# Boundary condition for Schrödinger equation at metal-semiconductor interface

Suppose I want to solve the $$1\text{D}$$, time-independent Schrödinger-equation for a metal-semiconductor junction.

In the metal region $$0 \leq x \leq x_{0}$$ the Schrödinger equation reads:

$$\left(-\frac{\hbar^2}{2m} \frac{d^2}{dx^2}+V(x)\right)\psi = E\psi$$

In the semiconductor region $$x_{0} the Schrödinger equation reads:

$$\left(-\frac{\hbar^2}{2m^{*}} \frac{d^2}{dx^2}+\Delta E_{SM} + V(x)\right) \psi = E\psi$$

Here $$\Delta E_{SM}$$ is the offset between the conduction band edges in the metal and semiconductor region. My question is: Is there a way to "integrate out" the metal region or replace it with an effective boundary condition at the semiconductor-metal interface?

Using the condition that the boundary condition must be linear, self adjoint, and involve no more than $$\psi$$ and its first derivative (these requirements follow from the full Schroedinger equation) the most general boundary condition for an interface between effective mass $$m_L$$ and $$m_R$$ is $$\left(\matrix{ \psi_{2L}\cr \partial_x \psi_{2L}}\right) = \left(\matrix{ a& b\cr c&d }\right) \left(\matrix{ \psi_{2R}\cr \partial_x \psi_{2R}}\right).$$ where $$\left(\matrix{ a& b\cr c&d }\right) = e^{i\phi}\sqrt{\frac{m_L}{m_R}}\left(\matrix{ A& B\cr C&D }\right).$$ Here $$A$$, $$B$$, $$C$$, $$D$$, and the phase $$\phi$$ are real and $$(AD-BC)=1$$. These conditions are used in