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ECE539 - Advanced Theory of and Devices Numerical Methods and Simulation / Umberto Ravaioli

Introduction to Models

1 Schr¨odingerEquation

One of the goals of is to give a quantitative description on a micro- scopic scale of individual elementary objects (e.g. electrons, , , etc.) which behave both like and , exhibiting diffraction and interference phenom- ena. Propagation cannot be described by a trajectory (t) as for a classical , but by a field-like quantity ψ(r, t) called wavefunction. The simplest wavefunction is a plane-

ψ = Ψ0 exp[i(k · r − ωt)] (1) The wavevector k and the angular ω are properties of the wave and are related to and (particle properties) according to the postulates expressed by the relations

p =h ¯k (2)

E =hω ¯ (3) The wave properties ω and k and the the particle properties E and p are linked by dispersion relations ω = ω(k) and E = E(p). For an electron in , the energy dispersion relation assumes the well known parabolic form which gives the kinetic energy as

p2 ¯h2k2 E = = (4) 2m0 2m0 −31 where m0 = 9.1×10 [kg] is the rest mass of the electron. A plane-wave infinitely extends in both space and time and represents a mathematical limit. To describe realistic situations, more complicated wavefunctions can be constructed as superposition of plane waves. ZZZ 3 ψ(r, t) = Ψ0(k) exp [i(k · r − ω(k)t)] d k (5)

When the wavefunction has an appreciable magnitude only over a relatively small volume, we have a wavepacket. At a sufficient distance the wavepacket can be considered a point- like object, propagating through space along a particle-like trajectory. The wavefunction ψ(r, t) plays the role of while |ψ(r, t)|2 is a probability density, such that ZZZ |ψ(r, t)|2 d3r = 1 (6)

The above relation is used to normalize the wavefunction, to determine the normalized distribution of the coefficients Ψ0(k). The evolution of the wavefunction is described by the time-dependent Schr¨odingerequa- tion

1 ∂ψ(r, t) ¯h2 ¯h = − ∇2ψ(r, t) + V (r)ψ(r, t) (7) ∂t 2m0 Note that the wavefunction is complex, as clearly indicated by the example in (1), and the Schr¨odingerequation is equivalent to two equations for the real and imaginary parts of ψ = <(ψ) + =(ψ):

∂<(ψ) ¯h2 ¯h = − ∇2=(ψ) + V (r)=(ψ) (8) ∂t 2m0

∂=(ψ) ¯h2 ¯h = ∇2<(ψ) − V (r)<(ψ) (9) ∂t 2m0 The above equations indicate that the rate of change in time of the real part of the wave- function depends on the imaginary part, and vice versa. It should also be pointed out that the first term on the right hand side of (7) represents the kinetic part of the Hamiltonian of the system and originates from the parabolic dispersion relation (3). In operator notation, the solution to Schr¨odinger equation is  ψ(r, t) = exp (− Ht) ψ(r, 0) (10) ¯h where

¯h2 H = − ∇2 + V (r) (11) 2m is the Hamiltonian of the system. Equation (10) represents a formal solution of the Schr¨odingerequation as a first order differential equation in time. For a numerical solution, we need a discretized form  ψ¯(t + ∆t) = exp(− H∗∆t)ψ¯(0) (12) ¯h where H∗ is a matrix representing the discretized Hamiltonian and ψ¯(t) is the solution vector defined on the mesh points of the discretization.  ∗ The term exp(− ¯h H ∆t) represents the exponential of an operator, and cannot be used directly. The space discretization approaches for the Schr¨odingerequation (7) must yield a solution which behaves like (12), providing a suitable approximation for the exponential term. The quality of the approximation is reflected in two important properties of the solution: (a) stability of the time iteration; (b) conservation of the wavefunction probability.

2 1–D Time–Dependent Schr¨odingerEquation

In this section we examine several finite difference approaches for the solution of the 1–D time dependent Schr¨odingerequation

∂ψ(, t) ¯h2 ∂2ψ(x, t) ¯h = − + V (x)ψ(x, t) (13) ∂t 2m ∂x2 assuming a uniform mesh discretization of the domain, with mesh size ∆x.

