dnA: 2 2 ða=cÞ þ ð2nÞ ð8:6Þ
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
251
The g0 2 , g3, and g3 are independent of s as g0 2 ¼ 0:5191, g3 ¼ 0.6352, and g3 ¼ 0.2732. The third term in Eq. (8.1) can be expressed as a sum of four terms as m X
V ADD ðri Þ ¼ V ADD ðr0 Þ þ V ADD ðr00 Þ þ 2
i ¼ m;0
m X
þ
m1 X
V ADD ðri > 0 Þ
V ADD ðri > 0 Þ
i ¼ 2;even
0
ð8:7Þ
i ¼ 1;odd
where the factor of 2 comes from the relationship of VADD (ri) ¼ VADD(ri). The first two terms in Eq. (8.7) can be simplified as V ADD ðr0 Þ þ V ADD ðr00 Þ 9 8 c 2 b2 c > > > > > > þ 2½g5 ðuÞ þ g6 ðuÞ þ ln > > > > a 2u þ 1 2að2u þ 1Þ > > > > = < q vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 0 12 0 1 u ¼ u > 4peeo c > > > 2 > > 6u > > @cð2u þ 1ÞA @cð2u þ 1ÞA7 t ln 1 þ þ > > 4 5 > > > > 2u þ 1 a a ; :
ð8:8Þ
where
g5 ðuÞ ¼
0 N=2 BX lim @
N=2 ! 1
1 nð2u þ 1Þ n¼1
N=2 ð
1 1 C dnA nð2u þ 1Þ
ð8:9Þ
1
and 0
1 N=2 ð N=2 X 1 1 B C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dnA: g6 ðuÞ ¼ lim @ N=2 ! 1 2 2 2 2 n¼1 ða=cÞ þ n2 ð2u þ 1Þ ða=cÞ þ n2 ð2u þ 1Þ 1 ð8:10Þ The terms given in Eq. (8.9) and Eq. (8.10) are shown in Figure VIII.1. The last two terms in Eq. (8.7) need further consideration. Summing over even i, the first term is given by,
252
J. H. YOU AND H. T. JOHNSON
(b) 0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
g5
g5
(a)
0.4 0.3
0.3
0.2
0.2
0.1
0.1 0 0.5
0 0
2
4
u
6
6
10
(c)
0.6
0.7
0.6
0.7
s
0.8
0.9
0.8
0.9
1
(d) 0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5 0.4
g6
g6
0.4
0.4 0.3
0.3
0.2
0.2
0.1
0.1 0 0
2
4
u
6
6
10
0 0.5
s
1
FIG. VIII.1. g5 as function of (a) u and (b) s. g6 as function of (c) u and (d) s.
u X
V ADD ðri>0 Þ
i ¼ 2;even
2
0 1 N=2 X 1 1 1 @ A qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ nð2u þ 1Þ i nð2u þ 1Þ þ i ða=cÞ2 þ i2 n ¼ 1;odd 0
3
6 7 6 7 6 7 6 7 q 6 17 ¼ 6 7: 7 4peeo c i ¼ 2;even6 N=21 X B C7 6 1 1 Brffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC 7 6þ @ 4
2
2 A 5 n ¼ 2;even ða=cÞ2 þ nð2u þ 1Þ i ða=cÞ2 þ nð2u þ 1Þ þ i u X
ð8:11Þ Replacing n ¼ 2m 1 (m ¼ 1, 2, . . .(N þ 2)/4 for odd n, and n ¼ 2m (m ¼ 1, 2, . . .(N 2)/4 for even n, and i ¼ 2j ( j ¼ 1, 2,. . . u/2) for even i, and introducing additional modified Euler’s constants, the expression takes the form as
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
u X
253
V ADD ðri > 0 Þ ¼
i ¼ 2;even
3 2 u=2 q X6 1 7 4g ðu; jÞ þ g8 ðu; jÞ þ g9 ðu; jÞ þ g10 ðu; jÞ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 4peeo c j ¼ 1 7 2 2 ða=cÞ þ ð2jÞ 3 2 2 2 2lnð2c=aÞ þ 2ln½ðb=2Þ ð2jÞ ln½ð2u þ 1Þ2 ð2jÞ2 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 37 6 6
7
2 c 7 6 c u=2 6 þln4 1 þ 2ð2u þ 1Þ 2j 5 7 X a 2ð2u þ 1Þ 2j q 1 7 6 a þ 7 6 7 4peeo c 2ð2u þ 1Þ j ¼ 1 6 2 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 7 6 2
7 6 c 4 þln4 1 þ c 2ð2u þ 1Þ þ 2j 55 2ð2u þ 1Þ þ 2j a a
ð8:12Þ where 0 g7 ðu; jÞ ¼
2Þ=4 BðNX 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lim B @
2ffi ðN 2Þ 2 m¼1 4 !1 ða=cÞ þ 2mð2u þ 1Þ 2j
ðN ð2Þ=4
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi dm
1
2
ða=cÞ þ 2mð2u þ 1Þ 2j
2
Þ
ð8:13Þ
0 g8 ðu; jÞ ¼
2Þ=4 BðNX 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lim B @
2ffi ðN 2Þ 2 m¼1 4 !1 ða=cÞ þ 2mð2u þ 1Þ þ 2j
ðN ð2Þ=4
1
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi dmÞ ða=cÞ2 þ 2mð2u þ 1Þ þ 2j
2
ð8:14Þ
254
J. H. YOU AND H. T. JOHNSON
0 g9 ðu; jÞ ¼
ðNX þ 2Þ=4
B lim @
ðN þ 2Þ 4 !1
m¼1
1 ð2m 1Þð2u þ 1Þ 2j
ðN þð2Þ=4
1 1 C dmA ð2m 1Þð2u þ 1Þ 2j
1
ð8:15Þ 0 g10 ðu; jÞ ¼
ðNX þ 2Þ=4
B lim @
ðN þ 2Þ 4 !1
m¼1
1 ð2m 1Þð2u þ 1Þ þ 2j
ðN þð2Þ=4
1 1 C dmA: ð2m 1Þð2u þ 1Þ þ 2j
1
ð8:16Þ In the same manner, uX 1
V ADD ðri > 0 Þ ¼
i ¼ 1;odd
2 3 u=2 q X 4 1 5 g11 ðu; jÞ þ g12 ðu; jÞ þ g13 ðu; jÞ þ g14 ðu; jÞ þ 4peeo c j ¼ 1 2j 1 2 3 2lnð2c=aÞ þ 2ln½ðb=2Þ2 ð2j 1Þ2 6 7
2 6 7 ln½ 2ð2u þ 1Þ ð2j 1Þ2 6 7 6 7 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 7 2v 3 0 1 u 6 2 7 u 6 7
u=2 X6 7 6u q 1 @c 2u þ 1 ð2j 1Þ A c 2u þ 1 ð2j 1Þ 7 t þln 1 þ 6 7 4 5 þ 7 a a 4peeo c 2ð2u þ 1Þ j ¼ 16 6 7 6 7 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 7 2v 3 0 1 u 2 6 7 u 6
7 c c 6u 77 6 4 þln4t1 þ @ 2u þ 1 þ ð2j 1Þ A 2u þ 1 þ ð2j 1Þ 5 5 a a
ð8:17Þ where 0 g11 ðu; jÞ ¼
B lim @
ðN 2Þ !1 4
ðNX 2Þ=4 m¼1
1 2mð2u þ 1Þ ð2j 1Þ
ðN ð2Þ=4
1 1 C dmA 2mð2u þ 1Þ ð2j 1Þ
1
ð8:18Þ
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
0 g12 ðu; jÞ ¼
ðNX 2Þ=4
B lim @
ðN 2Þ !1 4
m¼1
1 2mð2u þ 1Þ þ ð2j 1Þ
ðN ð2Þ=4
255
1 1 C dmA 2mð2u þ 1Þ þ ð2j 1Þ
1
ð8:19Þ 0
ðNX þ 2Þ=4
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ða=cÞ þ ½ð2m 1Þð2u þ 1Þ ð2j 1Þ2
1
C B C B m¼1 C B C B g13 ðu; jÞ ¼ lim B ðN þ 2Þ=4 C ð ðN þ 2Þ C B 4 !1B 1 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dm A @ 2 2 ða=cÞ þ ½ð2m 1Þð2u þ 1Þ ð2j 1Þ 1 ð8:20Þ 0
ðNX þ 2Þ=4
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða=cÞ2 þ ½ð2m 1Þð2u þ 1Þ þ ð2j 1Þ2
1
B C B C m¼1 B C B C g14 ðu; jÞ ¼ lim B ðN þ 2Þ=4 C: ð ðN þ 2Þ B C ! 1B 4 1 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dm A @ 2 2 ða=cÞ þ ½ð2m 1Þð2u þ 1Þ þ ð2j 1Þ 1 ð8:21Þ Finally, we obtain the electron–electron energy per electron from Eq. (8.1) as 9 8c > > þ ð2u þ 1ÞC1 þ C2 ðuÞ þ C3 ðuÞ þ C4 ðuÞ > > > >a > > > > > > > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 2
> > > > 1 4ðu þ 1Þ = < 2 2 Eo ð2u þ 1Þ þ ðc=aÞ ðc=aÞ þ lnðbÞ ln þ Ee ðuÞ ¼ ð2u þ 1Þ 2u þ 1 > 2u þ 2 > 0 1 0 1 > > > > > > 2 > > 2 pffiffiffi 2ðu þ 1Þ > > 8u þ 12u þ 5 c > > @ A @ A > > ln ln 2 þ > > ; : 2u þ 1 2u þ 1 a ð8:22Þ
256
J. H. YOU AND H. T. JOHNSON
where C1 ¼ 12 g1 þ g0 2 þ 12 ln
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ ðc=aÞ2 c=a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 þ ð2c=aÞ2 2c=a C2 ðuÞ ¼ g3 þ g4 þ g5 ðuÞ þ g6 ðuÞ þ ln 2 0 1 u=2 14 X X 1 1 C B gk ðu; jÞ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ C3 ðuÞ ¼ @ A 2j 1 2 2 j¼1 k¼7 ða=cÞ þ ð2jÞ
