The Maxwell-Boltzmann distnbutlOn function

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The Maxwell-Boltzmann distnbutlOn function

 

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Given,

fE=e-E-Ef/KTDcE = AE-Ec1/2 

a) Carrier distribution taken as f(E) in conduction band is directly proportional to the distribution of E,

         Distribution of (E) α E-Ec12.e-E-Ef/KT.............................................1

    Differentiate the above eq and equate to zero ,

    ddEE-Ec1/2.e-E-Ef/KT=0 121E-Ec1/2 e-E-Ef/KT+E-Ec1/2-1KTe-E-Ef/KT=0e-E-Ef/KT121E-Ec+E-Ec-1KT=0121E-Ec =-E-Ec-1KT121E-Ec = +E-Ec 1KTE-Ec-1/22=E-Ec1/2KTKT2=E-Ec1/2E-Ec-1/2KT2 =E-Ec1/2.E-Ec1/2KT2 = E-EcE = Ec+KT2

 

b) Conduction band electron concentration is,

     n = C.BDcE fE dE                                    Where C.B = conduction band

        =  Ectop of C.B DcE fE dE

    assume loop of conduction band be infinite,

      n=EcDcE fE dE

  Put the values,

  n = EcAE-Ec1/2. e-E-Ef/KT

Assume,

E-EcKT=x E=KTx+Ec

1KTdE=dx dE=dxKT  where  x 0 to

n=0AEc+KTx-Ec1/2.e-KTx+Ec-Ef/KT.KTdx   = 0AKTx .e-KTx+Ec-Ef/KT .KT dx   = AKT3/2 e-Ec-Ef/KT0x e-KTx/KT dx   = AKT3/2 e-Ec-Ef0xe-x dx

By Integrating ,

n =AKT3/2 e-Ec-Ef/KT.0x32-1e-x dx                                                  0x32-1e-x dx = gamma function32   = AKT3/2 e-Ec-Ef/KT gamma func32          gamma32=π2

n=π2A(KT)3/2e-(Ec-Ef)/KT

 

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