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