Silicon is doped to with a shallow donor that is fully ionised at

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

Silicon is doped to ND=3.0×1019 cm−3N_D=3.0\times10^{19}\,\mathrm{cm^{-3}} with a shallow donor that is fully ionised at 300 K300\,\mathrm{K}, so n=NDn=N_D by charge neutrality. Someone argues: "the law of mass action np=ni2np=n_i^2 always holds in thermal equilibrium, so I can use the non-degenerate formula n=Ncexp⁡[(EF−EC)/kBT]n=N_c\exp[(E_F-E_C)/k_BT] to place the chemical potential, with Nc=2.8×1019 cm−3N_c=2.8\times10^{19}\,\mathrm{cm^{-3}} for silicon at 300 K300\,\mathrm{K}." This gives η=(EF−EC)/kBT=ln⁡(ND/Nc)=0.070\eta=(E_F-E_C)/k_BT=\ln(N_D/N_c)=0.070. The exact Fermi-integral inversion for this same density instead gives η=0.44\eta=0.44. ((a)) Using η=0.44\eta=0.44, evaluate what the Boltzmann formula n=Nceηn=N_c e^\eta would predict for nn, and compare it with the actual density NDN_D. ((b)) Explain why the person's reasoning is wrong.

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