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Homework Help: Discretization of the Poisson Equation across Heterointerface

  1. Oct 1, 2007 #1
    1. The problem statement, all variables and given/known data
    Consider a 1D sample, such that for x < xb the semiconductor has a dielectric constant [tex]\varepsilon_{1}[/tex], and for x > xb has a dielectric constant [tex]\varepsilon_{2}[/tex]. At the interface between the two semiconductor matierials (x = xb) there are no interface charges. Starting from the condition
    [tex]\varepsilon_{1}\frac{\partial\psi}{\partial x}\mid_{x=xb} = \varepsilon_{2}\frac{\partial\psi}{\partial x}\mid_{x=xb}[/tex]

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

    2. Relevant equations
    Poisson Equation (In the form I think matters for the above problem)

    [tex]\nabla^2\phi = -\frac{\rho}{\varepsilon}[/tex]

    3. 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 [tex]\varepsilon_{1}\frac{\partial\psi}{\partial x}\mid_{x=xb} = \varepsilon_{2}\frac{\partial\psi}{\partial x}\mid_{x=xb}[/tex] and a dirichlet boundary at [tex]\rho = 0[/tex] when x=xb.

    Now for the finite difference approximation I get

    [tex]\frac{d}{dx}\psi(x) = \frac{\psi(x + \Delta x) - \psi(x - \Delta x)}{2\Delta x}[/tex]

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

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

    Any suggestions?
    Last edited: Oct 1, 2007
  2. jcsd
  3. Apr 16, 2009 #2
    Yes in the equation you started Epsilon 1 = epsilon 2
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