2 2.1 Simple explicit method A simple discretization of the Schr¨odingerequation on a uniform mesh can be attempted by direct application of finite differences to the time and space differential operators

 ¯h ψ(i − 1; n) − 2ψ(i; n) + ψ(i + 1; n) V (i)  ψ(i; n + 1) = ψ(i; n) + ∆t − ψ(i, j) (14) 2m ∆x2 ¯h In operator notation, this method can be written as

   ψ¯(n + 1) = I − H∗∆t ψ¯(n) (15) ¯h where I is the unity matrix and H∗ is the tridiagonal matrix for the discretized Hamiltonian obtained from the second term on the right hand side of (15). This simple approach is numerically unstable. The exponential operator in (13) is not correctly approximated by  ∗ the transformation matrix (I − ¯h H ∆t), which is not unitary and has eigenvalues with modulus larger than 1.

2.2 Three level explicit method A stable and unitary discretization is obtained by applying a centered time difference which to a three-step explicit method

 ¯h ψ(i − 1; n) − 2ψ(i; n) + ψ(i + 1; n) V (i)  ψ(i; n + 1) = ψ(i; n − 1) + 2∆t − ψ(i; n) 2m ∆x2 ¯h (16) For this three time level scheme, an approximate form of the operator notation, linking the solution at times (n − 1) and (n + 1) separated by 2∆t, can be obtained from (16) by using the linear interpolation

ψ¯(n + 1) − ψ¯(n − 1) = ψ¯(j) (17) 2 to eliminate the intermediate solution ψ¯(j)

I − ∆tH∗/¯h ψ¯(n + 1) = ψ¯(n − 1) (18) I + ∆tH∗/¯h The matrix approximating the exponential provides a unitary transformation, which makes the method stable and physically acceptable.

2.3 Fully implicit method The fully implicit method is obtained by discretizing the left hand side of Schr¨odinger equation at timestep (j + 1), to obtain the following algorithm

∆t ¯h −  (ψ(i − 1; n + 1) − 2ψ(i; n + 1) + ψ(i + 1; n + 1)) ∆x2 2m V (i) + ∆t ψ(i; n + 1) + ψ(i; n + 1) = ψ(i; n) (19) ¯h

3 This method is numerically stable. However, the corresponding operator form of the solution is  ψ¯(n + 1) = (I + H∗∆t)−1ψ¯(n) (20) ¯h from which one can readily see that the norm R |ψ(x)|2dx = 1 is not conserved, i.e. the scheme does not correspond to a unitary transformation and leads to unphysical results.

2.4 Semi-implicit method The right hand side of the Schr¨odingerequation is discretized as an average of the operator at both timesteps, similarly to the Crank-Nicholson approach for the diffusion equation.

ψ(i; n + 1) − ψ(i; n) 1 ¯h = [RHS(n + 1) + RHS(n)] (21) ∆t 2 In the literature, the technique is also known as Cayley method. The resulting algorithm is

! 2mh2 ψ(i − 1; n + 1) + Co − V (i) − 2 ψ(i; n + 1) + ψ(i + 1; n + 1) = ¯h2 ! 2mh2 −ψ(i − 1; n) + Co + V (i) + 2 ψ(i; n) − ψ(i + 1; n) (22) ¯h2 where

4m∆x2 C = (23) o ¯h∆t The solution in terms of operator transformation can be written as

I − ∆tH∗/(2¯h) ψ¯(n + 1) = ψ¯(n) (24) I + ∆tH∗/(2¯h) which is practically equivalent to (18) for the three level explicit approach. Therefore, this method is also stable and unitary. From the analysis above, it follows that numerical solutions of the 1–D time–dependent Schr¨odingerdiscretized with finite differences can be accomplished using the three level explicit scheme or the Crank–Nicholson implicit scheme. The explicit approach requires more memory storage, since the solution at three time levels must always be known. The implicit scheme requires the solution of a tridiagonal matrix at each timestep, as seen in the case of the 1–D diffusion equation. The analysis of stability conditions is left as an exercise for the reader.

