Electron Interactions in Solids

Abstract

In this chapter we consider two extensions of R-matrix theory which describe interactions of electrons in solids. In Sect. 12.1 we consider an extension of R-matrix theory by Michiels et al. [648, 649] and Jones et al. [511] which describes low-energy electron collisions with transition metal oxides. This theory enables recent electron energy loss spectroscopy (EELS) experiments to be analysed and we show that calculations using this theory are in reasonable agreement with electron spin-flip measurements of Müller et al. [670] and with spin-averaged differential cross section measurements by Gorschlüter and Merz [406]. We also note that work has been initiated by Higgins et al. [469, 186] extending R-matrix theory to describe low-energy electron collisions with surface adsorbates.

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In this chapter we consider two extensions of R-matrix theory which describe interactions of electrons in solids. In Sect. 12.1 we consider an extension of R-matrix theory by Michiels et al. [648, 649] and Jones et al. [511] which describes low-energy electron collisions with transition metal oxides. This theory enables recent electron energy loss spectroscopy (EELS) experiments to be analysed and we show that calculations using this theory are in reasonable agreement with electron spin-flip measurements of Müller et al. [670] and with spin-averaged differential cross section measurements by Gorschlüter and Merz [406]. We also note that work has been initiated by Higgins et al. [469, 186] extending R-matrix theory to describe low-energy electron collisions with surface adsorbates.

In Sect. 12.2, we consider an extension of R-matrix theory which describes electron transport in two-dimensional semiconductor devices in the presence of an external field. This extension was introduced by Jayasekera et al. [500, 501], in an analysis of experiments by Goel et al. [384], which showed that R-matrix theory could be used to describe the transmission of electrons in four-terminal devices. Further developments and applications to solid-state devices have also been made by Jayasekera et al. [502].

We conclude this introduction by observing that a closely related “embedding method” of solving Schrödinger’s equation for solid-state systems, in which space is divided into sub-regions, has been considered by Inglesfield [491]. For example, in the study of surfaces using the embedding method a surface potential is derived which can be added to Schrödinger’s equation for a limited region of space embedded in an extended bulk substrate. This potential, which is energy dependent and non-local, is added to the Hamiltonian for the surface region and is determined from the Green’s function for the bulk substrate. The embedding method was reviewed by Inglesfield [492] who considered two surface applications of the method. The first was the study of Rydberg series of electron states bound by the image potential [674], and the second was the study of Xe adsorbed on the surface of Ag [220]. Inglesfield also discussed a quite different application of the embedding method to the problem of electrons confined by a hard wall potential [234]. He then considered its relation to the R-matrix method, where the links with the R-matrix method are not just in the division of space into two or more regions but also in the mathematical structure of the method.

1 Electron Collisions with Transition Metal Oxides

In this section we consider an extension of R-matrix theory by Michiels et al. [648, 649] and Jones et al. [511] which describes electron transport in transition metal oxides using electron energy loss spectroscopy (EELS) experiments.

1.1 Introduction

Low-energy electron collision experiments with solid-state targets provide an important probe of the electronic structure of solids, yielding information on the momentum and energy transfer associated with excitations (see, for example, Fuggle and Inglesfield [354]). At high incident electron energies this process can be described in the Born approximation by a dielectric loss function. However, recently there has been increasing interest in low-energy EELS (LE-EELS) in which incident electrons, with energy typically in the range 20–100 eV, excite non-dipole-allowed transitions including electron exchange effects which can give rise to multiplicity-changing transitions. These LE-EELS experiments, which show a wealth of angle, spin polarization and energy-dependent structure (Fromme et al. [352, 353] and Gorschlüter and Merz [406]), have been used to study the localized 3d–3d excitations in transition metal compounds, such as NiO and CoO, and also the even more localized 4f–4f excitations in rare earth metals, such as Gd (Matthew et al. [646] and Porter et al. [750]). However, while the energy loss spectra, measured in this way, can be described by parametrized crystal field models, the R-matrix approach described in this section is one of the first ab initio procedures which can explain the energy loss spectra and their dependence on incident energy, angle of scattering and spin polarization.

