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Electron wavefunctions at crystal interfaces Patitsas, Stathis Nikos 1990

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ELECTRON WAVEFUNCTIONS AT CRYSTAL INTERFACES By Statins Nikos Patitsas B. Sc. (Physics) Laurentian University A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July 1990 © Stathis Nikos Patitsas, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract A one dimensional analysis of the boundary conditions of the electron energy eigenfunc-tion at a sharp interface between two crystals was made. An attempt to evaluate these conditions in terms of known band structure was made. It was concluded that this cannot be done in general. It was shown, however, that if the interface has the proper symmetry properties, the boundary conditions can be expressed in terms of only one unknown, energy-dependent parameter. It was concluded that setting this parameter equal to one gives boundary conditions which, though more general, are equivalent to the commonly used effective mass boundary conditions when they are applicable. It was concluded from numerical results for the transmission coefficient of the symmetric interface, that in general, these boundary conditions, which depend only on known band structure, do not give a good approximation to the exact answer. Since the energy dependence of the parameter mentioned above is described quite well qualitatively using the nearly free electron approximation or the tight-binding approximation, the applicability of any boundary conditions depending only on band structure can be predicted using these simple theories. The exact numerical results were calculated using the transfer matrix method. It was also concluded that the presence of symmetry in the interface either maximizes or minimizes the transmission coefficient. A tight-binding calculation showed that the transmission coefficient depends on an interface parameter which is independent of band structure. The transmission coefficient is maximized when this parameter is ignored. It was concluded that the effective mass equation is of little use when applied to this problem. Some transfer matrix results pertaining to the barrier and the superlattice were obtained. n Table of Contents Abstract ii List of Figures vi Acknowledgement xii 1 Introduction 1 1.1 Independent Electron Approximation 2 1.2 Single Channel Scattering and Specular Reflection 3 1.3 The Interface Matrix 4 1.4 One Dimensional Analysis 9 2 The Transfer Matrix Method 12 2.1 The Infinite Homogeneous Crystal 12 2.1.1 Transmission Matrix 12 2.1.2 Transfer Matrix 16 2.1.3 | X(a) |< 1 : Allowed Band 17 2.1.4 | X(a) \> 1 : Forbidden Band . . 19 2.2 The Interfere 21 2.2.1 Interface Matrix 21 2.2.2 Transmission Coefficient 24 2.2.3 Symmetry Properties 26 2.3 The Barrier 30 iii 2.3.1 Barrier Matrix 30 2.3.2 Transmission Coefficient 32 2.3.3 Bound States 35 2.4 The Superlattice 36 3 Analysis of Sharp Interface 40 3.1 Simplified Interface 42 3.2 Properties of the Transmission Coefficient: Maxima and Minima 47 3.3 Symmetry Properties of the Transmission Coefficient 48 3.4 Symmetry Results for a General Sharp Interface 50 3.5 The Ratios r and s 52 4 The Square Well Potential 55 4.1 Transmission Matrix 55 4.2 Results pertaining to the Barrier and the Superlattice 61 4.3 The Interface 62 4.4 The Barrier 65 5 Numerical Results 66 5.1 Elementary Numerical Results For Infinite Homogeneous Crystal 67 5.2 Numerical Results With the Interface 70 5.2.1 70 5.2.2 76 5.2.3 79 5.2.4 82 5.2.5 87 5.2.6 90 iv 5.2.7 93 5.2.8 96 5.3 Numerical Results for the Barrier 99 5.3.1 99 5.3.2 102 6 Tight-Binding Approximation 104 7 Effective Mass Formalism 115 7.1 Slowly Varying Potential 115 7.2 Sharply Varying Potential 117 8 Conclusions 127 Bibliography 131 Appendices 134 A General Constraints on the Interface Matrix 134 B Derivation of Results of Section 2.1 139 B.l Case 1: \X\ < 1 139 B. 2 Case 2: |X| > 1 142 C Derivation of Transmission Coefficients 145 C. l Transmission Coefficient of Interface 145 C.2 Transmission Coefficient of Special Barrier 146 D On the Independence of Energy Origin 149 v List of Figures 1.1 Schematic picture of a two dimensional interface divided into three regions; the interface region and the two bulk regions. The case of single channel scattering and specular reflection is shown 6 2.1 An arbitrary periodic potential, V(x), which is broken up into unit cells labelled by j 13 2.2 The potential V'(x) 15 2.3 The potential V"{x) 16 2.4 Schematic picture of the interface. The potentials on either side are peri-odic but otherwise arbitrary. The partitioning into unit cells labelled by j is shown 21 2.5 In (a), we have an interface with the unit cells on both sides being sym-metric. In (b), the position of the interface is moved left a distance x' relative to the crystals. In (c), the interface is moved right a distance x'. The transmission coefficients for cases (b) and (c) are the same 28 2.6 In (a), we have an interface with the unit cell on the right side being symmetric. To simulate lattice distortion, the "ions" on the left side are pushed to the right a distance x' in (a). In (b) the "ions" on the left side are pushed to the left the same distance x'. The transmission coefficients for the two cases are the same 29 2.7 A schematic of a barrier with N = 3. A mismatch is present (q ^  0). . . 31 vi 2.8 A schematic of a superlattice with Nx = 3 and = 2. The structure shown here is simply repeated to form the superlattice. The superlattice spacing is s. The potentials of both materials are periodic but otherwise arbitrary. The unit cells are labelled by j. Mismatches are present. . . . 36 3.1 Schematic picture of the interface composed of a periodic potential Vi(a;) plus the potential Vb 0(x — xo) 43 4.1 The square well periodic potential, V(x), which is broken up into unit cells labelled by j 56 4.2 The potential, V'(x), of a unit cell, for which we determine the transfer matrix. The shifting of the x axis is not important 56 4.3 Schematic picture of the interface. The potentials on either side are square and periodic. The partitioning into unit cells labelled by j is shown. . . . 63 4.4 A schematic of a barrier with N = 3. The potentials on either side are square and periodic. The partitioning into unit cells labelled by j is shown. A mismatch is present (q ^ 0) 65 5.1a A plot of X vs. (aa)2/2. Parameters are Vfa2 = 60h2/m , Vi = Vj , d/a = 0.75 69 5.1b A bandstructure plot of ka/n vs. (aa)2/2 for a homogeneous crystal. . . . 69 5.2.1a Bandstructures of the two crystals forming the interface. Parameters are ax = a2 = a, Vjxa2 = 8h2/m, Vf2a2 = 9.5h2/m, = Vi2 = V/1 5 di/a = d2/a = 0.4 73 5.2.1b Transmission coefficient of interface vs. energy. The interface parame-ters are b\ja = 0.3 (curve 1), 6x/a = 0.15 (curve 2), bi/a = 0 (curve 3), where b\ + b'2 = 0.6a in all three cases. Curve 4 is the PFEFM calculation. 73 vii 5.2.1c Transmission coefficient of interface vs. interface position. b\ + b'2 is kept at the constant value of 0.6a. mEa2/h2 =5.75 for curve 1 and 6.8 for curve 2 74 5.2.Id The ratios r and s vs. energy. The solid (dashed) curve represents r (s). 74 5.2. le Energy eigenfunction near the interface. Parameters are those of Fig-ure 5.2.1b, curve 1, at energy, mEa2/tt2 = 6.8. The particle is incident from the left. The solid (dashed) curve is the real (imaginary) part of the wavefunction and the interface is at x = 0 75 5.2.If Energy eigenfunction near the interface. Parameters are those of Fig-ure 5.2.1b, curve 1, at energy, mEa2/h2 = 6.8. The particle is incident from the left. Curve 1 (2) is the real (imaginary) part of the incident u(x), curve 3 (4) is the real (imaginary) part of the reflected u(x), and curve 5 (6) is the real (imaginary) part of the transmitted u(x) 75 5.2.2a Bandstructures of the two crystals forming the interface. Parameters are ax = a2 = a, V^a2 = 8h2/m, V/2a2 = 16ft2/m, = V{2 = V/ , , di/a = d2/a = 0.4 77 5.2.2b Transmission coefficient of interface vs. energy. The interface parame-ters are bi/a = 0.3 (curve 1), b\/a = 0.15 (curve 2), b\/a = 0 (curve 3), where bi + b'2 = 0.6a in all three cases. Curve 4 is the P F E F M calculation. 77 5.2.2c Transmission coefficient of interface vs. interface position. &i + 6 2iskept constant at 0.6a. mEa2/h2 =12.5 for curve 1 and 14.0 for curve 2. . . . 78 5.2.2d The ratios r and s vs. energy. The solid (dashed) curve represents r (s). 78 5.2.3a Bandstructures of the two crystals forming the interface. Parameters are ax = a2 = a, Vha2 = 8/i 2/m, Vha2 = 25.257i2/m, Vj, = V,-2 = Vfl1 di/a = d2ja = 0.4 80 viii 5.2.3b Transmission coefficient of interface vs. energy. The interface parame-ters are bx/a = 0.3 (curve 1), bx/a = 0.15 (curve 2), bi/a = 0 (curve 3), where bi + b'2 = 0.6a in all three cases. Curve 4 is the P F E F M calculation. 80 5.2.3c The ratios r and s vs. energy. The solid (dashed) curve represents r (s). 81 5.2.4a Bandstructures of the two crystals forming the interface. Parameters are ax = a2 = a, Vha2 = 11.85ft2/m, Vha2 = 8h2/m, Vh = V-2 = 12fc 2/ma 2, dx/a = 0.4, d2/a = 0.25 84 5.2.4b Transmission coefficient of interface vs. energy. The interface param-eters are bx/a = 0.3 (curve 1), bx/a = 0.15 (curve 2), bx/a = 0 (curve 3), where bx + b'2 = 0.675a in all three cases. Curve 4 is the P F E F M calculation. A l l four curves practically overlap 84 5.2.4c Transmission coefficient of interface vs. interface position, bx is kept constant at 0.3a. mEa2/h2 = 9 for curve 1 and 10.5 for curve 2 85 5.2.4d The ratios r and s vs. energy. The solid (dashed) curve represents r (s). 85 5.2.4e Energy eigenfunction near the interface. Parameters are those of Fig-ure 5.2.4b, curve 1, at energy, mEa2/%2 = 9. The particle is incident from the left. Curve 1 (2) is the real (imaginary) part of the incident u(x), curve 3 (4) is the real (imaginary) part of the reflected u(x), and curve 5 (6) is the real (imaginary) part of the transmitted u(x) 86 5.2.5a Bandstructures of the two crystals forming the interface. Parameters are ax = a2 = a, Vh - — 8^ 2 /ma 2 , V / 2 a 2 = 14ft2/m, Vi2a2 = 12/i 2/m, dx/a = 0.4, d 2 /a = 0.25 88 5.2.5b Transmission coefficient of interface vs. energy. The interface parame-ters are bx/a = 0.3 (curve 1), bx/a = 0.15 (curve 2), bx/a = 0 (curve 3), where bx + b'2 = 0.675a in all three cases. Curve 4 is the P F E F M calculation. 88 5.2.5c The ratios r and 5 vs. energy. The solid (dashed) curve represents r (s). 89 ix 5.2.6a Bandstructures of the two crystals forming the interface. Parameters are a2 = 1.5a!, Vh = Vh = 31h2/mal, Vha\ = 25.2h2/m, Vi2a\ = 23k2/m, dx = 0.6ai, d2 = 0.4a2 91 5.2.6b Transmission coefficient of interface vs. energy. The interface parame-ters are b\ = 0.2ai (curve 1), b\ = 0.05ai (curve 2), where b\ + b'2 = 0.65ai in both cases. Curve 3 is the PFEFM calculation 91 5.2.6c The ratios r and s vs. energy. The solid (dashed) curve represents r (s). 92 5.2.7a Bandstructures of the two crystals forming the interface. Parameters are a2 = 1.5oi, Vh = Vh = 31h2/rna2, Vha\ = 28.78fi2/ro, Vi2a2 = 23K2/m, di = 0.6ai, d2 = 0.4a2 94 5.2.7b Transmission coefficient of interface vs. energy. The interface parame-ters are b\ = 0.2ai (curve 1), &i = 0.05ai (curve 2), where b\ + b'2 = 0.65ai in both cases. Curve 3 is the PFEFM calculation 94 5.2.7c The ratios r and s vs. energy. The solid (dashed) curve represents r (s). 95 5.2.8a Bandstructures of the two crystals forming the interface. Parameters are a2 = 1.5a!, Vh = Vh = 3lh2/ma2, Vha\ = 29.78/L2/m, Vi2a2 = 23%2/m, da = 0.6ai, d2 = 0.4a2 97 5.2.8b Transmission coefficient of interface vs. energy. The interface parame-ters are &i = 0.2ai (curve 1), &i = 0.05ai (curve 2), where b\ + b'2 = 0.65ai in both cases. Curve 3 is the PFEFM calculation 97 5.2.8c The ratios r and s vs. energy. The solid (dashed) curve represents r (s). 98 5.3.1a Transmission coefficient of barrier vs. energy. The bulk parameters are d = a2 = a3 = a, Vh = V t l = Vi2 — V}3 = Vl3 = SU2/ma2, Vf2ma2 — 16ft2, di = d2 = dz = 0.4a. The interface parameters are &i = b2 = b3 = 0.3a and q = 0. N = 3. The solid curve is the actual value and the dashed curve is the PFEFM calculation. 101 x 5.3.1b Transmission coefficient of barrier vs. energy. There is now a mismatch; 5.3.2a Transmission coefficient of barrier vs. energy. The bulk parameters are G l = a2 = a3 = a, Vh = Vh = 11.85ft2/ma2, Vh = Vit = 8h2/ma2, Vf2 = Vi2 = I2h2/ma2, d\ = d3 = 0.4a, and d2 = 0.25a. The constant interface parameters are b\ = 63 = 0.3a and q = 0.25a. iV = 3. Curve 1 is for b'2 = 0.375a and curve 2 is for b'2 = 0.5a. The dashed curve is the PFEFM calculation which is the same for both values of b'2 103 6.1 Schematic picture of the interface composed of a periodic potential V(x) plus the potential Uo 0(x — xo). The partitioning into unit cells labelled by m is shown. This choice is arbitrary and does not depend on the value 7.1 An example of a barrier composed of an arbitrary periodic potential, V(x), q = 0.4a 101 of X Q 105 plus the barrier potential U(x) 117 xi Acknowledgement I would like to thank Dr. R. Barrie for his supervision and guidance throughout this study. Thanks are also due to Dr T. Tiedje for stimulating my interest in this research and to my wife Cathy for all her support. My daughter Elizabeth had an influence also. xn Chapter 1 Introduction This thesis is a study of the electronic properties of interfaces between crystals. The study is motivated by the recent technological advances made in the fabrication of inter-faces, in particular the technique of molecular beam epitaxy. Using this technique one is able to grow material on a surface one monolayer at a time. A common example is the growth of GaAs and AlGaAs. Interfaces between these materials can be produced which are perfectly abrupt and in which the lattice matching is preserved. The interface is defined as the meeting place of the two materials. The new technology makes this definition quite precise. By producing two or more parallel interfaces one produces what is in general called a heterostructure. A quantum well is produced by two parallel inter-faces and has the capability of confining an electron between the interfaces so that the electron is essentially confined to two dimensions. By creating a series of many parallel interfaces, evenly spaced, and of alternating materials one produces what is called a su-perlattice. It is clear that the interface plays a key role in the electronic properties of all these structures. Some of the problems of interest are; determining the current-voltage characteristic across a heterostructure, determining the band offsets across an interface, determining the discrete "subband" spectrum of a quantum well, and determining the "mini" bandstructure of a superlattice. In order to solve these problems satisfactorily a proper theoretical treatment of the electronic wavefunction at the interface is essential. Consider a single interface. We divide the physical system into three regions with boundaries parallel to the interface. See Figure 1.1. The x-axis is perpendicular to the 1 Chapter 1. Introduction 2 interface while the y-z plane is parallel to the interface. There is an "interface" region, containing the interface and there are two regions of bulk material on either side. The boundary between a bulk region and the interface region is denned as the point where a charge carrier, approaching the interface from a bulk region, begins to feel a substantial potential or interaction which is not present in the bulk material. For example, in a p-n junction the boundary is the edge of the depletion zone, which can be quite far from the interface. We are primarily concerned here with the "sharp" interface where the width of the interface region is about a lattice constant or less. In fact, for the model of the interface used in this study, the interface region has zero width, that is, the potential change due to the interface is modelled as a perfectly abrupt step function. Unless indicated otherwise, all the interfaces dealt with here are sharp. 1.1 Independent Electron Approximation All the electronic properties of the interface can be determined if one knows the electronic wavefunction of the system. Throughout this thesis, the independent electron assumption is made. That is, we treat the interaction of an electron with all the other electrons in the system as an averaged, "smeared out", potential. The electron-electron interaction can be important near an interface when there is charge transfer across the interface. This suggests that a better treatment of the electron-electron interaction might be necessary for some interfaces. This is not dealt with here, however. Even with the independent electron approximation one cannot solve for the electron wave function since the actual form of the independent electron potential is unknown. Although this means that the wavefunction in the interface region is completely unknown, we do know that in the bulk regions the wavefunctions must behave like Bloch functions. These Bloch functions may be generalized to evanescent Tamm states, having complex Chapter 1. Introduction 3 wavevector component perpendicular to the interface, as long as they are bounded.1 1.2 Single Channel Scattering and Specular Reflection In general an energy eigenstate of the Hamiltonian of the interface system can be written as an infinite sum of travelling waves and evanescent states with varying wave vectors but constant energy. It is reasonable to assume that lattice deformations in the interface region, in the y and z directions, are small so that the system is periodic along any line parallel to the interface. This means that crystal momentum parallel to the interface must be conserved. Throughout this thesis the energy eigenstate in the bulk regions will be written as2 *s.k1,k2(r) = < where A A Aipkiir) + ^Vk'i(r) in bulk region 1 (11) B$k2 (r) + -B'^k^ ( r) m bulk region 2 kj = kjxx + kjyy + kizz (1.2) Kj = -kja± + khy + kJMz , j = 1,2 (1.3) (see Figure 1.1). The components kjx are complex in general. Crystal momentum con-servation implies that hyy + klzz = k2yy + k2zi. (1.4) The assumption that there be only two independent eigenstates on each side of the inter-face is the single scattering channel assumption. For the sharp interface this assumption implies that the Bloch functions just on either side of the interface, have almost the 1This can be generalized to the case when the two bulk materials are not crystalline. The above applies not to the Bloch waves but to the incoming and outgoing waves (or evanescent states) near the interface, whatever they may be. 2The band indices for the Bloch functions are unnecessary and perhaps confusing here, so they are omitted. Chapter 1. Introduction 4 same functional dependence in the y and z directions; see [1]. If this is not the case, then more states on either side must be included such that the wave function is matched at the interface. It is straightforward to include more scattering channels, however this complication is not dealt with here. Although this seems to be a reasonable assumption, it may certainly break down in some systems. For more on this see [1,2,3,4,5,6]. The assumption that K. = (1-5) is the specular reflection assumption. That kj and k'j are compatible channels relies on some crystal-interface symmetry that is usually present. For an incident wave with wave vector, kj, one would expect the main reflection channel to be the one with wave vector, k'j. 1.3 The Interface Matrix The problem of the electronic properties of the interface reduces to finding the interface, or junction, matrix defined as / B \ 1 A \ = J V B' ) \ A' / The elements of the interface matrix, J ' , are functions of energy and wave vectors. In terms of the envelope function which is explained in Chapter 6, the equivalent problem is to find the boundary conditions for the envelope function across the interface. It is shown in Chapter 2 and Appendix A that there are only three independent parameters in the interface matrix. This is not surprising. If for example, one considers a wave of given energy incident on an interface3, then the electronic properties of the interface are 3The transmission channel is assumed to be known. Chapter 1. Introduction 5 completely characterized by three parameters; the magnitude of the transmitted wave, and the phase shifts of the reflected and transmitted waves. There are several methods i n the literature used to derive the interface matr ix or its equivalent. The ones the author is aware of are the transfer matrix method, tight-binding, effective mass equation ( E M E ) , and numerical pseudopotential. In this thesis the first three methods are used to study the sharp interface. The transfer matrix method is used for a one dimensional system. A good introduction is presented in [7]. For other examples of this type of method see [8,9,10]. One uses a two by two "transfer" matrix to relate the wave function in one unit cell to a neighbouring unit cell. It is an exact method. Although the exact form of the one-dimensional potential is i n general unknown, some general results without using this knowledge can be determined and are presented in Chapter 2. Not surprisingly an expression for the interface matr ix involving three unknown functions of energy is obtained. Information on band structure is obtained. Also, it is shown that the transmission coefficient of an interface depends on the position of the interface. Moving the interface can be accounted for by redefining the unit cells of both crystals. This implies that any expression for the transmission coefficient which depends only on band structure can only be correct in certain cases. The interface matrix cannot, in general, be expressed in terms of the band structures of the two materials alone. It depends on properties specific to the interface. Some results on the symmetry of the interface are arrived at. In particular, the transmission coefficient