3 Schr¨odingerequation for a semiconductor

In a , momentum space is periodic and the true wavefunction is approximately the product of a periodic Bloch function and an envelope function. Now, k is the so called , and the energy dispersion relation, expressed as E = E(k), is usually known as the bandstructure. The Schr¨odingerequation can be used to study the evolution of the envelope wavefunction for an electron in the conduction band, provided that the

4 effective mass m∗ is used in the Hamiltonian. A parabolic dispersion relation as in (4) is valid in correspondence of momentum space regions (valleys) surrounding specific values kvi, not necessarily zero, at which the kinetic velocity and the associated mechanical momentum vanish. Dispersion relations like (4) can be still be used in each valley, provided that an effective momentum ki = k − kvi is used. When the Schr¨odingerequation is applied to semiconductors in the effective mass ap- proximation, the potential V (r) is assumed to be only the electrostatic potential, since the effect of the periodic crystal potential is accounted for by the effective mass itself. Equation (7) can again be used for relatively low close to the bottom of the conduction band, where a parabolic dispersion relation is a good approximation. In semiconductors, some of the most interesting applications of Schr¨odingerequation involve spatially varying material compositions and heterojunctions. The effective mass approximation can still be used with some caution. Since the effective mass is a property of a bulk, it is not well defined in the neighborhood of a sharp material transition. In the hypothesis of slow material composition variations in space, one can adopt the Schr¨odinger equation with a spatially varying effective mass, taken to be the mass of a bulk with the local material properties. However, it can be shown that the Hamiltonian operator in (7) is no longer Hermitian for varying mass. A widely used Hermitian form brings the effective mass inside the differential operator as

¯h2  1  − ∇ · ∇ψ (25) 2 m∗ This approach is extended to abrupt heterojunctions, as long as the the materials on the two sides have similar properties and bandstructure, as in the case of the GaAs/AlGaAs system. One has to keep in mind that very close to the heterojunctions the effective mass Schr¨odinger equation provides a reasonable mathematical connection between the two regions, but the physical quantities are not necessarily well defined. For instance, in the case of a narrow potential barrier obtained by using a thin layer of AlGaAs surrounded by GaAs, it is not obvious what effective mass should be used for the AlGaAs, since such a region cannot be certainly approximated by a bulk. Assuming a uniform mesh size ∆x, the Hamiltonian (25) can be discretized in 1–D by introducing midpoints in the mesh intervals on the two sides of the generic grid point i. First, we evaluate the outer derivative at point i with centered finite differences, using quantities defined at points (i − 1/2) and (i + 1/2) " # ¯h2 ∂  1 ∂ψ  ¯h2  1 ∂ψ   1 ∂ψ  − ∗ ≈ − ∗ − ∗ /∆x (26) 2 ∂x m ∂x 2 m ∂x i+1/2 m ∂x i−1/2 and then the derivatives defined on the midpoints are also evaluated with centered differ- ences using quantities on the grid points

¯h2 ψ(i + 1) − ψ(i) ψ(i) − ψ(i − 1) − − /∆2x (27) 2 m∗(i + 1/2) m∗(i − 1/2) The effective mass is the only quantity which must be known at the midpoints. If an abrupt heterojunction is located at point i, the abrupt change in effective mass is treated without ambiguity. It can be shown that the box integration procedure yields the same result. Another hermitian Hamiltonian operator proposed for variable mass has the form

5 ¯h2  1  1  − ∇2ψ + ∇2 ψ (28) 4 m∗ m∗ which is the linear combination of two nonhermitian operators. It is instructive to compare the formulations (25) and (28). In 1–D, the operators can be rewritten as follows " # ¯h2 ∂  1 ∂ψ  ¯h2 1 ∂2ψ ∂ψ ∂  1  − = − + (29) 2 ∂x m∗ ∂x 2 m∗ ∂x2 ∂x ∂x m∗

" # " # ¯h2 1 ∂2ψ ∂2  1  ¯h2 1 ∂2ψ ∂ψ ∂  1  1 ∂2  1  − + ψ = − + + ψ (30) 4 m∗ ∂x2 ∂x2 m∗ 2 m∗ ∂x2 ∂x ∂x m∗ 2 ∂x2 m∗

The second operator has an additional term involving the second order derivative of the effective mass. For smoothly varying mass, the two approaches are approximately equiv- alent. If one were to use the form on the right hand side of (29) for discretization of the operator, it is easy to see that a direct application of finite differences is awkward. The proper procedure is to apply box integration to the interval [i − 1/2; i + 1/2] " # ¯h2 Z i+1/2 1 ∂2ψ ∂ψ ∂  1  − dx ∗ 2 + ∗ (31) 2 i−1/2 m ∂x ∂x ∂x m Integration by parts of the first term yields

" # ¯h2  1 ∂ψ i+1/2 Z i+1/2 ∂ψ ∂  1  Z i+1/2 ∂ψ ∂  1  − ∗ − dx ∗ + dx ∗ (32) 2 m ∂x i−1/2 i−1/2 ∂x ∂x m i−1/2 ∂x ∂x m The two integrals cancel, and if the result is divided by the integration length ∆x, we recover Eq. (26).