1.2 R-Matrix Theory

The generalization of R-matrix theory to describe low-energy electron collisions with transition metal oxides was made by Michiels et al. [648, 649] and Jones et al. [511] where the localized 3d–3d excitations in the transition metal compound NiO were studied. Following Jones et al. [511] we now briefly describe how R-matrix theory of electron collisions with atoms and atomic ions, discussed in Chap. 5, can be extended to describe the electronic transitions of Ni²⁺ ions which are situated in a crystal field. In contrast to electron collisions with a free Ni²⁺ ion, which can be treated using R-matrix theory presented in Chap. 5, we will see that the crystal field potential has a strong effect on the interaction between the scattered electron and the target ion.

We consider first the target states of the Ni²⁺ ion in an octahedral crystal field, where we limit our discussion to states associated with 3d–3d excitation, although other transitions can be treated by a straightforward extension of the theory considered here. The $\textrm{Ni}^{2+}\:3\textrm{d}^8$ configuration gives rise to the following five terms in the spherical environment of the free ion

$$\;^1\mathrm{S^e,\;^3P^e, \;^1D^e,\;^3F^e,\;^1G^e, }$$

((12.1))

where ³F^e is the ground term. Also the crystal field potential has the form [894]

$$\begin{array}{rcl}V_{\mathrm{C}}(r,\theta,\phi)&=&\left( \frac{7}{12}\right)^{{1/2}}\beta r^4 \left[Y_{40}(\theta,\phi)+\left( \frac{5}{14}\right)^{{1/2}} [Y_{44}(\theta,\phi)+Y_{4-4}(\theta,\phi)]\right]\nonumber\\ &&\;+\;V_{\mathrm{M}},\end{array}$$

((12.2))

where the Madelung potential $V_{\mathrm{M}}$ is the electrostatic shift at the origin due to the neighbouring ions which is fitted to Hartree–Fock band structure calculations [929]. This potential splits the 5 spherical terms into 11 target states as follows

$$\begin{array}{rcl}\;^1\mathrm{S^e}& \rightarrow& \;^1\mathrm{A_{1g}}\nonumber\\ \;^3\mathrm{P^e}& \rightarrow& \;^\mathrm{3T_{1g}}\nonumber\\ \;^1\mathrm{D^e}& \rightarrow& \;^\mathrm{1E_g\;+\;^1T_{2g}} \nonumber\\ \;^3\mathrm{F^e}& \rightarrow& \;^\mathrm{3A_{2g}\;+\;^3T_{1g}\;+\;^3T_{2g} }\nonumber\\ \;^1\mathrm{G^e}& \rightarrow&\;^1\mathrm{A_{1g}\;+\;^1E_g\;+\;^1T_{1g}\;+\;^1T_{2g} },\end{array}$$

((12.3))

where the labelling of the states on the right corresponds to the irreducible representations of the octahedral O_h symmetry group [928]. These states are determined as linear combinations of the five spherical states, determined by a Hartree–Fock calculation of Ni²⁺.

Having determined the target states, we are now in a position to construct the configuration interaction basis in the internal region, corresponding to expansion (5.6) in electron collisions with atoms and atomic ions. The channel functions are formed by coupling the target states, having symmetries defined by (12.3), with the spin–angle function of the scattered electron. The angular functions, which are appropriate to cubic symmetry are constructed from spherical harmonics, as follows

$$X_{h\ell}^{p\mu}(\theta_{N+1},\phi_{N+1})= \sum_m Y_{\ell m}(\theta_{N+1},\phi_{N+1})b_{h\ell m}^{p\mu},$$

((12.4))

where p denotes the irreducible representation (IR) and μ its component. Also h labels the different possible linear combinations of the spherical harmonics with angular momentum ℓ that transform according to the pth IR. The radial continuum basis orbitals $u_{ij}^0(r)$ in (5.6) representing the scattered electron are chosen to satisfy a zero-order differential equation corresponding to (5.75). The additional quadratically integrable functions, which are included in (5.6), come from the 3d⁹ configuration of Ni⁺ with spherical $^2\;\mathrm{D^e}$ symmetry which splits into $^2\;\mathrm{E_g}$ and $^2\;\mathrm{T_{2g}}$ symmetries in the octahedral crystal field. Finally, the coefficients corresponding to $a_{ijk}^{\varGamma}$ and $b_{ik}^{\varGamma}$ in (5.6) are obtained by diagonalizing the ($N+1$)-electron crystal field Hamiltonian and Bloch operator $H_{N+1}^{\mathrm{CF}}+{\cal L}_{N+1}$ in the internal region where

