7.2.2 Open System

In our application the environment (electrodes) absorbs outgoing particles and injects particles into the device with a Fermi-Dirac equilibrium distribution which is fixed by the demand of local charge neutrality at the boundary. Adding boundary conditions can change the character of the model and in general a separation ansatz will no longer work. However, if we have an open system and incoming waves are mixed states built from plane waves of the form $e^{\imath kx}e^{-\imath E t/\hbar}$ then this type of boundary conditions can still be treated by the separation approach.

In this way the von Neumann equation with absorbing boundary conditions is equivalent to a countable system of Schrödinger equations with transparent boundary conditions. In practice we solve the open Schrödinger equation for a set of values for $k$. Then the solution is built from the pure states $\vert\psi_k \rangle \langle \psi_k\vert$ which are summed with the correct weight given by the equilibrium Fermi-Dirac distribution in the electrodes. Solving the von Neumann equation by the separation method is known as the quantum transmitting boundary method. It has been introduced in [LK90b]. It has been applied for device simulations in [BA00], and [Fre87]. For an extension see [LKF02].

The reflection and transmission coefficients for pairs of energy-related incoming waves from left and right are not independent [Akt99] which can be used to reduce the numerical costs by a factor of 2.

Ballistic transport is only a good approximation if the influence of scattering can be neglected. The catch here is that in such cases the assumption of injection with an equilibrium distribution is doubtful.