4 2–D Time–dependent Schr¨odingerequation

The unitary finite difference schemes described earlier can be readily extended to the 2–D time–dependent case. Assuming uniform mesh ∆ = ∆x = ∆y, the three-level scheme gives the following 2-D algorithm

∆t ¯h ψ(i, j; n + 1) = ψ(i, j; n − 1) +  [ψ(i − 1, j; n) + ψ(i, j − 1; n) ∆2 m (33) 2∆tV (i) −4ψ(i, j; n) + ψ(i + 1, j; n) + ψ(i, j + 1; n)] − ψ(i, j; n) ¯h which is solved by simple recursion. The 2-D Crank–Nicholson algorithm is

∆t ψ(i, j; n + 1) − ψ(i, j; n) = − [C (ψ(i − 1, j; n + 1) + ψ(i, j − 1; n + 1) 2¯h 0 −4ψ(i, j; n + 1) + ψ(i + 1, j; n + 1) + ψ(i, j + 1; n + 1)) + V (i, j)ψ(i, j; n + 1) +

C0(ψ(i − 1, j; n) − ψ(i, j − 1; n) − 4ψ(i, j; n) + ψ(i + 1, j; n)) + ψ(i, j + 1; n)) + V (i, j)ψ(i, j; n)]

6 2 2 where C0 = −¯h /(2m∆ ). The algorithm can be rewritten in a form suitable for matrix implementation by moving terms at timestep (n + 1) to the left and terms at timestep n to the right. This yields a linear system with a 5–diagonal symmetric matrix. In order to simplify the matrix solution, it is possible to use an A.D.I. approach, similar to the one used for Poisson and diffusion equations. The time step is split into two steps of duration (∆t)/2, and the scheme is as follows: 1) Horizontal sweep

1 ∆t 1 1 ψ(i, j; n + ) +  [C (ψ(i − 1, j; n + ) − 2ψ(i, j; n + ) 2 2¯h 0 2 2 1 V (i, j) 1 +ψ(i + 1, j; n + ) + ψ(i, j; n + )] = (34) 2 2 2 ∆t ψ(i, j − 1; n) −  [C (ψ(i, j; n) − 2ψ(i, j; n) + 2¯h 0 V (i, j) ψ(i, j + 1; n) + ψ(i, j; n)] 2 2) Vertical sweep

∆t ψ(i, j; n + 1) +  [C (ψ(i, j − 1; n + 1) − 2ψ(i, j; n + 1) 2¯h 0 V (i, j) +ψ(i, j + 1; n + 1) + ψ(i, j; n + 1)] = (35) 2 1 ∆t 1 1 ψ(i, j; n + ) −  [C (ψ(i − 1, j; n + ) − 2ψ(i, j; n + ) + 2 2¯h 0 2 2 1 V (i, j) 1 ψ(i + 1, j; n + ) + ψ(i, j; n + )] 2 2 2 The A.D.I. procedure decouples adjacent rows and columns of mesh points and for every sweep a set of tridiagonal linear systems is solved.

5 1–D Time–independent Schr¨odingerequation

If we consider stationary states

 E  Ψ(r, t) = ψ(r) exp(−ωt) = ψ(r) exp − t (36) ¯h it is easy to see that the space dependent part of the solution ψ(r) satisfies the time inde- pendent Schr¨odingerequation

¯h2 − ∇2ψ(r) + V (r)ψ(r) = Eψr (37) 2m0 which is obtained by substituting (36) in the time dependent equation (7). For a system confined by a potential (e.g. a ) Eq. (37) is an eigenvalue problem, of which the eigenvalues are the allowed energy levels E and the eigenvectors are the corresponding wavefunction solutions ψ(r). For an open system, we have a boundary value problem where the energy of the wanted solution is specified, and one has to determine the space distribution of ψ(r).

