1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Discretization of the Poisson Equation across Heterointerface
Loading...