ð8:23Þ ð8:24Þ
ð8:25Þ
C4 ðuÞ ¼
9 8 h i
2 > > 2 2 2 > > ln ð2u þ 1Þ ð2jÞ ð2j 1Þ ln 2ð2u þ 1Þ > > > > > > > > > > " # ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r > > > >
> > 2 > > 2 > > þln > > 1 þ ðc=aÞ 2ð2u þ 1Þ 2j ðc=aÞ 2ð2u þ 1Þ 2j > > > > > > > > > > > > " # ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi r > > > >
u=2 < 2 = X 2 1 þln 1 þ ðc=aÞ 2ð2u þ 1Þ þ 2j ðc=aÞ 2ð2u þ 1Þ þ 2j : > 2ð2u þ 1Þ j ¼ 1 > > > > > " # > rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > >
2
> > > > > 2 > > > > 2u þ 1 ð2j 1Þ þln 1 þ ðc=aÞ ðc=aÞ 2u þ 1 ð2j 1Þ > > > > > > > > > > > "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #> > > > >
> 2 > > > 2 > > > 1 þ ðc=aÞ 2u þ 1 þ ð2j 1Þ ðc=aÞ 2u þ 1 þ ð2j 1Þ > ; : þln
ð8:26Þ IX. Appendix B
Appendix B proves that the Es(s) in Eq. (3.11) is equal to the Es(s) in Eq. (3.19) at s ¼ 0.5. Substituting s ¼ 0.5 into Eq. (3.11) gives qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8 2 = < 2 ½ 1 þ ðc=aÞ c=a 1 1 1 3 a cpðND NA Þ 1 ¼ Eo g1 þ g 2 þ þ ln Es 2 :2 4pac ðND NA Þ ; 2 2 4 4 2 ð9:1Þ
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
257
where g2(s) in Eq. (3.7) becomes 1 0 ðN þð2Þ=4 ðNX þ 2Þ=4 1 1 1 C B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dnA g2 ¼ lim @ 2 ðN þ 2Þ=4 ! 1 2 2 2 2 n¼1 ða=cÞ þ ð2n 1Þ ða=cÞ þ ð2n 1Þ 1
ð9:2Þ because of d ¼ c/(2s). From Eq. (3.19), substituting s ¼ 0.5 leads to 1 pa2 cðND NA Þ 1 1 3 ln 2 : ð9:3Þ ¼ Eo C1 þ þ ln Es 2 4 2 pac2 ðND NA Þ 4 Substituting C1 from Eq. (8.23) in Appendix A, Es in Eq. (9.3) can be rewritten as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8 2 = < 2 1 þ ðc=aÞ c=a 1 1 pa cðND NA Þ 1 3 : ¼ Eo g 1 þ g0 2 þ þ ln ln 2 Es ; :2 pac2 ðND NA Þ 2 4 2 4 ð9:4Þ Since g2(s ¼ 1/2) in Eq. (9.2) is equal to g0 2 Eq. (8.3), and
1 ln 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ðc=aÞ2 c=a pac2 ðND NA Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 ½ 1 þ ðc=aÞ c=a ln 2 ¼ ln 4pac2 ðND NA Þ 2
ð9:5Þ
Es(1/2) in Eq. (9.4) is equal to Eq. (9.1). Therefore, Es(s) from Eq. (3.11) and Es(s) from Eq. (3.19) are continuous at s ¼ 0.5. Figure IX.1 shows Es(s) and sF(s)as function of s at dislocation density at ndis ¼ 109 cm2 and carrier concentration at n ¼ 1017 cm3. The solid line indicates Es(s) from Eq. (3.11) and the dashed line indicates Es(s) from Eq. (3.19). Figure IX.1B shows that sF (s) has its minimum at s ¼ 0.4292.
258
J. H. YOU AND H. T. JOHNSON
(a)
3 2.5
Es (s)
2 1.5 1 0.5
0 −0.5
(b)
0
0.2
0.4
0
0.2
0.4
s
0.6
0.8
1
0.6
0.4
sF (s)
0.2 0 −0.2 −0.4 −0.6 −0.8
s
0.6
0.8
1
FIG. IX.1. (a) Es(s) and (b)sF(s) as function of s at dislocation density at ndis ¼ 109 cm2 and carrier concentration at n ¼ 1017 cm3. The solid line indicates from Eq. (3.11) and the dashed line indicates Es(s) from Eq. (3.11) and the dashed line indicates Es(s) from Eq. (3.19). sF(s) has its minimum at s ¼ 0.4292.
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES
259
X. Appendix C
TABLE X.1. GAN MATERIAL PROPERTIES
FOR
ELECTRON SCATTERING CALCULATIONS
Parameter
Symbol
Values
Dielectric constant (high frequency) Dielectric constant (low frequency) Polar phonon Debye temperature Mass density Speed of sound Piezoelectric constant Acoustic deformation potential Electron effective mass
e1 (F m1) e (F m1) Tpo (K) r (kg m3) s (m s1) hpz (or e14) (C m2) adp (eV) m* (kg)
5.35e0118 9.5e0118 1044119 6.10 103 120 6.59 103 119 0.5121 11.81221 0.20 m0122
118
A. S. Barker Jr. and M. Ilegems, Phys. Rev. B. 7, 743 (1973). D. C. Look, J. R. Sizelove, S. Keller, Y. F. Wu, U. K. Mishra, and S. P. DenBaars, Solid State Commun. 102(4), 297 (1997). 120 D. K. Schroeder, Semiconductor Material and Device Characterization, Wiley, New York (1990). 121 A. D. Byhovski, V. V. Kaminski, M. S. Shur, Q. C. Chen, and M. A. Khan, Appl. Phys. Lett. 68, 818 (1996). 122 I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001). 119
260
J. H. YOU AND H. T. JOHNSON
TABLE X.2. WURTZITE GAN MATERIAL PROPERTIES Parameter
GaN
˚) Lattice constant122 (A A 3.189 C 5.185 Energy parameters97 Egap (eV) 3.44 D1 (meV) 16 D2 (meV) 4 D3 (meV) 4 Conduction band effective mass122 m e? 0.20 me// 0.20 Valence band effective-mass parameters97 6.56 A1 A2 0.91 A3 5.65 A4 2.83 A5 3.13 A6 4.86
123
FOR
OPTICAL PROPERTY CALCULATIONS
Parameter
GaN
Deformation potentials (eV) a1 (eV) 6.5122 a2 (eV) 11.8122 D1 3.0123 D2 3.6123 D3 6.6123 D4 3.3123 D5 4.0123 pffiffiffi D6 ¼ ðD3 þ 4D5 Þ= 2 6.65123 Elastic stiffness constant122 C11 (GPa) 390 C12 (GPa) 145 C13 (GPa) 106 C33 (GPa) 398 C44 (GPa) 105 Static dielectric constant118 e?/e0 on basal plane 9.5
T. Ohtoshi, A. Niwa, and T. Kuroda, J. Appl. Phys. 82, 1518 (1997).
EFFECT OF DISLOCATIONS ON ELECTRICAL AND OPTICAL PROPERTIES TABLE X.3. ZINC-BLENDE GAAS MATERIAL PROPERTIES Parameter
GaAs
Lattice constant122 ˚) a (A Energy parameters122,126 a Egap (eV) D (eV) Ep (eV) Luttinger parameters124 gL1 gL2 gL3 Others125 A0 B a
1.519 (0 K) 1.425 (300 K) 0.33 25.7
0 0
OPTICAL PROPERTY CALCULATIONS
Parameter
5.653
6.85 2.1 2.9
FOR
261
GaAs
Deformation potentials ac (eV) 9.3124 0 b (eV) 0125 av(eV) 0.7124 b (eV) 2.0124 d (eV) 5.4124 Elastic stiffness constant122 C11 (GPa) 1221 C12 (GPa) 566 C44 (GPa) 600 Static dielectric constant102 e/e0 13.1
The band gap of GaAs is known to be dependent on temperature as,126,127 Egap ðTÞ ¼ Egap ð0 KÞ
5:5 104 TðKÞ2 TðKÞ þ 225
ð10:1Þ
in unit of eV.
124
C. Pryor, Phys. Rev. B 57, 7190 (1998). T. B. Bahder, Phys. Rev. B 45, 1629 (1992). 126 P. Lautenschlager, M. Garriga, S. Logothetidis, and M. Cardona, Phys. Rev. B 35, 9174 (1987). 127 M. Cardona, T. A. Meyer, and M. L. W. Thewalt, Phys. Rev. Lett. 92, 196403 (2004). 125
Author Index
Numbers in parentheses are reference numbers and indicate that an author’s name is not cited in the text.