$$H_{N+1}^{\mathrm{CF}}=H_{N+1}+V_{N+1}^{\mathrm{C}},$$

((12.5))

and where ${\cal L}_{N+1}$ is the Bloch operator, defined by (5.8). Also in (12.5), $H_{N+1}$ is the non-relativistic Hamiltonian in the absence of the crystal field defined by (5.3) and

$$V_{N+1}^{\mathrm{C}}=\sum_{i=1}^{N+1}V_{\mathrm{C}}(r_i,\theta_i,\phi_i),$$

((12.6))

where V _C is the crystal field potential defined by (12.2).

The radius a ₀ of the internal region in Fig. 5.1 should, in principle, extend further out than the distance between neighbouring atoms in the crystal. Typically, for electron collisions with Ni²⁺ ions in free space $a_0\sim 7$ a.u. However, in a solid-state environment the scattered electron “feels” the full Coulomb potential of the ion over a much shorter distance, typically the atomic sphere radius, and beyond this radius it interacts predominantly with the neighbouring atoms. This corresponds to the muffin-tin or atomic sphere approximation that is frequently made in band structure calculations (e.g. Gonis [390]). The atomic sphere radius of Ni²⁺ in NiO is taken to be 2.58 a.u. from conventional band structure calculations, and the scattering amplitude is determined at this radius. In a full multiple-scattering calculation, scattering by all the atomic spheres would be included. However, the calculations carried out so far use a single-scattering approximation taking a constant potential outside the atomic sphere radius.

In order to determine the scattering amplitude at the smaller radius, the internal region calculation is first carried out using a radius $a_0= 7$ a.u. The R-matrix on this boundary is then propagated backwards from $r= 7$ a.u. to $r= 2.58$ a.u., using the propagator equation (E.28), where the Green’s function in this equation is calculated using the same one-electron Hamiltonian used to calculate the radial continuum basis functions $u_{ij}^0(r)$ in (5.6). Given the R-matrix on the boundary $r= 2.58$ a.u., the corresponding K-matrix and S-matrix can be determined by fitting to an asymptotic region solution as in Sect. 5.1.4. Hence the differential and total cross sections for transitions between the target states defined by (12.3) can be calculated. Results from this calculation are presented in Sect. 12.1.3, where this model is compared with experiment.

Fig. 12.1

Comparison of calculated spin-flip spectra (top) with experimental data (bottom) for electrons scattered from NiO as a function of scattering angle α and energy loss. The scattering angles are in the range $73^\circ\leq \alpha \leq 123^\circ$ in intervals of 3.125° with 73° being the lowest line (Figs. 13 and 14 from [511])

1.3 Illustrative Example

In this section we illustrate the theory described in Sect. 12.1.2 by comparing R-matrix LE-EELS calculations for NiO, carried out by Jones et al. [511], with experiment. We show in Fig. 12.1 calculations of the spin-flip spectra of electrons scattered by NiO compared with experimental data obtained by Müller et al. [670]. In this experiment, the energy loss of polarized electrons incident at an angle of 45° to the normal to the surface with an incident energy of 33 eV and undergoing a spin-flip was measured for 17 scattering angles α in the range $73^\circ\leq \alpha \leq 123^\circ$ at an interval of 3.125°. (The scattering angle α is related to the angle to the normal θ _f by $\theta_f=135^\circ -\alpha$.) In the theoretical model the elastic peak is ignored as is the 0.6 eV loss peak, which is due to a surface excitation. The small 1.05 eV loss peak is due to $\;^3\mathrm{T_{2g}}$ excitation and the big peak at 1.7 eV is due to two overlapping transitions $\;^1\mathrm{E}_{\textrm{g}}$ (1.70 eV) and $\;^3\mathrm{T_{1g}}$ (1.75 eV). The 3.2 eV loss peak is due to two overlapping transitions $\;^3\mathrm{T_{1g}}$ (3.13. eV) and $\;^1\mathrm{T_{1g}}$ (3.28 eV) and appears to be too narrow due to the contributing states being closer together than the experiment. It also lies slightly above experiment. Most significantly, at 2.7 eV we see a small shoulder at $\alpha = 73^\circ$ that increases and becomes dominant at 123°. This peak is of interest as it is a combination of two triplet–singlet excitations to the $\;^1\mathrm{A_{1g}}$ (2.80 eV) and $\;^1\mathrm{T_{2g}}$ (2.70 eV) states for which spin-flip dominates. Overall these results, taken together with comparisons made with spin-averaged differential cross section measurements by Gorschlüter and Merz [406], show good agreement with experiment.

