Discretization of the Poisson Equation across Heterointerface

[JasonW]

Homework Statement

Consider a 1D sample, such that for x < xb the semiconductor has a dielectric constant \varepsilon_{1}, and for x > xb has a dielectric constant \varepsilon_{2}. At the interface between the two semiconductor matierials (x = xb) there are no interface charges. Starting from the condition
\varepsilon_{1}\frac{\partial\psi}{\partial x}\mid_{x=xb} = \varepsilon_{2}\frac{\partial\psi}{\partial x}\mid_{x=xb}

and using Taylor series expansion for \varepsilon around x = xb (for x < xb and x > xb), calculate the finite difference approximation of the Poisson equation at x = xb.

Homework Equations

Poisson Equation (In the form I think matters for the above problem)

\nabla^2\phi = -\frac{\rho}{\varepsilon}

The Attempt at a Solution

First let me start off that I haven't been in school for nearly 20 years and I'm very rusty with math. I think there are two boundary conditions to this problem, the newmann boundary is the \varepsilon_{1}\frac{\partial\psi}{\partial x}\mid_{x=xb} = \varepsilon_{2}\frac{\partial\psi}{\partial x}\mid_{x=xb} and a dirichlet boundary at \rho = 0 when x=xb.

Now for the finite difference approximation I get

\frac{d}{dx}\psi(x) = \frac{\psi(x + \Delta x) - \psi(x - \Delta x)}{2\Delta x}

\frac{d^2}{dx^2}\psi(x) = \frac{\psi(x + \Delta x) - 2\psi(x) + \psi(x - \Delta x)}{2\Delta x^2}

Now I am planning on generating an Ax=b matrix but I'm not sure how to apply the neumann boundary.

Any suggestions?

 

[JANm]

Homework Statement

Consider a 1D sample, such that for x < xb the semiconductor has a dielectric constant \varepsilon_{1}, and for x > xb has a dielectric constant \varepsilon_{2}. At the interface between the two semiconductor matierials (x = xb) there are no interface charges. Starting from the condition
\varepsilon_{1}\frac{\partial\psi}{\partial x}\mid_{x=xb} = \varepsilon_{2}\frac{\partial\psi}{\partial x}\mid_{x=xb}

and using Taylor series expansion for \varepsilon around x = xb (for x < xb and x > xb), calculate the finite difference approximation of the Poisson equation at x = xb.

Homework Equations

Poisson Equation (In the form I think matters for the above problem)

\nabla^2\phi = -\frac{\rho}{\varepsilon}

The Attempt at a Solution

First let me start off that I haven't been in school for nearly 20 years and I'm very rusty with math. I think there are two boundary conditions to this problem, the newmann boundary is the \varepsilon_{1}\frac{\partial\psi}{\partial x}\mid_{x=xb} = \varepsilon_{2}\frac{\partial\psi}{\partial x}\mid_{x=xb} and a dirichlet boundary at \rho = 0 when x=xb.

Now for the finite difference approximation I get

\frac{d}{dx}\psi(x) = \frac{\psi(x + \Delta x) - \psi(x - \Delta x)}{2\Delta x}

\frac{d^2}{dx^2}\psi(x) = \frac{\psi(x + \Delta x) - 2\psi(x) + \psi(x - \Delta x)}{2\Delta x^2}

Now I am planning on generating an Ax=b matrix but I'm not sure how to apply the neumann boundary.

Any suggestions?

Yes in the equation you started Epsilon 1 = epsilon 2