In an n-type silicon sample the electron mobility is limited by two independent…

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

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

Problem

In an n-type silicon sample the electron mobility is limited by two independent scattering mechanisms whose rates add by Matthiessen's rule. At a reference temperature T0=100 KT_0=100\,\mathrm{K} the ionised-impurity-limited mobility is μI(T0)=2000 cm2 V−1s−1\mu_I(T_0)=2000\,\mathrm{cm^2\,V^{-1}s^{-1}} and the phonon-limited mobility is μph(T0)=6000 cm2 V−1s−1\mu_{\rm ph}(T_0)=6000\,\mathrm{cm^2\,V^{-1}s^{-1}}, with μI(T)=μI(T0) (T/T0)3/2\mu_I(T)=\mu_I(T_0)\,(T/T_0)^{3/2} and μph(T)=μph(T0) (T/T0)−3/2\mu_{\rm ph}(T)=\mu_{\rm ph}(T_0)\,(T/T_0)^{-3/2} over the temperature range of interest. ((a)) Starting from the energy dependence of the screened-Coulomb (Rutherford-like) scattering time τ(E)∝E3/2/NI\tau(E)\propto E^{3/2}/N_I for an ionised-impurity density NIN_I, and using the thermal average appropriate to a non-degenerate carrier gas, derive the temperature dependence μn,I∝T3/2/NI\mu_{n,I}\propto T^{3/2}/N_I quoted above. ((b)) Find the temperature T1T_1 at which μI(T1)=μph(T1)\mu_I(T_1)=\mu_{\rm ph}(T_1), and the combined mobility, by Matthiessen's rule, at that temperature.

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