The intrinsic Fermi level does not lie exactly at the middle of t
The intrinsic Fermi level does not lie exactly at the middle of the energy band gap.
Which of the following is a correct explanation for the following statement?A. <span lang="EN-IN" style=" line-height: 107%; background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;">The effective density of state at the conduction band edge (N<sub style="">c</sub>) ≠ The effective density of state at the valence band edge (N<sub style="">v</sub>)</span>
B. <span lang="EN-IN" style=" line-height: 107%; background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;">The effective mass of electron ≠ The effective mass of hole</span>
C. <span lang="EN-IN" style=" line-height: 107%; background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;">Electron concentration and hole concentration are not equal for an intrinsic semiconductor.</span>
D. <span lang="EN-IN" style=" line-height: 107%; background-image: initial; background-position: initial; background-size: initial; background-repeat: initial; background-attachment: initial; background-origin: initial; background-clip: initial;">No two electrons occupy the same energy state.</span>
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Right Answer is:
SOLUTION
Concept:
For an intrinsic semiconductor:
Electrons and holes are generated in pairs.
∴ Electron concentration = Hole concentration
\({n_i} = {N_c}\;{e^{ - \frac{{\left( {{E_c} - {E_F}} \right)}}{{KT}}}}\)
\({p_i} = {N_v}\;{e^{ - \frac{{\left( {{E_F} - {E_v}} \right)}}{{KT}}}}\)
∴ \({N_c}\;{e^{ - \frac{{\left( {{E_c} - {E_F}} \right)}}{{KT}}}} = {N_v}\;{e^{ - \frac{{\left( {{E_F} - {E_v}} \right)}}{{KT}}}}\)
\(\frac{{{N_c}\;{e^{ - \frac{{\left( {{E_c} - {E_F}} \right)}}{{KT}}}}}}{{{N_v}\;{e^{ - \frac{{\left( {{E_F} - {E_v}} \right)}}{{KT}}}}}} = 1\)
∴ \(\frac{{{N_c}}}{{{N_v}}}\;{e^{\frac{{ - {E_c} + {E_F} + {E_F} - {E_v}}}{{KT}}}} = 1\)
Taking log on both sides,
\(\ln \left( {\frac{{{N_c}}}{{{N_v}}}} \right) + \left( {\frac{{ - {E_c} + 2{E_F} - {E_v}}}{{KT}}} \right) = 0\)
\(KT\ln \left( {\frac{{{N_c}}}{{{N_v}}}} \right) = {E_c} + {E_v} - 2{E_F}\)
∴ \({E_F} = \frac{{\left( {{E_C} + {E_v}} \right)}}{2} - \frac{1}{2}KT\ln \left( {\frac{{{N_c}}}{{{N_v}}}} \right)\)
\({N_c} = 2{\left( {\frac{{2\pi {m_n}KT}}{{{h^2}}}} \right)^{\frac{3}{2}}}\)
and
\({N_v} = 2{\left( {\frac{{2\pi {m_p}KT}}{{{h^2}}}} \right)^{\frac{3}{2}}}\)
As effective mass of an electron is less than that of a hole so Nv > Nc and it affects the position of intrinsic Fermi level