is either maximized or minimized when the interface has symmetry. In Chapter 3 the one-dimensional sharp interface is again treated i n general. The analysis is done directly in terms of the unknown Bloch functions. A simplified interface is studied, leading to simple integral expressions for the transmission coefficient i n terms of the unknown Bloch functions. A n important symmetry result for the general sharp interface is obtained which sheds light on the validity of the various envelope function Chapter 1. Introduction 6 Figure 1.1: Schematic picture of a two dimensional interface divided into three regions; the interface region and the two bulk regions. The case of single channel scattering and specular reflection is shown. Interface Bulk Region 2 Interface Region Chapter 1. Introduction 7 matching schemes discussed in Chapter 7. In particular, it is shown that in the symmetric case the interface matrix can be expressed in terms of only one unknown (not determined from band structure) parameter. In Chapter 4 the square well potential is used as a specific form of the periodic potential. Although a numerical solution could be determined for any given form of the potential, the advantage of the square well potential is that exact analytical expressions are determined. The disadvantage, of course, is that this form of the potential is not very realistic. Numerical results for the square well potential are presented in Chapter 5. This includes results for the transmission coefficient of several interfaces and barriers. Some plots of the wave function near an interface are also presented. The results are exact. No numerical solutions of Schrodinger's equation were performed. The results shown in Chapters 2 and 3 are verified. In particular, the results show that the exact form of the electron potential near the interface is essential in determining the interface matrix. It is also shown that the symmetry of the interface plays an important part in determining the transmission coefficient. A comparison of a parameter free calculation depending only on band structure, which is introduced in Chapter 7, is made with the exact results. In general this calculation is not successful. Some insight into why this is so is gained using the simple nearly free electron and tight binding approximations. In the tight-binding method, studied in Chapter 6 it is assumed that the electron wave function is well localized around the ions of the crystal. The method is an approx-imate one. The advantage to this method is that the interface, and hence the electronic properties of the interface, are described by a small number of energy and wave vector independent parameters. In the simple analysis (single band, nearest neighbour cou-pling) of Chapter 6 there is just one significant parameter describing the interface. This parameter does not depend on the band structures of either material on each side of the Chapter 1. Introduction 8 interface. It is shown that the exact value of this parameter is essential in determining the interface matrix. It is also shown that choosing this parameter in the most symmetric way, maximizes the transmission coefficient. Although the calculation done in Chapter 6 is one dimensional, this method is readily applicable in three dimensions. The disadvan-tage to this method is that these parameters are difficult to calculate and are usually determined empirically. Also, the band structure of the bulk materials is oversimplified. If a more accurate analysis is required one must introduce more parameters, including parameters which mix bands. Since the bulk bandstructures are known one can deter-mine the bulk parameters. However a more accurate description of the interface would require more parameters that are not well known. If the number of these parameters gets too large the method loses its value. In fact if one is only interested in a small energy range then the presence of three or more unknown parameters makes the method useless from the practical point of view. For some of the many examples of the tight binding method applied to interfaces see [2,3,4,11,12,13,14]. The effective mass formalism is studied in Chapter 7. This method is ideally suited for slowly varying (compared to the lattice constant) applied4 potentials. For a more exact statement see, for example, [15]. This method has been studied extensively; see [16] for example. The prime advantage in this case is that an equation, (the EME), similar to the Schrodinger equation is obtained for the envelope function which does not contain the crystal potential itself but only the band structure which is well known. Solutions can easily be found which give a lot of information about the system. The effective mass formalism describes a slowly varying junction very well; see [15,17] for example. As beautiful as the effective mass equation is, its derivation is not valid for a sharply varying system such as the abrupt interfaces described above. Attempts, [18,19,20,21], to apply this method to such systems persist however. This warrants the study in Chapter 7. 4This is any other potential besides that of the unperturbed bulk crystal. Chapter 1. Introduction 9 We conclude that the effective mass formalism is of little use when applied to the sharp interface. Various envelope function matching conditions are discussed. These schemes are not justified. Conditions for their validity are obtained. The pseudopotential method is not discussed in this thesis. It is a numerical method that is well suited for three dimensional calculations and easily incorporates multichannel scattering. For some of the many examples of its use with interfaces see [5,6,22,3]. 1.4 One Dimensional Analysis All of the analysis in this thesis is done in one dimension except in Appendix A where some general three dimensional results are shown. If we only consider states that have no component of k parallel to the interface then this simplification is justified. Under the specular reflection assumption, this simplification is also justified for states with nonzero components of k parallel to the interface. The three dimensional problem may then be mapped onto an equivalent one dimensional problem. As stated in Section 1.2, under the specular relection assumption, the Bloch functions just on either side of a sharp interface, have almost the same functional dependence in the y and z directions. This means that if ^ E l ^ k ^ r ) and dtfE.k-j^C1*)I&x a r e continuous at an arbitrary point r = (xo, yo, ZQ) on the interface, then they are also continuous at all other values of y and z on the interface (XQ constant). Thus the appropriate one dimensional eigenfunction (a function of x only) is defined by 5 where the fixed values of y and z are understood. The one dimensional Bloch function of the material on the left side is similarly defined as (1.7) 0k! (a) = 0k1(*»yo,«o). (1.8) 8The two different functions are distinguished by their arguments. Chapter 1. Introduction 10 In order to give this function a more one dimensional appearance it is relabeled as ^  (x). It is understood that it is also a function of k\y and k\x. The three dimensional dispersion relation is £(ki). In Appendix A it is shown that the x component of the probability cur-rent must be conserved at the interface. This means that the one dimensional dispersion relation must be defined as S{hx) = £(ki) (1.9) where k\y and kiz are kept constant so that • g J - , . ^ , - ^ . (U0) Note that £(k\x) is not equal to £(kixSt) in general. Similar results apply to the right side as well. All the one dimensional dispersion relations used throughout this thesis are denned as in equation (1.9). The crystal potential for the one dimensional problem, with the dispersion relation of equation (1.9), is different for different values of y0 and z0. This is not a problem in practice since the three dimensional potential is not known anyway. For given yQ, z0, ki, k2 and E, the one dimensional version of equation (1.1) is T , \ \ A^hx(x) + A'^-klx(x) in bulk region 1 *E{X) = \ _ (1-11) . B4>hx (x) + B'$-kix (a) in bulk region 2 where we remember that these functions and hence the interface matrix depend on the parallel components of the wave vectors also. This shows up in the dispersion relations given by 6(klx) = 5(kx) (1.12) €(k2x) = £(k2). (1.13) Although not done here, one may account for non-specular reflection of a wave by allowing the dispersion relations for the incident and reflected waves to be different. Specular reflection implies that the one dimensional incident and reflected states are a Chapter 1. Introduction 11 time-reversed pair, with wavenumbers ± kx. This is not true for non-specular reflection. Also, one may account for multi-channel scattering by having more than two states on each side, each having its own dispersion relation. The interface matrix would then grow in dimension. Keeping all this in mind we proceed with the one dimensional analysis. Chapter 2 The Transfer Matrix Method 2.1 The Infinite Homogeneous Crystal The problem of the one dimensional homogeneous crystal amounts to solving the Schrodinger equation for a particle of mass m in a potential V(x) that is periodic with period a: " 2 ^ a & * ( x ) + V(XWX) = E ^ (2J) See Figure 2.1 for an arbitrary example of such a potential. A useful technique of analyzing this problem without actually solving for the eigen-functions is the transfer matrix technique. 2.1.1 Transmission Matrix This involves (see Figure 2.2) studying the Schrodinger equation for the potential: V'(x) = 0[x - ja]8[(j + l)o - x]V(x) (2.2) where 9[x-ja]0[(j + l)a-x]=< 0 if x < ja 1 if ja < x < (j + l)a (2.3) 0 if x > (j + l)a and j is some integer. The Schroedinger equation for this potential may be written as - + ^V'{x)^x) = aV(*) (2.4) 12 Chapter 2. The Transfer Matrix Method 13 Figure 2.1: An arbitrary periodic potential, V(x), which is broken up into unit cells labelled by j. 0 "V J = - 2 i = - i V j = 0 i = i i = 2 / " —3a --2a a ) < a c la where1 a = sJlmEltf. (2.5) It is useful to spend some time studying this equation before coming back to the case of the periodic potential. For x < ja, the plane waves, e±xax, are solutions to this equation. Following Cohen-Tannoudji, Diu and Laloe ([7]), we call va(x) the solution to equation (2.4) which equals f*°{*-i*-*l'>) for x < ja. Although the form of va(x) can be complicated for ja < x < (j + l)a, it must be a linear combination of plane waves for x > (j + l)a. Thus, f e««(*-i«*-<»/2) if x < ja va(x) = { (2.6) [ i?(a)e'«(*-J°-"/2) + Gfa)e-ia(x-ja-a/2) J f x > + where F(a) and G(a) are complex quantities depending on the potential V'(x). The centering of these plane waves at x = ja + a/2 is for convenience. We can write the general eigenfunction of equation (2.4) for energy, E, as 4>a{x) = A va(x - ja - a/2) + A'v*a(x - ja - a/2) (2.7) 1The energy zero will always be set such that E > 0. Chapter 2. The Transfer Matrix Method 14 so that in the zero potential regions, Aeia(x-ja-a/2) + Ale-ia(x-ja-a/2) jf x < ja  Aeic[x-ja-a/2) + £ e-ia(x-ja-a/2) jf x > fj + ^ a (f>a(x) = i (2.8) This means that where = M(a) I (2.9) M(a) = (2.10) F(a) G*(a) \ G{a) F*(a) M(a) is called the transmission matrix for the potential V'(x). The probability current for the eigenstate <j>a{x) is given by fi J(x) = —Re m = — [| A |2 - | A' I2] = —[| i |2 - | A' |2]. m m which is independent of x. A consequence of this is the relation DetM(a) =| F(a) | 2 - | G(a) |2= 1. (2.11) (2.12) (2.13) To obtain the actual value of the probability current one would need to normalize <f>a(x) first. A normalization of the eigenfunction, \?(x), of equation(2.1) is done in Section 2.1.3. A useful result is derived by looking at the potential V"(x) obtained by reflecting V'(x) about the point x = ja + a/2; see Figure 2.3. V"(x + ja + a/2) = V'(ja + a/2 - x). (2.14) The general eigenfunction for the potential V"(x) is written as, i4e-ia(x-jo-o/2) _|_ £leia(x-ja-a/2) jf x < ja  Ae-ia(x-ja-a/2) _j_ Afeia(x-ja-a/2) [f x > (j + 1 ) a > 4>a{2ja + a — x) = < (2.15) Chapter 2. The Transfer Matrix Method 15 V'(x) Figure 2.2: The potential V'(x). £eict(x-ja-a/2) J^l e-ia(x-ja-a/2) J^eia(x-ja-a/2) j^re-ia(x-ja-a/2) ja (j + l)a X Note the directions of all the waves have changed so that the transmission matrix for V"(x) is defined as, (2.16) It is clear by symmetry that Figures 2.2 and 2.3 are consistent, that is, equation (2.9) holds in both cases. Inverting equation (2.9), using equation (2.13), gives ('A') = M/(a) (*) (Fl G*i ^ (A'\ KA J KAJ Fi J which is rewritten as, (A'\ K A J ( ( \ F* -G -G F -G -G* F* / A A' (2.17) Thus V A J Fj(a) = F(a) and Gj(a) = -G*{a). (2.18) (2.19) This result will be used when studying the interface. A corollary to this is that G(a) is pure imaginary for a symmetric unit cell. A unit cell is symmetric if V'(x) = V"(x). Chapter 2. The Transfer Matrix Method 16 V"(x) Figure 2.3: The potential V"(x). ^/g!a(x—ja—a/2) e-t 'a(x-ja-a/2) j^reia(x-ja-a/2) e-ia(x-ja-a/2) ja (j + l)a X 2.1.2 Transfer Matrix Coming back to the infinite crystal we see that in the region ja < x < (j + l)a the general eigenfunction of equation (2.1) may be written as, ^E(X) = Ajva(x - ja - a/2) + A'jV^x - ja - a/2) (2.20) for all j. By equation (2.7), the eigenfunction, ^E(X), at the point x = (j + l)a is smoothly matched (up to first derivative) to the function ^.eio<(^-i"—a/2) _j_ £l e-ia(x-ja-a/2) 3 3 (2.21) and is also smoothly matched to the function ^. + i e »Qf (a ; -0 ' - | - l )o-a/2) _|_ ^ / ^ e - i ' a ( x - ( j + l ) a - a / 2 ) (2.22) This implies that A'; D(a) •i+i Ai A'j where D(a) etaa 0 0 e-iaa (2.23) (2.24) Chapter 2. The Transfer Matrix Method 17 Defining Q(ct) = D{cx)M(a) = ( eiaaF(a) eiaaG*(a) ^ e~iaaG(a) e~taa F*(a) ) gives ( A \ \ 4 + i / = Q(«) (2.25) (2.26) We call Q(ct) the transfer (or iteration) matrix. It will be useful for later results to present some properties of the matrix Q(a). All of the following results are shown in detail in Appendix A. An understanding of the wave propagation of the solutions ^(x) comes from diagonalising Q. The two eigenvalues of Q are denoted A and 7 with eigenvectors A and T respectively. If the two eigenvectors are linearly independent we can write The band structure is determined by cA + eT. (2.27) X(a) = Re[eiaaF(a)] = -TrQ(a). £1 (2.28) If the energy E is such that | X(ct) \< 1 then E lies in an allowed band and if | X(a) |> 1 then E lies in a forbidden band. We consider the two cases separately. 2.1.3 I X(a) \< 1 : Allowed Band The eigenvalues of Q(a) are e±,k^a where k(a) is defined by X(a) = cos[k(a)a] 0 < k(a) < ?r/a (2.29) Equation (2.29) defines the band structure where k is the crystal momentum. This shows that the band structure does not contain all the information about the crystal since X(a) does not contain all the information in the matrix Q. If desired, the E(k) Chapter 2. The Transfer Matrix Method 18 dispersion relation can be extended to the range — wfa < k < %ja where E(—k) = E(k). The results for the eigenvectors are first of all that if A = \ f l then2 We define and g_ = r 9' /*' r = \ 9' J Y(a) = lm[eiaaF(a)] so that the 1-1 element of Q is X + iY. Equation (2.13) now becomes3 X2(a) + Y2(a)-\G(a) | 2 = 1 (2.30) (2.31) (2.32) (2.33) The sign of Y is the same as that of the slope of E vs.k. Thus if Y is negative then the eigenvalue etka represents a wave moving to the left. We want A always to represent a wave moving to the right4. Defining Y e = we have and The fact that eika L f A = e Y — e sin(A;a) ieiaaG* ' | Y | - sin(fca) | Y | + sin(fca) < 1 (2.34) (2.35) (2.36) (2.37) 2 This ensures the two eigenvectors represent time reversed states since the wavefunctions they rep-resent are complex conjugates. 3Equation (2.32) is also true for | X(a) |> 1 since the definition for Y is valid in this case also. 4By the above footnote this means V represents a wave moving to the left. Chapter 2. The Transfer Matrix Method 19 verifies that A', = A represents a wave moving to the right since the probability current is 1— | fj | 2 within a positive constant factor. The probability current of the general energy eigenstate of equation (2.27) in an allowed band is j = _ | / 1 2 ] [ | C | 2 _ | e n = - l / | 2 | V | f , f c Q, [lc[2 - |e|2]. m m \Y I + sm ka (2.38) The current splits into two; one part (c term) represents a wave moving to the right, and the other part (e term) represents a wave moving to the left. The normalization of a Bloch wave moving to the right (e = 0), for example, is done by using the relation, 1 dE ha, 2TT% dk m 2.1.4 | X(a) |> 1 : Forbidden Band We can write E-m\f\2-\m (2.39) X(a) = e' cosh[p(a)] p(a) > 0 where £ = X \x\' The eigenvalues are given by A = e'epa and 7 = e'e~pa. The respective eigenvectors are given by /' %Y — e'smh(pa) and g iY — e' s'mh.(pa) (2.40) (2.41) (2.42) (2.43) / -e'aaG* g' e~iaaG Thus A represents a wavefunction increasing exponentially to the right and T represents a wavefunction decreasing exponentially to the right. The fact that 9 / 9' = 1 (2.44) Chapter 2. The Transfer Matrix Method 20 shows by equation (2.12) that the probability current is zero for these states.5 The two cases \X\ < 1 and \X\ > 1 are formally related. Equations (2.29), (2.36), (2.40) and (2.43) are consistent with the transformation p —• ik and e' —• e. Of course a linear combination of A and V can have a nonzero probability current. Chapter 2. The Transfer Matrix Method 21 Figure 2.4: Schematic picture of the interface. The potentials on either side are periodic but otherwise arbitrary. The partitioning into unit cells labelled by j is shown. -3ai —2a\ — a,\ 2.2 The Interface We model the sharp interface as being perfectly abrupt. The potential might look like that of Figure 2.4. This is an important simplification since the actual potential near a real interface is not well known. We also assume very little lattice deformation so that the crystal potentials remain periodic up to the step on both sides. 2.2.1 Interface Matrix The most appropriate way to analyze this problem with the transfer matrix method is to make the step in the potential (at x = 0) the boundary of the unit cells as well. Following Section 2.1 we write the general solution in the region jax < x < (j + l)cti , j < —1, on the left side, as ^(x) = Ajva(x - jax - d/2) + A'jV*a(x - jax - ax/2) (2.45) Chapter 2. The Transfer Matrix Method 22 and in the region (j — l)a2 < x < ja2 , j > 1, on the right side, as $(x) = AjWa(x - ja2 - a2/2) + A!jW*a(x - ja2 - a2/2). (2.46) We thus have the transition matrix Mi (a) = Fi{a) G{(a) VGi(a) FJ(a) for the potential 9[x + ai]0[—:r]V(a:) on the left side and the transition matrix (2.47) M2{a) = 1 F2{a) G*2(a) ^ (2.48) \G2(a) F*(a)/ for the potential 0[a:]0[a2 — x]y(x) on the right side. The transfer matrices are given by Qi(a) = Dx(a)M1{a) and Q2(a) = D2(a)M2(a) where V 0 e —«aoi and Do e , M 2 0 0 e~iaa2 (2.49) (2.50) Let Ai(a) and 71(a) denote the eigenvalues of Mi (a) with eigenvectors Ai(a) = fi \ fi) and Ti(a) = 9i 9'i (2.51) respectively. Similarly let \2(ct) and 72(a) denote the eigenvalues of M2(a) with eigen-vectors Aa(a) = \ and F2(ct) = 92 \ 92 J (2.52) respectively. Just as in equation (2.27), we write generally, = ci(a)Ai(a) + ei(a)ri(a) / A_!(a) N (2.53) Chapter 2. The Transfer Matrix Method 23 and \ = c2(a)A2(a) + e2(a)r2(a). In order to relate these two vectors we use equation (2.9), U w v A'-i (2.54) (2.55) and equation (2.23). The distance from the center of the j = — 1 unit cell to the center of the j = 1 unit cell is (ax + a2)/2. Thus e i a (a i+ 0 2 ) / 2 Q 0 e - « ' a ( a i + a 2 ) / 2 ^ e - i a ( o 1 - a 2 ) / 2 Q 0 e « ' a ( a i - a 2 ) / 2 / \ - » a o i \ \ 0 I e-i<t> o ^  ^ 0 e1'* , e t a ( o 1 - a 2 ) / 2 [ciAxAi + e^iTj] where we have defined a (2.56) (2.57) Equations (2.55) and (2.56) are equivalent to h 92 \ fi 92 J \ ( r \ c2 This gives \ e 2 J V e2 / Aici \ 7i d / (2.58) / A 1 C 1 \ (2.59) Chapter 2. The Transfer Matrix Method. 24 where J(ct) is the interface matrix and is given by (2.60) In this expression we have chosen the normalization of the eigenvectors of the two transfer matrices such that f\ = /2 = g[ = g'2 = 1. The interface matrix is defined for all energies except at band edges of the material on the right. There are four energy-dependent parameters on the right side of equation (2.60). Using the current conservation relation derived in Appendix A reduces to three the number of parameters needed to specify the interface matrix 2.2.2 Transmission Coefficient We would like to find the transmission coefficient of a Bloch wave from the left side of the interface to the right side of the interface. We first, as in Section 2.1, define the quantities Xt(a) = Me iaaiFi(<*)] = 7 p Qi(<*) (2-61) and X2{a) = Re[e ia^F2{a)} = ^Tr Q2(cc). (2.62) For the problem to be well defined the energy, E , must be such that it lies in an allowed band on the left side.6 It is clear that the transmission coefficient will equal zero if the energy lies in a forbidden band on the right side. This means we consider only those energies for which |Xi(a)| < 1 and7 | X2(a) |< 1. The boundary condition for this problem is that there be no incoming wave from the right side of the interface. This 6Actually the energy can't lie at the band edge either since k = 0 there. One cannot define the problem in this case since there is no wave motion. 