Past exam of the condensed matter physics course of the University of Oxford / 2019 / 3 / c / Solution

At $300\mathrm{K}$, the two-dimensional effective density of states values areUsing $E_g=0.36\mathrm{eV}$ and $k_{B}T\simeq 0.02585\mathrm{eV}$ givesFor the doped sample, estimate the intrinsic-to-extrinsic crossover by $n_i(T_{\ast })\simeq n_d=10^9{\mathrm{cm}}^{-2}$. With the masses and gap held fixed, $n_i(T)=AT{e}^{-B/T}$, whereSolving $A{T}_{\ast }{e}^{-B/T_{\ast }}=10^9{\mathrm{cm}}^{-2}$ givesIt is a smooth carrier crossover, not a thermodynamic phase transition. Donors are essentially ionized near this temperature because their binding energy, calculated next, is much smaller than $k_B{T}_{\ast }$. At substantially higher temperatures intrinsic pairs dominate; below the crossover and above appreciable donor freeze-out, the majority density approaches $n_d$.