2.4.3.1 SCHRÖDINGER Equation and SCHRÖDINGER-POISSON Solvers

At the heart of quantum device simulation stands SCHRÖDINGER's equation [63]

$$-\frac{\hbar}{\imath} \frac{\partial \Psi(\mathbf{r},t)}{\partial t} = \ensuremath{{\underline{H}}}\Psi(\mathbf{r},t)\ ,$$ (2.12)

where $\ensuremath{{\underline{H}}}$ denotes the HAMILTONian^2.9 of the system. For the stationary case, the SCHRÖDINGER equation can be written as [64]

$$\left(-\frac{\hbar^2}{2m} \ensuremath{{\mathbf{\nabla}}}^2 + W(\mathbf{r}) \right) \Psi(\mathbf{r}) = {\mathcal{E}}\Psi(\mathbf{r})\ ,$$ (2.13)

where $W(\mathbf{r})$ is an external potential energy. The central quantity is the wave function $\Psi(\mathbf{r})$. It is related to the probability $P_\mathcal{V}$ of finding an electron within a volume $\mathcal{V}$ by

$$P_\mathcal{V} = \int_{\mathcal{V}}\Psi(\mathbf{r})\Psi(\mathbf{r}... ... = \int_{\mathcal{V}} \rho(\mathbf{r}) \, \ensuremath {\mathrm{d}}\mathbf{r}\ ,$$ (2.14)

where $\rho(\mathbf{r})$ denotes the probability density. The probability to find the electron somewhere must be unity, so

$$\int_{-\infty}^{\infty}\Psi(\mathbf{r})\Psi(\mathbf{r})^* \, \ensuremath {\mathrm{d}}\mathbf{r} = 1\ .$$ (2.15)

From the wave function, the current density can be calculated via

$$\mathbf{J}(\mathbf{r}) = \frac{\imath \hbar\ensuremath {\mathrm{q... ...{\mathbf{\nabla}}}\Psi^* - \Psi^* \ensuremath{{\mathbf{\nabla}}}\Psi \right)\ ,$$ (2.16)

which obeys the continuity equation

$$\ensuremath{{\mathbf{\nabla}}}\cdot \mathbf{J}(\mathbf{r}) = -\ensuremath {\mathrm{q}}\frac{\partial \rho(\mathbf{r})}{\partial t}\ .$$ (2.17)

Common approaches to couple SCHRÖDINGER's equation to BOLTZMANN's transport equation perform a SCHRÖDINGER-POISSON self-consistent loop: The carrier concentration is calculated quantum-mechanically and used in POISSON's equation to obtain the electrostatic potential which is again used in SCHRÖDINGER's equation until convergence is reached. The resulting quantum-mechanical carrier concentration is used to derive correction factors for the solution of BOLTZMANN's transport equation [65,66].

A. Gehring: Simulation of Tunneling in Semiconductor Devices