7A limiting procedure from the allowed side of the band edge is used to find the transmission coefficient when the energy lies at a band edge on the right side. Chapter 2. The Transfer Matrix Method means e 2 = 0. (2 The transmission and reflection coefficients are given respectively by Ti = Jt/Ji (2 Ri = 1 — T j = —Jr I Ji (2 where Ji = ^ [ \ h \ 2 - \ f i \ 2 \ \ c l \ 2 (2 is the incident probability current from the left, ha Jr = — [ | fx V - I fi I ] I ex | 2 (2 m is the reflected probability current to the left, and ha Jt = — [I /2 T ~ I /2 I ] I C2 | 2 (2 m is the transmitted probability current to the right. The problem reduces to finding T _ [i h i2 -1 n i2] i c 21 2 [| / i i2 -1 n i2] i ex i2 v or ^ - w (2 The answer is found in Appendix B to be j, ( E ) = 2(sin feiai)(sinfe2a2)  n | l i ^ | +(sinfeia1)(sinfc2a2) - crRe(GjG2) 1 where as in Section 2.1 Yi=Jm[eiaaiFi(a)] and Y2 = Im[eiaa> F2(a)] (2 Chapter 2. The Transfer Matrix Method 26 and a is the sign of Y\Y-i'-(2.73) One of the main reasons for this analysis of the interface is to see what effect the position of the potential jump (the interface) relative to the adjacent ions has on the transmission coefficient. For example one might ask if T/ is maximized at all by having the potential jump exactly halfway between the adjacent ions of the left and right crystals. One would expect some kind of effect since changing the position of the potential jump changes the choice of unit cell which changes the transfer matrix, that is, F and G. One would also like to account for any deformation of the crystals near the interface. The simplest way to do this is to change the distance between the two ions at the interface maintaining the periodicity of the crystals up to the interface. Again this amounts to changing the unit cells. 2.2.3 Symmetry Properties In this section we assume that symmetric unit cells can be found for both crystals. In the notation of Section 2.1 these would be unit cells for which V"(x) = V'(x). This is often the case if there is one atom per unit cell. Suppose the position of the potential jump is such that both unit cells are symmetric as shown in Figure 2.5a. As Figures 2.5b and 2.5c clearly demonstrate, the unit cells defined after moving the interface to the left an arbitrary distance x' are the reflections about the center of the cells of those unit cells defined after moving the interface to the right the same distance x'. As discussed in Section 2.1, equation (2.19), the transformation of the transfer matrix when going from one shifted unit cell to its reflection is just G —» — G* , while F stays the same. It is easy to see that the transformation F\ —» Fi , Gi —> —Gl , F 2 —*• F 2 , G2 —> —Gl (2.74) Chapter 2. The Transfer Matrix Method 27 leaves the expression (2.71) for Tr(E) unchanged. Thus we have the result that T/ , as a function of the position of the potential jump, is even about the position where both unit cells are symmetric. Another result which pertains to changing the distance between the ions on either side of the interface is that the transformation8 Fi -> Ft , Gi -Gl , F2-*F2 , G2 -» G2 (2.75) also leaves Ti(E) unchanged when G\ is pure imaginary, that is, the unit cell on the right is symmetric. An example of this transformation is shown in Figures 2.6a and 2.6b. Thus TT, as a function of the distance from the potential jump to the nearest ion on its right, is even about the position where the unit cell on the right is symmetric. 8The same result holds upon switching the subscripts 1 and 2. Chapter 2. The Transfer Matrix Method 28 Figure 2.5: In (a), we have an interface with the unit cells on both sides being symmetric. In (b), the position of the interface is moved left a distance x' relative to the crystals. In (c), the interface is moved right a distance x'. The transmission coefficients for cases (b) and (c) are the same. -3c?i —2ai —ai -4ai —3ai —2a 4a2 5a2 (c) X-i = -4 i = -3 -4a -3a J = -2 -2a i = - i V(x) l\T i = i V i = 2 V i = 3 V i = 4 2a 3a 4a 5a V i = 5 x Chapter 2. The Transfer Matrix Method 29 Figure 2.6: In (a), we have an interface with the unit cell on the right side being sym-metric. To simulate lattice distortion, the "ions" on the left side are pushed to the right a distance x' in (a). In (b) the "ions" on the left side are pushed to the left the same distance x'. The transmission coefficients for the two cases are the same. (a ) V j = -4 i = -3 V J = -2 *-x' V(x) "V i = i V i = 2 V i = 3 V i = 4 \T j = 5 (b 4ai ) i = -4 3ai i = -3 2ai i = -2 ~x' i = - i V(x) i = i * 2 2 V i = 2 V i = 3 a2 4 V i = 4 J=5 a2 4ax — 3ax — 2ai D 122 S !a2 2 a2 4 a2 I )a2 Chapter 2. The Transfer Matrix Method 30 2.3 The Barrier Here we study the system shown in Figure 2.7 composed of two succesive interfaces. The barrier consists of the material between the interfaces. The number of full unit cells in the barrier is N. The material on either side of the barrier is not necessarily the same. The assumptions about the form of the interface of Section 2.2 apply to each of the two interfaces here. The boundaries of the unit cells in Regions 1 and 2 are determined at the left interface at x = 0. The unit cell in Region 3 is determined by the right interface. Unless the distance between interfaces is an integer multiple of a2 then there will be in general a mismatch in the barrier at the right interface. 2.3.1 Barrier Matrix For a general eigenstate ^E(X) of the system at energy E the solution in the region ja\ < x < (j + l)a\ , j < — 1 , of region 1 is VE(x) = AjVa[x - (j + l)ai - aa/2] + Aya[x - (j + l ) a i - ai/2]. (2.76) The solution in the region (j — l)a2 < x < ja2 , 1 < j < N , of region 2 is V(x)E = Ajwa[x - (j - l)a2 - a2/2] + A'jW*a[x - (j - l)a2 - a2/2] (2.77) and the solution in the region (j — l)ax + q < x < ja\ + q,j>N-\-l,oi region 3 is V{x)E = Ajua[x - (j - l)a3 - q - a3/2] + A'jU*a[x - (j - l)a3 - q - o3/2]. (2.78) The functions va(x),wa(x),ua(x) all have the properties described in Section 2.1. Maintaining the same notation as in Section 2.2 we write quite generally: = c1(a)A1(a) + e1(a)r1(a), (2.79) Chapter 2. The Transfer Matrix Method 31 Figure 2.7: A schematic of a barrier with N = 3. A mismatch is present (q ^  0). V(x) j = -3j = -2j = -l x —3ax —2ai —a\ 0 and Ai(«) I Alia) AN+1(a) = c2(a)A2(a) + e2(a)r2(a) A'N+1{a) ) = c3(a)A3(a) + e3(a)r3(a). Also c2 \e2 ) l = Ji{«) Aicx V 7 i e i where Ji(a) is the interface matrix for the left interface and is given by fi92 where (2.80) (2.81) (2.82) (2.83) a. fa = Tj-(°i ~ fl2)- (2.84) Chapter 2. The Transfer Matrix Method 32 At the right interface we have V e3 J - Jr(ct)Qm(q,a) A 2 c 2 (2.85) \ 72 e2 where Jr(a) is the interface matrix for the right interface and is given by 1 Jr(a) = -where <?^r = — (02 - A3)-(2.86) (2.87) (5m(g,a) is the transfer matrix for the mismatch region at the right side of the barrier.9 Thus if we define the barrier matrix B(a) by (2.88) C31 ( A i e i \ = B(a) e3 j \ 7 i e i / then \N Q B(a) = J r(a)gm( 9 ,a) j 2 | J,(a). 0 7 2 N (2.89) In these expressions we have, as in Section 2.2, chosen f\ = /2 = /3 = g[ = g'2 = g3 = 1. 2.3.2 Transmission Coefficient We wish to find the transmission coefficient of a wave from the left side to the right side. For the same reasons as in Section 2.2 we consider only those energies for which I Xi(a) \< 1 and | -X"3(a) |< 1. The boundary condition for this problem is that there be no incoming wave from the right side of the barrier. This means e3 = 0. (2.90) 'Note that Qm is the identity matrix when q = 0. Chapter 2. The Transfer Matrix Method 33 Just as in Section 2.2 the transmission coefficient is given by [1- I fi I2] I ci | 2 [ i - I A l 2 ] 1 (2.91) [1- I fi I2] I SJi1 I2 where Bxx is the 1 — 1 element of the inverse10 of the barrier matrix B(ct). An explicit form for the transmission coefficient similar to equation (2.71) for the interface is complicated and not very illuminating in general. It is however easy enough to compute so this is what is done. A simple explicit form can be derived however, if we eliminate the mismatch problem in the barrier by setting q = 0, and by making the material in region 3 the same as in region 1. We also assume there is no applied bias. The answer is found in Appendix B to be 1 TB{E) 1 - sin^A^c^) 1 - y 1y 2-Re(GiG*) (sin2 k\ at) (sin2 foai) (2.92) for E lying in an allowed band of the barrier medium11 and 1 TB(E) = 1 + sinh2(AT/o2a2) 1 + Y1Y2-Re(G1G2-) (sin2 fci0i)(sinh2 p^a^) (2.93) for E lying in a forbidden band of the barrier medium.12 The expression for the transmis-sion coefficient of this idealized barrier, when E lies in an allowed band of the barrier, can be related quite simply to the transmission coefficient of an interface13. Equation (2.71) is written as o [YXY2 - Re(GiG2] = sin kxax sin k2a2 - lj (2.94) 1 0It is shown in Appendix B that B~l exists everywhere except at the band edges of the barrier. It is also shown that Bxx cannot equal zero. 11A;2 is the crystal momentum (Bloch wave vector) in the barrier. 12The two expressions (2.92) and (2.93) are simply related by p2 = ik%. 13The transmission coefficients of both interfaces are the same in this case. Chapter 2. The Transfer Matrix Method 34 or [TO - Re(G*G2]2 = sin2 fclfll sin2 k2a2 {^-^ + • (2.95) Substitution into equation (2.92) gives TB = ? 1 • (2.96) 1 + 4 f ! = £ ) sin2 Nk2a2 This expression may be analytically continued to the case where E lies in a gap of the barrier, even though the actual transmission coefficient of each interface is zero. This expression is exactly the same as that of a free particle incident on a square barrier of length Na2 except Tj would be the transmission coefficient of a step, and k2 would be the plane wavenumber in the barrier (see Section 4.2). This simplicity is due to the fact that the barrier consists of an exact integer multiple of unit cells. By Bloch's theorem a Bloch function simply changes by a factor of elNk2a2 upon crossing the barrier. The presence of a mismatch would complicate things considerably. Equation (2.96) shows quite nicely the dependence of the barrier transmission coefficient on two effects: 1) How the particle gets through each interface and 2) the resonance (or decay) of the particle in the barrier. This result can be derived quite easily actually using elementary optics. Considering multiple reflections in the barrier and summing a geometric series gives equation (2.96). In this special case one might use the reflection symmetry about the middle of the barrier (if present) to restrict the choice of unit cells. We would like to keep the choice arbitrary however. To do this we imagine that material 3 is not the same as material 1 but it happens to have the same electron potential as material 1 except it is raised or lowered by some constant amount of energy. We then apply a voltage across the system so that the potentials in materials 1 and 3 are the same. Because of the applied potential one cannot use symmetry to restrict the choice of unit cells. Chapter 2. The Transfer Matrix Method 35 2.3.3 Bound States Here we would like to find the energy spectrum of bound states in the barrier. The bound eigenstates must go to zero far away on both sides of the barrier. Thus we must have | Xi(a)\ > 1 , | X3(a) |> 1 and choose c3 = ei = 0. (2.97) From equation (2.88) the condition14 on E is Bn = 0. (2.98) For the specific case of no mismatch (t = 0) and material 3 the same as material 1, the condition is cosNk2a2 - , . f ^ ^ l , \YiY3 - ReGiGJ] = 0 (2.99) (smh pidi J(sm k2a2) for E lying in the an allowed band of the barrier medium and e'2 cosh Np2a2 - . ^ s m h i V ^ [YlY2 - ReG1G*2] = 0 (2.100) (smhpiai)(sinh p2a2) for E lying in a forbidden band of the barrier medium. 1 4For | .X"i(a) |> 1 and | Xz(a) |> 1, Bn is real, so there is only one condition. Chapter 2. The Transfer Matrix Method 36 Figure 2.8: A schematic of a superlattice with Nx = 3 and A^ 2 = 2. The structure shown here is simply repeated to form the superlattice. The superlattice spacing is s. The potentials of both materials are periodic but otherwise arbitrary. The unit cells are labelled by j. Mismatches are present. V(x) ; = - 3 ; = - 2 ; = - l N2a2 ^xK. Nxax 3 = 1 i = 2 i = 3 i = 4 j = 5 j = 6 J = 7 -3ax —2ax —ax 0 a2 2a2 s + a2 2.4 The Superlattice Here we study the system shown in Figure 2.8 composed of repeating barriers of the kind discussed in Section 2.3. There are two materials present. In a given region of material 1 we have Nx unit cells of material 1 plus some mismatch. This is followed by a region of N2 unit cells of material 2 plus some mismatch. This is then repeated indefinitely in the idealized case. This problem is very similar to that of the barrier. We write the analogue of equa-tions (2.79),(2.80) and (2.81), \ = cx{a)Ax(a) + ex(a)Tx(a), (2.101) Chapter 2. The Transfer Matrix Method 37 AN2+I(OS) 4v2+i(a) / ANi+Nl(a) ^ = c2(a)A2(a) + e2(a)T2(a) = c3(a)Ax(a) + e3(a)Ti(a) (2.102) (2.103) = c4(a)Ai(a) + e4(a)r1(a). (2.104) V Ajva+JVl(a) y In going from j = —1 to j = Ni + N2 one has traversed a supercell, that is, one unit cell of the superlattice. We define the superlattice matrix as c4 e 4 = S(a) \ e * y (2.105) Using the results of Section 2.3.1 we write {*)-' \ e 3 ) where \ 7i ei j B(a) = Qm 1 (?i ,Qf)Jr(a)Qm a (92 ,a) J/(a). (2.106) (2.107) The matrices Qmi and Qm2 are the transfer matrices for the mismatches. Thus S(a) = 1 V o (2.108) Going through N super unit cells just involves finding SN. Thus we can treat S as a generalized transfer matrix. In order to find the superlattice band structure one would simply diagonalize 5 at a given energy, E. In general this would be very tedious to do analytically, but easy to do numerically. Since the determinant of S is unity, the product of the eigenvalues of S must be unity. If the eigenvalues are of the form e±%KS where s = NiO! + N2 + a2 + qt + q2 (2.109) Chapter 2. The Transfer Matrix Method 38 is the superlattice constant then E lies in an allowed superlattice band (mini-band). If not then E lies in a mini-bandgap. All the results derived for the diagonalization of the transfer matrix Q depended only on the fact that Q is of the form Q = 1 F G*^ where Note that \G F* ) detQ = \F\2 - \G\2 = 1. (2.110) (2.111) pi Qi* ^ \ G' F'* / (2.112) FF' + G*G' FG'* + (F'G)* ^ F*G' + F'G (FF1)* + GG'* , Thus the UQ form" is preserved under multiplication. If E lies in a bandgap of either material then some of the matrix factors in S do not have the UQ form". If however, E lies in allowed bands of both materials, then S does have the "Q form". The mini-band structure is thus given by ReQii(a) = cos KS , 0 < KS < ic. (2.113) We can obtain a simple expression in the special case where there is no mismatch, that is, c/i = c/2 = 0. By symmetry this requires all the unit cells of material 1 to be the same and similarly for material 2. The supertransfer matrix is then S(a) = I „ » ' e i M N i - l ) a i Q e - i e i M N i - l ) a i \ 0 \ 0 e-*«lfcloi (2.114) where B is given by equation (C.10). By equations (2.113) and (C.18) the mini-band structure is given by cos KS — Re cos N2k2a2 + —\ , y (YXY2 - Re(G1G*2)) (sm Kiaij^sm K2a2J (2.115) Chapter 2. The Transfer Matrix Method 39 / 7 AT w AT » \ (siniV2A:2a2)(sin./ViA;iai) ._ , T , . _ (cos An Mai)(cos N2k2a2) - i \ yi*2 - Re(GiG; (smfciai)(sinA:2a2) . (2.116) In terms of the transmission coefficient of an interface, using equation (2.94) we have T 2 1 cos KS = (cos k\Ariai)(cos N2k2a2) — o(sm N2k2a2)(sm Nikxai) ——1 . (2.117) III J This simple result is the same as the dispersion relation of equation (2.29) in the case of square well potentials (see Section 4.2) except, as in Section 2.3.2, Tj would be the transmission coefficient of a step, and k\, k2 would the plane wave numbers in the two materials. The reason for the simplicity is the same as for the simplicity of equation(2.96). The presence of any mismatches would complicate things. It is interesting to examine two extreme cases of equation(2.117). When Tj = 1 we get cos KS = (cos &iiViOi)(cos N2k2a2) — a(s'm N2k2a2)(sin N\kxai) = cos [kxN\ai + ok2N2a2] (2.118) When ki and k2 represent waves moving in the same direction then a = 1. Thus K is simply a weighted average of k\ and k2. The electrons see an "average" or "virtual" crystal. There are no interference effects from the interfaces. When Tj is very small we see that for E to be in an allowed mini-band we must have (sin N2k2a2) or (sin iV i&ia i ) very small. The interfaces act very much like the nodes of a vibrating string. The mini-bands will be very thin and will be centered about the "resonant" energies of the "loops". This is a tight-binding superlattice. Chapter 3 Analysis of Sharp Interface In this chapter an exact expression for the interface matrix and the transmission coeffi-cient for a sharp interface is derived in a more direct way than in Chapter 2. Some sym-metry results are then derived. The system consists of two semi-infinite one-dimensional crystals on either side of the point x = x0. They are both assumed to be periodic right up to the point x = XQ. An example of this is shown in Figure 2.4 where XQ = 0. Schrodinger's equation is written as -h2 d2 2m dx2 VE(X) + [0(xo - x)Vi(x) + 6(x - X0)V2(X)]VE(X) = E^E{x) (3.1) where both V\{x) and V2(x) are the periodic potentials of the media on the left and right sides respectively. The dispersion relation for medium 1 is given by £n{k\) and that of medium 2 is £n{k2).1 The solution to equation (3.1) is VE(x) = [Aj>niM(x) + A'$nu-kl(x)]e(x0-x) + [B$r»te(x) + B (3.2) where £»»(*!) = Sni(-h) = £n2(k2) = £n2(-k2) = E (3.3) and h2 d2 . . -$nukAX) + (3)0*1! , * ! ( & ) = ^ii(*l)0n 1,*i(a!) (3.4) 2m dx2 -h2 d2 2m dx2 0n2,fc2(z) + Vl{X)i>n2,k2{x) = £„2(k2)4>n2,k2(x). (3.5) 1 n is the band index. 40 Chapter 3. Analysis of Sharp Interface 41 We assume the energy E lies in an allowed band on both sides. As long as the periodic potentials are well behaved we must match $E(X) and its first derivative at x = x0. These conditions are written as AipmMM + A'i>nu-kl(xo) = Bipn2ik2(x0) + B'ipn2^k2(xo) M n u k l M + A'^_kl{xQ) = Bi>'n2ik2(xQ) + B'4>'n2i_k2(x0). The interface matrix is then given by (3.6) (3.7) / B \ ' A \ \ B' ) \ ^ ' / where j = ^ n u k A X o W n 2 - k 2 i . X o ) ~ 0n1,fc1(a:o)0n2,-fc2(xo) ^ n 2 , k 2 ( X o W n 2 , - k 2 ( X o ) ~ ^MM^-^M r, (scQfe.-fe fop) ~ 0n1,-fc1(a;o)0na,-iba(go) J\2 — —* ~ * ~ 4 > n 2 , k 2 ( X o W n 2 - k 2 ( X o ) ~ 0n2lfc2(a;o)0n2,-fc2(a:o) (3.8) (3.9) (3.10) J21 = Jl*2 (3.11) J22 = Jn- (3.12) In order to obtain the transmission coefficient we assume without any loss of generality that 0m,fci(aj) and ^^^{x) represent waves moving to the right.2 Thus we set The solutions are B' = 0. A_ = - t n u k i M f t n t t o M + V'n1,fc1(a:o)0n2,fc2(xo) A 0ni,_fc1(xo)0|laiifca(a:o) - {xo)4>n2,k2(x0) (3.13) (3.14) 2We have chosen the two time-reversed eigenstates as our fundamental solutions to the Schrodinger equation. If for example, ^>ni,ki(x) represented a wave moving to the left then we would just switch B and B' in our equations. Chapter 3. Analysis of Sharp Interface 42 (3.15) and A ^ n u - k A X o W n 2 < k 2 M ~ ftm^hM^kzixo) These two expressions agree with those of Grinberg and Luryi ([10]). The reflection coefficient is given by R It is shown below (equation (3.43)) that the transmission coefficient is given by A' (3.16) T = 1 - R = d£n2(k2) dk2 afci B2 A (3.17) It is difficult to go further without knowing what the quantities tj>n,k(xo) and ^ n.fc(xo) actually are. Some results can be obtained by simplifying the interface. We will return to the general case in the last two sections of this chapter. 3.1 Simplified Interface We make the simplification, V2(x) = V1(x) + V06(x-x0) (3.18) where Vo is a constant. We are left with a periodic potential and a step function potential added to it; see Figure 3.1. Dropping the hats and tildes, equations (3.3),(3.4) and (3.5) become eni(kx) = 4 (fci) = €n2{k2) = £n2{k2) + V0 (3.19) and -h2 d2 2m dx2 -h2 d2 2m dx2 ^n2,k2(X) + Vl{x)^n2,k2(X) = [£n2(h) ~ V0}^n2tk2{x). (3.20) (3.21) Chapter 3. Analysis of Sharp Interface 43 Figure 3.1: Schematic picture of the interface composed of a periodic potential Vi(x) plus the potential Vo 0(x — xo). V(x) Vo —i— 2a - 1 — 3a i— 4a —4a —3a —2a —a x0 0 i a If we multiply equation (3.20) by ^n2Mi.x)i equation (3.21) by i>ni,ki(x) a n < ^ subtract the two we get •^W(ni,kun2,k2,x) = ^ ^ ^ ^ ( z ) ^ , ^ ) = v0ipnukl(x)ij>n2ik2(x) (3.22) where we have defined W(n, k, n', k', x) = ^ntk(x)^'n,tk,(x) - *l>n.,k,(x)Vnk(x) (3.23) and Thus and 2m Vp 2m A^ _ W(n2,k2,ni,ki,xp) A W(ni,-ki,n2,k2,xo) B_ _ W(n\,—k-y,n\,ki,XQ) A W(ni,-ki,n2,k2,x0)' (3.24) (3.25) (3.26) Chapter 3. Analysis of Sharp Interface 44 It is worthwhile to spend some time studying W(n, k, n', k', x). It is easy to show that if a is the period of the periodic potential then W(n,k,n',k',x + a) = e^k+k')aW(n,k,n',k',x). (3.27) This shows that R and T are unchanged when the potential jump is moved a distance a, which is an expected result. By equations (3.22) and (3.27) we have W{n,k,n',k',x + a)-W{n,k,n',k',x) = [e^k+k')a - l)W(n,k,n',k',x) rx+a = V0 dx'^ntk(x')ll)n,lki(x'). Jx (3.28) We consider the following two cases. Case 1: k + k' is not a reciprocal lattice vector G. We have by equation (3.28) VQ rx+a W(n,k,ri,k',x) = ^.(fc+fc))a _ ^ dx'$n,k{x')il>n>,k'{x') = (e.-oJ?