3.5.4 Quantum Transmitting Boundary Method

An alternative method to solve the SCHRÖDINGER equation has been proposed by FRENSLEY and EINSPRUCH [154] which is based on the tight-binding quantum transmitting boundary method (QTBM) introduced by LENT [155]. It has been used to simulate electron transport in resonant tunneling diodes [153]. The method is based on the finite-difference approximation of the stationary one-dimensional SCHRÖDINGER equation (3.61) on an equidistant grid with an effective mass $m_j$ and a grid spacing $\Delta$

$$\ensuremath{{\underline{H}}}\Psi_j = -s_{j-1}\Psi_{j-1} + d_{j}\Psi_{j} - s_{j+1}\Psi_{j+1} = {\mathcal{E}}\Psi_j \ ,$$ (3.84)

where $s_j = \hbar^2/(2m_j\Delta^2)$ and $d_j = \hbar^2 / (m_j\Delta^2) + W_j$. For the evaluation of the transmission coefficient it is necessary to assume open boundary conditions. They are introduced by writing the wave functions at the boundaries of the simulation domain as

$$\Psi_1 = a_1 + b_1$$ (3.85)

$$\ensuremath{\Psi_\mathrm{N}} = \ensuremath{a_\mathrm{N}}+ \ensuremath{b_\mathrm{N}}$$ (3.86)

and relate them to the wave functions outside of the simulation domain by

$$\Psi_0 = a_1\exp(-\imath k_1\Delta) + b_1\exp( \imath k_1\Delta) \ ,$$ (3.87)

$$\Psi_\mathrm{N+1} = \ensuremath{a_\mathrm{N}}\exp(-\imath \ensuremath{k_\mathrm{N}}\Delta) + \ensuremath{b_\mathrm{N}}\exp( \imath \ensuremath{k_\mathrm{N}}\Delta) \ .$$ (3.88)

This introduces four unknowns and two equations into the system. Setting

$$a_1 = \zeta_1 \Psi_0 + \xi_1 \Psi_1$$ (3.89)

$$\ensuremath{a_\mathrm{N}} = \zeta_\mathrm{N} \Psi_\mathrm{N+1} + \xi_\mathrm{N} \ensuremath{\Psi_\mathrm{N}}$$ (3.90)

eliminates the unknown values of $b_1$ and $b_\mathrm{N}$ and gives a linear system for the $\mathrm{N}+2$ complex values $\Psi_j$

$$\left( \begin{array}{ccccccc} \zeta_1 & \xi_1 & & & & & \\ -s_1 &... ...ray}{c} a_1 \\ 0 \\ 0 \\ \cdots \\ 0 \\ a_\mathrm{N} \\ \end{array} \right) \ .$$ (3.91)

Setting $a_1=1$ and $a_\mathrm{N}=0$ yields the values of the wave function in the whole simulation domain for an incident wave from the left side like in the transfer-matrix method. The method is easy to implement, fast, and more robust than the transfer-matrix method. A further advantage of this method is its suitability for two- and three-dimensional problems. It thus represents a much more powerful method than the transfer-matrix based methods which are limited to one-dimensional problems only. Note that the QTBM is closely linked with the non-equilibrium GREEN's function formalism (NEGF, see Section 2.4.3.4): The matrix in expression (3.91) is the inverse of the retarded GREEN's function (2.25) for an open system without scattering. However, the values of $\zeta$ and $\xi$ are complex, so the matrix admits complex eigenvalues and complex solving routines are necessary.

A. Gehring: Simulation of Tunneling in Semiconductor Devices