A Abell, J., 244(110) Abermann, R., 60(125) Abraham, F., 57(113) Adamson, A. W., 26(53) Ade, G., 245(116) Adesida, I., 149(21) Adkins, C. J., 1(8) Adzic, R., 107(46) Adzic, R. R., 96(23), 119(82), 125(111, 112), 129(124), 132(135, 136), 135(141, 143), 136(143), 137(144) Ahern, J. E., 20(38) Akasaki, I., 148(12), 149(19), 151(36) Akyu¨z, C. D., 198(100) Albu, T. V., 129(126, 128, 129) Aleksiejunas, R., 151(39) Alexander, J. I., 38(68) Alle`gre, J., 245(114) Amano, H., 149(19), 151(36) Amano, M., 148(12) Anderson, A. B., 129(128, 129) Anderson, D. M., 26(46) Andre, C. L., 203(104) Andreasen, G., 103(31) Angelo, A. C. D., 126(113, 114) Antoine, O., 112(70) Appleby, A. J., 111(62), 112(68, 69), 116(79) Arif, M., 114(73) Arslan, I., 150(29) Arvia, A. J., 103(31), 123(99) Askgaard, T., 122(92) Atanassov, P., 120(84) Austin, L. G., 82(8) Aziz, M. J., 138(146)
B Bacon, D. J., 145(3) Bagotsky, V. S., 110(55) Bahder, T. B., 192(98), 261(125) Baker, B. S., 111(62) Bakker, G., 46(83) Balbuena, P. B., 109(54), 117(80), 129(130), 131, 131(131, 133), 139(150) Bard, A. J., 93(17) Barker, A. S., 259(118) Barton, S. C., 140(153) Bashyam, R., 140, 140(154) Bastard, G., 244(113) Baylin, M., 1(12) Beaumont, B., 149(17), 151(33, 35) Beckman, S. P., 152(52) Beden, B., 123(99) Bejan, A., 20(39) Belkhir, M. A., 164(71) Bensahel, D., 152(51) Bernardini, F., 244, 244(112) Biegler, T., 99(27) Binyamin, G., 140(153) Birringer, R., 53(109) Bliek, L., 154(62) Bligaard, T., 94(20), 101(30), 106(42), 123(98) Bockris, J. O’M, 93(16) Boeckl, J. J., 203(104) Bogdanovskaya, V. A., 140(152) Bo¨hm, K., 152(49) Bonch-Bruevich, V. L., 159(67) Bonevich, J. E., 53(107) Borup, R. L., 120(84) B ttiger, J., 53(108) Bouchoule, S., 152(51)
263
264
AUTHOR INDEX
Bour, D. P., 150(26) Brankovic, S. R., 119(82), 125(111, 112) Brantley, W. A., 244(108) Brazel, E. G., 149(20) Briddon, P. R., 163(69), 164(73) Brooks, H., 51(93) Broudy, R. M., 154(61) Browning, N. D., 149(17), 150(29) Bru¨ckner, P., 149(23) Brusic, V., 108(48) Bu, Y., 101(29) Buckius, R. O., 20(37) Buff, F. P., 44(71) Bultel, Y., 112(70) Bus, V., 124(107) Byhovski, A. D., 259(121) C Cahn, J. W., 13, 13(25, 26), 18(33), 28(55, 56), 36, 38, 47, 49, 49(88), 50(91), 51, 53(106), 71 Cai, J., 245(115) Caillard, A., 88(10) Calabrese, C. M., 114(73) Caldi, D. G., 26(52) Calleja, E., 151(33) Callen, H. B., 6(17) Callister, W. D., Jr., 146(10) Calvert, C., 137(145) Cammarata, R. C., 22(43), 29(60), 45(78–80), 46(81), 52(99), 53(100, 101), 53(104), 53(107), 53(110), 60(124, 128), 74(131) Campidelli, Y., 152(51) Car, R., 128(119) Caram, R., 126(114) Cardona, M., 261(126, 127) Carr, E. C., 241(107) Caruso, R., 245(117) Chang, C. S., 192(97), 193 Chang, H. H., 111(63) Chang, L. L., 244(113) Chason, E., 60(124) Chatenet, M., 120(83, 87) Chaudhuri, T., 113(71) Chen, J., 150(27) Chen, M., 115(76) Chen, Q., 89(11) Chen, Q. C., 259(121)
Chen, S., 117(81) Chen, Y., 171(88) Cherns, D., 149(22), 150, 150(30), 171, 171(86), 245 Chevallier, J., 53(108) Chevary, J. A., 128(118) Chialvo, A. C., 97(24), 104(33, 34) Chichibu, S. F., 244(111), 245 Chin, D.-T., 111(63) Chorkendorff, I., 122(92) Chriqui, Y., 152(51) Christian, J. W., 57(118) Christner, L. G., 106(44, 45) Chrzan, D. C., 152(52) Chua, S. J., 151(35) Chuang, S. L., 192(97), 193, 202(102) Claude, E., 120(83) Clausen, B. S., 122(92), 123(98) Clouser, S. J., 111(59, 60) Conwell, E. M., 185, 185(94) Corcella, A., 120(87) Coutanceau, C., 88(10) Crabtree, G. W., 78(1) Craford, M. G., 152(40), 153(54) Cui, S., 90(15) D Dai, T., 150(27) Damilano, B., 245(114) Damjanovic, A., 108(48, 49), 108(50–52), 110 Dannehy, C. S., 114(73) Dapkus, P. D., 244(108) Darken, L. S., 15(30) Datye, A. K., 152(42) Dave, B., 112(68) Davies, J. H., 202(101) Dawson, L. R., 245(117) Dawson, R., 152(42) DeHoff, R. T., 1(11) DenBaars, S. P., 149(20), 244(111), 259(119) Derouin, C. R., 116(78) Desai, S. K., 132(134) deTacconi, N. R., 123(99) Dexter, D. L., 145(4) Dhar, H. P., 106(44, 45) Di Felice, R., 171(89) Diat, O., 89(12) Dijilali, N., 105(35)
AUTHOR INDEX Dimitrov, N., 138(146) Ding, Y., 115(76), 123(97), 138(148) Doerner, M. F., 60(122) Domen, K., 149(18) Donnan, F. G., 3(13), 26(47) Doppalapudi, D., 149(24), 188(96) Doremus, R. H., 57(115) Dorsch, W., 170(82, 83) Doverspike, K., 150(28), 171(87) Dresselhaus, M. S., 78(1) Dudley, M., 170(84) Dumesic, J. A., 123(95) Dumsic, J. A., 123(93) Durand, R., 112(70) E Eastman, L. F., 149(24), 188(96) Eby, R. K., 53(100), 53(101) Eckstein, R., 170(82, 83) Edger, J. H., 165(79) Edwards, B., 90(15) Eichler, A., 129(127) Eisman, G. A., 114(73) Eiswirth, M., 125(109) Ekedahl, L.-G., 101(28) Elsner, J., 163, 163(69), 164, 164(73), 171, 173(92) Elstner, M., 163(69) Enayetullah, M. A., 116(79) Erickson, A. N., 149(20) Erikson, H. I., 241(107) Eriksson, M., 101(28) Erlebacher, J., 115(76), 115(77), 138(146), 138(147, 148) Ertl, G., 125(109) Esaki, L., 244(113) Esquivel, A. L., 152(47) Evans, J., 123(94) Evans, U. R., 110(57, 58) Ewing, J. A., 145(1) F Fareed, Q., 151(39) Faulkkner, L. F., 93(17) Faurie, J. P., 149(17) Feibelman, P. J., 94(19) Fermi, E., 1(6)
265
Ferreira, P. J., 117(81), 120(88) Fiedenhans’l, R., 53(108) Filhol, J., 106(40) Fini, P. T., 244(111) Finn, C. B. P., 1(10) Fiolhais, C., 128(118) Fiorentini, V., 244(112) Fischer, B., 152(49) Fischer, F. D., 29(63) Fischer, R., 244(113) Fisher, H. P., 53(101) Fitzgerald, E. A., 203(104) Floro, J. A., 60(124) Fowler, B., 135(140) Frank, F. C., 52(94), 169(81) Frauenheim, T., 164(71) Frauenheim, Th., 163(69), 164(73), 173(92) Freund, L. B., 198(100) Friesen, C., 124(102) Fritz, G., 115(77), 138(147) Fuhrman, D., 245(116) Fuller, C. S., 153(53) Furthmu¨ller, J., 165(77) G Gallart, M., 245(114) Ganti, S., 49(89) Gao, H., 43(70) Gao, S., 129(125) Garnier, J.-P., 88(10) Garriga, M., 261(126) Gaska, R., 151(39) Gaskill, D. K., 150(28), 171(87) Gast, A. P., 26(53) Gasteiger, H. A., 94(21), 105(38), 110(56), 120(88), 125(110) Gatto, V., 148(15) Gauthier, Y., 124(107) Ge, S., 114(74) Gebel, G., 89(12) Gennero de Chialvo, M. R., 97(24), 104(33, 34) Genshaw, M. A., 108(49) Gerthsen, D., 149(23) Gervasio, D., 109(53) Ghoneim, M. M., 111(60) Gibart, P., 149(17), 151(35) Gibbs, D., 45(75, 76)
266
AUTHOR INDEX
Gibbs, J. W., 1(1–4), 1–4, 6(14–16), 7(19, 20), 8(21), 10(22), 15(28, 29), 18(32), 22(41, 42), 24(44), 25(45), 28(58), 29(64), 31(65), 32(66, 67), 38, 57(121), 69(129) Gil, A., 124(104) Gil, B., 245(114) Girolami, G., 241(107) Glasko, V. B., 159(67) Godfrey, M. J., 45(77) Gokhale, A., 123, 123(93) Gomer, R., 26(49) Gong, H. M., 224(106) Goodrich, F. C., 46(86) Gottesfeld, S., 80, 80(6, 7), 112(65, 67) Go¨tz, W., 150(26), 151(36) Gra´na´sy, L., 57(119) Grandjean, N., 245(114) Greeley, J., 106(42), 139(151) Greer, A. L., 57(114) Grgur, B. N., 98(26), 110(56), 123(100) Grinberg, I., 124(105) Grove, W. R., 79(5) Grubel, G., 45(76) Gruber, Th., 149(23) Gu, W., 105(38) Gu, Z., 139(150) Guggenheim, E. A., 46(84, 85) Guilminot, E., 120(83, 87) Gutierrez, R., 173(92) H Haas, A., 3(13), 26(47) Haasen, P., 154(62) Hacke, P., 149(18) Hafner, J., 129(127), 165(74–76) Haiss, W., 29(62) Hamada, I., 106(41) Hammer, B., 124(106) Hammershoi, B. S., 122(92) Hangleiter, A., 245(116) Hansen, P. J., 149, 149(20) Hanson, J. C., 125(112) Harms, F., 46(83) Harten, U., 45(74) Hartman, R. L., 152(44) Haruta, M., 123(96) Hashimoto, S., 149(16)
