Silicon is doped with phosphorus donors

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 with ND=5×1015 cm−3N_D=5\times10^{15}\,\mathrm{cm^{-3}} phosphorus donors, ionisation energy ΔED≡EC−ED=45 meV\Delta E_D\equiv E_C-E_D=45\,\mathrm{meV} below the conduction band edge, degeneracy factor g=2g=2. At T=60 KT=60\,\mathrm{K}, deep in freeze-out, charge neutrality reduces to n≈(NDNcg)1/2exp⁡ ⁣(−ΔED2kBT)n\approx\left(\dfrac{N_DN_c}{g}\right)^{1/2}\exp\!\left(-\dfrac{\Delta E_D}{2k_BT}\right), where Nc(T)=Nc(300 K)(T300 K)3/2N_c(T)=N_c(300\,\mathrm{K})\left(\dfrac{T}{300\,\mathrm{K}}\right)^{3/2} and Nc(300 K)=2.8×1019 cm−3N_c(300\,\mathrm{K})=2.8\times10^{19}\,\mathrm{cm^{-3}} for silicon. ((a)) Calculate NcN_c at 60 K60\,\mathrm{K}. ((b)) Calculate the free-electron density nn at 60 K60\,\mathrm{K} and the fraction of donors ionised, n/NDn/N_D.

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