In conclusion, we have shown that the single inelastic scattering approximation, adopted in the above theory, provides an overall understanding of the main features of the experiment. However, a complete R-matrix theory of LE-EELS must also include a treatment of multiple-elastic scattering events that occur before and after the inelastic scattering event. In order to do this Jones et al. [511] suggested that the results from the theory of diffuse low-energy electron diffraction (LEED), used to treat an additional scatterer by Pendry et al. [725, 807], could be used with the further simplification that the cross section for inelastic scattering is small compared with elastic scattering, so that we need to only consider a single inelastic scattering with no multiple events. Jones et al. [511] also pointed out that it is possible to include the damping of the propagating electrons due to the mean free path effects in the electron gas, in this multiple-scattering formalism. The inclusion of multiple scattering and damping in the R-matrix formalism is thus a challenge for future work on LE-EELS from NiO and other transition metal oxides.

2 Electron Transport in Semiconductor Devices

In this section we consider a recent extension of R-matrix theory which describes electron transport in two-dimensional semiconductor devices in the presence of an external magnetic field perpendicular to the device. This theory was introduced by Jayasekera et al. [500, 501] in an analysis of experiments by Goel et al. [384] who observed significant bend resistance in InSb four-terminal devices. The theory has enabled the transmission coefficients in these devices to be calculated, and further developments and applications have been made to solid-state devices by Jayasekera et al. [502].

2.1 Introduction

Modern experiments can fabricate two-dimensional semiconductor devices in which the mean free path of an electron is larger than the size of the device. As a result, the electron transport properties of these devices have been of interest both theoretically and experimentally for several years. Studies of magnetotransport have led to many advances, such as the quantum Hall effect [724], as well as to applied devices, such as magnetic field sensors and spin-based devices.

We consider a two-dimensional device illustrated in Fig. 12.2 which consists of a rectangular internal region and four leads which has been used in a negative bend resistance (NBR) experiment by Goel et al. [384], analysed by Jayasekera et al. [500, 501]. In this experiment, a current is injected from lead 2 to lead 3 (I ₂₃) in Fig. 12.2 and the voltage, V ₁₄, between leads 1 and 4 is measured. The bend resistance is defined as $R_B=V_{14}/I_{23}$. If the electron transport is ballistic, charges tend to go forward into lead 4 giving a negative bend resistance. When an external magnetic field is applied perpendicular to the device electrons tend to deflect into lead 3 which suppresses the bend resistance. The bend resistance therefore decreases as a function of the applied magnetic field.

Fig. 12.2

Schematic diagram of a four-terminal two-dimensional junction device. It consists of an internal region and an external region made up of four leads, where the dashed lines indicate the boundary between the internal region and the external region and where a magnetic field B is applied perpendicular to the plane of the device. A local coordinate system (x,y) is introduced in the internal region, and local coordinate systems ($x_p,y_p), p=1,\dots,4$, are introduced in each lead, where $x_p=0,\;p=1,\dots,4,$ is where the pth lead meets the internal region. Finally, in the rectangular internal region the coordinates x and y satisfy $-a\leq x\leq a$ and $-b\leq y\leq b$

2.2 R-Matrix Theory

We now consider the generalization of R-matrix theory to describe the transport of electrons through a two-dimensional device in the presence of an applied perpendicular magnetic field B as illustrated in Fig. 12.2. In this analysis we assume that the electron transport is ballistic and we model the transport using a single-electron picture. We commence from the time-independent Schrödinger equation

$$H\varPsi_j=E\varPsi_j,$$

((12.7))

where Ψ _j is the two-dimensional wave function describing the scattered electron and j labels the linearly independent solutions of (12.7). Also in (12.7), E is the total collision energy and the Hamiltonian H is given by

$$H=\frac{1}{2\textrm{m}^\ast}(\textbf{P}-{\textrm{e}}\textbf{A})^2+V,$$

((12.8))

where m^* is the effective mass of the electron and A is the vector potential. We observe that for devices made of InSb the electron has a small effective mass $\textrm{m}^{\ast}=0.0139{\textrm{m}}$, where m is the free electron mass. We also note that the potential V in (12.8) is set zero in the calculations undertaken, although this is not essential in the R-matrix formalism.