Haskell, B. A., 244(111) Haszko, S. E., 244(108) Haug, A. T., 120(86) Haugk, M., 164(73), 173(92) Hawkridge, M. E., 150, 150(30) He, T., 121(90) Heaton, Th., 124(102) Heggie, M. I., 163(69), 164(73), 173(92) Heindl, J., 170(82, 83) Heinke, W., 145(7) Heinzel, A., 78(2) Heller, A., 140(153) Henderson, J. R., 44(73) Henkelman, G., 129(121) Hermann, A., 113(71) Herring, C., 26(48, 49), 36, 44 Herzog, A. H., 152(40) Hickner, M. A., 89(13) Hillert, M., 16(31), 57(120) Hilliard, J. E., 28(56, 57) Hino, T., 149(16) Hinze, P., 245(116) Hirth, J. P., 205(105) Hiscocks, S. E. R., 152(41) Hitzel, F., 245(116) Hofmann, D., 170(82) Hogarth, M. P., 78(4) Hohenberg, P., 127(116) Holby, E. F., 117(81) Horino, K., 149(18) Howell, J. R., 20(37) Hrbek, J., 123(94) Hsueh, K.-L., 111(63) Hu, J., 52(98) Huang, J. C., 111(59) Huang, K. G., 45(75, 76) Huang, Q., 161(68) Huang, X., 170(84) Hudson, J. B., 26(54) Hull, D., 145(3) Hussey, D. S., 114(73) Hu¨tter, M., 53(102) Hwang, Y. G., 164(71) Hyman, M. P., 131(132) I Ibbetson, J. P., 149(20) Ikeda, M., 149(16)
AUTHOR INDEX Ikeshoji, T., 106(41) Ilegems, M., 259(118) Innocente, A. F., 126(113) in’t Veld, P. J., 53(102) Iojoiu, C., 120(83) Ito, M., 124(103) Ito, R., 152(46) Iwasa, N., 148(13, 14) Iwasita, T., 125(109) Iwaya, M., 149(19) J Jackson, K. A., 128(118) Jacobson, D. L., 114(73) Janssens, T. V. W., 123(98) Jaramillo, T. F., 101(30) Jensen, M. K., 114(73) Jia, Q. J., 150(27) Jiang, D. S., 150(27), 224(106) Jiang, J., 105(36) Jiao, C. G., 149(22), 245 Jin, R. Q., 150(27) Johnson, H. T., 153(55–58), 198(100) Johnson, N. M., 151(36) Johnson, R., 137(145) Johnson, W. C., 12(24), 38(68), 50(92) Johnston, W. D., Jr., 152(45) Jones, R., 164(73), 173(92) Jonsson, H., 94(20), 106(42), 129(121, 123) Jorne, J., 105(38) Josell, D., 53(107) K Kadjo, A. J.-J., 88(10) Kaminski, V. V., 259(121) Kamiyama, S., 149(19) Kang, T. W., 152(48) Karan, K., 105(35) Karlberg, G. S., 101, 101(30), 106(42) Karma, A., 138(146) Kaschiev, D., 57(117) Keenan, J. H., 1(5), 20(35) Keffer, D., 90(15) Keller, S., 244(111), 259(119) Kelton, K. F., 57(114, 116) Kermarrec, O., 152(51) Keune, D. L., 152(40)
267
Keyes, B. M., 203(104) Khan, M. A., 259(121) Kim, H.-H., 140(153) Kim, T. W., 152(48) Kim, W. B., 123(95) Kim, Y. J., 138(148) Kim, Y. S., 90(14) Kingston, W. E., 26(48) Kinoshita, K., 107(47), 125(108) Kirkwood, J. G., 44(71) Kitamura, F., 124(103) Kitchin, J. R., 94(20) Kiyokku, H., 148(14) Kleinman, D. A., 154(60) Klie, R., 136(143) Kobayashi, T., 123(96) Koch, H., 116(79) Koch, R., 60(127) Kocha, S., 120(88) Koh, S., 134(139) Kohn, W., 127(116), 128, 128(117) Kohnstamm, Ph., 46(82) Koide, T., 148(12) Koike, S., 148(12) Kosevich, A. M., 53(103) Kosevich, Yu. A., 53(103) Koyama, T., 244(111) Kozodoy, P., 149(20) Kramer, R., 60(125) Kreidler, E., 121(90) Kresse, G., 124(104), 124(107), 165(74–77) Kucernak, A., 105(36) Kuhr, T. A., 170(85) Kuramata, A., 149(18) Kuroda, T., 260(123) Kush, A. K., 106(44, 45) Kuznetsova, E. M., 145(9) L la O’, G. J., 120(88) Labusch, R., 154(63), 155(66) Lahee, A. M., 45(74) Lamas, E. J., 117(80) Lamy, C., 123(99) Larche´, F. C., 13, 13(25, 26), 49, 49(88), 51 Largeau, L., 152(51) Laugier, M., 60(126) Lautenschlager, P., 261(126)
268 Lee, H. I., 152(48) Lee, J., 115(77), 138(147) Lee, K. Y., 151(38) Lee, M. L., 203(104) Lee, S. M., 164(71) Lee, Y. H., 164(71) Lefebvre, P., 245(114) Leisch, J., 134(139) Lekner, J., 44(73) Leo, P. H., 38(69), 52(98) Lester, S. D., 153(54) Leung, K., 163(70) Li, F. H., 161(68) Li, G., 151(35) Li, J. C. M., 15(30) Li, P., 151(35) Li, T., 129(130) Li, X., 224(106) Li, X. Y., 224(106) Liang, J. W., 224(106) Liliental-Weber, Z., 171(88), 173 Lin, W. F., 125(109) Lin, W. N., 152(47) Lindqvist, L., 94(20) Litt, M. H., 111(61) Liu, J., 90, 90(15) Liu,, J. P., 150(27) Liu, P., 123(94), 132(135) Liu, Z., 111(61) Liu, Z. S., 224(106) Liu, Z.-S., 106(43), 127(115) Logadottir, A., 94(20) Logan, R. A., 154(60) Logothetidis, S., 261(126) Look, D. C., 149, 149(25), 185(95), 259(119) Lopez, N., 123(98) Lorimor, O. G., 244(108) Lothe, J., 205(105) Lu, J.-Q., 153(55, 58) Lucas, C. A., 123(100), 135(140) Ludlow, D. J., 114(73) Ludowise, M. J., 241(107) Lundgren, E., 124(107) Lundquist, J., 103(32) Lundstrom, I., 101(28) Lupis, C. H. P., 1(9) Luttinger, J. M., 197(99)
AUTHOR INDEX M Ma, S., 123(94) MacLellan, A. G., 44(72) Maillard, F., 120(83, 87) Makharia, R., 120(88) Malvern, L. E., 12(23) Mani, P., 139(149) Manko, D. J., 116(79) Mansfield, M., 45(77) Marek, T., 170(83) Margetis, D., 56(111) Markovic, N. M., 98(26), 110(56), 123(100), 125(110), 134(138), 135(140) Martemianov, S., 88(10) Martins, M. E., 103(31) Maru, H. C, 106(44) Ma¨ser, J., 60(125) Mason, S. E., 124(105) Massies, J., 245(114) Masumoto, Y., 244(109) Mathieu, H., 245(114) Mathur, A., 115(77), 138(147) Matsushita, T., 148(14) Matthews, J. W., 52(95) Mavrikakis, M., 123(93), 123(98), 129(124), 135(141, 142), 137(144) McBreen, J., 114(74) McFadden, G. B., 26(46) Medlin, J. Will, 131(132) Mei, X. B., 161(68) Mello, R. M. Q., 105(37) Mendez, E. E., 244, 244(113) Meng, S., 129(125) Meyer, J. R., 259(122) Meyer, T. A., 261(127) Mickelson, L., 124(102), 151(39) Mickevicius, J., 151 Mikel, S. E., 129(126) Mills, G3, 129(122) Minsky, M. S., 203(103) Mishra, U., 149(20) Mishra, U. K., 259(119) Mittal, V. O., 120(84) Miyajima, T., 149(16) Mo, Y., 136(143) Mochrie, S.G. J., 45(75), 45(76) Morel, A., 245(114)
AUTHOR INDEX Morgan, D., 117(81), 120(88) Morikawa, Y., 106(41) Morin, F. J., 154(59) Morkoc¸, H., 151(38), 244(113) Mostow, G. D., 26(52) Mott, N., 49(87) Motupally, S., 120(86) Moustakas, T. D., 149(24), 188(96), 244(110) Mukerjee, S., 133(137) Mu¨ller, E., 149(23) Mu¨ller, St. G., 170(82, 83) Mullins, W. W., 14(27), 26(50) Mun, B. S., 135(140) N Nabarro, F. R. N., 49(87) Nagahama, S., 148(13, 14) Nakamura, R., 149(19) Nakamura, S., 148(13, 14), 171(86) Nakasa, O., 152(46) Nakashima, H., 152(46) Narayanamurti, V., 149(20) Needs, R. J., 45(77) Nellist, P. D., 149(17) Netzel, C., 245(116) Neugebauer, J., 164, 164(72), 171(89) Neurock, M., 105(39), 106(40), 132(134) Neyerlin, K. C., 105(38) Ng, H. M., 149(24), 188(96) Nielsen, M. M., 53(108) Nilekar, A. U., 135(141), 137(144) Niwa, A., 260(123) Nix, W. D., 43(70), 60(122) Norskov, J. K., 94(20), 101(30), 106(42), 122(92), 123(98), 124(106), 129, 131, 139(151) Northrup, J. E., 171(89), 172, 172(90, 91) Northwood, D. O., 113(72) Nye, J. F., 75(132) O ¨ berg, S., 163(69), 164(73), 173(92) O Ohring, M., 60(123) Ohtoshi, T., 260(123)
269
Okamoto, Y., 106(41) Omne`s, F., 149(17) Onuma, T., 244(111) Oriani, R. A., 15(30) Otani, M., 106(41) Ovesen, C. V., 122(92) P Padovani, S., 124(107) Panels, J. E., 94(21) Park, S. S., 151(38) Parrinello, M., 128(119) Parthasarathy, A., 112(68, 69) Patriarche, G., 152(51) Paulus, U. A., 134(138) Pearson, G. L., 154(59, 60) Pederson, M. R., 128(118) Pennycook, S. J., 149(17) Perdew, J. P., 128(118) Perez, M., 123(94) Petroff, P., 152(44) Pinto, L. M. C., 126(114) Pippard, A. B., 1(7) Pitera, A. J., 203(104) Piu, P., 123(97) Pivovar, B. S., 89(13), 90(14) Po¨do¨r, B., 184, 184(93) Ponce, F. A., 150(26), 153(54), 171(86), 245(115) Porezag, V. D., 163(69) Pourbaix, M., 95(22) Prater, K. B., 78(3) Protsailo, L. V., 120(86) Pryor, C., 261(124) Q Qian, W., 150, 150(28), 171(87) Qiu, X. G., 151(34) Quaino, P. M., 97(24), 104(34) Queisser, H. J., 145(6–8), 153(53) R Radmilovic, V., 134(138) Raistrick, I. D., 112(65) Ralph, T. R., 78(4)
