A silicon sample is doped so heavily that the electron gas is degenerate

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Semiconductor physics and devices · Semiconductor physics and devices

Solid State Physics, Ch Semiconductors: masses, statistics, defect levels and mobility

Problem

A silicon sample is doped so heavily that the electron gas is degenerate, with (EF−EC)/kBT=η=1.5(E_F-E_C)/k_BT=\eta=1.5. Silicon's effective density-of-states constant at 300 K300\,\mathrm{K} is Nc=2.8×1019 cm−3N_c=2.8\times10^{19}\,\mathrm{cm^{-3}}, and at this η\eta the Fermi integral evaluates to F1/2(1.5)=1.903F_{1/2}(1.5)=1.903. ((a)) Calculate the true electron density n=Nc2πF1/2(η)n=N_c\dfrac{2}{\sqrt\pi}F_{1/2}(\eta). ((b)) Calculate the density the non-degenerate Boltzmann formula nB=Nceηn_B=N_ce^{\eta} would give instead, and state by what factor it overestimates nn.

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