In order to solve (12.7) in the internal region in Fig. 12.2 it is convenient to transform this equation so that the Hamiltonian is dimensionless [500]. We measure the lengths in terms of a characteristic length in the device; typically the width of the input lead, $w_0=2b$, is chosen. We measure the energies in terms of $E_0={{\mathrm {\hbar}}}^2/\textrm{m}^{\ast}w_0^2$ and define ${\cal E}=E/E_0$. Also a quantity $l_B^2={{\mathrm {\hbar}}}/{\textrm{e}}B$ is introduced, where l _B , which has the dimension of length, is called the “magnetic length” and is the average radius of the lowest Landau level of the system. Finally we define the dimensionless magnetic field ${\cal B}=w_0^2/l_B^2$. The Schrödinger equation (12.7) then becomes

$$\left[-{\frac{1}{2}} \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right) +{{\mathrm {i}}} {\cal B}\left({\cal A}_x\frac{\partial}{\partial x} +{\cal A}_y\frac{\partial}{\partial y}\right) +\frac{{\cal B}^2}{2}\left({\cal A}_x^2+{\cal A}_y^2\right)\right] \varPsi_j ={\cal E}\varPsi_j,$$

((12.9))

where x and y are dimensionless coordinates and ${\mbox{\boldmath${\cal A}$}}$ is the dimensionless vector potential.

In the following discussion of the solution in the internal region we will consider the general solution of (12.9). However, as discussed by Jayasekera et al. [500, 501] different gauges can be used in the solution. In the symmetric gauge we have $\textbf{A}^{\mathrm{sym}}=(-y/2,x/2,0)$ and in the asymmetric gauge $\textbf{A}^{\mathrm{asym}}=(-y,0,0)$. While both gauges produce the same magnetic field the choice of gauge is important when the problem is solved approximately. We have to choose the gauge such that the solution will satisfy the boundary conditions of the system and the use of the appropriate gauge will give faster convergence. We will see when we discuss the solution in the external region that the asymmetric gauge is used. Finally, for future reference we rewrite (12.9) as follows

$${\cal H}\varPsi_j={\cal E}\varPsi_j,$$

((12.10))

and we consider in turn the solutions of this equation in the internal and external regions in Fig. 12.2.

2.2.1 Internal Region Solution

We consider first the solution of (12.10) in the internal region defined in Fig. 12.2. Following our discussion of electron–atom collisions in Sect. 5.1.2, we expand the wave function Ψ _j in terms of energy-independent basis functions ψ _k which we write here as

$$\varPsi_{ j}(x,y)=\sum_k \psi_k(x,y)A_{kj}({\cal E}),$$

((12.11))

where $A_{kj}({\cal E})$ are energy-dependent expansion coefficients, which depend on boundary conditions satisfied by the wave function Ψ _j at the energy ${\cal E}$. We then expand the basis functions ψ _k in terms of non-orthogonal energy-independent functions $\phi_i(x,y)$ as follows

$$\psi_k(x,y)=\sum_{i=1}^n \phi_i(x,y) a_{ik}.$$

((12.12))

Finally, we determine the coefficients a _ik in (12.12) by diagonalizing ${\cal H}+{\cal L}$ in this basis giving

$$\langle\psi_k|{\cal H}+{\cal L}|\psi_{k^{\prime}}\rangle_{\mathrm{int}} ={\cal E}_k\delta_{kk^\prime},$$

((12.13))

where ${\cal H}$ is defined by (12.9) and (12.10) and ${\cal L}$ is a Bloch operator which ensures ${\cal H}+{\cal L}$ is hermitian in the internal region, where the integration in this equation is carried out over this region.