- - 1) j ^ d x ' e ^ ^ ) ^ ^ ) (3.29) where by Bloch's theorem, ^n,k{x) = elkxunik(x). The function un<k(x) is periodic with period a. Case 2: k + k' = G. Under this condition it is extremely unlikely that VQ = 0 for n ^ n'. Thus we have the orthogonality condition f X + a dx'i>n,k{x')^n,,-k{x') = 0 (3.30) Chapter 3. Analysis of Sharp Interface 45 for n ^  n'. For n = n' we have £n(k') = Sn(G -k) = £n(-k) = €n(k) (3.31) so that Vo = 0 is implied by the condition k + k' = G. Thus we must take a limit, W(n, G — k, n, k, x) = W(n, — k, n, k, x) = limW(n, — k, n, k + e, x) 2m , . [Sn(k) -Sn(k + e)] /•»+« , , = —7T lim h2 «-o (ei£a - 1) 2midSn(k) /•*+» , /•s-t-a / dxVn,-fc(a; /)V'n,fc-r £(« /) px+a %2a dk Ix dx'u"-k(x')un,k(x>) 2mi dSn(k) j j x t u (xt}u rxi} ri a Ok Jo 2midSn(k) fa A f ( ,M2 _ m * dSn(k) h2a dk so that W(n, —k, n, A;, x) is independent of x. We note that O777? W(n, -k, n', k', x) = —(rbn>k\K\^n.lk.) (3.33) where K is the operator corresponding to the probability current density (see [7], page 239), K(x) = ±[\x)(x\P + P\x)(x\] (3.34) and we have used the fact that tf„,-fc(*) = #U(aO. (3.35) P = —ih(d/dx) is the momentum operator. The probability current density3 for the state ipnyk(x) is % 1 OS J = {tl>n,k\K\il>n,k) - — W(n,-ktn,k,x) = (3-36) 'Actually, this is the expectation value of the probability current density. Chapter 3. Analysis of Sharp Interface 46 where we have used equation (3.32). Since *&E{X) is a stationary state the probability current is independent of position. Evaluated just on the right side of x = XQ it is 3 = J t = I'm]" K 2 . * 2 ( * ° ) ^ 2 , f c 2 ( Z o ) - 0n 2 , fc 2 (Zo )C 2 , * 2 ( *o) ] = ^fbPn2,-k2(x0)^'n2<k2(x0) - ^n2,k2{XoWn2,-k2(xo)} Just on the left side of x = XQ we have a - r [ A ^ n i , f c l ( x 0 ) + A'4>ni<-kl(x0)}*[Ail>'niM(x0) + A'^nu_kl(x0)] jWnuk&o) + A'ip'nu_kl(x0)]*[Ai>nukl(x0) + A V n i , _ f c l ( x 0 ) ] J = 2mi h Lmi Ji -\- JT (3.37) (3.38) (3.39) where Ji Jf — W ( n i , - * ! , r»i,A;i, x 0 ) = 2mi W(ni , - fc i , re i , &i ,x 0 ) . = 27T7I dki \Af dSni (3.40) (3.41) 2mi v " i ' u'- 2x71 dfci J,- and J r are the incident and reflected currents respectively. The splitting of the current into two opposite currents is a general result which only used the time reversal property of the eigenfunctions, equation (3.35). The reflection and transmission coefficients are given by R -Jr Ji A and T = — = Ji W(n2,-k2,n2, k2,x0) W ( n i , - & i , r c i , ki,x0) dSn7(k2) 8k2 9£nj (ki) 9*1 B2 A (3.42) (3.43) Chapter 3. Analysis of Sharp Interface 47 3.2 Properties of the Transmission Coefficient: Maxima and Minima We already know that the transmission coefficient as a function of the interface position, XQ, is periodic with period a. We would like to know more about T(xo), in particular where it has extrema. We note that on the right hand side of equation (3.26) the only term with x0 dependence is W(n\, — k\,n2, k2,xo). We write where c is a positive constant and we have omitted the band indices for convenience. Using equations (3.22) and (3.23) gives, dT -2c d dx0 \W(—ki,k2,x0)\4dx0 -2c \W(-h,k2,x0)\' | W ( - * i , * 2 , * o ) | 4 d W(-k1,k2,x0)-—W*(-k1,k2,x0) CLXQ d + W*(-kl,k2,x0)—W(-k1,k2,x0) CLXQ —2c r -^ yj7 [WkiixoWkiM ~ K ( x o ) ^ * a ( « o ) ] [ t ; o^fc,(2:0)^(2:0)] + [ ^ 1 ( ^ 0 ) ^ ( ^ 0 ) - i>k1(xo)^k2(xo)}[voipi1(x0)i>k2(xo)]] [[^(xoWk^o) - ^M^k^oWvo^ixoWk^Xo)] + [ ^ ( ^ 0 ) ^ ( ^ 0 ) - ^(xoWk2(xQ)}[vo^kl{xo)^k2(x0)]] \W{-~kx, k2,x0)\4 -2c \W(--h, k2,x0)\4 -icv0 \W(--ku k2,x0)\4 ^IM^MWiKk^xo)]. (3.45) Thus T(xe) will have some kind of extreme value when M^M^UMWik^k^x,)] = 0. (3.46) Chapter 3. Analysis of Sharp Interface 48 To find out what kind of extreme value we have we must evaluate the second derivative at that point: o?T{xe) -2c d2 dxl \W(-kuk2,x0)\4dx -Acv0 \W(-kuk2,xe)\4 Re \W{-kuk2,xe)\2 (3.47) ^(fc 1 ,A: 2 ,X e )[^ 1 (x e )V'r 2 (Xe)+0r i (^)^ 2 (Xe)] +V0\lJ)k1(Xe)lI>k2(Xe)\2]] ^ ( - k ^ k l X e ) | 4 R e [ i W ^ k t e e ) ! 2 ~ Wh{*e)i>k2{Xe)\2 +2ilm[i)kl(xe)^k\(xe)if>k2(xe)il>l2(xe)] + v0\ipkl(xe)^k2(xe)\2] = | ^ ( - ^ T 2 0 , X e ) | 4 [W^X'W>»(X'W - i^*-)iM*«)i a +V0\^kl(Xe)^k2(Xe)\2} • If Vo is large and \ipkl(xe)il>k2(xe)\2 is not small, then = M = & ^ ' ^ ( l l ) f e ( x ' ) | 2 ( 3 ' 4 8 ) which implies that T is a maximum. If v0 is large the only way T can be a minimum is if |^fe1(xe)^j.2(xe)|2 is small and \ipkl(xe)il?k2(xe)\2 is not small. This condition must occur somewhere since T(xo) must have a minimum somewhere. 3.3 Symmetry Properties of the Transmission Coefficient Suppose the potential V(x) has a symmetric unit cell and let's position the crystal so that V(—x) = V(x). Thus, ij)k(—x) is a solution to the Schrodinger equation for the homogeneous crystal, equation (3.20). This means that if)k(—x) is a linear combination Chapter 3. Analysis of Sharp Interface 49 of ipk(x) and ij)-k(x). Since i>k{x + a) = eikaMx) (3-49) then ihk[-{x-")] = eikaM-x)- (3-50) This means that tpk(—x) is proportional to ^-k(x): 1>k{-x) = J+*ib_k(x) - e^l(x). (3.51) Thus 0,(0) = es'^ Vfc(0) (3.52) for all k. Also note that = -e-»*jj^k{-x) = -e^*ipk{-x). (3.53) Thus #(0) = -e»>tft(0) (3.54) for all A;. From the definition of W(ki,k2,xo) we see that equation (3.46) is satisfied at Xo = 0. We can say more than this just by looking at equation (3.44) for T(XQ). Since W(-k1,k2,-x0) = i>-kl(-x0)fk2(-x0)-^'_ki(-x0)il>k2(-x0) = -e^+^ipl^XoWZixo) + e ^ + ^ V - f e l (*o)^(x0) = -e^+^W^-fc^fca.so) (3.55) then T(—x0) = T(xo) for all jr0, that is, T(x0) is symmetric about all the symmetry positions of the crystal. Note that the half-integer multiples of a are also points of symmetry. One can eliminate the phase factor, e^k, by redefining the Bloch functions. Making the transformation, ipk(x) —> e~l^kl2tl)k(x) will do this. Chapter 3. Analysis of Sharp Interface 50 3.4 Symmetry Results for a General Sharp Interface The above symmetry result is also valid for the general sharp interface with completely different crystals on both sides. We assume that the unit cells of both crystals are sym-metric and the two crystals can be positioned such that Vi(x) = Vi(—x) and V2(x) = V2(—x). Choosing the global phase of the Bloch functions appropriately gives the follow-ing results, 4 1, f c l(-x) = fc1,fclW (3.56) = (3-58) = (3-59) This means that V>m,*i(0), V*n2lfc2(0) are r e a ^ a n d that $^,(0), V^2,*2(0) are imaginary. It is easily verified that by equations (3.15) and (3.17) this implies that T(—x0) = T(xo). The expression for the probability current simplifies. By equations (3.36),(3.35) and (3.23) we have and S a ^ T 1 - ^ -«»*U(o). (3.6i) It is useful to rewrite the connection formulas, equations (3.6) and (3.7) when the interface is at the symmetric position XQ = 0. They are, (A + AOta*(0) = (* + #)K.te(0) (3.62) (A - A')<, f c l(0) - (B - £ ' )& 2 ,* 2 (0) . (3.63) Defining the real-valued ratios, r and s, as r=h*M (3.64) Vw 2 (0) Chapter 3. Analysis of Sharp Interface 51 and equations (3.62) and (3.63) become s = <,fc,(0) r{A + A') = B + B' s{A -A') = B - B'. Equations (3.60) and (3.61) imply that din-, dk2 In determining the transmission coefficient, the ratio, B/A, is given by B 2d£ni(k1)/dk1 A )dSnJdh+rd£nJdk2 so that the transmission coefficient is given by T = 4 d£ni(h) dh |4rs| d£n2{h) dk2 ($\dLjdkx\-rr\d£n2ldk2\)' (3.65) (3.66) (3.67) (3.68) (3.69) (3.70) (M + W Thus, in this symmetric case, the transmission coefficient depends on only one energy dependent parameter that is not specified by band structure. We note that as a function of |r| and |s|, the transmission coefficient is maximized, with a value of one, when 5 = \ dki d£n2 dk2 Chapter 3. Analysis of Sharp Interface 52 3.5 The Ratios r and s One would like to know, even approximately, what the ratios r and s are without having to solve Schrodinger's equation exactly. In a real system the points of symmetry that we would be interested in are the points midway between the ions. We assume that the point of symmetry, x = 0, is one of these points, that is, it is not the position of an ion. In determining how the ratios r and s behave near a band edge, equations (3.60) and (3.61) are very useful. For example, at a band edge of the material on the left side, d£ni(ki)/dki = 0 so that either 0ni,A:i(O) or 0 ,^^ (0) must equal zero. This means that either r or s equals zero at the band edge in this example. It turns out that the two simple theories discussed below, predict quite well whether it is r or s which equals zero at the given band edge. They cannot however be expected to accurately predict, in general, the value of the nonzero ratio at the band edge. In the nearly free electron approximation (NFEA) it is assumed that the crystal potential is small (apart from a constant). If the wave vector is not close to a band edge (Bragg plane) then we have essentially the free electron case (see [23]). The modulus of the plane wave normalized Bloch function is \J\[2~K. To express what happens near a Bragg plane one needs to define the Fourier coefficients of the crystal potential V\{x) on the left side as and similarly for the right side. These coefficients are real since V\(—x) = Vi(a;). At the band edge on the left side between the nth and n + 1th bands the normalized wave function4 is given by5 Ashcroft and Mermin (see [23]) as, (3.71) (3.72) 4The wave function has an arbitrary phase factor. 5The wave number k\ is omitted for convenience. Chapter 3. Analysis of Sharp Interface 53 at the top of the nth band and 0 n + i ( x - fll/2) = 4 = k W a i - sgn(14n)e-^01] (3.73) at the bottom of the n + 1th band. The reason for the shift of ai/2 in the x origin is that they define the origin to be at an ion. As an example, if VXl is positive, then within a sign, 0i (x) = —-= sin7rx/ai, (3-74) 0 2 (x) = —= cos nx/ai, (3.75) V71" 0i(x) = cos irx/di (3.76) ai and 02(x) = —^—sin7rx/ci. (3.77) O l At x = 0, 0i and 02 are zero, while 02 and 0^  are larger in magnitude than the free electron values by a factor of As an example of estimating r and s, consider an interface in which the bottom of a nearly free electron band (the n + 1th one) on the left is matched with the middle of a nearly free electron band on the right. If V\n is positive, then in that energy range one would expect that \r\ ~ 1.4 and s ~ 0. On the other hand, in the tight-binding approximation (TBA) it is assumed that the Bloch functions are well localized about the ions of the crystal; see Chapter 6. This means that if the TBA is valid then 0m,*i(O) a n d 4>n2,k2(fy a r e s n i a U - The better the approximation is the smaller the function values are. Expressions for the Bloch function and its derivative evaluated at x = 0 are given by equations (6.55) and (6.56). It is assumed the Bloch funciLns are very well localized about the ions. For the left side these expressions are 0ni,fci(O) = /ii.ru cos fciai/2 (3.78) Chapter 3. Analysis of Sharp Interface 54 and a^'nukl(0) = ih'ltnismkiai/2 (3.79) with similar expressions for the right side. The wave numbers in these expressions are in the extended-zone scheme. The constants hit7l and h\ n are dimensionless and independent of k\. By equations (3.60), (3.61) and (6.40) they are related by, M i = 3 ^ 1 (3.80) where is the width of the nth band on the left side. There is a similar expression for the right side. Any calculation that uses equations (3.78) or (3.79) will be referred to as a tight-binding (TB or TBA) calculation. The value of h\}Tl is considerably less than I/1/7F (the NFEA value at k\ = 0) in the extreme tight-binding limit. Chapter 4 The Square Well Potential The analysis of Chapters 2 and 3 was quite general. We would like now to give a concrete example. Given a form of the periodic potential one could, easily enough, determine the transfer matrix of Chapter 2 by numerically solving Schrodinger's equation in a unit cell with the appropriate boundary conditions. We would like, however, to avoid numerical solutions by determining exact analytic expressions for the transfer matrix and hence the interface matrix. We choose the crystal potential to be a square well potential as shown in Figure 4.1 since the Schrodinger equation for this potential has simple (exponential) solutions. The potential of an ion is represented by a square well of depth V< and width d. Thus the crystal is characterized by the four parameters a, d, Vi, V/, (see Figure 4.2) where a is the lattice spacing. The only purpose in the value of V) is to define the zero of energy. The task in this chapter is to determine the transfer matrix, or what is almost the same thing, the transmission matrix, for the potential shown in Figure 4.2. 4.1 Transmission Matrix This problem would be considerably simplified by a more symmetric choice of unit cell and choosing the energy zero such that Vj = V. However for the case of a nonhomogeneous system, such as an interface, it is necessary to do the unsimplified problem. 55 Chapter 4. The Square Well Potential 56 Figure 4.1: The square well periodic potential, V(x), which is broken up into unit cells labelled by j . V(x) i = -2 j = o i = 2 —3a —2a —a 0 a 2a 3a Figure 4.2: The potential, V'(x), of a unit cell, for which we determine the transfer matrix. The shifting of the x axis is not important. V'{x) V V, -a/2 a/2 x Chapter 4. The Square Well Potential 57 The energy eigenfunction consists of plane waves everywhere. We write for the eigen-function with energy E: Aeiax + A'e-iax for x <-a/2 Bxe^x + B[e-tt* for - a/2 < x < -a/2 + V <j)a(x) = < B2e*iX + B'2e~^x for - a/2 + V < x < a/2 - b B3e^x + B'3e-tJx for a/2 - b < x < a/2 Aeiax + A'e-iax for x > a/2 (4.1) where \2mE a = h2 0 6 = 2m(Vf - E)  " V h2 l2m{Vf - Vi -~E)" h2 (4.2) (4.3) (4.4) We consider only E > 0 and V} > V,-. £/ and can be real or imaginary. We want to find the iteration matrix, M, defined by equation (2.9), M (4.5) We do this in steps. To relate B\ and B\ to B2 and B2, for example, we match <f>a{x) and its first derivative at a; = —a/2 + b' = x'. This gives Solving this gives ( e*«' -dx' \ \ 6e' -de B 2 \ B'2J (4.6) < B 2 \ -1 2& J_ 26 1 BX \ B i { (6 - tf)e^x' + 0 ) e ^ - ^ ^ ' ) \ B i ) (4.7) Chapter 4. The Square Well Potential 58 Similarly fJh\ 1 ' (&+e/)e«'-w-" (0 " fc)^'*0*" ' "20 (6 + 0)e(^°*" , a/2 - b. Also £ 2 2 / ^ 3 > where Mu = 26 ' ( 0 + ia)eUf-iaW2 ( 0 - ia)e«f+iaW2 > ^ ( 0 - ;a)e-tf/+i<*)0/2 ( 0 + za)e-^ -ia)°/2 ym _ 1 / ( 0 + » a ) e « / - ' a W 2 - ( 0 - ia)e-«/+la)a/2 ^  2 i a \ - ( 0 - m)e«/+ia)a/2 ( 0 + ict)e-ttf-ia>l2 , If the matrix, M,-, is defined by / f t \ ( Bx \ = Mi ^ 3 \ / then by equations (4.7) and (4.8) { M i ) u = i f e + O ) 2 ^ - ^ " - * 0 - (6 - O ) 2 ^ ^ * " - * ' ) ] £ 2 4. £? cosh (id + , ' sinh £,d 20£» and (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) (4.14) (M,-) i2 = 406 t i l 206 (4.15) Chapter 4. The Square Well Potential 59 To get the other two matrix elements we note that if £/ is imaginary then M,- is a transmission matrix having the form of equation (2.10). Using (£/ —• —£/) gives the other two matrix elements so that \ . (4.16) Mi = cosh £id + 2 ^ ' sinh £,d \ -c-«/(*-6')^|f- sinh t2 +Z2 cosh £,d — ^ £ sinh £,d / We want to calculate the matrix, M = MdM{Mu. We proceed directly. First define A = b - b'. (4.17) We use the following identities a = d + b + b' -d + a/2 = & ' - ( d - A ) / 2 and 6 - 6 ' - a / 2 = -b'-(d-A)/2 (4.18) (4.19) (4.20) to get (MiMu)n e-iaa/2e-l,(d-A)/2 eW\is + »a)(cosh £id + S l i L sinh &d) 2?K« + e - ^ ' ( £ / _ ^ ) ^ ^ s i n h £ i d 2 €/& (4.21) (M,-M u ) 1 2 = eiaa / 2 e-4/(d-A ) / 2 20 e«/6'(0 - m)(cosh&d + 4TT^ S I N H ^ ) +e-^6'(6 + m ) ^ f ^ - s i n h e ^ (4.22) ( M ; M U ) 2 1 = -iaa/2etf(d-A)/2 £2 _|_ £2 e_4/6'(0 - taO(cosh&d - * ' sinh&d) 2S/& -e^'(0 + ^) | f^sinh6d (4.23) Chapter 4. The Square Well Potential 60 (MiMu)22 = e«ao/2 e4 /(d-A)/2 20 e-Wfa + ta)(coshfcd - &±f- sinhfcd) Now we use the identities (4.24) -b' + (a + A - d ) / 2 = A and 6' + (a + A-d)/2 = a - d (4.25) (4.26) to obtain M n = (M d) n((M,-M u)n + ( M d ) 1 2 ( ( M , M u ) 2 1 e — I d a ——— 2(£? — a 2) cosh£,-dsinh£/(a — d) + 4i£/a cosh (idcosh £/(a — d) 4«£/Q; l (£2 _ Q/2)(£ 2 -f ^?) £2 + £2 H— * — sinh£,dcosh£/(a — d) + 2ia-^——- sinh£,dsinh£/(a — d) (4.27) In most cases we deal with, £t- is imaginary. Letting = id and noting equations (2.28) and (2.32) gives Mn = F(a) = e-iaa[X + iY] where X = cos dd cosh £/(a — d) + sin/?dsinh£/(a — ci!) 2?/ P y = " 2 ^ r + — ^ — s m ^ c o s h ' We have denned for convenience £2 -S = cos /?dsinh£/(a — d) + sin /3d cosh (a — d). 2 ? / P (4.28) (4.29) (4.30) (4.31) Chapter 4. The Square Well Potential 61 Similarly M 1 2 = (Md)n((M,-Mu)i2 + {Md)u((MiMu)22 1 2cosh&dsinh£/(a — d) + sinh&c?cosh£/(a — d) £ 2 — £2 + { f sinh&d [(ff - a2) cosh£/A + 2t£ /asinh£ /A] = G*(a) = U + iW (4.32) where U = - ^ ^ sinftdBinhfrA (4.33) W = + " ( Q 2 " f2 Kf + ^ Bin/gdcoBhOA. (4.34) Equations (4.3) and (4.4) can be used to rewrite Y, U and W as, Y(a) = ^-^S(a) + ^ ^ ( s i n 3d) cosh £/A (4.35) 20 oc h *£}afJ U{a) = Re[G*(a)] = -^(sin/ta) sinh^A (4.36) W(a) = Im [GT(a)] = ^ ^ ? ( « ) - ^ ^ | ^ ( s i n ^ ) cosh^A. (4.37) 4.2 Results pertaining to the Barrier and the Superlattice Here we derive some simple results which pertain to the results of Sections 2.3.2 and 2.4. We would first like to relate the dispersion relation, X = cos ka, in terms of the trans-mission coefficient of one of the steps of Figure 4.1. Consider, for example, the step in Figure 4.2 at x = —a/2 -f b'. We consider only the case where we have plane waves (E > Vf and E > V/ — V) so that £/ is imaginary, that is, 0 = id'. The transmission coefficient of a plane wave coming in from the left of this step is found by setting B'2 = 0 Chapter 4. The Square Well Potential 62 in equation (4.7). This gives T ° - B > B I - ( 8 + B>y  (4-38) In this case the dispersion relation for the crystal becomes by equation (4.29), r 2 l cos ka = cos 8d cos 8'{a — d) — —— 1 sin 8d sin 8'(a — d) (4.39) L J s J which is of the same form as the dispersion relation, equation (2.117), for the special superlattice. Similarly we consider the well (barrier) of width d and depth Vi (height —Vi) of Figure 4.2. The transmission coefficient of the well (barrier) is found by setting B'z = 0 in equation (4.13). This gives1 A828'2 T = -i—i- (A 40) w 4/?2/?'2 + (/?2 - /?'2)2 sin213'(a - d)' K ' Using equation (4.38) we rewrite this expression in terms of Ts as, Tw = / (4.41) l + 4 ( ± ^ ) s i n 2 / ? ' ( a - < Z ) ^ ; which is of the same form as the transmission coefficient, equation (2.117), of the special barrier. 4.3 The Interface The square well model of the interface is shown in Figure 4.3. Each of the two media are described by four parameters, a1,til,Vi1,V}1 for the left side and a2id2,Vi2,Vf2 for the right side. The interface is characterized by two more parameters, b\ and b'2. The unit cells will not in general be symmetric. can be real or imaginary in this case. Chapter 4. The Square Well Potential 63 Figure 4.3: Schematic picture of the interface. The potentials on either side are square and periodic. The partitioning into unit cells labelled by j is shown. V(x) b'2 d2 62 V »2 V h i = -3 j = ~2 i = - i i = 2 i = 3 —dai -2ax a2 2a2 3a2 The quantities X, F, (7 and W are now labelled by subscripts; 1 refers to the left side and 2 refers to the right side. The transmission coefficient is given by equation (2.71)2. This expression is bothersome since lijV^Wi,!^ all depend on the arbitrary choice of the energy origin. This means that they depend on the absolute value of the energy. Specifically they depend on the quantities E^y^,V/2. A physically relevant quantity such as Ti must not depend on these quantities. For example, by equation (4.29), X\, which represents the band structure of the material on the left of the crystal, does not depend on the choice of energy origin. This means, by equation (2.29), that kx is also energy-origin independent. After some algebra (see Appendix D) T/ is shown to be explicitly independent of the energy zero. The expression is: TAE) = Msinfciai)(sinfc2a2) denominator 'Note that Re{G\G2) = UXU2 + WXW2. Chapter 4. The Square Well Potential 64 where denominator = cr(sin&iax)(sin A;2a2) h SIS2 " " (sxn/^Xcosh^A^ffx + , 2 " / (sin ftdi) (cosh foAi)^ ^V!lVRR (sin/Mi)(sin/M2)x (sinhc:/lA1)(sinh02A2) - % ^ ( c o 8 h & A i ) ( c o B h £ A A 2 ) The unit cells are symmetric when Ai and A 2 are zero. Thus we see that general symmetry results of Section 2.2.3 hold here. Tj is invariant under the transformation Ax -+ -Ax A 3 (4.43) (4.44) T is also invariant under the transformation A 2 -> - A 2 (4.45) when Ax = 0. As a verification, setting Vix = V{2 = 0 gives £/j = iB\ , £ / 2 = ifa > -^1 — cos/?xax , X 2 = cos /?2a2 , 5x = i sin /?iai , S2 = i sin /?2a2 and 4/?i& T(£) = (4.46) (A + &)2 which is the transmission coefficient for a free particle (no lattice) in a potential step of height Vh-Vh. Chapter 4. The Square Well Potential 65 Figure 4.4: A schematic of a barrier with N = 3. The potentials on either side are square and periodic. The partitioning into unit cells labelled by j is shown. A mismatch is present (q ^ 0). A V{x) i = -3 i = -2 —3aj — 1a\ —o-\ do &'3|d3| 63| i = 2 j = 3 i = 4 j = 5 j = 6 02 2a2 3ao 3a2 + q +2a3 4.4 The Barrier For these numerical computations the potential shown in Figure 4.4 is used. The param-eters describing the potential are ,^1 ,^ ,^ for the left side, 02,02,Vi2,V/2 for the barrier and az,d3,Vi3,Vf3 for the right side. The left interface is characterized by the parameters, 61 and b'2 and the right interface is parametrized by b2,b'3 and the mismatch parameter q. All these parameters (except q) are defined just as in Figure 4.3. For the special case when the material in region 3 is the same as that of region 1 and g = 0 the transmission coefficient of the barrier is given by equation (2.96) where Tj is given by equation (4.42). Chapter 5 Numerical Results In this chapter, numerical results for the case of the square well crystal potential studied in Chapter 4 are presented. A l l the results are exact. Nowhere is an approximate numerical solution of Schrodinger's equation performed. In Section 5.1 some elementary numerical results for the infinite homogeneous crys-ta l are presented. This amounts to plugging in specific values of the parameters into equations (2.29) and (4.29). Numerical results for the interface are presented in Section 5.2. This amounts to plugging i n specific values of the parameters into equations (2.71),(4.30),(4.33) and (4.34). A variety of examples wi th different values of the input parameters are presented. A n attempt was made to cover al l of the possible kinds of band matchups. A n example of a k ind of band matchup is when the bottom part of a given band of the material on the left side matches wi th the top part of a given band of the material on the right side. A different band matchup is easily made by shifting one side of the crystal potential up by a constant value. It was found that a single choice of the crystal potentials on the two sides did not adequately exhaust all the possible kinds of band matchups. Thus, other choices are used in Section 5.2. This variety also helps to verify some of the general results presented earlier. The procedure of Section 5.2 amounts to choosing a set of parameters which characterize the bulk crystal potentials of both sides to give a certain band matchup. Results are given for the transmission coefficient as a function of energy and interface parameters. Some plots of the wavefunction near the interface are also 66 Chapter 5. Numerical Results 67 made. Results for the transmission coefficient using the parameter free envelope function matching (PFEFM) method (see Chapter 8) are also presented and comparisons are made to the correct results. A good way to do this is to plot the ratios r and s introduced in Section 3.4 as functions of energy. In Section 5.3 results for the transmission coefficient of the barrier as a function of en-ergy are presented. This is done for three different barriers. The transmission coefficient is evaluated using equation (2.91). The barrier matrix is calculated simply by multiply-ing the appropriate two by two matrices. Comparisons to the PFEFM calculations are made. 5.1 Elementary Numerical Results For Infinite Homogeneous Crystal In Figure 5.1a, X(ct) is plotted as a function of energy in units of %2 /(ma2) for parameters, V/a2 = Q0h2/m, Vi = Vf and d/a = 0.75; see Figure 4.2. The corresponding band structure defined by equations (2.29) and (4.29) is shown in Figure 5.1b. The allowed bands are given by \X\ < 1. There are three bands "in the wells", that is, with energy less than Vf. These bands are narrow with large bandgaps, indicating they are of a tight-binding nature. The discrete spectrum of an individual well of infinite depth is K ~ 2md* { 5 A ) where n = 1,2,3,... and these energies are measured from the bottom of the well. In Figure 5.1b the depth of the well is not really sufficient to warrant this approximation. In a more realistic, yet standard, calculation (see [7], page 76) for the spectrum of an isolated well of finite depth, the corresponding energies are approximately given by 2md2En/ft'2 = 0.709,2.823,6.212 for n = 1,2,3 respectively. In Figure 5.1b these energies lie at about (aa)2/2 = 6.2,24.8 and 57.1. These energies lie close to but just above the first three corresponding bands. The presence of the other wells not only splits the discrete energies Chapter 5. Numerical Results 68 into continuous bands but also causes the middle of the bands to lie lower in energy than the corresponding discrete energy value. The third and fourth bands represent a transition from tightly bound electrons to nearly free electrons. From about the fifth band on, the dispersion relation is about the same as for a free electron. The gap between the fifth and sixth bands is negligible. Chapter 5. Numerical Results 69 Figure 5.1a: A plot of X vs. ( aa ) 2 /2 . Parameters are V ) a 2 = 60h2/m , Vi = Vj d/a = 0.75. 