270
AUTHOR INDEX
Ram-Mohan, L. R., 259(122) Rand, A. J., 99(27) Rappe, A. M., 124(105) Rasmussen, F. B., 53(108) Rasmussen, P. B., 122(92) Razaq, A., 109(53) Razaq, M., 109(53) Read, W. T., 154(59) Read, W. T., Jr., 145(5), 147(11) Rebane, Y., 150(31) Reddy, A., 93(16) Redinger, J., 124(107) Redondo, A., 116(78) Ren, B., 101(29) Reschikov, M. A., 151(38) Rice, J. R., 26(47) Ringel, S. A., 203(104) Rodriguez, J. A., 123(94) Rodriguez-Rivera, G. J., 123(95) Roedel, R. J., 245, 245(117) Rohrer, S., 150(28) Rolle, K. C., 20(36) Romano, L. T., 149(21) Rosenhain, W., 145(1) Rosner, S. J., 241(107) Ross, P., 103(32) Ross, P. N., 98(26), 110(56), 112(64), 123(100), 125(108, 110), 134(138), 135(140) Rossinot, E., 120(83) Rossmeisl, J., 94(20), 101(30), 106(42) Rossow, U., 245(116) Rowland, L. B., 150(28), 171(87) Rusanov, A. I., 18(34), 46(86) Rutledge, G. C., 53(102) Ruud, J. A., 53(105) Ruvimov, S., 171(88) S Sagnes, I., 152(51) Saint-Girons, G., 152(51) Sakai, H., 148(12) Sakalauskas, S., 151(39) Sakurai, T., 151(34) Salvarezza, R. C., 103(31) Sanchez, E. K., 170(85) Sanchez, J.-Y., 120(83) Sa´nchez-Rojas, J. L., 151(33) Sandy, A. R., 45(76)
Sano, H., 123(96) Sano, T., 149(19) Sasaki, K., 135(141), 136(143), 137(144) Sato, N., 20(40) Saul, R. H., 244(108) Sautet, P., 124(104) Savinell, R. F., 111(61) Scarpellino, A. J., 125(108) Scherer, G. G., 134(138) Schmid, M., 124(107) Schmidt, T. J., 134(138) Schmidt-Rohr, K., 89(11) Scholz, F., 149(23) Schroeder, D. K., 259(120) Schro¨ter, W., 154(62, 63), 155(66) Schulz, H., 165(78) Schweitz, K. O., 53(108) Secanell, M., 105(35) Segawa, Y., 151(34) Seitz, F., 145(4) Sekerka, R. F., 14(27), 38(69) Selvan, M., 90(15) Sen, S., 152(47) Senoh, M., 148(13), 14 Sepa, D. B., 108(50–52) Sethuraman, V. A., 120(86) Sham, L. J., 128, 128(117) Shao, I., 53(107) Shao, M., 135(141) Shao, M.-H., 132(135) Shao-Horn, Y., 117(81), 120(88), 121 Sharma, P., 49(89, 90) Shen, J., 106(43) Sheng, W. C., 117(81) Shi, J. Y., 151, 151(32) Shi, Z., 127(115) Shin, M. W., 148(15) Shreter, Y., 150(31) Shur, M. S., 151(39), 259(121) Shuttleworth, R., 29(59) Sieradzki, K., 45(79), 52(99), 53(104), 53(110), 74(130, 131), 120(89), 138(146) Silva, E. R., 126(114) Simha, N. K., 29(63) Singh, D. J., 128(118) Sitch, P. K., 163(69), 164(73) Sivananthan, S., 149(17) Sizelove, J. R., 149, 149(25), 259(119) Skowronski, M., 150(28), 170(85), 171(87)
AUTHOR INDEX Skulason, E., 101(30), 106(42) Sleradzki, K., 46(81) Smith, A. J., 122(91) Smith, C. S., 26(49) Song, C., 88(9) Song, D., 106(43) Spaepen, F., 29(61), 52(99), 53(105), 57(119) Spagnol, P., 113(71) Speck, J. S., 244(111) Springer, T. E., 96(23), 112(67) Srinivasan, S., 111(63), 112(65, 66, 68, 69), 116(78, 79), 133(137) Srivastava, R., 139(149) Srolovitz, D. J., 60(124, 128) Stamenkovic, V., 134(138) Stamenkovic, V. R., 135(140) Stechel, E. B., 163(70) Steeds, J. W., 171(86) Steele, B. C. H., 78(2) Steele, W., 90(15) Steffensen, G., 122(92) Steigerwald, D. A., 153(54) Stonehart, P., 103(32), 112(64), 125(108) Strasser, P., 134(139), 139(149) Strausser, Y. E., 149(20) Street, R. A., 151(36) Streitz, F. H., 53(110) Strunk, H. P., 170(82, 83) Su, J., 123(97) Sugimoto, Y., 148(14) Sugino, O., 106(41) Suleman, A., 105(35) Sun, Q., 150(27) Suzuki, H., 148(12) Suzuki, T., 244(109) T Tajima, M., 152(50) Takahashi, M., 124(103) Takanami, S., 149(19) Taliercio, T., 245(114) Tamulaitis, G., 151(39) Tanahashi, T., 149(18) Tang, Y., 88(9) Tarasevich, M. R., 110(55), 140(152) Tarsa, E. J., 149(20) Taylor, C. D., 105(39)
271
Taylor, G. I., 145, 145(2) Terao, S., 149(19) Thewalt, M. L. W., 261(127) Thiemann, K. H., 165(78) Thompson, C. V., 57(114) Tian, F., 123(97) Tian, Z.-Q., 101(29) Ticianelli, E. A., 105(37), 116(78) Toba, R., 152(50) Toennies, J. P., 45(74) Tomiya, S., 149(16) Toney, M. F., 134(139) Tremiliosi-Filho, G., 126(114) Trew, R. J., 148(15) Trimble, T. M., 45(79, 80), 60(128) Trimm, D. L., 122(91) Tryk, D., 109(53) Tsai, C. T., 152(43) Tsao, J. Y., 52(97) Tschoegl, N. W., 6(18) Tse, J. S., 128(120) Turnbull, D., 57(112) U Uberuaga, B. P., 129(121) Uedono, A., 244(111) Ukai, T., 149(19) Uribe, F, 135(141) Uribe, F. A., 112(67) V Valerio, J. A., 135(141) Van de Walle, C. G., 164, 164(72) van der Merwe, J. H., 52(94), 52(96) van der Waals, J. D., 46(82) Vanamu, G., 152(42) Vanderbilt, D., 244(112) Varfolomeev, S. D., 140(152) Varga, P., 124(107) Vhakraborty, A., 244(111) Vielstich, W., 125(109) Villega, I., 124(101) Vineyard, G. H., 129(122) Vogel, W., 103(32) Voitl, T., 123(95) Vojnovic, M. V., 108(50–52) Vollath, D., 29(63)
272
AUTHOR INDEX
VonNeida, A. R., 245(117) Voorhees, P. W., 12(24) Vosko, S. H., 128(118) Vracar, L. M., 108(51, 52) Vukmirovic, M. B., 129(124), 135(141, 142), 136(143), 137(144) Vurgaftman, I., 259(122)
Woll, Ch., 45(74) Woo, Y. D., 152(48) Wood, D. L., III, 120(84) Woods, R., 99(27) Wright, A. F., 163(70) Wright, P. J., 150(26) Wu, D. T., 57(119) Wu, Y. F., 259(119)
W X Waag, A., 149(23) Wainright, J. S., 111(61) Waitz, T., 29(63) Wang, C.-Y., 114(74) Wang, E. G., 129(125) Wang, G., 135(140) Wang, H., 88(9), 106(43), 127(115) Wang, J. F., 150(27) Wang, J. X., 96(23), 119(82), 125(111, 112), 132(136), 135(141) Wang, K. L., 152(48) Wang, W., 151(35) Wang, Y., 109(54), 113(72), 117(80), 131(131, 133) Wang, Y. T., 150(27) Wang, Y. Z., 151(32) Warahina, M., 152(50) Washburn, J., 171(88) Watanabe, S., 203(103) Wayne, D. M., 120(84) Weaver, M. J., 124(101) Webber, H. C., 152(41) Weidner, J. W., 120(86) Weimann, N. G., 149, 149(24), 182, 188(96) Weisskopf, V. F., 185, 185(94) Weissmu¨ller, J., 53(106) Wessels, B. W., 151(37) Wheeler, A. A., 26(46) Wheeler, L. T., 49(90) Widom, B., 26(52) Wien, W., 46(83) Wilkinson, D. P., 106(43), 127(115) Will, F. G., 97(25) Wilson, M. S., 80, 80(6, 7) Wilt, D. M., 203(104) Winnacker, A., 170(82, 83) Wipf, D. O., 121(90) Witrouw, A., 53(105) Wokaun, A., 134(138)
Xian, H., 101(29) Xin, Y., 149(17) Xu, C., 123(97) Xu, S. J., 151(35) Xu, X., 101(29), 123(97) Xu, Y., 123(98), 129(124) Xue, Q. K., 151(34) Xue, Q. Z., 151(34) Y Yamada, N., 123(96), 203(103) Yamada, T., 148(14) Yamaguchi, H., 148(12) Yamasaki, M., 148(12) Yan, S. G., 94(21) Yang, H., 224(106) Yang, Y., 161(68) Yaropolov, A. I., 140(152) Yeager, E., 109(53), 111, 111(59, 60) Yi, G.-C, 151(37) You, J. H., 153(55–58) Young, W. T., 171(86) Youtsey, C., 149(21) Yu, P., 151(32) Yu, S. H., 114(73) Z Zaidi, S. H., 152(42) Zangwill, A., 26(51) Zaslavsky, A., 198(100) Zawodzinski, T. A., 120(84) Zehner, D. M., 45(75, 76) Zei, S., 125(109) Zeis, R., 115(77), 138(147) Zelanay, P., 140, 140(154) Zhang, G. Y., 151(32)
AUTHOR INDEX Zhang, H., 151(32) Zhang, J., 88(9), 106(43), 127(115), 129(124), 132(136), 135(141), 135(142), 136(143), 137(144) Zhang, J. C., 150(27) Zhang, S. M., 224(106) Zhang, Y., 140(153) Zhao, D. G., 224(106) Zhao, H., 123(97)
Zhou, J. M., 161(68) Zhou, Y. Q., 161(68) Zhu, J. J., 224(106) Zhu, X. A., 152(43) Zhu, X. Y., 164(71) Zhu, Y., 125(112) Zimmer, P., 53(109) Zinola, C. F., 103(31) Zu, Q., 121(90)
273
Subject Index
