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Homework Help: QM - Etop of electron distribution of a semiconductor

  1. Nov 14, 2009 #1
    1. The problem statement, all variables and given/known data

    I'm trying to find energy level above Ec where electron distribution makes a peak for a nondegenerate semiconductor. For this case we may take GaAs having Eg = 1.42eV at T = 300K.

    2. Relevant equations
    [tex]m_e[/tex]=single isotrophic effective mass or [tex]m_0[/tex]
    energy states, [tex]g_{c}(E) = \frac{m_{e}\ast\sqrt{2m_{e}(E-E_{c})}}{pi^2 * hbar^3}[/tex]
    fermi function for a nondegenerate semiconductor, [tex]f(E) = exp((E_f-E)/kT)[/tex]
    electron distribution, [tex]n=N_{c}*exp((Ef-Ec)/kT)[/tex] and [tex]N_{c}=4.21\ast10^{17} cm^-3[/tex]

    3. The attempt at a solution
    I think I'll give a fermi energy level equal to 3kT above Ec where semi.con. is still nondegenerate. Then I'll calculate n. Afterwards I'll equate n to [tex]\int g_{c}(E)*f(E)*dE [/tex] taking a limit to 99 % of n. By that I intend to find top limit of the integral which must be the Etop.
    But i do not how to evaluate a integral such as [tex]\sqrt{E}*exp(c*E)[/tex]
    ps: partial integral is not working.
    Is there another {easy :( }approach?
  2. jcsd
  3. Nov 17, 2009 #2
    I found the answer:
    derivative of
    [tex]g_{c}(E) * f(E) [/tex]
    gives the minimum points of electron distribution
    one of them is [tex]E_{c}[/tex] and the other is [tex]E_{top}[/tex] which is asked by the question ;)
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