10-Figure 5.1b: A bandstructure plot of kaf-K vs. ( aa ) 2 / 2 for a homogeneous crystal. ka 0.5 0 20 80 100 160 Chapter 5. Numerical Results 70 5.2 Numerical Results W i t h the Interface 5.2.1 We first consider the interface between two crystals with bulk parameters given by, a\ = a2 = a, Vha2 = 8h2/m, Vha2 = 9M2/m, Vh = Vh = Vfl, dxja = d2/a = 0.4; see Figure 4.3. All the Figures 5.2.1a through 5.2.If present results using the above values of the bulk parameters. These bulk parameters will not be changed until the next section (Section 5.2.2). This style of presentation is used for the rest of this chapter. The two crystals are the same except the right side potential is higher than the left side potential. The band structures of the two materials are shown in Figure 5.2.1a. The bands are lined up so that roughly the top half of the first band on the left side overlaps with the bottom half of the first band on the right side. In Figure 5.2.1b the transmission coefficient of the interface is plotted as a function of energy in the range of the overlap of the bands mentioned above. The interface parameters are b\/a = 0.3 (curve 1), b\ja = 0.15 (curve 2), and b\/a = 0 (curve 3), where b\ + b'2 = 0.6a in all three cases. The transmission coefficient always goes to zero at the band edges. This is because the group velocity is zero there. Curve 1 corresponds to the most symmetric case (Ai = A2 = 0) where the interface is halfway between the square wells to either side. Figures 5.2.1b and 5.2.1c show that, as a function of interface position (while keeping &i + b'2 constant), Tj is a minimum at the symmetric position. Note that plots of Tj versus energy for interface parameters b\/a = 0.45 and bi/a = 0.6 (&i + b'2 = 0.6a in both cases) exactly overlap curves 2 and 3 respectively. The values of the right side of equation (3.47) are indeed positive at xe = 0 at the energies mEa2/h2 = 5.75 and 6.8. One can understand this in a qualitative way by using the nearly free electron approximation (NFEA) and tight-binding approximation (TBA) results at the end of Chapter 3. Both approximations predict that near the top of the first Chapter 5. Numerical Results 71 band on the left side,1 01,^ (0) is small and 0^(0) is not small. Near the bottom of the first band on the right side one expects 0i,jfc2(O) not to be small and 0^(0) to be small. Using these results in equation (3.47) indicates that the second derivative of Tj is positive at xe = 0.2 Since both approximations give the same qualitative results, it is reasonable to assume that the same holds for bands intermediate to the two extremes. This is indeed the case here. The symmetry of the plots of Figure 5.2.1c verifies the results of Section 3.3 and the more general results of Section 2.2.3. Curve 4 of Figure 5.2.1b is the parameter free envelope function matching calculation (PFEFM) discussed in Chapter 7, equation (7.42). This calculation would give the same transmission coefficient as curve 1 of Figure 5.2.1b if \r\ or |s| was equal to one. The PFEFM calculation overestimates the transmission coefficient in this case. At Ea2 ~ 6.7h2/m the PFEFM calculation gives a very poor value (too small) for the reflection coefficient. Since the PFEFM calculation uses only band structure information it is independent of the choice of unit cells for the crystals on either side of the interface. Thus the PFEFM calculation gives the same answer for all values of b\ and b'2 such that 0 < & i < 0.6a and 0 < b'2 < 0.6a. Since the transmission coefficient is not independent of these interface parameters, we conclude that the PFEFM calculation can only be correct in certain cases. In this example the PFEFM calculation agrees best with the most asymmetric case shown by curve 3. The ratios r and s are plotted versus energy in Figure 5.2.Id. At energies near the bottom of the right-side band, s gets very large and at energies near the top of the left-side band, r goes to zero. This agrees with both the TBA and the NFEA results which predict that 0x (^0) is small near the bottom of the right-side band and that 01 (^0) is small near the top of the left-side band. As the energy approaches the bottom of the right-side band, r approaches the value 0.828. At this energy the TB calculation, using 1In this chapter, the evaluation of the Bloch functions at x = 0 is done for the symmetric case only (Ax = A 2 = 0). 2In this case VQ = 3 is positive and not too large. Chapter 5. Numerical Results 72 equation (3.78), gives the value3 of 0.58 for r. The discrepancy indicates that the Bloch functions are not that well localized. As the energy approaches the top of the left-side band, s approaches the value 1.64. The TB calculation gives 2.01. Figure 5.2.le is a plot of the real and imaginary parts of an energy eigenfunction (not normalized) near the interface. The energy is mEa2/h2 = 6.8 and the particle is incident from the left. The interface parameters are those of Figure 5.2.1b, curve 1 which means the square wells are centered at x = ±a/2, ±3a/2, On the left side the superposition of incident and reflected waves gives a wavefunction which is well localized at the square wells. On the right side, the transmitted wave function is much less localized. The modulus of the wave function at the symmetry points x = 0, a, 2a,... midway between the centers of the square wells is 52% of that at the centers of the wells. In Figure 5.2.If the incident and reflected waves are shown separately at the same energy of mEa2 j%2 = 6.8. Only the periodic parts of the Bloch functions are shown. More explicitly, if the wavefunction is given by A\j)nukl (x) + A'4>nu.kl (x) for x < 0 B^n2tk2(x) for x > 0 Aeik^unukl(x) + A'e-ik*xunu-kl(x) for x < 0 Beik>xun2ik2(x) for x > 0 then the function u(x) is defined by u(x) = (5.2) Aunukl(x) + A'unit_kl(x) for x < 0 Bun2tk2(x) for x > 0. Figure 5.2.If shows graphically how much of the incident wave is transmitted across the interface. 3Since the two crystal potentials are the same within an additive constant, h\ „ = h-} n-Chapter 5. Numerical Results 73 Chapter 5. Numerical Results 74 Figure 5.2.1c: Transmission coefficient of interface vs. interface position. b\ + b'2 is kept at the constant value of 0.6a. mEa2/H2 =5.75 for curve 1 and 6.8 for curve 2. 0.8-0.6-0.4-0.2-Figure 5.2.Id: The ratios r and s vs. energy. The solid (dashed) curve represents r 7* and s 3-2-mEa? h2 Chapter 5. Numerical Results 75 Figure 5.2.le: Energy eigenfunction near the interface. Parameters are those of Fig-ure 5.2.1b, curve 1, at energy, mEa2/h2 = 6.8. The particle is incident from the left. The solid (dashed) curve is the real (imaginary) part of the wavefunction and the interface is at x = 0. 3-, — — , . Figure 5.2.If: Energy eigenfunction near the interface. Parameters are those of Fig-ure 5.2.1b, curve 1, at energy, mEa2/h~2 = 6.8. The particle is incident from the left. Curve 1 (2) is the real (imaginary) part of the incident u(x), curve 3 (4) is the real (imaginary) part of the reflected u(x), and curve 5 (6) is the real (imaginary) part of the transmitted u(x). 9X '» . i »v • ' ' • It i \ > 1 I ky \j \f \y v / \ / \ A A A a w WW W V W 5 ' I 1 1 1 1 1 1 - 6 -4 - 2 0 2 4 6 x Chapter 5. Numerical Results 76 5.2.2 Next we consider the same system except the potential on the right side is raised further; Vf2a2 = 16h2/m. The bottom of the first band on the right side is lined up just higher than the bottom of the second band on the left side; see Figure 5.2.2a. As Figures 5.2.2b and 5.2.2c show, the transmission coefficient depends very much on the symmetry of the interface. A t energies near the bottom of the bands the transmission coefficient is almost one in the symmetric case (curve 1), and falls off rapidly as the interface is moved from the symmetric position. This effect is less pronounced further away from the bottom of the bands. The strong dependence of the transmission coefficient on the position of the interface enforces the above assertion that the P F E F M calculation can only be correct in certain cases. In this example the P F E F M calculation (curve 4) agrees best wi th the symmetric case (curve 1). A t the bottom of the bands, V^.^CO) a n d ^i,fc2(0) are small while V>2,fc1(0) and ^i,fc2(0) are not small. In equation (3.47) the only significant term is the one with the Vo factor which implies that the transmission coefficient is indeed a maximum when the interface is symmetric. The ratios r and s are plotted i n Figure 5.2.2d. The sharp divergence of s near the band bottoms is due to the right-side band min imum being slightly higher than that of the left side. Thus, as the energy approaches the bottom of the right-side band, V>i fca(0) approaches zero. In the same l imit r approaches the value 1.765. The N F E A predicts a value of about one or a litt le greater since the bottom of the bands are not perfectly matched. The T B A expression is given by ^1,2/^2,1- Since the band on the right side is more of a tight binding band than that of the left side, one expects this ratio to be greater than one. A t the top of the right-side band, r is large since f/>i,A;2(0) is small, and s approaches the value of —0.913. Again , this value is probably larger in magnitude than the N F E A value of —0.707 because of the tight binding nature of the right-side band. Chapter 5. Numerical Results 77 Figure 5.2.2a: Bandstructures of the two crystals forming the interface. Parameters are a x = «2 = a, V ^ a 2 = 8h2/m, Vf2a2 = I6h2/m, V t l = K2 = Vfiy di/a = d2/a = 0.4. 16-r— 1 , Figure 5.2.2b: Transmission coefficient of interface vs. energy. The interface pa-rameters are b\/a = 0.3 (curve 1), bi/a = 0.15 (curve 2), b\/a = 0 (curve 3), where bi + b'2 = 0.6a in all three cases. Curve 4 is the P F E F M calculation. 15.5 mEa2 h2 Chapter 5. Numerical Results 78 Figure 5.2.2c: Transmission coefficient of interface vs. interface position. & i + b'2 is kept constant at 0.6a. mEa2/h2 =12.5 for curve 1 and 14.0 for curve 2. i i i 1 i i i i i 1 1 12 12.5 13 13.5 14 U.5 15 15.5 mEa2 Chapter 5. Numerical Results 79 5.2.3 The band structure on the right side is raised even further now (V/ 2 = 25.25) so that the top of the first band on the right side is slightly higher than the top of the second band on the left side; see Figure 5.2.3a. Figure 5.2.3b shows that the transmission coefficient again depends very much on the symmetry of the interface. In the symmetric case (curve 1) the transmission coefficient is the largest, approaching the value one at Eo? ~ 23ft 2/m. The value of TT falls off rapidly as the interface is moved from the symmetric position. This is due primarily to the large value of v0 = 34.5 in equation 3.47. The P F E F M calculation of Figure 5.2.3b, curve 4, does not match well with any of curves 1,2 or 3. The P F E F M calculation seems to be affected considerably by the kink in the top of the left-side band. The correct calculations (curves 1,2 and 3) are not affected by this kink. This is an example of how band structure may not be important in determining the transmission coefficient. In Figure 5.2.3c we see that near the bottom of the right-side band, |s| becomes large as expected and r approaches the value 1.51 which is quite close to the N F E A value of 1.41. The value of V>2,fci(0) at this energy is 0.46 which is fairly close to the free electron value of 0.4. Near the top of the bands, r gets large as expected. However the behaviour of s as the energy approaches the top of the left-side band is rather odd. There is a kink corresponding to the kink at the top of the left-side band. After the kink, s rises sharply to zero. One would expect the N F E A to be very good on the left side near the kink. For energies just below the kink, the N F E A value for s is near4 —1.4 as it more or less is. The Fourier coefficient V\2 defined by equation (3.71) turns out to be negative in this case. This means that at the kink, (0) goes to zero, which results in the kink in the plot of s versus energy. 4 A factor of two must be included because the band on the left side is the second one; see equa-tion (3.72). Chapter 5. Numerical Results 80 Figure 5.2.3a: Bandstructures of the two crystals forming the interface. Parameters are ax = a2 = a, Vfla2 = 8h2/m, Vha2 = 25.25h2/m, Vh = Vi2 = Vfl, dx/a = d2/a = 0.4. 20 H 1 1 1 1 1 1 1 -1 - 0 . 5 0 0.5 1 —ky a k^a Figure 5.2.3b: Transmission coefficient of interface vs. energy. The interface pa-rameters are bi/a = 0.3 (curve 1), b\ja = 0.15 (curve 2), b\/a = 0 (curve 3), where bi + b'2 = 0.6a in all three cases. Curve 4 is the PFEFM calculation. mEa2 Chapter 5. Numerical Results 81 Figure 5.2.3c: The ratios r and s vs. energy. The solid (dashed) curve represents r 4 -2 -4 * -* ^ ^ -m * * — — ^ — " „ - » • • * * 2 1 I 21.5 I I 22 22.5 i 23 i 23.5 1 24 24 .5 mEa2 Chapter 5. Numerical Results 82 5.2.4 We now consider an interface with a different form of the crystal potential on the right side, with deeper and thinner wells. The bulk parameters are a2 = a, V/2a2 = Vi2a2 = I2h~2/m. The crystal potential on the left side is unchanged within an additive constant; VfrCt2 = 11.85/i2/m. The band structures are shown on Figure 5.2.4a. The bottom of the first band on the left side is lined up just higher than the bottom of the first band on the right side. Although the two crystal potentials are substantially different, the two band structures are very similar. Figure 5.2.4b shows that the value of the transmission coefficient is almost equal to 1 throughout almost the whole energy range of the the first band on the right side. The transmission coefficient is practically independent of the interface position since curves 1,2 and 3 coincide within one percent. Since V/j and V/2 are about the same one would expect little dependence of the transmission coefficient on the interface position (as long as b\ -f b'2 remains constant). This example indicates the importance of the band structure in determining the transmission coefficient. In fact the transmission coefficient seems to depend only on the band structure. This statement is supported by the fact that the PFEFM calculation, which uses only band structure information, gives almost perfect agreement in this case. However, when the distance between the square wells on either side of the interface is changed, as is done in Figure 5.2.4c, the transmission coefficient falls off to values considerably less than 1. The case of b\ + b'2 = 0.675a is a "resonance" condition, in which the distance between the centers of all the square wells is a. The PFEFM calculation cannot take this effect into account. Figure 5.2.4d shows that near the bottom of the bands, %>[ *2(0) becomes small as expected and r approaches the value 1.04 which is very close to the NFEA value of 1. Near the top of the bands, 0i,fc2(O) gets small as expected and s approaches the value 0.98 which again is very close to the NFEA value of 1. The fact that one of r Chapter 5. Numerical Results 83 and s remains close to the value 1 throughout the energy range of Figure 5.2.4d verifies the validity of the PFEFM calculation; see Chapter 7. The plot of u(x) in Figure 5.2.4e indicates that most of the wave function incident from the left is transmitted through the interface (at energy Ea2 = 9%2/m). Chapter 5. Numerical Results 84 Figure 5.2.4a: Bandstructures of the two crystals forming the interface. Parameters are ax = a2 = a, Vha2 = 11.85h2/m, V^a2 = 8h2/m, Vh = Vi2 = 12h2/ma2, di/a — 0.4, d2/a = 0.25. 20-Figure 5.2.4b: Transmission coefficient of interface vs. energy. The interface pa-rameters are bi/a = 0.3 (curve 1), bi/a = 0.15 (curve 2), bi/a = 0 (curve 3), where bx + b'2 = 0.675a in all three cases. Curve 4 is the PFEFM calculation. All four curves practically overlap. 0.5-Chapter 5. Numerical Results 85 Figure 5.2.4c: Transmission coefficient of interface vs. interface position. b\ is kept constant at 0.3a. mEa2/h2 = 9 for curve 1 and 10.5 for curve 2. "I 1 1 1 1 : 1 1 8 8.5 9 9.5 10 10.5 11 mEa2 h2 Chapter 5. Numerical Results 86 Figure 5.2.4e: Energy eigenfunction near the interface. Parameters are those of F ig -ure 5.2.4b, curve 1, at energy, mEa2/h2 = 9. The particle is incident from the left. Curve 1 (2) is the real (imaginary) part of the incident u(x), curve 3 (4) is the real (imaginary) part of the reflected u(x), and curve 5 (6) is the real (imaginary) part of the transmitted u(x). 2 - MAAA 4 A :'\ A A ;\ w » / V w 2 l l 1 \ / \ \ \ \ \ 5 y i\ \J i\ W'» \J '\ \J '» ' » i v i \ v ' i v ' » v ' , it i i / i * i ' i # i ' i i i » i » i » • ' » ' i ' i * i fi >' »' » 1 i i i i 1 1 1 1 »n 1 I • I 1 I 1 i 1 1 1 1 -6 -4 -2 0 x 2 4 6 Chapter 5. Numerical Results 87 5.2.5 We want now to match the the bottom part of the second band on the left side of Figure 5.2.4a wi th the top part of the the first band on the right side of Figure 5.2.4a; see Figure 5.2.5a. The new values of V^ma2/h2 and Vf2ma2/h2 are 8 and 14 respectively. As curves 1,2 and 3 of Figure 5.2.5b show, the transmission coefficient for this interface is quite small in the energy range shown. Even in the most symmetric case (curve l ) , i n which Tj is maximum, Tj is always less than 0.33. The P F E F M calculation (curve 4) gives very poor results in this case. This is supported by the fact that the values of | r | and |s| are so far from one as shown in Figure 5.2.5c. As the energy approaches the bottom of the left-side band, s goes to zero since T/>2,fci(0) does. The value of r approaches 3.06. Since V>i,fc2(0) g ° e s to zero at the top of the right-side band, it is s t i l l small at the bottom of the left-side band which gives the large value of r . The T B expression for r at this energy is 2.67/11,2/^ 2,1- The ratio 1^,2/^ 2,1 should be larger than one since the left-side band has more tight binding characteristics. Similarly, the T B expression for s at the top of the right-side band is — 0.25/ii,2/^2,1- The actual value for s is -0.495. Moving the bands down on the right side so that the band overlap is decreased would have the effect of making r larger and s smaller. This is a "mismatch" condition and leads to a very small transmission coefficient. Chapter 5. Numerical Results 88 Figure 5.2.5a: Bandstructures of the two crystals forming the interface. Parameters are a x = a 2 = a, Vjx = = 8h2/ma2, Vf2a2 = 14/i2/m, Vi2a2 = 12h2/m, di/a = 0.4, d2/a = 0.25. 12.8-Figure 5.2.5b: Transmission coefficient of interface vs. energy. The interface pa-rameters are b\/a = 0.3 (curve 1), bi/a — 0.15 (curve 2), bi/a = 0 (curve 3), where h + b'2 = 0.675a in al l three cases. Curve 4 is the P F E F M calculation. 1-Ti 0.5-12.7 mEa2 h2 Chapter 5. Numerical Results 89 / Figure 5.2.5c: (s). r and s The ratios r and s vs. energy. The solid (dashed) curve represents r 12.7 iEa2 T2— Chapter 5. Numerical Results 90 5.2.6 Next it is shown that making the lattice constants of the two crystals different does not have a major effect. The new bulk parameters are a2 = 1.5ai, V}, = = 31ft2/ma2, V/2a2 = 25.2ft2/m, Vi2a\ = 23ft2/m, di = 0.6ai, d2 = 0.4a2. The band structure is shown in Figure 5.2.6a. The bottom of the second band on the left is matched just below the second band on the right. In Figure 5.2.6b the transmission coefficient is plotted for the symmetric case (curve 1) with interface parameters given by b\ = 0.2ax, b'2 = 0.45ax and a nonsymmetric case (curve 2) with interface parameters given by b\ = 0.05a!, b'2 = 0.6ai. The PFEFM calculation (curve 3) agrees quite well with the symmetric calculation. Figure 5.2.6c shows that either r or s is always quite close to one. The limiting value of r as the energy approaches the bottom of the left-side band is 1.13. The NFEA value is 1 and one would expect /*i i 2 and h2>2 to be about the same in this case. The limiting value of s as the energy approaches the top of the right-side band is 0.92. The TBA value, assuming hit2/h2,2 = 1, is 0.615. The value of 02,fe2(O) does go to zero at the top of the right-side band. Chapter 5. Numerical Results 91 Figure 5.2.6a: Bandstructures of the two crystals forming the interface. Parameters are a2 = 1.5oi, Vh = Vh = 31ft2/mai> vha\ = 25.2ft2/m, Vi2a\ = 23^2/m, dx = 0.6au d2 = 0Aa2. 