A Ab initio molecular dynamics, 109, 128, 131, 132f, 158 for edge dislocations, 152 of GaN, 165–169 Acoustic deformation potential, 259t Activation losses, 90 ORR and, 94 Adsorption, 30 of CO, 106 enthalpy of, 100 equilibrium and, 100 Gibbs adsorption equation, 30 surface energy and, 31–32 Langmuir, 103 ORR and, 108–109 for solid surfaces, 48 Temkin, 101, 108 Al. See Aluminum Aluminum (Al), 122 Amorphous solids, 13, 36 Anisotropic dislocation fields, 207–213 Anodic overpotential, 90, 94, 102 Arrhenius expression, 93 Availability, 19–22 free energy and, 23 in open systems, free energy and, 60 of solid-solid interfaces, 54–56 for surfaces, 39–42 in Lagrangian coordinates, 42–43 nonequilibrium and, 56 physical origin of, 43–46 B Band edge peaks, 150–152, 204, 225. See also Inhomogeneous band edge shifts band gap and, 228–235 carrier concentration and, 248 dislocation density and, 226–228 Band gap, 192 band edge peaks and, 228–235 Bessel function, 159, 183 B-G. See Bonch-Bruevich and Glasko Binary alloys, 133–135
Bipolar plates, 113 BL. See Blue luminescence Bloch wave function, 203 Blue luminescence (BL), 152 Boltzmann’s constant, 180, 203 Bonch-Bruevich and Glasko (B–G), 159 dislocations and, 174 electrostatic potential and, 213 mobility and, 192 Born-Oppenheimer assumption, 128 Burgers vectors, 54 edge dislocations and, 146, 206f, 208, 211 open-core screw dislocations and, 171 screw dislocations and, 237 Butler-Volmer equation (B-V), 92–94, 107 B-V. See Butler-Volmer equation C Cahn-Hilliard model, 28 Capillary effects, 32–33 elasticity and, 49 with fluids, 71 with multicomponent solids, 36–39 with solid-fluid surfaces, 45 with solids, 33–36 Carbon in catalyst layer, 116 Nafion and, 116–117 Pt and, 115–116 Carbon dioxide (CO2), 120 Carboxyl, 123 Carrier concentration, 147, 149, 151, 154 band edge peaks and, 248 dislocation density and, 230f electrostatic potential and, 229 Hall mobility and, 174 Catalyst layers carbon in, 116 curvature and, 120–121 degradation of, 119–122 in MEA, 114–117 Nafion in, 116 for PEM, 113–122 Pt and, 116 Cauchy stress tensor, 12, 68
275
276
SUBJECT INDEX
CH. See Crystal-field split-off holes Chemical equilibrium for fluids, 25–26 for multicomponent solids, 36–39 in solid-liquid interface, 33 surface energy and, 35 of thin films, 63 Chemical potential, 40f in fuel cells, 85 for multiphase systems, 40f nonhydrostatic stress state and, 18 in nucleation, 59 of solids, 17 surface stress and, 35 of surfaces, 56 Chemical potential of component i, 7 lattice sites and, 14 Chemical reactions, 3 equilibrium and, 9 Clausius’ relation, 5–6, 19 Closed systems availability for, 22 free energy for, 22–23 mechanical equilibrium in, 24 surface stress in, 41 thermal equilibrium in, 23–24 CO, 120 adsorption of, 106 HOR and, 124–127 Pt and, 123–124 Ru and, 125 Co, Pt and, 134 CO2. See Carbon dioxide Coal gasification, 123 Coherence of interfaces, 51–52 thin films and, 64 Common-ion effect, 134 Conduction bands, 192, 230f effective mass of, 260t Hamiltonian for, 194 Copper (Cu) Pt and, 119 WGS from, 122 Corrosion, 137 Crystal anisotropy, 74–75 Crystal-field split-off holes (CH), 194, 214, 215f screw dislocations and, 238–242 Crystalline solids, 13 interfacial region of, 36 Crystallographic surface orientation effect, 97 Cu. See Copper
Curvature catalyst layers and, 120–121 fundamental equation and, 69–71 D Damjanovic model, 110 Norskov’s model and, 131 d-band, 124 DFT and, 135 Dealloying, 134, 137–139 Pt and, 138–139 Debye screening length, 159, 183 Deformation plastic, 145 in solids, 64–69, 65f Deformation potential acoustic, 259t dislocation density and, 232 electrostatic potential and, 231 Deformation tensor, 12 Density functional method based on tight binding (DF-TB), 163 Density functional theory (DFT), 80 d-band and, 135 GGA and, 124, 128 ORR and, 127, 129 UPD and, 101 DFT. See Density functional theory DF-TB. See Density functional method based on tight binding Dielectric constant, 259t Diffusion potential, 14 lattice sites and, 14–15 of solids, 17 Dirichlet condition, 205 Dislocation. See also Edge dislocations; Screw dislocations B-G and, 174 GaAs and, 143–261 GaAs vs. GaN, optical properties of, 248–249 GaN and, 143–261 electrical properties of, 246–247 optical properties of, 247–248 in III-V materials, 244–249 misfit, 52, 55, 147f, 161t, 162t mixed, 146 optical properties and, 192–204 in semiconductors, 145 single electron model and, 192–204 ZB GaAs, 152–153 Dislocation density, 186
SUBJECT INDEX band edge peak and, 226–228 carrier concentration and, 230f deformation potential and, 232 drift mobility for, 190f, 192, 247 GaAs edge dislocations and, 217–220 GaN edge dislocations and, 217–220 Hall mobility and, 192, 247 screw dislocations and, 248 Dividing surface construction, for fluid surfaces, 69–71 d-orbitals, 131 Drift mobility, 182, 184 for dislocation density, 190f, 192, 247 scattering model and, 186 E EAM. See Embedded atom method Edge dislocations, 145, 146–148 ab initio molecular dynamics of, 152 Burgers vectors and, 146, 206f, 208, 211 as electron acceptors, 147–148 electron scattering and, 174–192 electrostatic potential of, 158–159 filling fraction of, 155–158 for GaAs, 204–236 dislocation density and, 217–220 for GaN, 204–236 ab initio molecular dynamics for, 165–169 atomistic structure of, 162–165 dislocation density and, 217–220 SO and, 216 negative charge of, 225 optical properties and, 204–236 piezoelectric effects and, 149 in semiconductors, 154–159 strain fields and, 149, 205–213 in WZ GaN, 149–150, 174–182 in ZB GaAs, 159–162 Effective surface pressure, 64 Eigenvalue problem, 197, 202 Elasticity capillary effects and, 49 surface strain and, 53 ZB GaAs and, 208–209 Electrical properties, of GaN, dislocations and, 246–247 Electrode reactions, kinetics of, 88–96 Electron effective mass, 259t Electron mobility, scattering model for, 182–186 Electron scattering, edge dislocations and, 174–192
277
Electron-electron energy, 249–256 Electrostatic potential B-G and, 213 carrier concentration and, 229 deformation potential and, 231 of edge dislocations, 158–159 Hall mobility and, 192 Hamiltonian for, 214 Embedded atom method (EAM), 45–46 Energy. See also Free energy conservation of, 5 equilibrium and, 8 internal, 4 entropy and, 6 minimum criteria for, 24 surface chemical equilibrium and, 35 for fluid surfaces, 44 Gibbs adsorption equation and, 31–32 Enthalpy, of adsorption, 100 Entropy, 6 availability and, 20 equilibrium and, 8 free energy and, 23 internal energy and, 6 Epitaxial stress, of thin films, 61–64 Equations of condition, 9 Equilibrium. See also Chemical equilibrium; Mechanical equilibrium adsorption and, 100 chemical reactions and, 9 criteria for, 8–12 energy and, 8 entropy and, 8 of fluids, 15–19 general deductive theory of, 2 for multiphase systems, 39t of solids, 15–19 thermal, in closed systems, 23–24 Euler equation, 7–8 Eulerian coordinates, 51, 67 Euler’s constant, 178, 179f Excess quantities, with surfaces, 25–26, 50 Exergy. See Availability External load, 83 F FEM. See Finite element method Fermi level, 155 Fermi–Golden rule, 246 Filled-core screw dislocations, 173t
278
SUBJECT INDEX
