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How to solve steady state excess hole density

  1. Nov 16, 2011 #1
    1. The problem statement, all variables and given/known data

    A hole current of 10^(-5) A/cm2 is injected into the side (x=0) of a long N-silicon. Assuming the holes flow only by diffusion and that at very large values of x, the distribution of excess holes decays to zero, Determine

    a)The steady-state excess hole density at x=0
    b)Hole current density at x=100μm
    c)the rate of generation and recombination of electron-hole pair at x=100μm

    Answers:
    a)2.794*10^(10)cm-3
    b) 1.667*10^(-6) A/cm2


    Given:
    J=10^(-5) A/cm2
    μp=480 cm2/Vs
    μn= 1350 cm2/Vs
    lifetime of hole, τ=2.5μs


    2. Relevant equations

    J= -q*D*(dp/dx) D= k*T*μp/q

    continuity equation:
    dp/dt = -(p-p0)/τ- div(J)/q

    where p: hole density; p0= hole density at equilibrium.
    p-po=Δp

    3. The attempt at a solution

    dp/dx= J/(-q*D)= -5*10^(12)

    I think Δp=p-p0=0 when t=τ=2.5μs
    and Δp is at max. when t=0, and x=0

    I tried to solve the continuity equation, which is 1 ODE

    dp/dt = -(p-p0)/τ- div(J)/q
    p'=p0/τ-p/τ-div(J)/q

    p= p0-τ*div(J)/q+C*exp(-t/τ)

    Since i think Δp=p-p0=0 when t=τ

    p=p0-τ*div(J)/q+τ*e*div(J)/q*exp(-t/τ)


    at t=0;
    Δp=p-p0=τ*div(J)/q*(e-1)

    but i can't find div(J)

    b) i think J is constant, how it can be a function of x.
    c) generation rate: p0/τ= (ni/τ)=4*10^(15)
    recombination rate: Δp/τ= (p(x=100μm)-p0)/τ
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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