40-mEa2 30-20-10--1 -0.5 0.5 -fci ai 7T Figure 5.2.6b: Transmission coefficient of interface vs. energy. The interface parame-ters are bx = 0.2ax (curve 1), bx = 0.05ai (curve 2), where bx + b'2 = 0.65ai in both cases. Curve 3 is the PFEFM calculation. Chapter 5. Numerical Results 92 Figure 5.2.6c: The ratios r and 5 vs. energy. The solid (dashed) curve represents r (a). 4-and s 22 22.2 22.8 23.2 23.4 23.6 IEO,2 Chapter 5. Numerical Results 93 5.2.7 The right side is now moved up (V/2af = 28.78ft2/m) so that the top of the second band on the right side lines up just under the top of the second band on the left side; see Figure 5.2.7a. From Figure 5.2.7b we see that the PFEFM calculation (curve 3) does not agree well with the correct calculations (curves 1 and 2). The transmission coefficient does not depend much on the position of the interface in this case. Note the spike in curves 1 and 2 near the top of the right-side band. Both 02,fei (0) and 02,fc2 (0) a r e small in this energy region and as the energy approaches the top of the right-side band, they approach zero at the same rate (constant r) before the spike. When the energy reaches the top of the right-side band r shoots up sharply. The spike in the plots of the transmission coefficient occurs because at one point r and s are equal. By equation (3.70) this means that curve 1 of Figure 5.2.7b actually reaches the value of one when this happens. At this energy s has a value of 1.8. The NFEA value is 1. At the bottom of the right-side band the value of r is 0.74 which is quite close to the NFEA value of 0.707. Chapter 5. Numerical Results 94 Figure 5.2.7a: Bandstructures of the two crystals forming the interface. Parameters are a2 = 1.5ai, Vh = Vh = 31h2/ma2, Vha\ = 28.18U2/m, Vi2a\ = 23h2/m, dx = 0.6aa, d2 = 0.4a2. 27.5-T — — 1 -, 27-mEai —pr-26.5-26-25.5 -0.5 0.5 -fci a 101 Figure 5.2.7b: Transmission coefficient of interface vs. energy. The interface parame-ters are b\ = 0.2ai (curve 1), b\ = 0.05ax (curve 2), where b\ + V2 = 0.65ai in both cases. Curve 3 is the PFEFM calculation. Chapter 5. Numerical Results 95 Figure 5.2.7c: The ratios r and s vs. energy. The solid (dashed) curve represents r (s). r and s 1-25.4 25.6 25.8 I 26 T T 26.2 26.4 26.6 26.8 27 27.2 mEa2 Chapter 5. Numerical Results 96 5.2.8 The right side is now moved up again (Vf2a\ = 29.78ft2/m) so that the bottom part of the second band on the right side lines up with the top part of the second band on the left side; see Figure 5.2.8a. From Figure 5.2.8b we see that the PFEFM calculation (curve 3) agrees poorly with the correct calculations (curves 1 and 2). The transmission coefficient depends little on the position of the interface in this case. This is because Vft and V/2 are almost the same in this case (and the last case). This case is quite similar to that of Figures 5.2.5a to 5.2.5c. The value of 02,fci(O) is small near the top of the left-side band where r = 0 and s = 3.3. The TBA value for s, assuming h\t2 = h2,2, is 1.36 and the NFEA value is 7.8. The value of •02,fc2(O) is small near the bottom of the right-side band where r = 0.48. The TBA value for r, assuming hit2 — is 0.28 and the NFEA value is 1.37. Chapter 5. Numerical Results 97 Figure 5.2.8a: Bandstructures of the two crystals forming the interface. Parameters are a 2 = 1.5au Vh = Vix = 31^ 2 /maf , Vha\ = 29.78ft 2/m, Vi2aj = 23f t 2 /m, dx = 0.6ai, d2 = 0.4a 2 . 27.2 27-mEaj 26.8-26.6-26.4 - | 1 f- 1 1 1 1 1 -1 -0.5 0 0.5 1 —fei ai kvao. IT 7T Figure 5.2.8b: Transmission coefficient of interface vs. energy. The interface parame-ters are b\ = 0.2a\ (curve 1), b\ = 0.05ai (curve 2), where b\ + b'2 = 0.65ai in both cases. Curve 3 is the P F E F M calculation. Chapter 5. Numerical Results Figure 5.2.8c: The ratios r and s vs. energy. The solid (dashed) curve represents r ( r and s 2 -1-26.5 26.6 27.2 mEa2 Chapter 5. Numerical Results 99 5.3 Numerical Results for the Barrier In this section we present the numerical reults for some barriers. For a l l cases materials 1 and 3 are the same. 5.3.1 We first consider the barrier wi th bulk parameters given by ax = a2 = a 3 = a, V/ , = Vh = Vi2 = Vh = Vi3 = 8h2/ma2, Vhma2 = 16ft2, dx = d2 = d3 = 0.4a; see Figure 4.4. The barrier material is the same as that of the right side of Figure 5.2.2a while the surrounding material is the same as that of the left side of Figure 5.2.2a. The interface parameters are b\ = b2 = 6 3 = 0.3a and q = 0. There are three unit cells in the barrier and there is no mismatch. In this case the transmission coefficient of the barrier is given by the simple expression of equation (2.96). The transmission coefficient, as a function of energy, is given by the solid curve of Figure 5.3.1a. Note the oscillations characteristic of a barrier and the nonzero value in the bandgap of the barrier. The P F E F M calulation is given by the dashed curve. This calculation gives the correct position of the two resonance peaks at mEa2/ft2 = 12.7 and 13.9, and gives good agreement for mEa2/ft2 greater than 13.9. However, the P F E F M calculation completely misses the resonant peak at mEa2/h2 = 12.3 and gives poor agreement in the region near mEa2/ft2 = 13.2. A mismatch given by q = 0.4a is now introduced while al l the other parameters are kept the same. In this case the transmission coefficient of the barrier is not given by the simple expression of equation (2.96). The transmission coefficient, as a function of energy, is given by the solid curve of Figure 5.3.1b. There is a drastic change from the case of no mismatch. There is a resonance peak at mEa2ft? = 13.3 which was a min imum in the q = 0 case. The other peaks have been suppressed to the form of "shoulders" on either side of the main peak. The P F E F M results have not changed much from the q = 0 case. Chapter 5. Numerical Results 100 The extra peak at mEo?/h2 = 14.9 is simply due to the extra length of the barrier in this case. Thus there is very poor agreement with the solid curve. Chapter 5. Numerical Results 101 Figure 5.3.1a: Transmission coefficient of barrier vs. energy. The bulk parameters are a\ = a2 = a3 = a, Vjx = — V{2 = Vj3 - K'3 = 8ft2/ma2, V/2ma2 = 16ft2, ^ = ^ 2 = ^ 3 = 0.4a. The interface parameters are bi = b2 = b3 = 0.3a and q = 0. JV = 3. The solid curve is the actual value and the dashed curve is the PFEFM calculation. mEa2 Figure 5.3.1b: Transmission coefficient of barrier vs. energy. There is now a mismatch; q = 0.4a. mEa2 Chapter 5. Numerical Results 102 5.3.2 In the next example we consider the barrier with bulk parameters given by 01 = 02 = a3 = a, Vh = Vh = 11.85ft2/ma2, Vh = Vh = 8h2/ma2, Vh = Vk = 12h2/ma2, d\ = 0I3 = 0.4a, and d2 = 0.25a. The barrier material is the same as that of the right side of Figure 5.2.4a while the surrounding material is the same as that of the left side of Figure 5.2.4a. The constant interface parameters are bx = 63 = 0.3a and q = 0.25a. The barrier contains three complete unit cells in the barrier. In Figure 5.3.2a, curve 1 is for b'2 = 0.375a and curve 2 is for b'2 = 0.5a. Curve 1 corresponds to the case where the interface on the left side of the barrier is symmetric but the interface on the right side is not symmetric due to the mismatch. The resonant peaks have been almost completely washed out in this case. Curve 2 corresponds to a symmetric barrier. The electron potential has reflection symmetry about the center of the barrier. Three resonance peaks are now present which reach up to the value of one. The dashed curve is the PFEFM result which is practically the same as the transmission coefficient when no mismatch is present (q=0). The PFEFM result is not close to either of the solid curves. Chapter 5. Numerical Results 103 Figure 5.3.2a: Transmission coefficient of barrier vs. energy. The bulk parame-ters are aa = a2 = a3 = a, -- Vf3 = 11.85/i2/raa2, — Vi3 -- 8ft2/ma2, V}2 = Vi2 = I2h2/ma2, d\ = d3 = 0.4a, and d2 = 0.25a. The constant interface pa-rameters are b\ = 63 = 0.3a and q = 0.25a. iV = 3. Curve 1 is for 62 = 0.375a and curve 2 is for b'2 = 0.5a. The dashed curve is the PFEFM calculation which is the same for both values of 62. mEa2 ~1T Chapter 6 Tight-Binding Approximation Here we study the sharp interface in the tight-binding approximation. In particular we would like to determine the dependence of the interface matrix and the transmission coefficient, on a parameter that characterizes the interface. We need to first define the Wannier functions. To avoid complications wi th the definition of the Wannier functions (see [13,24]) we assume the potential for our system consists of a one-dimensional homogeneous, periodic potential, V(x), perturbed by the step potential The ions are positioned at the points ( . . . , —3a/2, —a/2, a /2 ,3a /2 , . . . ) and the unit cell boundaries are at the points ( . . . , —2a, —a, 0, a, 2a , . . . ) . The lattice spacing is a and we restrict xQ to lie in the range (—a/2, a/2); see Figure 6.1. The solutions to the unperturbed problem are the Bloch functions, r^n<k(x), and they satisfy the equation Roughly speaking the Wannier functions are localized functions centered at a given unit cell. There is one Wannier function for each unit cell whose position is labelled by /. We wi l l center the Wannier functions at the center of the unit cells so that U(x) = UQ9(x - x0). (6.1) -h2 d2 2m dx2 ^n,k(X) + V(x)tpn<k{x) = En(k)lpn,k{x). (6.2) / = (m - l / 2 ) a , m is an integer. (6.3) 104 Chapter 6. Tight-Binding Approximation 105 Figure 6.1: Schematic picture of the interface composed of a periodic potential V(x) plus the potential Uo 9(x — x0). The partitioning into unit cells labelled by m is shown. This choice is arbitrary and does not depend on the value of x0. V{x) + U(x) Chapter 6. Tight-Binding Approximation 106 (x|an(/)> = a„0r, /) = f 7° dke~ikl ^n,k{x) (6.4) This relation between / and m allows ns to use them interchangeably below. We assume an infinite crystal and define our Wannier functions as1 fir/a T / O The integral over k is over the first Brillouin zone (BZ). The Wannier functions are roughly speaking, Fourier transforms of the Bloch functions, which explains why they are localized. Using the time reversal property of the Bloch functions, the Wannier functions may also be written as 12a „ r7° Za rKla an(x,l) = \—Re dke-tkli>nik(x) (6.5) V 7T JO which shows that these functions are real. The inverse of equation (6.4) is ^n,k(x) = \ffj2eik'an(x,l) (6-6) V Z7T i where the sum is an infinite sum over all the unit cells of the crystal. This is easily verified by evaluating Sfl dke-ikli>nik(x) = ; f E / dke-ikleikl'an(x,l') V 2ir JBZ Z7r ^ JBZ = Yl h,i'an(x, I) = an(x, I). I' The Wannier functions are orthonormal, (a n(Z)|M0> = I dk [ dk'eikle-ik'v^n,k\^n,,kl) ZTT JBZ JBZ = [ dk ( dk'e{kle-ik'l'S(k-k')Snnl Z7T JBZ JBZ = £JBzdkeik«-l'K,n> = 6,j.6n,n. (6.7) 1There should be no confusion between the symbols for the Wannier function, a„(/) and the lattice constant a. Chapter 6. Tight-Binding Approximation 107 since the Bloch functions are orthonormal. We can rewrite equation (6.2) as W » J O = ^ (W».*> (6-8) We want to solve the eigenvalue equation, {H + U)\V) =E\V). (6.9) Since the Wannier functions form an orthonormal basis we can make the expansion tt(aO = £ / » ( 0 « » ( * > 0 - (6-10) n,l /„(/) is usually referred to as the envelope function. Taking the inner product of equa-tion (6.9) with (an(/)| gives £ /n ' ( / ' )M0l(#o + U- E)\an,{l')) = 0 (6.11) n',l' for all n, I. Now (an{l)\E\an,{l')) = E8n,n,8l,ll. (6.12) Also by equations (6.4) and (6.8), (a n(/)|ff„|aN,(0) = T- I dk I dk'eikle-ik'''Wnik\HoWn>,k>) ATT JBZ JBZ = 7T- I dk I dk'eikle-ik'l'£n(k)S(k-k')6nnl 2TT JBZ JBZ = ^ Jbz dkeikV-l')£n(k)8n,n, = Sn<n,Bn{l - 0 (6.13) where £ n is the Fourier transform of £„, that is, = d k e i k l £ n ( k ) . (6.14) Z7r JBZ Chapter 6. Tight-Binding Approximation 108 The inverse of this is £n(k) = '£e-ihlSn(l). (6.15) Equation (6.11) is rewritten as, £ / n ( / % ( / " I') + E/n'(O<*»(0|tfMO> = Efn(l) (6.16) I' n',l> To get a handle on the matrix elements of the step potential U(x) we need to make the tight-binding approximation. We assume that the Wannier functions are sufficiently localized so that the overlap between Wannier functions at different sites is negligible un-less they are nearest neighbours. For simplicity we restrict ourselves to a one-band model by assuming the matrix elements of U(x) between different bands is zero. Specifically this all means that (an(l)\H0\an(l')) (an(l)\U\an,(l')) (an(l)\U\an(l)) (an(l)\U\an(l')) = £n(l-l') = Q if \l-l'\>a (6.17) = Sn,n>(an(l)\U\an(l')) (6.18) 0 for m < —1 rin for m = 0 UQ — pn for m = 1 UQ for m > 2 (6.19) = 0 for|/-/'|>o (6.20) {an(l)\U\an(l + a)) = r 0 for m < — 1 * tn for m — 0 0 for m > 2 (6.21) Chapter 6. Tight-Binding Approximation 109 Since the Wannier functions are real-valued the parameters r)n, Hn, tn are real. These parameters characterize the effects of the interface in this method. One might assume nn and nn are negligible since most of the contribution to the matrix element (an(/)|an(/)) which equals one, occurs near the peak of the Wannier function at x = I. The "tails" don't contribute much. In order for the matrix element (dn(l)\U\an(l + a)) to equal zero however, the "tails" of the Wannier functions play an important role. Thus we cannot assume tn is very small. For m < — 1 equation (6.16) becomes -lnfn(l-a) + /3nfn(l)-lnfn(l + a) = Efn(l) (6.22) where for convenience we have defined 8n = £{0) (6.23) 7n = -£n(a) = -£n(-a). (6.24) A solution2 is /»(/) = e'*1' (6.25) which gives E = /3n — 2~fn cos kia = £n{ki) (6.26) The more spread out the Wannier functions are the more Fourier components are present in £n{k). For m > 2 equation (6.16) becomes - 7 » / n ( / - a) + Bnfn(l) - 7 n / B ( Z + a) + U0fn(l) = Efn(l) (6.27) A solution is /n(0 = eik>' (6.28) 2We are concerned only with energies lying in allowed bands of both sides in this analysis. Chapter 6. Tight-Binding Approximation 110 which gives E = UQ + 0n — 2 7 „ cos k2a — UQ + £„(fc2)-We write generally, ' Aeiklt + A'e~ikl1 for m < 0 ^ Beik*1 + B'e-ik*1 for m > 1. We have two equations to make the connection across the interface. For m tion (6.16) is (6.29) /n(0 = (6.30) 0 equa-7 n / n(-3a / 2 ) + 0nfn(-a/2) - 7 n / n (a /2 ) + <»/„(a/2) - Efn(-a/2) (6.31) or -7„Ae- ' f c l 3 a / 2 - ynA'eikl3a/2 + 0nAe~ik^2 + 0nA'eikia/2 - ^nBeik2a/2 - lnB'e~ik^2 +r}nAe-,kia/2 + rjnA'eikia/2 + tnBeik*a/2 + tnB'e~ik2a/2 = EAe~ikial2 + EA'eikia'2. Re-organizing this gives (tn - ln)eik*a/2B + ( i n - ln)e~ik^2 B' (6.32) = (7ne-ifcl° - /?n - r)n + E)e-'ha<2A + (7ne,'*ia -0n--qn + E)eik^2A' = (-7ne,fcl° - 7 /„ )e - i / : i a / 2 A + ( - 7 ^ - ^ - r!n)eik*a'2A'. Similarly, the other equation for m = 1 is - 7n/n(-3a/2) + 0nfn(-a/2) - 7nfn(a/2) + *„/„(a/2) = Efn{-a/2) (6.33) or -7 n Ae - , " f c i a / 2 - 7 nA'e t' f c i a / 2 + 0nBeik^2 + 0nB'e-ik2a/2 - ^nBeik^al2 - ^nB't~ik^al2 +{U0 - pn)Beik*a'2 + (t/0 - pn)B'e-ik*al2 + tnAe~ikia/2 + tnA'eik*a>2 = EBeik*al2 + EB'e-ik2a/2. Chapter 6. Tight-Binding Approximation 111 Re-organizing this gives (lne-ik2a - fin)eik2a/2B + (lneik*a - fxn)e-ik^2B' = ~(tn ~ 7n )e- , f c l 0 / 2A - (tn - ln)eik^2A'. We can rewrite these equations in the form of an interface matrix: e-ik2a/2Bf (6.34) = J \ I where J\2 = -(7e'-*.» + rj)(7eik>a - fi) + (7 - t)2 2ry(tf — 7) sin A;2a _ ( 7 e - « * . q + n)( 7e'^° - )^ + (7 - t)2 2^ 7(2 — 7) sin k2a J2I — J\2 (6.35) (6.36) (6.37) (6.38) (6.39) J22 — J\\-The band indices have been dropped for convenience. To find the transmission coefficient from the left side to the right side we impose the boundary condition that there is no incoming wave from the right side of the interface. Since dE -^j- = 27„a sin ka (6.40) which is true on both sides of the interface we see that the sign of k determines the direction of propagation of the wave (see equation 3.36). We will assume that k\ and k2 are positive but all the following results are true for k\ or k2 negative if we replace all the fci's and A:2's with their absolute values. The above boundary condition is JS' = 0. (6.41) Chapter 6. Tight-Binding Approximation 112 Solving equations (6.32) and (6.34) gives eikia/2A, (7eikia + ^ ) ( 7 e - « f c 2 a _ ^ _ ( 7 _ ty det (6.42) eik2a/2B (2ij(~f -t)smkia) g—ikia/2y^ det where det = ( 7 - t)2 - ( 7e- l' f c l° + 7/)(7e- i fc2a - p). (6.43) (6.44) We now make the simplifying approximation that 77 = p = 0. We do this to focus in on the dependence of R and T on the more significant parameter, t. The reflection coefficient is given by R A' 7 4 + (7 ~ t)4 - 2 7 2 (7 ~ t)2 cos(fc1 - fc2)q 7 4 + (7 - t)4 - 272(7 - t)2 cos(kx + k2)a 74 + (7 — £)4 — ^72(7 — t)2[cos kxa cos fc2a + sin kxa sin fc2a] 74 + (7 — t)4 — 272(7 — £)2[cos kxa cos fc2a — sin kxa sin fc2a] _ 472(7 — i)2(sin fcia)(sin fc2a) ~ 74 + (7 - *)4 - 272(7 - i)2 cos(ibi + k2)a (6.45) and the transmission coefficient is given by 472(7 - t)2sin2 kxa T = dE dk2 dE dk! 47 (7 — t) sin fci a y4 + (7 - t)4 ~ 272(7 - t)2 cos(fci + k2)a 472(7 — t)2(sin fcia)(sin k2a) (6.46) 7 4 + (7 - t)4 - 2 7 2 (7 - t)2 cos(fci + k2)a' We have also verified that R + T = 1. The only parameter in the right side of equation (6.46) which depends on the interface is t. We would like to know how T depends on t. Equation (6.46) gives T = 0 when t = 7. This however is to be taken seriously only if 7 is small compared to the bandgap Chapter 6. Tight-Binding Approximation 113 of the material. Otherwise the effect of the other bands must be included since t = 7 means that UQ is of the order of the bandgap. To find extrema of T(t) we take derivatives of equation (6.46), f = I" - [ 8 7 > ( 7 - t ) . - 8 / ( 7 _ «)] . (6.47) Setting this equal to zero gives three possible solutions. The first is t = 7 for which T = 0, a minimum. The other solutions are t = 0 and t = 2j both of which give = ( s i n ; M ( s i n M t = Q o r ( 6 < 4 g ) s in 2 (Jb i + fc2)a/2 ' ' v ' This value is close to one3 when k\ and k2 are close so we expect it to be a maximum. The second derivative at the extreme points, te, is T"(t • U) = ^ [ - ^ - I.)' + 7*1 (6.49) which is positive for t = 7 and negative for 2 = 0 and t = 27. Thus the transmission coefficient is a maximum when t = 0. Although the methods found in the literature quoted above vary, the common (see [2,13,14]) choice of description, when made arbitrarily, of the interface in tight-binding calculations is the most symmetric one. In this calculation this choice is equivalent to t = 0. This arbitrary choice is not justified and may give erroneous results. Unfortunately the dependence of T on the position of the interface itself is impossible unless the form of the Wannier functions are known. It is useful to consider the infinite homogeneous crystal (by setting UQ = 0) in order to derive a result used in Chapters 3 and 5. We desire a tight binding expression for the value of the wave function and its derivative halfway between two ions at a unit cell boundary. In particular we desire the expression evaluated at x = 0. The Bloch 3That the value is always less than one is due to the convexity of sinx in the interval 0 < x < 7r. Chapter 6. Tight-Binding Approximation 114 function for the nth band is given by equation (6.6). In the nearest-neighbour coupling approximation considered in this chapter we have (6.50) 0n,*(O) = J — [e-ika'2an(0,-a/2)eika/2an(0,a/2)] . where k is not necessarily in the first Brillouin zone, that is, we use the extended-zone scheme. If ipn,k(—x) = i>n,-k{x) then by equation (6.4) (6.51) and a'n(-x,l) = -a'n(x,-l) (6.52) where the prime denotes differentiation with respect to x. Defining the real valued, dimensionless quantities, K = J—an(0,a/2) (6.53) V 7T and K = ^—a'n^al2) (6.54) gives in this case, 0n,fc(O) = hn cos ka/2 (6.55) and arJi'ntk(0) = ih'nsmka/2. (6.56) Chapter 7 Effective Mass Formalism The effective mass formalism is ideally suited for potentials that are slowly varying on the scale of a lattice constant. This problem has been studied extensively; see for exam-ple ([16,17]). Here we study the the effective mass formalism applied to a potential that is not slowly varying. The formalism is similar to that of the tight-binding approximation, so the notation of Chapter 6 is used. The idea behind the effective mass formalism is to derive an equation for the envelope function, /n(/), similar to the Schroedinger equation. Ideally, this equation is independent of the actual forms of the periodic crystal poten-tial and the actual wavefunction, ty(x), and depends only on the band structure of the material. 