The Bloch operator ${\cal L}$ in (12.13) has four components. The first two correspond to the kinetic energy terms in (12.9) and the second two to the magnetic field terms involving ${\cal B}$ in (12.9). The remaining terms in (12.9) involving ${\cal B}^2$ are hermitian in the internal region. We consider first the kinetic energy terms in (12.9). We find that

$$-{\frac{1}{2}} \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right) +L_x+L_y$$

((12.14))

is hermitian in the internal region where

$$L_x={\frac{1}{2}}\left[\delta(x-a)\frac{\partial}{\partial x} -\delta(x+a)\frac{\partial}{\partial x}\right]$$

((12.15))

and

$$L_y={\frac{1}{2}}\left[\delta(y-b)\frac{\partial}{\partial y} -\delta(y+b)\frac{\partial}{\partial y}\right].$$

((12.16))

Next we consider the magnetic field terms in (12.9). We find that

$${{\mathrm {i}}} {\cal B}\left({\cal A}_x\frac{\partial}{\partial x} +{\cal A}_y\frac{\partial}{\partial y}\right)+{\cal L}_x+{\cal L}_y$$

((12.17))

is hermitian in the internal region where

$${\cal L}_x=-{\frac{1}{2}} {{\mathrm {i}}} {\cal B}{\cal A}_x\left[\delta(x-a)-\delta(x+a)\right]$$

((12.18))

and

$${\cal L}_y=-{\frac{1}{2}} {{\mathrm {i}}} {\cal B}{\cal A}_y\left[\delta(y-b)-\delta(y+b)\right].$$

((12.19))

It follows that the Bloch operator ${\cal L}$ in (12.13) is defined by

$${\cal L}=L_x+L_y+{\cal L}_x+{\cal L}_y,$$

((12.20))

where the boundaries of the internal region in the above definitions of the Bloch operators L _x , L _y , ${\cal L}_x$ and ${\cal L}_y$ correspond to $x=\pm a$ and $y=\pm b$ in Fig. 12.2.

We can now solve (12.10) in the internal region by including the Bloch operator ${\cal L}$, defined by (12.20), on both sides of this equation giving

$$\left({\cal H}+{\cal L}-{\cal E}\right)\varPsi_j={\cal L}\varPsi_j.$$

((12.21))

This equation has the formal solution in the internal region

$$\varPsi_j=\left({\cal H}+{\cal L}-{\cal E}\right)^{-1}{\cal L}\varPsi_j,$$

((12.22))

where the spectral representation of the Green’s function $\left({\cal H}+{\cal L}-{\cal E}\right)^{-1}$ can be obtained in terms of the energy-independent basis functions ψ _k defined by (12.12) and (12.13). Equation (12.22) then becomes

$$|\varPsi_j\rangle=\sum_{k=1}^n |\psi_k\rangle\frac{1}{{\cal E}_k-{\cal E}} \langle\psi_k|{\cal L}|\varPsi_j\rangle.$$

((12.23))

Equation (12.23) can now be evaluated on the four boundaries between the internal and the external regions in Fig. 12.2 and hence enables an R-matrix to be defined which relates the wave functions to the normal derivatives on these boundaries. By projecting this equation onto the transverse eigenfunctions for each lead in the external region, the transmission amplitudes between the leads can be determined. It has been found by Jayasekera et al. [500] that expanding the wave function ψ _k in (12.12) in terms of functions φ _i which are non-orthogonal and which satisfy arbitrary boundary conditions avoids the use of a Buttle correction, required when homogeneous boundary conditions are used (see Sect. 5.3.2), and improves the convergence.

2.2.2 External Region Solution

We consider next the solution of (12.10) in the external region defined in Fig. 12.2. As in the internal region we assume that the leads are rectangular. For each lead we define a local coordinate system $(x_p, y_p)$ as in Fig. 12.2, where y _p is the transverse coordinate and x _p is the longitudinal coordinate in the pth lead. In the absence of a magnetic field, the transverse confining potential is an infinite square well with $V=0$ in the lead. The transverse lead eigenfunctions are therefore sine functions. However, these functions are not applicable if a magnetic field is present, so we seek transverse functions $f_{p\nu_p}(y_p)$ and wave numbers $k_{p\nu_p}$ in the pth lead, where ν _p is the transverse quantum number. Following [501] we choose the asymmetric gauge to describe the vector potential where $\textbf{A}=(-By_p,0,0)$, which we note is different for different leads. Also, as in (12.9), we measure the length in terms of the width $w_0=2b$ of the input lead and we define ${\cal E}=E/E_0$, $l_B^2={{\mathrm {\hbar}}}/{\textrm{e}}B$ and the dimensionless magnetic field ${\cal B}=w_0^2/l_B^2$. The Schrödinger equation in the pth lead then becomes

$$\left[-{\frac{1}{2}}\frac{{{\mathrm {d}}}^2}{{{\mathrm {d}}} y_p^2}+{\frac{1}{2}}\left(k_{p\nu_p}^2+y_p{\cal B}\right)^2 \right]f_{p\nu_p}(y_p)={\cal E}f_{p\nu_p}(y_p).$$