Filling fractions, 175 of edge dislocations, 155–158 Finite element method (FEM), 153 of k-p Hamiltonian, 198–202 Schro¨dinger equation with, 197 First law of thermodynamics, 4–5 Fluids capillary effects in, 71 chemical equilibrium for, 25–26 equilibrium of, 15–19 solids and Laplace pressure with, 33 mechanical equilibrium between, 32–33 surfaces of, 29 curved, 29 dividing surface construction for, 69–71 planar, 29 stability in, 31–32 surface energy for, 44 surface stress for, 44 Fourier transform, Schro¨dinger equation and, 195 Frank model, for open-core screw dislocations, 169–170 Frank-van der Merwe growth, 61–62 Free energy, 22–24, 180 availability and, 23 for closed systems, 22–23 entropy and, 23 in fuel cells, 83 minimum criteria for, 24 nucleation and, 59 open system availability and, 60 ORR and, 108–109 Pt and, 131 water and, 88 Fuel cells, 78–140. See also Oxygen reduction reaction; Proton exchange membrane ammeter in, 84 chemical potential in, 85 free energy in, 83 resistors in, 84 thermodynamics of, 82–112 water in, 87 Full-core screw dislocations, 173t LDF for, 171 Fundamental equation, 2, 7 curvature and, 69–71 volume and, 68–69 G GaAs band edge peaks and, 226–228
dislocations and, 143–261 edge dislocations and, 204–236 dislocation density and, 217–220 vs. GaN, optical properties of, 248–249 HH and, 216f LH and, 216f optical properties of, 204–236 photoluminescence in, 152–153 SO and, 216f spontaneous emission spectrum calculation for, 225 Gallium, 166–169 screw dislocations and, 172 GaN. See also Wurtzite dislocations and, 143–261 edge dislocations and, 204–236 ab initio molecular dynamics for, 165–169 electrical properties of, dislocations and, 246–247 electron scattering calculations for, 259t hexagonal open-core screw dislocations in, 170–174 optical properties of, 204–236 dislocations and, 247–248 photoluminescence in, 151–152 spontaneous emission spectrum calculation for, 220–224 YL in, 162 Gas diffusion layer (GDL), 104–105, 114 Gases, equilibrium state of, 3 Gaussian distribution, 203, 242 General deductive theory, 2 Generalized gradient approximation (GGA), 166 DFT and, 124, 128 GGA. See Generalized gradient approximation Gibbs adsorption equation, 30 surface energy and, 31–32 Gibbs-Duhem equation, 7–8 nucleation and, 59 pressure and, 31 Gibbs-reversibility. See Reversible variation Gibbs-Thomson effects, 120–121 on solids, 71–74 Gibbs-Thomson-Freundlich effects, 56 GL. See Green luminescence Gold, oxides and, 123 Gouy-Chapman model, 92 Gradient thermodynamic methods, 28 Graphite, 113 Green luminescence (GL), 152 Griffiths model, 109, 109f
SUBJECT INDEX H H2O2. See Peroxide H3Oþ. See Hydronium ion Hall mobility, 174, 184, 190f carrier concentration and, 174 dislocation density and, 192, 247 electrostatic potential and, 192 Hamiltonians, 192. See also k-p Hamiltonian for conduction bands, 194 for electrostatic potential, 214 for ZB GaAs, 196–198 hcp. See Hexagonal close-packed Heat exchange, 16 reversible processes with, 25 Heavy holes (HH), 194, 214, 215f GaAs and, 216f open-core screw dislocations and, 248 screw dislocations and, 238–242 Helmholtz free energy. See Free energy Hexagonal close-packed (hcp), 147 Hexagonal open-core screw dislocations in GaN, 170–174 strain fields and, 237 Heyrovsky-Volmer sequence, 96, 104–105 HH. See Heavy holes HOR. See Hydrogen oxidation reaction Hydrogen, 82 Pt and, 102–107 UPD and, 103 weakly bound, 103 Hydrogen oxidation reaction (HOR), 78 anodic overpotential and, 102 CO and, 124–127 improvement of, 122–127 kinetics of, 96–97 Pt and, 106, 127 Tafel slope and, 127 Hydronium ion (H3Oþ), 89 Hydrophobic perflorinated hydrocarbon, 88–89 Hydroxides, 108 ORR and, 131–132 water and, 111 I III-V materials, dislocation in, 244–249 Inhomogeneous band edge shifts in screw dislocations, 238 in WZ GaN, 213–215 in ZB GaAs, 215–217
279
Interface coherence, 51–52 Interface stress physical origin of, 54–56 surface work and, 55 Interfacial region, 26 of amorphous solids, 36 of crystalline solids, 36 pressure in, 29, 46 Internal energy, 4 entropy and, 6 minimum criteria for, 24 Intrinsic stress, 60–61 Ir, 134 Isolated system, 8 Isotropic dislocation fields, in WZ GaN, 205–207 Isotropic surface, surface stress of, 32 K Kinetics of electrode reactions, 88–96 of HOR, 96–97 of ORR, 107–112 Kohn-Sham equations, 128 Korteweg-van der Waals model, 28, 29 k-p Hamiltonian, 193–198 FEM of, 198–202 WZ GaN and, 193–195 z-direction and, 218 Kronecker delta, 42 k-space, Schro¨dinger equation and, 195 L Lagrangian coordinates, 67, 128 surface availability in, 42–43 Langmuir adsorption, 103 Laplace pressure, 29, 34 with solid-liquid interface, 33 Lattice constant, 260t, 261t Lattice sites, 13 chemical potential of component i and, 14 diffusion potential and, 14–15 reversible variation and, 16 Layer quantities, 46–48 LCAO. See Linear combination of atomic orbitals LDA. See Local density approximation LDF. See Local-density functional Leonard-Jones potentials, 90 LH. See Light holes
280
SUBJECT INDEX
Light holes (LH), 194, 214, 215f GaAs and, 216f open-core screw dislocations and, 248 screw dislocations and, 238–242 Linear combination of atomic orbitals (LCAO), 154 Linear defects, 144–145, 146–147 Liquids equilibrium in, 3 solids and chemical equilibrium in, 33 mechanical equilibrium between, 32–33 vapors and, 28 Local density approximation (LDA), 128 Local-density functional (LDF), 163 for full-core screw dislocations, 171 Luttinger parameters, 261t M Mass, 13 density, 259t MD. See Molecular dynamics MEA. See Membrane electrode assembly Mechanical equilibrium, 5, 16 in closed systems, 24 between solid and fluid, 32–33 for surfaces, 26 Membrane electrode assembly (MEA), 79, 79f catalyst layer in, 114–117 in PEM, 113 MEP. See Minimum energy path Methanol, 122–123 Minimum energy path (MEP), 129 Minimum free energy criteria, 24 Minimum internal energy criteria, 24 Misfit dislocations, 52, 55, 147f, 161t, 162t Mixed dislocations, 146 Mixtures, 3 Mn, 125 Mobile components, 15 Mobility. See also Drift mobility; Hall mobility B–G and, 192 of electrons, scattering model for, 182–186 Read potential and, 186 scattering model and, 187f Molecular dynamics (MD), 80 Multicomponent solids capillary effects with, 36–39 chemical equilibrium for, 36–39 Multiphase systems, 3
equilibrium for, 39t pressure of, 40f temperature of, 40f N Nafion, 88–92, 111–112 carbon and, 116–117 in catalyst layer, 116 degradation of, 119–120 PEM and, 112 Pt and, 120 NEB. See Nudged elastic band Nernst potential, 94 Neutron scattering, 114 Ni, 130 Pt and, 134 Non-diffuse fluid surfaces, 3 Nonequilibrium flow processes, 22 surface availability and, 56 Nonhydrostatic stress state chemical potential and, 18 of solids, 12 Nonuniform systems, 2 Norskov’s model, Damjanovic model and, 131 Nucleation chemical potential in, 59 free energy and, 59 Gibbs-Duhem equation and, 59 reversible processes in, 58 during solidification, 57–60 Nudged elastic band (NEB), 128 O O2. See Oxygen OCV. See Open circuit voltage OH, 130 ORR and, 131 Ohmic over-potential, 87 OOH, 131–132 Open circuit voltage (OCV), 87–88, 92 Open systems, availability for, 21 free energy and, 60 Open-core screw dislocations, 173t Burgers vectors and, 171 Frank model for, 169–170 GaN optical properties and, 236–244 hexagonal in GaN, 170–174
SUBJECT INDEX strain fields and, 237 HH and, 248 LH and, 248 Optical polar phonon scattering, scattering relaxation time for, 185 Optical properties calculations for, for ZB GaAs, 261t dislocation and, 192–204 edge dislocations and, 204–236 of GaAs, 204–236 of GaN, 204–236 dislocations and, 247–248 open-core screw dislocations and, 236–244 ORR. See Oxygen reduction reaction Oxides, 108 gold and, 123 Pt and, 110 Oxygen (O2), 130 binding of, 130f ORR and, 131 Pt and, 108, 122 reduction of, 130f Oxygen reduction reaction (ORR), 78 activation losses and, 94 adsorption and, 108–109 anodic overpotential and, 94 calculations for, 129–133 development of, 139–140 DFT and, 127, 129 hydroxides and, 131–132 improvement of, 127–139 kinetics of, 107–112 new metallic catalysts for, 133 O2 and, 131 OH and, 131 PEM and, 111 pH and, 108 properties of, 107–108 Pt and, 130 P Pauling model, 109f PAW. See Projector-augmented wave method Pb, 125 PEM. See Proton exchange membrane Perflorinated hydrocarbon, 88–89 Periodic density functional theory, 105 Peroxide (H2O2), 120 pH, 94 ORR and, 108 Phase transitions, 3