7.1 Slowly Varying Potential The derivation of the effective mass equation (EME) starts with the form of the Schrodinger's equation given by equation (6.16). The first step in deriving the EME is to deal with the crystal potential (HQ) term in equation (6.16) which is £/»(0W-0. (7.1) Consider the following Taylor series identity: el&fn(x) = fn(l + x) (7.2) 115 Chapter 7. Effective Mass Formalism 116 where the domain of the envelope function has been extended from the discrete set of ion sites to the whole x axis. Using this identity and equation (6.15) gives £ / n ( / ' ) £ n ( / " 0 = £/„(/ + l')£n{l') = £ Bn^V ^ fn{x)\ . = £n (i-f) /»(*) . /' v v x~' \ axj x = l (7.3) In this expression we have assumed that fn(x) is analytic on the whole x axis since the radius of convergence of the Taylor series is assumed to be infinite and we want this to be valid at all the lattice sites, I. To get any practical use out of this expression one must make the effective mass approximation that in the region of energy we are interested in we have £n{k) = £n(0) + 2 7,2 2m* This gives us the nice result, £/n<(OMOI#o|<V(0) = n',l' £n(0) + -h2 d2 2m* dx2 (7.4) (7.5) x=l The second step in deriving the EME is dealing with the U term of equation (6.16). If U(x) is so slowly varying that it is practically constant over the spatial extent of a Wannier function, which is of the order of a, then (an{l)\U\an\l'))~8n>n,8ltllU(l). (7-6) This result is exact when U(x) is constant. Thus we obtain the EME (see [16,17]) '-n2 d2 + £n(0) + U(x)-E Ux) = 0. (7.7) x=l 2m* dx2 One should take some time to appreciate this result. It is notoriously difficult to improve upon. The usefulness of this result is that one can obtain for certain problems the energy spectrum and information about the wave function without knowing anything about the crystal potential. One does not have to determine the actual wave function. The link between the H0 term and the U term is that U(x) in the EME must be analytic so that /n(a;) is analytic, for equation (7.3) to be valid. Chapter 7. Effective Mass Formalism 117 Figure 7.1: An example of a barrier composed of an arbitrary periodic potential, V(x), plus the barrier potential U(x). V(x) -f U(x) l 1 1 • l-i—i 1 , ,—I , 1 1 x —5a —3a —a XQ <L 2<l ija 1SL X \ 9a 11a 13a 2 2 2 2 2 2 2 2 2 2 7.2 Sharply Varying Potential The problem with the sharp interface is that the assumption of equation (7.6) is invalid. Without any approximations the U term of equation (6.16) would result in a non-local, band-mixing potential in the EME. Worst of all this potential would depend on the actual wavefunction which would defeat the purpose of the whole method. It is instructive to study a simple example similar to that of Chapters 3 and 6, that of a barrier shown schematically in Figure 7.1. The problem is that of a periodic homogeneous crystal perturbed by the potential, U(x) = 6{x - x0)8(Xl - x)U0. (7.8) We want to calculate the U term in equation (6.16). Using equations (6.4) and (6.5) C h a p t e r 7. E f f e c t i v e M a s s F o r m a l i s m 118 this is written as, E MnMl)\U\an(l')) = / dkdk'eikU-ik'VUl') T dxrn,k{x)^Ax) ni \i ZTT N , JBZ JBZ JXQ = z Mr, ["''[•'•** 7T n,j, Jo Jo Re \eikle-ik'1' f ' dx^*n>k(x)^n,}kl(x) . L Jx0 ' . (7.9) As an example we will evaluate this term assuming the Bloch functions are free waves. One might expect the effective potential to be the same as U(x) for this case. This will be shown not to be the case however. We assume ^k(x) = ^ =e*'(fc+*) , k > 0 (7.10) where n = 1,2,3,... and1 < r = _ ( _ l ) » - 2 L n / 2 j ^ L [ _ n / 2 j . ( 7.H) For example, i>3,k{x) = -^ =et(A;+2ir/'aK The Bloch function for negative k is obtained by taking the complex conjugate of equation (7.10). Thus f 1 dxPnik(x)ipn,,k,(x) = ±- f 1 dxe-**-*V* ,-H*' JXQ ZTT JXQ leHk'-k+c'-c)Xm sm[(k'-k + a'-cr)A'a/2} 7r k' - k + a' - a \ • ) where and xm = (x0 + Xi)/2 (7.13) A' = (an - x0)/a. (7.14) 1 [x\ is the greatest integer less than or equal to x for x real. Chapter 7. Effective Mass Formalism 119 The U term is equal to 2aU0 n',i £/n ' (> ' )Re Jo Jo k'-k + cr' -a '-o-(7.15) This integral is difficult to do in general so it's appropriate to simplify it. We are interested only in the regions 0 < k < ir/a and 0 < k' < ir/a so that — ir/a < k' — k < ir/a. The function sin [(fc' - k + o' - g)A'q/2] k'-k + c-'-o ( 7 - 1 6 ) peaks at k' — k + o' — a = 0 and is mostly localized in the region -27r/A'o < k' - k + o' - a < 2ir/A'a. When A ' is much larger than one, this region is much smaller than the region of inte-gration. We will ignore the contribution from the terms where o' — a ^ 0. There are contributions from these "interband" terms. For example there is a contribution when2 <r' — a = 27r/a in the region where k' ~ 0 and k ~ n/a. However for A ' large this region is pointlike, whereas the contribution from the a' — a = 0 term comes from the whole "ridge" k ~ k'. One should realize that in certain situations, especially when A ' is not small, the interband terms might become important. The remaining integral is still complicated. An exact answer is not required, so to facilitate the integration we make the replacement sin p ' - f c )A'a /2 ] ^ k ' ~ k " [ 0 for \k'-k\>^-a which is exact when A ' is infinite. The factor v is of the order unity. The U term is approximately given by %± for =2ffli < jfc/ _ k < 4v a'a A'a 2An example is for n = 1 and n' = 2 so i int a = 0 and a' = —2ir/a. Chapter 7. Effective Mass Formalism 120 ^ a Up A J2fn(l') j dh r_,-(fc+2WA'aM«m-n J(k-2*v/&'a)(xm-l')] a2UoA' ^ £ / ; n [ e ' 2 ™ ( * " - ' ' ) / A ' ° - e - ' 2 ^(^- / ' ) /A 'a ] . J70A'a sin27ri/(xm - /)/(A'a) 2-7rv (xT O — /) The effective potential is given by U0A'a sin [27ri/(arm - l)/(A'a)] fn(l). (7.18) (7.19) 27TI/ (Xm — I) and is now local. As a verification, taking the width, A1 a, of the barrier to infinity gives a constant effective potential equal to UQ. For a finite barrier width the effective potential only vaguely resembles U(x). Although the two potentials are centered about the same point and have about the same width3, there is no sharp jump at x = x 0 or x = x\. Also, the effective potential has infinite extent. One would expect a similar effective potential, although more complicated, if we used the exact Bloch functions instead of plane waves. One would also expect the non-plane wave characteristic of the actual Bloch functions to inhibit the o"/,/  term in the U term thus making the effective potential slightly non-local. The actual Wannier functions are probably more localized than those for the free particle. This suggests that, ignoring the nonlocality, a realistic effective potential would more resemble U(x) than the one derived above. In the extreme case of the tight-binding approximation of Chapter 6 (see equations (6.19) and (6.21)) the local part of the effective potential is almost4 equal to U(x). However there is a considerable non-local part represented by the matrix ele-ment, tn. Thus, the tight-binding method and the local effective potential are consistent 3The closest zeros of the effective potential to the point xm are at xo and x\ when /i = 1 4They are equal if r}n = pn — 0-Chapter 7. Effective Mass Formalism 121 only if tn = 0. Let L be a rough measure of the extent of the Wannier function for the crystal. We conclude that outside of a distance L of a sharp interface, the effective potential is practically equal to U(x). Inside of this distance, however, the two potentials will not be similar. Another point worth mentioning is that the idea of discontinuities in the envelope function or its derivative is inconsistent with the E M E since it invalidates the derivation of equation (7.3). In order for equation (7.3) to be valid, / n(x) must be analytic everywhere. As an example of how the use of the E M E near an abrupt interface leads to trouble, consider the tight-binding example of Chapter 6. Near the interface at / = —a/2 the far left-hand side of equation (7.3), using equation (6.33), is rewritten as, £ fn(l')Sn{l ~ I') = - 7 n / n ( - 3 a / 2 ) + Bnfn(-a/2) - 7«/n(a /2) = £ / n ( - a / 2 ) - * n/ B(a/2) V so that by equation (7.4), h2 (7.22) 2lno? and £»(0) = pn- 2 7 n . (7.23) For k\ ~ 0 we have (7.24) Using equation (6.25), the far right side of equation (7. 3) is written as, £n{i-T~)fn(x) = £n(0) + h2k2 /»(-a/2). (7.25) x=—a/2 2m* Chapter 7. Effective Mass Formalism 122 Thus, equation (7.3) is correct only if tn = 0. This gives an example of how the abruptness of U(x) can invalidate the HQ term in the EME. Consider now the case of an interface with completely different materials on both sides. The lattice constants are the same on both sides although it is not necessary to make this restriction. The choice of the complete set of Wannier functions is not unique. For our purposes we require that an(x, I) = an(x, I) for / large and positive (7.26) an(x, I) = an(x, I) for / large and negative (7.27) where /) = Jf dke-ikl4>n<k(x) (7.28) V ZTT J-ir/a «n(x, /) = yff dke-M^k{x). (7.29) The function i>n,k(x) is a Bloch function for an infinite crystal composed of the material on the right side of the interface. Similarly, the function tpn,k(x) is a Bloch function for an infinite crystal composed of the material on the left side of the interface. It is not clear how to define the Wannier functions near the interface. All we require is that the entire set of Wannier functions is complete. For more on this see [24]. Equation (6.10) is still valid and remains as the definition of the envelope function. Outside of a distance L = max(Li, L2) from the interface, the envelope function is given by [AeiklX + A'e-ikl*] for x <x0 - L /»(*) where (7.30) [Beih2X + B'e-ihiX\ 8n,n2 for x > xQ + L E - Eni(ki) = Sn2(ki). (7.31) This is not a result of the EME, although the EME gives the same answer. Suppose we are far enough away from the interface on the left side so that the effect of all the Chapter 7. Effective Mass Formalism 123 Wannier functions near and on the right side of the interface is negligible. Then we can write *E(X) = £ / n ( 0 a n ( x , 0 ~ £ / n ( / ) a n ( x , / ) ~ £ [Aeik>* + A ' e " , f c i x ] n,/ = [ ^ n , , * ! (*) + A ' ^ m - f c i (x)] for x large and negative. Similarly we can write VE(X) = ^ [£0n2)fc2(x) + £'0„2,_*2(x)] (7.32) (7.33) for x large and positive. This is fully consistent wi th equation (1.11). Thus the coefficients of the envelope functions on either side of the interface are related by the same interface matrix that relates the coefficients of the Bloch functions on either side of the same interface. That is, • / R \ / A \ (7.34) A very simple set of boundary conditions which are sometimes used to relate the plane wave envelope functions are: f A \ =  J \ B > ) and / n i M = fn2(x0) (7.35) (7.36) where it is assumed that the envelope functions remain plane waves right up to the interface. These "parameter free" envelope function matching conditions have not been justified. Their desirable feature is that they contain no unknown parameters. Using equation (7.30) they are rewritten as, A + A' = B + B' (7.37) Chapter 7. Effective Mass Formalism 124 and h.(A-A>) = ^(B-B>) m rrio (7.38) so that the interface matrix is given by 1 = • 2k2 { \ k2 (7.39) If one wants to use this matrix at all energies, then in order to be consistent with the current conservation equation (A. 13), one must define the effective masses as d £ n i { h ) h2h dh (7.40) and similarly for m*,. Equation (A.13) takes the form ±(\A\'-\AT) = ±{\B\' m 1 rn2 The transmission coefficient is given by the very simple result, \B'\2)- (7.41) T = 4 d £ n i { k i ) d £ n 2 ( k 2 ) dk2 1 |4fci k2m*m*,\ ( \ d £ n i ( k i ) / d h \ + \ d £ n 2 ( k 2 ) / d k 2 \ ) ' (7.42) (\hm*2\ + \k2m\\y This analysis cannot be correct under all conditions. For example, it gives a transmission coefficient which is independent of the position of the interface. One would like to know under what conditions, if any, equations (7.35) and (7.36) are correct. Comparing these equations to equations (3.66) and (3.67) of Section 3.4, we see that equations (7.35) and (7.36) are correct if all of the following three sufficient conditions hold: 1) there exist symmetric unit cells in both crystals; 2) the interface is at a point of symmetry, that is, it lies at the boundary of the symmetric unit cells to each side; and 3) either |r| or \s\ equals one. The signs of r and s are irrelevant except that their product must satisfy equation (3.68). The condition \r\ — 1 is equivalent to the condition \s\ = 1. This is Chapter 7. Effective Mass Formalism 125 because the coefficients A' and B' have arbitrary phases. Changing the sign of these coefficients "switches" equations (3.66) and (3.67). The set of boundary conditions proposed (and not justified) by Harrison (see [1]) are a generalization of conditions (7.35) and (7.36). They involve two unknown, energy dependent, parameters which are equivalent to the ratios r and s. Thus Harrison's generalized boundary conditions are correct when the symmetry conditions 1) and 2) mentioned above are satisfied, but not otherwise. Attempts have been made to derive the matrix elements of the interface matrix using the EME. We have concluded above that within a distance L of the interface, the EME in the form of equation (7.7) is not valid. Morrow and Brownstein (see [18,19,20]), however, assume the EME to be valid near and at the interface. Boundary conditions similar to equations (7.35) and (7.36), for the envelope function are then made. This is an incorrect procedure and should be taken not as a serious derivation, but as an empirical adoption. Their analysis involves arbitrary parameters in their BC's, which don't seem to be universal from system to system and are not energy independent. In fact these arbitrary parameters can be determined in terms of r and s in the special case where the bands on both sides of the interface are parabolic. Any contraints imposed on the boundary conditions on the envelope function, or the number of parameters needed to characterize the interface, should not be taken seriously. The problem of using the EME to determinine the boundary conditions on the en-velope function, that is, determining the interface matrix, is very similar to solving a scattering problem, where the scattering potential has a finite extent of length L and is otherwise not well known. As shown in Appendix A, the problem amounts to determining three independent parameters at each energy. It seems that a derivation of these three values using the EME requires knowledge of the exact form of the effective potential near the interface. Since, for the case of the sharp interface, one must first know the real Chapter 7. Effective Mass Formalism 126 potential before obtaining the effective potential, the EME seems to be of little use for this case. Chapter 8 Conclusions In chapter 2 the transfer matrix method was used to write general (one-dimensional) expressions for the band structure of a crystal, the interface matrix, the barrier matrix and the superlattice matrix. This resulted in general expressions for the transmission coefficient of an interface and a barrier, the bound states in a well, and the band structure of a superlattice. Although the expression for the band structure of a crystal is well known (see [7] for example), the other results are new as far the author knows. For the case of the interface it is concluded that the interface matrix and hence the transmission coefficient depends on the position of the interface. Some general symmetry results for the interface are arrived at in Section 2.2.3. These results pertain to the dependence of the transmission coefficient on the position of the interface and are valid when there exist symmetric unit cells on both sides of the interface. The results are that Tj, as a function of the position of the interface, is even about the position of symmetry (when both unit cells are symmetric). Also, when the unit cell on one side is symmetric and is fixed, Tj, as a function of the distance from the interface to the nearest ion on the other side, is even about the symmetric position. The transmission coefficient of a barrier containing an exact number of unit cells and surrounded on both sides by the same material is expressed very simply in terms of only the transmission coefficient of the interfaces of the barrier and an interference term. Similarly for a two-component superlattice with an exact number of unit cells in each layer, the dispersion relation is given very simply by a term depending on the transmission coefficient of an interface and on two interference 127 Chapter 8. Conclusions 128 terms. The expression for the superlattice dispersion relation is very similar to that obtained by Bastard ([25]), using effective mass boundary conditions for the envelope function. These simplifications depend crucially on there being an exact number of unit cells in the barrier or each layer. The results for the interface and the barrier have all been verified by the numerical results of Chapter 5, where the square well form of the potential was used. Although similar results have been obtained in the literature ([8],[9] and [10],), there are some significant new results presented here. Trzeciakowski deals only with symmetric unit cells, thus missing out on the symmetry results mentioned above. Our results agree with his conclusions that the interface matrix cannot in general be described in terms of band structure alone. This is an important conclusion, since this means that the commonly used (see for example,[18,19,20,21,22,25,26,27]) effective mass boundary conditions can only be correct under certain conditions on the interface, if at all. Grinberg and Luryi deal with the delta function Kronig-Penney model which is only a special case of the square well potential considered here. Thus the results presented here are more general. Our results are in agreement with their conclusion that the transmission coefficient of the interface depends on the distance between the delta function potentials on either side of the interface. Their results indicate that the transmission coefficient is largest in the symmetric case. We conclude, however, that the symmetric case corresponds in general to either a maximum or a minimum in the transmission coefficient. We conclude in general that changing the form of the potential near the interface can drastically change the electrical properties of the interface. A precise knowledge of the potential near the interface is essential to obtaining accurate quantitative predictions related to the interface. In chapter 3 it is verified in a more direct, yet still general, way than that of Chapter 2, that the transmission coefficient of an interface, as a function of the position of the Chapter 8. Conclusions 129 interface, is even about the position of symmetry, when both unit cells are symmetric. In the case where the crystal potentials on both sides are the same, apart from a constant potential difference, a simple expression predicts whether the transmission coefficient at the point of symmetry is a maximum or a minimum. The expression depends on the value of the Bloch functions at the interface. It is concluded, however, that if the potential difference is large enough, the transmission coefficient must be a maximum. Coming back to the general interface in the symmetric case, it is concluded that the interface matrix can be specified in terms of the two ratios r and s. A current-conservation relation between r and s, involving band structure was arrived at. Thus the interface matrix is described by only one unknown parameter. This is a new result. Using this result, the boundary conditions introduced by Harrison ([1]) on the envelope function are justified in the symmetric case. Without the symmetry however, his boundary conditions are not correct. The behaviour of r and s as functions of energy can be described qualitatively quite well by the nearly free electron approximation and the tight binding approximation. In Chapter 6 the envelope function is introduced. A single-band, nearest-neighbour tight binding approximation is made for an interface in the case where the crystal poten-tials on both sides are the same, apart from a constant potential difference. The interface matrix depends on three unknown interface parameters, two of which are probably in-significant. The significant parameter, tn, depends only on the form of the interface and not on the band structure of the two materials. That the interface matrix depends on the position of the sharp interface verifies the results of Ando, Wakahara and Akera ([3]). This parameter is often ignored in the literature without justification. It is found that as a function of tn, the transmission coefficient of the interface is maximum when tn = 0. This is a new result. In Chapter 7 it is found that a derivation of an effective mass equation for the sharp interface leads to an equation that is more complicated to solve than the original Chapter 8. Conclusions 130 Schrodinger equation. The effective potential is nonlocal, is smoothed out, and depends on the form of the actual interface potential. It is concluded that such a procedure is not useful. It is also concluded that in the tight binding limit of Chapter 6, the EME and the tight binding method are consistent only if tn = 0. Any results which assume the validity of the EME for the sharp interface are suspect. A very simple set of boundary conditions on the envelope function which involve no unknown parameters (PFEFM) is introduced. These conditions are similar to the standard effective mass boundary condi-tions mentioned above. The difference, however, is that the PFEFM conditions are valid at all energies, not just where the bands are parabolic. It is shown that the interface matrix denned by these conditions is the same as the exact interface matrix in the sym-metric case if either |T - | or \s\ equals one. This is a new result. Grinberg and Luryi used an envelope function matching condition, which is valid only at band edges, to compare to their exact results. They concluded that the envelope matching condition gives the best approximation when, in their notation, 7 = 0. In our notation this is equivalent to °i + b'2 = b[ + b2; see Figure 4.3. This condition, however, is not sufficient. In order to have a symmetric interface the unit cells must be symmetric, that is, we must also have bi = b[. Evidence that the transmission coefficient is a function of the interface position even when bx -f b'2 = b[ + 62 is given by Figure 5.2.2c. Any calculation involving band structure only cannot account for this dependence on interface position. The envelope matching condition must be compared only to the symmetric case. When the interface is not symmetric it seems that no simple set of boundary conditions exists. Our numerical results showed that although the PFEFM sometimes gives reasonable approximations to the symmetric case, most often the approximation is poor. Bibliography [1] Walter A. Harrison. Tunneling from an independent-particle point of view. Physical Review, 123(1):,85, July 1961. [2] G.C. Osbourn and D.L. Smith. Transmission and reflection coefficients of carriers at an abrupt GaAs-GaAlAs (100) interface. Physical Review B, 19(4):2124, February 1979. [3] T. Ando, S. Wakahara, and H. Akera. Connection of envelope functions at semicon-ductor heterointerfaces. I. Interface matrix calculated in simplest models. Physical Review B, 40(17):11609, December 1989. [4] T. Ando and H. Akera. Connection of envelope functions at semiconductor heteroin-terfaces. II. Mixings of T and x valleys in GaAs/Al^ Gai_a; As. Physical Review B, 40(17):11619, December 1989. [5] A.C. Marsh and J.C. Inkson. Electronic properties of a (111) GaAs - A\x G&i-X As heterojunction. Solid State Communications, 52(12):1037, 1984. [6] A.C. Marsh and J.C. Inkson. Electron scattering from heterojunctions. Journal of Physics C: Solid State Physics, 17:6561, 1984. [7] Claude Cohen-Tannoudji, Bernard Diu, and Franck Laloe. Quantum Mechanics. Volume One, Hermann and John Wiley & Sons, Inc., 1977. [8] Witold Trzeciakowski. Boundary conditions and interface states in heterostructures. Physical Review B, 38(6):4322, August 1988. [9] Witold Trzeciakowski. Effective-mass approximation in semiconductor heterostruc-tures: one-dimensional analysis. Physical Review B, 38(17):12493, December 1988. [10] Anatoly A. Grinberg and Serge Luryi. Electron transmission across an interface of different one-dimensional crystals. Physical Review B, 39(11):7466, April 1989. [11] T. Ando and S. Mori. Effective-mass theory of semiconductor heterojunctions and superlattices. Surface Science, 113:124, 1982. [12] A. Ishibashi, Y. Mori, K. Kaneko, and N . Watanabe. A new connection rule of wave functions at a heterointerface and band discontinuity between GaAs and AlGaAs. Journal of Applied Physics, 55(12):4087, June 1986. 