((12.24))

This equation is then solved numerically for the transverse functions $f_{p\nu_p}(y_p)$, as discussed by Tamura and Ando [912], and the collision wave function in the pth lead is expanded in terms of these functions as

$$\varPsi_{jE}(x_p,y_p)=\sum_{\nu_p}\tau_{p\nu_p}\exp({{\mathrm {i}}} k_{p\nu_p} x_p)f_{p\nu_p}(y_p).$$

((12.25))

The transmission amplitudes $\tau_{p\nu_p}$ in (12.25) can then be determined by substituting the expression (12.25) for the collision wave functions for each lead into the internal region solution (12.23) evaluated on the boundary between the internal and the external regions. This enables the flux J _p in the pth lead to be determined using the result

$$\begin{array}{rcl}J_p\sim \int {{\mathrm {d}}} y_p \left[\varPsi_{jE}^{\ast}(x_p,y_p) \left(-{{\mathrm {i}}}\frac{{{\mathrm {d}}}}{{{\mathrm {d}}} x_p}-A_{x_p}\right)\varPsi_{jE}(x_p,y_p)\right.\nonumber\\ +\,\varPsi_{jE}(x_p,y_p) \left.\left({{\mathrm {i}}}\frac{{{\mathrm {d}}}}{{{\mathrm {d}}} x_p}-A_{x_p}\right)\varPsi_{jE}^{\ast}(x_p,y_p)\right].\end{array}$$

((12.26))

Finally, from the resulting transmission amplitudes $\tau_{p\nu_p}$, we can calculate the transmission coefficients $T_{ij}=J_i/J_j$, where J _i and J _j are calculated using (12.26).

In comparing this analysis with that adopted in electron collisions with atoms, ions and molecules we observe that in the present analysis the external and asymptotic regions required in electron collisions are combined into one external region. This is because we have been able to match the solution on the outer boundary of the internal region directly with a linear combination of asymptotic solutions defined by (12.25). Of course, this simplification is compensated for by the more complicated nature of the boundary between the internal and the external regions in semiconductor devices, as illustrated in Fig. 12.2.

2.3 Illustrative Example

We consider calculations of transmission coefficients carried out by Jayasekera et al. [501] for a four-terminal symmetric square device consisting of a sample of InSb with an electron concentration $1.90\times 10^{11}\;\mbox{cm}^{-2}$ which is slightly less than the experimental value. The Fermi energy at this concentration is 32.7 meV, which equals 60 in the units E ₀ used in this calculation. The width of the internal region in this device $w=2a=2b$, illustrated in Fig. 12.2, is 0.1 μm.

Fig. 12.3

Transmission coefficients for electrons injected into the four-terminal square device shown in Fig. 12.2. (a) shows the transmission coefficients at zero magnetic field. (b) shows the transmission coefficients when ${\cal B}=w_0^2/l_B^2=6$ (Fig. 2 from [501])

We show the transmission coefficients calculated for this device in Fig. 12.3. At zero magnetic field, shown in Fig. 12.3 a, T ₁₂ and T ₃₂ lie on top of one another and T ₄₂ is always larger than the transmission coefficients for the sidearms. Therefore, more electrons accumulate in the forward lead than in the sidearms giving a negative bend resistance. However, as shown in Fig. 12.3 b, we see that in the presence of a magnetic field, electrons are more likely to be deflected into the sidearms. Thus we see that at some energies the transmission coefficient T ₃₂ is larger than the forward transmission coefficient T ₄₂. In this case fewer electrons accumulate in lead 4, and the negative bend resistance (NBR), discussed in Sect. 12.2.1, is suppressed. The sign and magnitude of the bend resistance R _B depend on the ratio of these transmission coefficients.

Finally, we mention that R-matrix theory methods have also been used to calculate the cooling properties of several two-dimensional devices by Jayasekera et al. [502]. It is thus clear from the work reported in this section that R-matrix theory methods play an important role in the analyses of electron transport in semiconductor devices.