7 5.1 Confined potential systems 5.2 1–D open systems An open system consists of a finite region which we assume to be connected to reservoirs of particles. Waves (particles) flow between the system and the reservoirs, with the reflection and transmission properties depending on the potential profile. Waves entering from any separate reservoir contact can be studied with independent solutions. Assuming a 1–D domain x ∈ [0,L] with a uniform grid with mesh size ∆x, the time independent Schr¨odingerequation in 1-D is discretized with finite differences as

¯h2 ψ(i − 1) − 2ψ(i) + ψ(i + 1) − 2 + [V (i) − E]ψ(i) = 0 (38) 2m0 ∆ x E represents the total energy, so that V (i) − E is the kinetic energy. It is convenient to set the reference zero of the potential V (x) at the inflow boundary, so that E corresponds to the incident kinetic energy. The kinetic energy varies spatially according to the potential changes, and the wavenumber kx(x) varies accordingly. The knowledge of kx at the reservoir ends allows the specification of boundary conditions. The travelling wave at the specified energy can be assumed to be a plane wave of the form

ψ(x) = A(x) exp[±kx(x)x] (39) where the negative sign in the exponential is referred to waves travelling along the negative x–direction. Boundary conditions can be derived from (39) for both forward and backward waves. While the exponentials in the boundary conditions are completely specified from the knowledge of E and V (x), the prefactor A(x) is set to a conventional reference (for instance A = 1) at one of the boundaries and the value at the other end is an outcome of the solution. The solution for the prefactor is in general complex, and contains the information on reflection and transmission coefficient for the wave. In 1–D, all is needed for the solution is a recursion algorithm. For a forward travelling wave which enters the open system at x = 0, we can set A(L) = 1 at the output boundary, and obtain the boundary condition

ψ(N) = exp[kx(N)L] (40) Here, N is the index corresponding to the right boundary. Assuming that the domain is subdivided into N mesh intervals, the index for the left boundary is i = 0. Since a constant potential can be assumed in a perfect reservoir, with no change in the wavevector kx, i.e. kx(N) = kx(N + 1), the wavefunction can be specified on an extra mesh point outside the domain as

ψ(N + 1) = exp[kx(N)(L + ∆x)] (41) The knowledge of ψ(N) and ψ(N + 1) is sufficient to calculate the value of ψ(N − 1). The recursion can then be iterated until all the values of ψ(i) are obtained. The general recursion algorithm can be easily obtained by rewriting (38) " # 2m∗∆2x ψi−1 = 2 + (V (i) − E) ψ(i) − ψ(i + 1) (42) ¯h2

8 2 2 The potential is referenced to V (0) = 0, so that E represents the kinetic energy ¯h kx(0)/2m0, as well as the total energy, at the inflow point i = 0. Because of reflection due to the potential variations, throughout the device one can express the wavefunction as the superposition of an incident and a reflected wave

ψ(i) = I(i) exp[kx(i)i∆x] + R(i) exp[−kx(i)i∆x] (43) At the output, because of the constant potential in the reservoir, we assume that there is no reflection, being R(N) = 0 and I(N) = A(N). At the input we can express the total wavefunction as

ψ(0) = A(0) = I(0) + R(0) (44) where A(0) is the final result of the recursion procedure. In order to determine the am- plitudes of the incident wave I(0) and of the reflected wave R(0) we need an additional relation and to do so, the solution is also obtained at a mesh point inside the reservoir x = −∆x with index i = −1. Since we assume again that the potential does not vary inside the reservoir, we have kx(−1) = kx(0) and we can write

ψ(−1) = I(−1) + R(−1) = I(0) exp[−kx(0)∆x] + R(0) exp[kx(0)∆x] (45) From (44) and (45) one obtains

ψ(0) exp[k (0)∆x] − ψ(−1) I(0) = x (46) 2 sin(kx(0)∆x) Once I(0) is determined, the wavefunction throughout the device can be renormalized to make I(0) = 1. The procedure for inflow from the right boundary is essentially identical, with appropriate sign changes for the waves. The recursion relation becomes " # 2m∗∆2x ψi+1 = 2 + (V (i) − E) ψ(i) − ψ(i − 1) (47) ¯h2 and is started assuming ψ(0) = 1 and ψ−1 = exp[−kx(0)∆x] To find the total wavefunction from the renormalized results of the two recursion steps, one needs to specify the injection conditions. We will consider now the case of a semicon- ductor system with n–type reservoirs. Assuming that the reservoir is in equilibrium with Fermi level EF and including the effect of , the particle density is given by 1 ZZZ ∞ 1 n = 3 dkxdkydkz (48) 4π −∞ 1 + exp[(E − EF )/kBT ] The particle energy, assuming for simplicity an isotropic mass, can be decomposed as