281
Phosphoric acid, 111 Photoluminescence in GaAs, 152–153 in GaN, 151–152 Piezoelectric constant, 259t Piezoelectric effects edge dislocations and, 149 scattering relaxation time and, 185 strain fields and, 149 Planck’s constant, 183 Plastic deformation, 145 Platina, 79–80 Platinum (Pt), 79–80 carbon and, 115–116 catalyst layer and, 116 CO and, 123–124 Co and, 134 Cu and, 119 dealloying and, 138–139 electrochemistry of, 97–102 free energy and, 131 HOR and, 106, 127 hydrogen and, 102–107 Nafion and, 120 Ni and, 134 O2 and, 108, 122 ORR and, 130 oxides and, 110 for PEM, 117 skin electrocatalysts, 135–137 Tafel slope and, 105, 127 TEM of, 118 Point defects, 144 Poisson ratio, 205 Poisson’s equation, 178 Polar phonon Debye temperature, 259t Polarization, 203 Polymer electrolyte membrane. See Proton exchange membrane Pourbaix diagram, 95, 95f, 122 Pressure Gibbs-Duhem equation and, 31 in interfacial region, 29, 46 of multiphase systems, 40f PEM and, 85 of vapors, 74 Projector-augmented wave method (PAW), 165 Proton exchange membrane (PEM), 79f, 81 catalyst layer in, 113–122 development of, 139 loss in, 88–89 MEA in, 113
282
SUBJECT INDEX
Proton exchange membrane (PEM) (Continued ) Nafion and, 112 ORR and, 111 pressure and, 85 Pt for, 117 temperature and, 85 water and, 113 Pseudosplitting model, 110 Pt. See Platinum Pt3Ni, 134–135 Q Quasi-static, 5 R Read potential, 159, 182, 189f mobility and, 186 Resistors, in fuel cells, 84 Reversible processes, 5, 9–10 with heat, 25 in nucleation, 58 Reversible variation, 10–12 lattice sites and, 16 surfaces and, 28–29 thin films and, 62 Ru, CO and, 125 S Sb, 125 Scattering of electrons, edge dislocations and, 174–192 model for drift mobility and, 186 for electron mobility, 182–186 mobility and, 187f total drift and, 186 of neutrons, 114 relaxation time, 185 for optical polar phonon scattering, 185 piezoelectric effects and, 185 time, 184 Schro¨dinger equation, 128, 192 with FEM, 197 Fourier transform and, 195 k-space and, 195 Screw dislocations, 145. See also Open-core screw dislocations Burgers vectors and, 237 CH and, 238–242
dislocation density and, 248 filled-core, 173t full-core, 173t LDF for, 171 Ga and, 172 HH and, 238–242 inhomogeneous band edge shifts in, 238 LH and, 238–242 strain fields and, 237–238 in WZ GaN, 150–151 types of, 173t Second law of thermodynamics, 5 Semiconductors dislocations in, 145 edge dislocations in, 154–159 Si, 166 Single electron model, dislocation and, 192–204 Slip bands, 145 Small strain approximation, 67–68 Small strain tensor, 12 Sn, 126 SO. See Split-off Solid(s) amorphous, 13, 36 capillary effects with, 33–36 chemical potential for, 17 crystalline, 13 interfacial region of, 36 deformation in, 64–69, 65f diffusion potential of, 17 equilibrium of, 3, 15–19 Gibbs-Thomson effects on, 71–74 liquids and chemical equilibrium in, 33 Laplace pressure with, 33 mechanical equilibrium between, 32–33 multicomponent capillary effects with, 36–39 chemical equilibrium for, 36–39 nonhydrostatic stress state of, 12 surfaces of, adsorption for, 48 thermodynamics of, 12–19 Solid-fluid surfaces, capillary effects for, 45 Solidification, nucleation during, 57–60 Solid-liquid interface, surface stress of, 33 Solid-solid interfaces, 49–54 availability of, 54–56 Speed of sound, 259t Split-off (SO), 194. See also Crystal-field split-off holes GaAs and, 216f GaN edge dislocations and, 216
SUBJECT INDEX Spontaneous emission spectrum calculation, 202–203 experimental comparison of, 242–243 for GaAs, 225 for GaN, 220–224 Spring constant, 45 Strain fields, 12, 49–50 edge dislocations and, 149, 205–213 elastic, 53 hexagonal open-core screw dislocations and, 237 piezoelectric effects and, 149 screw dislocations and, 237–238 of surfaces, 53 Sulfonic acid, 89 Sulfuric acid, 111 Surfaces availability for, 39–42 in Lagrangian coordinates, 42–43 nonequilibrium and, 56 physical origin of, 43–46 chemical potential of, 56 energy of chemical equilibrium and, 35 for fluid surfaces, 44 Gibbs adsorption equation and, 31–32 excess quantities with, 25–26, 50 of fluids, 29 mechanical equilibrium for, 26 reversible variation and, 28–29 strain fields of, 53 stress, 29 chemical potential and, 35 in closed systems, 41 fluid surfaces for, 44 of isotropic surface, 32 physical origin of, 43–46 of solid-liquid interface, 33 on thin films, 60–64 thermal equilibrium for, 25 thermodynamics of, 3–4, 24–56 uniform phases and, 26 work of, interface stress and, 55 T Tafel slope, 93, 93n HOR and, 127 Pt and, 105, 127 water and, 113 Tafel-Volmer sequence, 96, 104–105 Teflon, 114 TEM. See Transmission electron microscopy
283
Temkin adsorption, 101, 108 Temperature of multiphase systems, 40f PEM and, 85 Thermal equilibrium, 16, 34 in closed systems, 23–24 for surfaces, 25 thin films and, 62 Thermodynamics of fuel cells, 82–112 laws of, 4–6 of solids, 12–19 of surfaces, 3–4, 24–56 systems of, 2 Thin films, 50f chemical equilibrium of, 63 coherence and, 64 critical thickness for, 74–75 epitaxial stress of, 61–64 reversible variation and, 62 surface stress on, 60–64 thermal equilibrium and, 62 Thomas-Fermi screening, 183 Total drift, 174 scattering model and, 186 Transfer coefficient, 93 Transmission electron microscopy (TEM), 117 of Pt, 118 U UHV. See Ultra-high vacuum Ultra-high vacuum (UHV), 135 Ultra-soft pseudopotentials (US-PP), 165 Ultraviolet photoemission spectroscopy (UPS), 135 underpotential deposition (UPD), 97–101 DFT and, 101 hydrogen and, 103 strongly bound, 103 Uniform phases, 24 surfaces and, 26 UPD. See underpotential deposition UPS. See Ultraviolet photoemission spectroscopy US-PP. See Ultra-soft pseudopotentials V Valence bands, 192, 230f effective mass parameters for, 260t Vapors
284
SUBJECT INDEX
Vapors (Continued ) liquids and, 28 pressure of, 74 VASP. See Vienna ab initio simulation package Vienna ab initio simulation package (VASP), 163 Volmer-Weber growth mode, 61 Volume, 5, 9, 13 fundamental equation and, 68–69 Vulcan XC72, 115
X XRD, 133–134 Y Yellow luminescence (YL), 151 in GaN, 162 YL. See Yellow luminescence
W Z Water free energy and, 88 in fuel cells, 87 hydroxides and, 111 management of, 114 PEM and, 113 Tafel slope and, 113 Water-gas shift reaction (WGS), 122–123 WGS. See Water-gas shift reaction Wurtzite (WZ), 147 GaN, 148–152 edge dislocations in, 149–150, 174–182 inhomogeneous band edge shifts in, 213–215 isotropic dislocation fields in, 205–207 k-p Hamiltonian and, 193–195 optical property calculations for, 260t screw dislocations in, 150–151, 173t WZ. See Wurtzite (WZ)
ZB. See Zinc-blende z-direction, 205 k-p Hamiltonian and, 218 Zeroth law of thermodynamics, 25 Zero-valued Dirichlet condition, 205 Zinc-blende (ZB), 147 GaAs anisotropic dislocation fields in, 207–213 dislocation in, 152–153 edge dislocations in, 159–162 elasticity and, 208–209 Hamiltonian for, 196–198 inhomogeneous band edge shifts in, 215–217 optical property calculations for, 261t
