Treat the electrons at a single GaAs/ heterojunction in the chapter's triangular-well…

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

Solid State Physics, Ch Heterostructures, quantum wells and the two-dimensional electron gas

Problem

Treat the electrons at a single GaAs/Al0.3Ga0.7As\mathrm{Al_{0.3}Ga_{0.7}As} heterojunction in the chapter's triangular-well approximation, with the average field F=ens/2ε0εsF=en_s/2\varepsilon_0\varepsilon_s (εs=12.9\varepsilon_s=12.9), m∗=0.067 mem^*=0.067\,m_e and levels En≈(ℏ2/2m∗)1/3[32πeF(n−14)]2/3E_n\approx(\hbar^2/2m^*)^{1/3}[\tfrac32\pi eF(n-\tfrac14)]^{2/3}, at low temperature. ((a)) Show that the subband spacing E2−E1E_2-E_1 grows as ns2/3n_s^{2/3}, while the filling EF−E1E_F-E_1 of the lowest subband grows as nsn_s. ((b)) Find the sheet density ns∗n_s^* at which the Fermi level reaches E2E_2 and a second subband starts to fill. ((c)) Find E2−E1E_2-E_1 at ns∗n_s^*, and say whether the approximation's neglect of the depletion charge and of self-consistency makes your ns∗n_s^* a firm number or an estimate.

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