¯h2k2 ¯h2(k2 + k2) E = E + E = x + y z (49) x ⊥ 2m∗ 2m∗

The integration over k⊥ = (ky, kz) is performed in polar coordinates

∞ 2π ∞ 1 Z Z Z |k⊥| n = 3 dkx dθ d|k⊥| (50) 4π −∞ 0 0 1 + exp[(E⊥ + Ex − EF )/KBT ] The integration over the angle θ provides a factor 2π and a change of variable is performed from |k⊥| to E⊥| using

9 √ 2m∗E |k | = ⊥ (51) ⊥ ¯h √ 2m∗ d|k⊥| = √ dE⊥ (52) 2¯h E⊥ which gives

m∗ Z ∞ Z ∞ dE n = dk ⊥ (53) 2 2 x 2π ¯h −∞ 0 1 + exp[(E⊥ + Ex − EF )/kBT ]

The integration over E⊥ in (53) can be solved exactly according to Z dx = − log[1 + exp(x)] (54) 1 + exp(x) from which

m∗k T Z ∞  E − E  n = B dk log 1 + exp F x (55) 2 2 x 2π ¯h −∞ kBT From the result above, one can deduce the corresponding to an incident momentum kx as " !# m∗k T E − ¯h2k2/2m∗ n(k ) = B log 1 + exp F x (56) x 2 2 2π ¯h kBT The expression in (56) is used as weight to combine the incident waves which are injected from the reservoir momentum distribution. The Fermi–Dirac distribution in the contacts is valid close to equilibrium. Far from equilibrium, one can still use this approach to get a qualitative solution, however, in a general situation, the boundary conditions for the injection distribution should be adapted selfconsistently to the bias. Once the wavefunction is determined, it is possible to evaluate the current flowing in the structure. Using a parabolic dispersion relation, the electron current associated with a specific momentum component is defined as

¯hk J = ρv(k) = −qψ∗ψ (57) m∗ where ρ is the density associated to the wavefunction. Since for a plane wave we have

∇ψ = kψ (58) we obtain another useful expression for the current

¯h J = −q =(ψ∗∇ψ) (59) m∗ The continuity equation for electrons has the form

∂ρ ∂|ψ|2 = −q = ∇ · J (60) ∂t ∂t where

10 ∂|ψ|2 ∂ψ ∂ψ∗ = ψ∗ + ψ (61) ∂t ∂t ∂t From Schr¨odingerequation

∂|ψ|2  ¯h 1   ¯h 1  = ψ∗ ∇2ψ + V (r)ψ − ψ ∇2ψ∗ + V (r)ψ∗ (62) ∂t 2m∗ ¯h 2m∗ ¯h ¯h = (ψ∗∇2ψ − ψ∇2ψ∗) (63) 2m∗ and since ∇ · (ψ∗∇ψ) = ψ∗∇2ψ + ∇ψ∗ · ∇ψ, we obtain

∂|ψ|2 ¯h = ∇ · (ψ∗∇ψ − ψ∇ψ∗) (64) ∂t 2m∗ From (60) and (64) we obtain the expression for the current

q¯h J =  (ψ∗∇ψ − ψ∇ψ∗) (65) 2m∗

In practice, one cannot select all possible values for the incident momentum kx, and the momentum axis is discretized. Assuming a uniform discretization step ∆k, we can take the mid–value of the mesh intervals, ki, as the average momentum for the step, so that the total electron concentration can be expressed as

X ∗ n(x) = n(ki)∆kψi (x)ψi(x) (66) i and the total current density is

q¯h  ∂ψ (x) J(x) = − X n(k )∆k= ψ∗(x) i m∗ i i ∂x i (67) q¯h  ∂ψ (x) ∂ψ∗(x) =  X n(k )∆k ψ∗(x) i − ψ (x) i 2m∗ i i ∂x i ∂x i Note that the results shown here for the current are general and can be applied to the time–dependent problem as well.