131 Bibliography 132 [13] Qi-Gau Zhu and Herbert Kroemer. Interface connection rules for effective-mass wave functions at an abrupt heterojunction between two different semiconductors. Physical Review B, 27(6):3519, March 1983. [14] P.A. Schulz and C.E.T. Gongalves da Silva. Simple model for resonant tunneling beyond the effective-mass approximation. Physical Review B, 35(15):8126, May 1987. [15] Thaddeus Gora and Fred Williams. Theory of electronic states and transport in graded mixed semiconductors. Physical Review, 177(3):1179, January 1969. [16] J.M. Ziman. Principles of the Theory of Solids. Cambridge University Press, 1965. [17] C M . van Vliet and A.H. Marshak. Wannier-Slater theorem for solids with nonuni-form band structure. Physical Review B, 26(12):6734, December 1982. [18] Richard A. Morrow and Kenneth R. Brownstein. Model effective-mass hamiltonians for abrupt heterojunctions and the associated wave-function-matching conditions. Physical Review B, 30(2):678, July 1984. [19] Richard A. Morrow. Establishment of an effective-mass hamiltonian for abrupt heterojunctions. Physical Review B, 35(15):8074, May 1987. [20] Richard A. Morrow. Effective-mass hamiltonians for abrupt heterojunctions in three dimensions. Physical Review B, 36(9):4836, September 1987. [21] Ian Galbraith and Geoffrey Duggan. Envelope-function matching conditions for GaAs/(Al,Ga)As heterojunctions. Physical Review B, 38(14):10057, November 1988. [22] S. Collins, D. Lowe, and J.R. Barker. On the accuracy of the effective mass approx-imation for electron scattering at heterojunctions. Journal of Physics C: Solid State Physics, 18:L637, 1985. [23] Neil W. Ashcroft and David N. Mermin. Solid State Physics. Saunders College, 1976. [24] C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James. Direct calculation of the tunneling current. Journal of Physics C: Solid State Physics, 4:916, 1971. [25] G. Bastard. Superlattice band structure in the envelope-function approximation. Physical Review B, 24(10):5693, November 1981. [26] M. Altarelli. Band structure, impurities and excitons in super lattices. In G. Allan, G. Bastard, N. Boccara, M. Lannoo, and M. Voos, editors, Heterojunctions and Semiconductor Superlattices, Springer-Verlag, 1986. Bibliography 133 [27] L. Eaves, D.C. Taylor, J.C. Portal, and L. Dmowski. Quantum tunneling of electrons through III-V heterostructure barriers. In G. Bauer, F. Kuchar, and H. Heinrich, ed-itors, Two-Dimensional Systems: Physics and New Devices, Springer-Verlag, 1986. [28] W. Kohn. Analytic properties of Bloch waves and Wannier functions. Physical Review, 115(4):809, August 1959. Appendix A General Constraints on the Interface Matrix Here some general constraints on the form of the interface matrix are derived in three dimensions. This includes the constraint of current conservation at a sharp interface. Also, equation (1.9), the definition of the one dimensional dispersion relation, is justified, thus justifying the reduction of the three dimensional problem to a one dimensional one. We proceed similarly to Section 2.1. Using the notation of Chapter 1 we consider the three dimensional energy eigenstate defined by . , . 0n 1 (k a(r) in bulk region 1 Cs,k(r) = t _ (A.l) k ^r(fciJV'n2,k2(r) + ^(A;iJV)„2)k2(r) in bulk region 2 where we have used equations (1.2) and (1.3). We have left out the dependence of T and Q on the components of ki parallel to the interface. This dependence is to be understood. It is also understood that k 2 is a known function of V.\. In general k\x may be complex. Since by the assumption of specular reflection the value of the energy, £n2{k2x), is independent of the sign of k2x we are free to assume k2x(—k\x) = — k2x(k\x). Another independent energy eigenstate is formed by changing the sign of k\x, A 0n,,kl(r) in bulk region 1 Cjs,k'(r) = \ „ (A.2) . ^(-^ix)0n 2 ,k 2(r) + <7(-fci,)0n2,ka(r) in bulk region 2. If one writes the general eigenstate as Ac^j^r) + A'(EM'(r) then by equations (1.1) and (1.6) the interface matrix must take the form 134 (A.3) Appendix A. General Constraints on the Interface Matrix 135 If there are no evanescent waves then the Bloch functions are waves which gives = (A.4) and H-hx) = F{ku). (A.5) Since T and Q are complex and current conservation gives one constraint, we conclude that in general the interface matrix is completely described by three independent func-tions of energy and parallel wave vector components. This result does not depend on the interface being sharp. We now derive explicit forms of the current conservation constraint for a sharp inter-face. With an argument similar to that of Chapter 3 one can show that the probability current density for the three dimensional Bloch state, ipntk(r), with all components of k real, is J = ^ W „ ( k ) . (A.6) This is proportional to the group velocity of a wave packet with its frequency spectrum peaked at k. All the analysis of Chapter 3 except the expressions for the reflection and transmission coefficients and right up to equation (3.32) continues to hold even if the energy lies in a bandgap. One simply replaces ik\ with p\ if the energy, E = £ni(pi), lies in a gap on the left side and similarly for the right side. The index n now labels the bandgaps. Equa-tion (3.39) no longer holds. Since the "wave" number p is real we can choose the Bloch functions to be real. Consider the one dimensional state, A\j>nitPl{x)-\-A'xj)ni-Pl(x), which requires that its domain be bounded by interfaces on both sides. Using equations (3.32) and (3.38), the current density for this state is easily shown to be j = ^-W{nupunu-Pux)\m{A*A') = z l d S ^ i m ( A * A') (A.7) m irn Op\ Appendix A. General Constraints on the Interface Matrix 136 In three dimensions, where the Bloch states can be evanescent in the ±x directions, the x component of ki is in general complex, written as k\x — ipix. This complicates matters. It is easier to be a little inaccurate and continue with a one dimensional argument, assuming a complex wavevector. It will be straightforward to generalize to three dimensions after. As a function of k\ = ki — ipi the function €ni (ki) is complex, having real and imaginary parts defined by = + <(*!)• (A.8) Since the energy must be real there is a constraint which relates k\ and p\ given by: S\h) = 0. (A.9) This is a property of the dimensionality of the system. In one dimension the condition is k\p\ = 0. Thus unlike in one dimension, one can have both k\ and p\ nonzero in three dimensions. This is due to the parallel components of ki. We conclude that except, perhaps for the case of glancing incidence, the product k\p\ is small, so the reduction of the problem to one dimension does not cause serious problems with the evanescent states. Since £ni(k\) is analytic (see [28]), we can evaluate, for example, the limit of equation (3.32) in any direction in the complex k\ plane. Using this idea and taking limits in the k\ and p\ direction generalizes equation (3.39) to give for the current: d£ni{ki) d£ni(ki) J = - L \\A\2e2"x - | A ' | 2 e - 2 H Re u ^ l > + J - I M T M( E- 2^A * A ' ) 2irn 1 J |_ oki J irn [ oki J - 2 7 r f t t | A | e | A | e ' dkx +nh dpi M e A A ) -(A.10) Except for notation, this expression is exactly the same as that for the a; component of the current in the three dimensional case, where kix = kix —ip\ is the x component of ki. Actually this expression is not quite correct in three dimensions. The partial derivatives Appendix A. General Constraints on the Interface Matrix 137 of the dispersion relations are averages over a unit cell of functions of position. The above expression would be the correct expression for the average over a unit cell of the x component of the current, except we have pulled the exponential factors out of the averaging. This is a good approximation if fci and p\ are small. Thus, unless we have glancing incidence and the product k\p\ is not small, equation (A. 10) is very good approximation for the x component of the probability current. For the case of the sharp interface it turns out that it is the x component of the current density that is conserved. This is because it is the partial derivative with respect to x that is matched at the interface. Another way to see this is to consider a wave packet incident on the interface from the left side with its frequency spectrum peaked at ki. When the wave packet crosses the interface it will stretch or shrink in the x direction by a factor of the ratio of the x components of the group velocities of the two bulk crystals. If the interface is sharp there will be negligible stretching or shrinking parallel to the interface. Thus the appropriate transmission coefficient would be £ 2|x-vgn2(k2) T = ( A . l l ) x - V £ n i ( k i ) This result agrees with that of Osbourn and Smith (see [2]). Thus the choice of the one dimensional dispersion relation, equation (1.9) is justified, since it is consistent with current conservation in the one dimensional problem. We conclude that the appropriate expression for current conservation at a sharp interface is [|A|2e2"* - lAfe"2"*] Re = [|ff | 2 e w - Ifffe"2"2*] Re + 2Im '94(ki)l . d k u . '0£n 2(k 2)' + 2Im '05„ 2(k 2)l L d l** L dk2l \ hn(e-2ikl*xAmA') Im(e-2ik3*xB*B'). (A.12) Any set of boundary conditions relating ff and ff' to A and A' across a sharp interface Appendix A. General Constraints on the Interface Matrix 138 must be consistent with equation (A.12). rlf there are no evanescent states present on either side, equation (A.12) reduces to m > _ \ A f ) ^ ^ - = (|I?|2 - i B f ) ^ ^ - . (A.13) In this case, equation (A.3) gives (W ~ \B'\2) = m2 - \g\2)(\A\2 -\Af) (A.14) so equation (A.13) is equivalent to a£n, (ki) Appendix B Derivation of Results of Section 2.1 Here we derive in detail the results that were just stated in Section 2.1. We continue with the notation of that section. First we diagonalize the transfer matrix Q(a). The eigenvalue equation is det(g - A/) = 0 (B.l) where I is the two by two identity matrix. Expanding gives A 2 - 2XX + 1 = 0 (B.2) A = X ± iVT-X*. (B.3) B . l Case 1: \X\ < 1 Let so that X = cos ka , Q<ka<* (B.4) \ = X ±ismka = e± . (B.5) We want A always to represent a wave moving to the right. This means the modulus of / ' must be smaller than that of / . To do this we set A = eeika. (B.6) 139 Appendix B. Derivation of Results of Section 2.1 140 This makes sense since the sign of Y changes sign from one band to neighbouring bands, as does the sign of dE/dk. Substituting this back into the top of equation (B.l) gives (X + iY - eicka)f + etaaG*f = 0 (B.7) or / ' cos ka + iY — cos ka — it sin ka f ~ -eiaaG* Y — e sin ka (B.8) ieiaaG* The other eigenvalue is 7 = A*. Substituting this into the bottom of equation (B.l) gives e~iaaGg + (X- iY - e-ieka)g' = 0 which is the complex conjugate of equation (B.7) so that 9' f* (B.9) We rewrite identity (2.33) as Y2 = \G\2 + sin2 ka (B.10) (B.ll) Using this we get L f (Y — esinfca)2 Y — esinfca —sinfca^^ Y2 — sin2 ka Y -fesin ka |Y| + sinfca — (B.12) thus verifying that A represents a state carrying current to the right. Other useful expressions are (B.13) f ~ ieiaa\G\2 Y" + esinfca 1 L f 2 sin ka \Y\ + sinfca (B.14) Appendix B. Derivation of Results of Section 2.1 141 1 + 2\Y\ 2Y \Y\ + sin ka Y + esin/ca 2 1 + 1 -Y (B.15) (B.16) 2 e sin /ca We would like now to verify that k is indeed the crystal momentum. This means that if we let (AA 3 = A then Bloch's theorem must hold: VE(x + a) = X$E(x) = eitka^E(x). By equation (2.20) we have WE(X) - fva(x - X Q - ja) + f'v*a{x ~ X Q - ja) (B.17) (B.18) (B.19) for ja < x < (j + l)a. By equation (2.26) and the fact that A is an eigenvector of Q with eigenvalue A we have Aj+i ^ f = A \ V Ai A'i (B.20) so that in the region (j + l)a < x < (j + 2)a : *E{x) = X[fva(x -XQ- [j + l)a) + f'v*a(x - X Q - (j + l)a)]. (B.21) This means that for (j + l)a < x + a < (j + 2)a : $E{x + a) = X[fva(x + a - x0 - (j + l)a) + f'v*a(x + a - x0 - (j + I)a)] (B.22) that is, $E{x + a) = X[fva{x -XQ- ja) + fv*a(x - x0 - ja)} (B.23) for ja < x < (j + l)a, thus verifying equation (B.18). Appendix B. Derivation of Results of Section 2.1 142 Now we find the probability current of a general energy eigenstate (in an allowed band). The state is written as \ A ' i J cA + er. (B.24) The current which is independent of position is given by equation (2.12) j = ^ f - K f (B.25) ha m [\c\2[\f\2 ~ |/f] + |e|2[M2 - \9'\2} + ce*[fg* - fg'*} + c*e[f*g - f*g'}} Using equations (B.10) and (B.14) gives J = ^ [kl2[l/l2 - l/T]'+ W W - W\2}} = ^ [ l / l 2 -\f\ 2][\c\ 2 - |e|2] ho>. 2 sin ka r. l 2 l 2 l "I/I i ^ , , , [\c\2~\e\2]- (B.26) m |F| + sin ka By equation (3.36) the current of a normalized Bloch wave carrying current to the right (e = 0) is given by J = 1 dE 2irh dk " Thus the normalization condition for the Bloch function is expressed as, I dE ha, 2irh dk m \c\2[\f\2 - l/T] (B.27) (B.28) B.2 Case 2: \X\ > 1 Let so that X = e' cosh pa , p > 0 (B.29) \ = X ±t' sinh pa = e'e±pa (B.30) Appendix B. Derivation of Results of Section 2.1 143 The two eigenvalues are A = e'epa and 7 = t'e pa. Substituting A in the top of the eigenvalue equation (B.l) gives /' X + iY — e'(cosh pa) — e'(sinh pa) Y +it'sinh pa f -eiaaG* ieiaaG* (B.31) Similarly for 7: g X — iY — e'(cosh pa) + e'(sinh pa) y + ie'sinhpa g' ~ -e-aaG _ -ie-aaG Equation (B.10) does not hold now. Identity (2.33) is written \G\2 = Y2 + sinh2 pa = (Y - it' sinh pa)(Y + it' sinh pa) With this we see that (B.32) Y2 + sinh2 pa \GJ2 and similarly (B.33) (B.34) (B.35) so that these states are not current carrying states. Some useful expressions are (Y + it' sinh pa)G -iGe-iaa / ieiaa\G\2 Y — it' sinh pa 9 (Y + it'sinh pa)G* iG*eiaa 9' -ie~aa\G\2 Y — j'e'sinh pa (Y + it' sinh pa)2 Y + it' sinh pa fg' \G\i Y — ie'sinh pa fg —2it' sinh pa fg' Y — it' sinh pa 2Y fg fg' Y — ie'sinh pa Y 1 ^ fg' _ 1 — 4^ —ie'sinh pa fa' r (B.36) (B.37) (B.38) (B.39) (B.40) (B.41) Appendix B. Derivation of Results of Section 2.1 144 Now we find the probability current for the general energy eigenstate (in a gap) of equation (B.24). Using equations (B.25),(B.34),(B.35),(B.31) and (B.32), the current is given by ha J [« W - / V I + c*e[r9 - /'*«/]] m 2fta: r .. Re ce*fg' m I —4t'ha sinh pa m -iY — e' sinh pa iY — e' sinh pa^ 0taaQ* -etaaG* Re ce*fg'* etaaG* (B.42) One can usually make a formal transition between the two cases, \X\ < 1 and \X\ > 1, simply by making the transformation p —• ik and e' —• e. Appendix C Derivation of Transmission Coefficients Here we derive in detail the expressions for the transmission coefficients of an interface (equation (2.71)), and the special barrier (equations (2.92) and (2.93)). We continue with the notation of Sections 2.2 and 2.3. C . l Transmission Coefficient of Interface To get the ratio, c 2 / c i , when e2 = 0 it is easiest to get the inverse of the interface matrix. By equation (2.59) we have c2 Aci J u -1 (C.l) where Jxx is the 1-1 element of J 1 . The determinant of J is Thus (1 - h9v (1 - 9ifj){l - toft) l-gifj 1 ( -e-Vgift + e* e^g2 - e - ' fy i 1 ~ f{9i (C.2) (C.3) \ e- '*/J - e«'*/{ - f[g2 This expression is valid everywhere except at band edges of the crystal on the left side of the interface. Thus C2 1 - f[gi 145 (C.4) Appendix C. Derivation of Transmission Coefficients 146 Now we use equations (2.31),(2.35) and (2.69) to get T = ( i - \ f f l ( i - \ m (1 - c-»*/f7J)(l - e**f[ft) 1 (1 - l / i | 2 )(l - \f'2\2)l + | / i / 5 | a _ 2Re[e-^/r/fl- ( ° ' 5 ) By equation (2.37), . , , , • 3 _ | +sinfeiai)(iy2| + sink2a2) + (|*i| - sinfciaiXftal - sinfc2a2) H l O T ( | l i | + sin fciai)(|F2| + sin k2a2) 2\Y\Y2\ -f 2sin ftiai sin k2a2 (|Yi| + sin fciai)(|F2| + sin k2a2)' By equations (B.13) and (2.57) (C.6) oG\G2 (|Fi| + sin Aia i ) ( | y 2 | + sinfc2a2) where a = t\t2. Putting this all together, and using equation (B.14) gives, (C.7) Ti 2 sin fciai |yi | + sin k\a\ 2 sin k2a2 |y 2 | + sin k2a2 (\YX \ + sin fciai)(|y2| + sinfc2a2) 2|yiy 2 | + 2 sin M i sin k2a2 - 2Re[aG*G2] 2(sin fciai)(sin k2a2) \Y\Y2 \ + sin fciai sin k2a2 — oRe[GlG2]' C.2 Transmission Coefficient of Special Barrier (C.8) We have set q = 0 and material 3 is the same as material 1. The two materials also have the same unit cell. Thus Jr -e-^'flfi/a* + t^g2 - e~^gx ^ -1 (C.9) Appendix C. Derivation of Transmission Coefficients 147 This means, ( ^  0 \ Ji. (CIO) \ 0 ^2 J B -1 = jr1 Calling Ji = J , <{>i = <t>, gives (1 - fe)U - fe) 7 2v (e^-e-^fe)( e -^-e^fe) +X?(e»g2-e-i*<n)(-e-»f2 + e»f1)] 1 [2ei<f,f[g2e'2 sinh Np2a2 + 2 e - * ' * f e 4 sinh Np2a2 ( l - f e ) ( l - f e ) +(e'2coshiVp2a2)(l - fe - fe + fefe) -(e'2 sinh Np2a2){\ + fe + fe .+ fefe)] where |X 2 | > 1 and we used ^2 ~l2 = 2 4 S i n h NP2®2 (C.12) ^2 +I2 = 24 cosh Np2a2. (C.13) For \X2\ < 1 we simply make the substitutions p2 = ik2 and e' = e, except that the e2 in front of the cosh is simply dropped. Using equations (B.13) and (B.37) gives g 2 i 0 r / _ G\G2 , . 1 (Yi + e i sin kiai)(Y2 — ^'2 s m n />2a2) e-2i<j> rl _ ^1^2  2 (Xi + £i sin hai)(Y2 - « 4 sinh p2o2)' Note that 1 ~ fe - fe + fefe = (1 - fe)(l - fe)- (C16) Appendix C. Derivation of Transmission Coefficients 148 iexe'2YxY2 Also by equations (B.16) and (B.41) 1 + ftg2 + fjgi + ffofjgi _ (1 - figi)(l - fe) (sin ki ax) (sinh p2a2) Putting this all together, using equations (B.14) and (B.39) gives 1 (C.17) &n ~ e2 c o s n Np2a2 + — sinh N p2a2 2ex sin kxax itxYxY2 —2ie'2 sinh p2a2 [2 s\nh N p2a2{GxG2 + G\G2)\ (sin fc!ai)(sinh p2a2) i i A T sinhNp2a2 = e'2 cosh Np2a2 - —— ~ r r [YXY2 - He{GxG2)\. (sin A;1aij(sinh p2a2) (C.18) Thus Ifiii1!2 = cosh2 Np2a2 + sinh2 N p2a2 (sin2 /ciai)(sinh2 /?2a2) Using the identity cosh2 x = 1 + sinh2 a; and the fact that 1 [YXY2 - Re(GxG*2)}2 (C.19) TB \BXX^ (C.20) gives equation (2.93). Equation (2.100) is derived by setting Bxx = 0 where px = ikx, e'x = ex. The matrix element Bxx is obtained by changing the sign of Af in Bxx . Appendix D On the Independence of Energy Origin Here it is verified that equation (2.71), the expression for the transmission coefficient for the interface, is independent of the choice of energy origin. An expression for the transmission coefficient will be derived which is explicitly independent of the quantities £,a,V/j and V}2. We continue with the notation of Sections 2.2, 4.1 and 4.3. By equations (4.29) and (4.33), Xi,X2,Ui, and U2 are all independent of the energy origin. This means sinfciai and sin k2a2 are also independent of the energy origin. Thus the expression we must work with in equation (2.71) is YiY2 — W1W2. Note that Si and 52 are independent of the energy origin. For shorthand we define p = 2m/h2. Using equations (4.35),(4.37) and (4.31) gives YXY2-WXW2 = a 2 L 2£/ 3a _2(ha '-pVh ^S2 + 4ft aft P2Vi2Vh Si + 4ft aft J -pViM2-eh). 52 + 4ft aft -pVUa2-ft) 4ft aft sin ftc?i cosh £/i Ai sin ftd2 cosh A2 sin ft di cosh Ai (D.l) sinftd2 cosh /^2^2 + + (y- f t )cy-f t ) P2Vf,vh 2(ha 2thcc 2 i h i h J vM<x2-eh)-vi2vh(*2-eh) S^ft^ft 8 0 2 f t « 2 f t S1S2 p2 sin ftd2 cosh £/2 A 2 p2 sin ftdi cosh <;/x Ai Si S2 149 Appendix D. On the Independence of Energy Origin 150 muicc^ P t i t 2 x sin 3\d\ sin 02d2 cosh £/j A i cosh £/ 2 A 2 . To simplify this we use equations (4.2) and (4.3) to obtain P2VhVh - ( a 2 - ft)(a2 - ( \ ) = 2a - 2 (£ / l + t h ) (D.2) ^ ( a 2 - ff2 - P V > 2 ) ( a 2 - ffj = 2a\(\ - ft)- (D.3) Thus we obtain l ^ a - W W a = _ f f i + ^ g l g 2 (D.4) . PK ' 2 (cj 2 2 - ft) . Q j i > A C + — , > 2 > o sin 02d2 cosh £ / 2 A 2 S i ffi sin /Vi cosh ^ A i f t p Vij V i 2 sin /3id a sin 02d2 cosh £/j A i cosh £/ 2 A 2 . This expression is clearly independent of the energy origin. Substitution of this ex-pression into equation (2.71) along wi th the fact that Re (G 1 (7 2 ) = UiU2 + WiW2 gives equation (4.42). 

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