Piecewise-Constant Potential

If an arbitrary potential barrier is segmented into N regions with constant potentials (see Fig. 3.9) the wave function in each region can be written as the sum of an incident and a reflected wave [93] $ \Psi_j(x)=A_j\exp(\imath k_j x)+B_j\exp(-\imath k_j x)$ with the wave number $ k_j=\sqrt{2m_j({\mathcal{E}}-W_j)} / \hbar$. The wave amplitudes $ A_j$, $ B_j$, the carrier mass $ m_j$, and the potential energy $ W_j$ are assumed constant for each region $ j$. With the interface conditions for energy and momentum conservation

$\displaystyle \Psi_j(x^-)$ $\displaystyle =$ $\displaystyle \Psi_{j+1}(x^+)\ ,$ (3.73)
$\displaystyle \frac{1}{m_j}\frac{\ensuremath {\mathrm{d}}\Psi_j(x^-)}{\ensuremath {\mathrm{d}}x}$ $\displaystyle =$ $\displaystyle \frac{1}{m_{j+1}}\frac{\ensuremath {\mathrm{d}}\Psi_{j+1}(x^+)}{\ensuremath {\mathrm{d}}x} \ ,$ (3.74)

the outgoing wave of a layer relates to the incident wave by a complex transfer matrix:

$\displaystyle \left( \begin{array}{c} A_{j} \\ B_{j} \\ \end{array} \right) = \...
...}{c} A_{j-1} \\ B_{j-1} \\ \end{array} \right) \qquad 2 \le j \le \mathrm{N}\ .$ (3.75)

The transfer matrices are of the form

$\displaystyle \ensuremath{{\underline{T}}}_j = \frac{1}{2} \left( \begin{array}...
...\ 0 & \gamma^{-k_{j - 1}} \end{array} \right) \qquad 2 \le j \le \mathrm{N} \ ,$ (3.76)

with the phase factor $ \gamma = \exp(\imath \Delta (j-2))$. The transmitted wave in Region N can then be calculated from the incident wave by subsequent multiplication of transfer matrices:

$\displaystyle \left( \begin{array}{c} \ensuremath{A_\mathrm{N}}\\ \ensuremath{B...
...\underline{T}}}_j \left( \begin{array}{c} A_1 \\ B_1 \\ \end{array} \right) \ .$ (3.77)

If it is assumed that there is no reflected wave in Region N and the amplitude of the incident wave is unity, (3.77) simplifies to

$\displaystyle \left(\begin{array}{c}\ensuremath{A_\mathrm{N}}\\ 0\end{array}\ri...
..._{22} \end{array}\right) \left(\begin{array}{c} 1 \\ B_1 \end{array}\right) \ ,$ (3.78)

and the transmission coefficient can be calculated from (3.54). The transfer-matrix method based on constant potential segments has the obvious shortcoming that, for practical barriers, the accuracy of the resulting matrix strongly depends on the chosen resolution. A more rigorous approach is to use linear potential segments.

A. Gehring: Simulation of Tunneling in Semiconductor Devices