6 Tight-binding formulation

The finite difference discretization of Schr¨odingerequation, which provides a mathematical approximation for the problem, can be reinterpreted from the point of view of physics with the so called formulation. The mesh points which substitute the space continuum can be taken to represent virtual atoms, so that the grid is like a crystal lattice. The tight binding lattice emulates the plane wave in the space continuum, if we assign an appropriate hopping potential be- tween nearest-neighbor sites of the virtual lattice. Assuming parabolic bands with constant effective mass m∗ and a uniform lattice spacing ∆, the hopping potential is

11 ¯h2 V = − (68) h 2m∗∆2 In 1-D the tight binding lattice is a chain of points. For this case, using the Dirac notation of quantum mechanics, we can write the tight binding Hamiltonian as

X X H = (i) |i >< i| + Vh (|i >< i + 1| + |i + 1 >< i|) (69) i i Here, (i) represents the energy of the site i and is given by

(i) = −2nVh + V (i) (70) where n indicates the space dimensions and V(i) is the . It is easy to see that the Hamiltonian in (69) corresponds to the finite difference discretization of the Schr¨odingerequation, and can be represented with the matrix

7 Applications

7.1 1-D time–dependent Schr¨odingerequation The following computer code provides the solution of the 1-D time-dependent Schr¨odinger equation using the Crank-Nicholson semi–implicit approach. The initial condition is a gaus- sian wavepacket with a specified kinetic energy corresponding to the motion of the packet centroid. At the boundaries the wavefunction is assumed to be zero, which is equivalent to perfectly reflecting boundary conditions. The code reads three input files which must be provided by the user: file INPUT1.D, which includes a number of simulation parameters, file EFMASS.D, which contains the relative effective mass, defined on the mesh intervals (so that EFMASS(i) = m∗(i + 1/2)), and file .D with the potential energy distribution in units, defined on the mesh points.

C PROGRAM SCHR1D C C Basic version in complex arithmetic of the 1-D time-dependent C Schroedinger equation, with space-dependent effective mass. C C Crank-Nicholson finite difference approach. Subroutine C TRIDIAG solves the resulting tridiagonal matrix. C C Initial condition is a Gaussian wavepacket moving from C left to right. C C No scaling used - M.K.S. units C------C DEFINITION BLOCKS C------REAL K0 COMPLEX PSI,C2,UIM,A,B,C,RHS,W,G

12 C------C COMMON BLOCKS C------C V = potential energy profile C PSI = wavefunction C COMMON /VAR1/ V(20000),PSI(20000),PACK(20000) COMMON /VAR2/ EFMASS(20000),C1(20000),C2(20000),EEE(20000) COMMON /MAT1/ A(20000),B(20000),C(20000),RHS(20000) COMMON /MAT2/ W(20000),G(20000) COMMON /MESH/ N C------C OPEN INPUT FILES C------OPEN(unit=8,file=’INPUT1.D’) OPEN(unit=9,file=’EFMASS.D’) OPEN(unit=10,file=’VOLT.D’) OPEN(unit=20,file=’OUT1.DAT’) OPEN(unit=30,file=’OUT2.DAT’) C------C PHYSICAL PARAMETERS C------C HBAR = ’s constant/2*pi C ELMASS = electron mass in vacuum C------HBAR = 1.0546E-34 ELMASS = 9.1E-31 C------C IMAGINARY UNIT C------UIM=(0.,1.) C------C INPUT - READ DISCRETIZATION AND ENERGY PARAMETERS C------C NITER = total number of time iterations C DX = mesh interval C DT = timestep C SIGFACT = spread of initial wavepacket in units of DX C VVT = energy of particle. C N = number of mesh points (including boundaries) C ICENT = initial mesh point location of wavepacket centroid C------READ(8,*) NITER READ(8,*) DX READ(8,*) DT READ(8,*) SIGFACT READ(8,*) VVT READ(8,*) N

