On the mobility of very pure semiconductors at very low temperatures

The mobility μ of a very pure semiconductor at very low temperatures is investigated in terms of a model where electrons are scattered by charged impurities distributed uniformly in space, and the electron-electron interaction is taken into account by the Debye-Hueckel screening in the interaction potential. The equation for the current relaxation rate Γ, derived previously by the proper connected diagram expansion, incorporates the quasi-particle effect in a self-consistent manner. The solution of this equation at high carrier concentrations n yields the so-called Brooks-Herring formula. At lower concentrations, the solution deviates significantly from the latter. The solution is in general smaller than the standard expression for the rate based on the Boltzmann equation; and this is consistent with the existing conductivity data available. At the very low concentrations e.g. n = n₃ = 10¹³cm^?3 or lower for Ge, the mobility calculated is inversely proportional to the square-root of the impurity concentration n_s, and has a $\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$-dependence (T: temperature).

$\text{μ = 0.3597\&z.xl;h}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{k(k}{}_{\mathrm{B}}\text{T)}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}\text{(ze)}{}^{\mathrm{?1}}\text{n}{}_{\mathrm{s}}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{m}{}^1{}^{\mathrm{<mtext>?3</mtext><mtext>4</mtext>}}$

, where k is the dielectric constant. The conductivity data directly comparable with this formula are not available at present. However, the quasi-particle effect which led to this peculiar concentration-dependence should also show itself in the cyclotron resonance width; there, experiment and theory both show the ${}_{\mathrm{n}}{}_{\mathrm{s}}$-dependence for very pure semiconductors.