13 READ(8,*) ICENT C------C INPUT - EFFECTIVE MASS DISTRIBUTION C------DO 10 I=1,N-1 READ(9,*) EFMASS(I) 10 CONTINUE C------C INPUT - POTENTIAL DISTRIBUTION C------DO 20 I=1,N READ(10,*) V(I) 20 CONTINUE C------C INITIALIZE TIME C------TIME = 0.0 C------C ZERO (REFLECTING) BOUNDARY CONDITIONS FOR WAVEFUNCTION C------PSI(1) = (0.,0.) PSI(N) = (0.,0.) C------C FORM INITIAL GAUSSIAN WAVEPACKET C------C K0=momentum of wavepacket centroid C SIG=spread of wavepacket C------K0 = SQRT(2.*EMS1/HBAR*VVT/HBAR) SIG = SIGFACT*DX C------DO 200 I = 2,N-1 EEE(I) = EXP(-((I-1)*DX-ICENT*DX)**2/2./SIG**2) PSI(I) = COS(K0*(I-1)*DX)*EEE(I)*10+ * SIN(K0*(I-1)*DX)*EEE(I)*10*UIM 200 CONTINUE C------C CHECK NORMALIZATION OF WAVEPACKET C------SUMO=0. DO 300 I = 1,N SUMO=SUMO+ABS(PSI(I))**2 300 CONTINUE C ------SUMO = SUMO*DX C------C NORMALIZE WAVEPACKET C------

14 DO 400 I = 1,N PSI(I) = PSI(I) / SQRT(SUMO) 400 CONTINUE DO 500 I = 1,N PACK(I) = ABS(PSI(I))**2 500 CONTINUE DO 550 I=1,N WRITE (20,*) PACK(I) 550 CONTINUE C------C SET-UP COEFFICIENTS FOR MATRIX CALCULATIONS C------C1 = DX**2/HBAR*2.*ELMASS/HBAR C2 = (2.*DX**2/DT*2.*ELMASS/HBAR)*UIM C------DO 600 I=1,N-1 EMINV(I) = 1./EFMASS(I) 600 CONTINUE C------DO 650 I=1,N-1 EMINV2(I) = EMINV(I) + EMINV(I+1) 650 CONTINUE C------C DO 2000 - OUTER TIME LOOP C------DO 2000 ITER=1,NITER C------C This routine defines the coefficients of the tridiagonal matrix C C B(i) = Main diagonal C A(i) = Lower diagonal C C(i) = Upper diagonal C C------DO 700 I=2,N-1 B(I)=-EMINV2(I-1)+C2-C1*V(I) C(I)= EMINV(I) A(I)= EMINV(I-1) 700 CONTINUE C------C This subroutine calculates the right hand side C------DO 800 I=2,N-1 RHS(I)=-PSI(I-1)*EMINV(I-1)+(C2+C1*V(I)+EMINV2(I))*PSI(I) # -PSI(I+1)*EMINV(I) 800 CONTINUE C------CALL TRIDIAG

15 C------C PACK=Wavefunction probability ( modulus) C------DO 900 I = 1,N PACK(I) = ABS(PSI(I))**2 900 CONTINUE C------C CHECK INTEGRAL OF WAVEFUNCTION C------SUM = 0. DO 1000 I = 1,N SUM = SUM+PACK(I)*DX 1000 CONTINUE C------C UPDATE TIME CLOCK C------TIME = TIME+DT C------2000 CONTINUE C------DO 3000 I=1,N WRITE (30,*) PACK(I) 3000 CONTINUE C------C END OF TIME LOOP C------STOP END C------C------C TRIDIAGONAL SYSTEM SOLVER C------SUBROUTINE TRIDIAG COMMON /VAR1/ V(20000),PSI(20000),PACK(20000) COMMON /MAT1/ A(20000),B(20000),C(20000),RHS(20000) COMMON /MAT2/ W(20000),G(20000) COMMON /MESH/ N COMPLEX V,PSI,A,B,C,RHS,W,G C------C C Gaussian Elimination C C------W(2)=C(2)/B(2) C ------DO 10 I=3,(N-1) W(I)=C(I)/(B(I)-A(I)*W(I-1)) 10 CONTINUE

16 C ------G(2)=RHS(2)/B(2) C ------C ------DO 20 I=3,N-1 G(I)=(RHS(I)-A(I)*G(I-1))/(B(I)-A(I)*W(I-1)) 20 CONTINUE C------C C Backsubstitution C C------PSI(N-1)=G(N-1) C ------DO 30 I=2,(N-2) K=N-I PSI(K)=G(K)-W(K)*PSI(K+1) 30 CONTINUE C------RETURN END C------C------

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