On mobility of electrons in a shallow Fermi sea over a rough seafloor

Several doped semiconductors, in contrast to heavily-doped silicon and germanium, host extremely mobile carriers, which give rise to quantum oscillations detectable in relatively low magnetic fields. The small Fermi energy in these dilute metals quantifies the depth of the Fermi sea. When the carrier density exceeds a threshold, accessible thanks to the long Bohr radius of the parent insulator, the local seafloor is carved by distant dopants. In such conditions, with a random distribution of dopants, the probability of finding an island or a trench depends on on the effective Bohr radius, a$_{B}^{*}$ and the carrier density, n. This picture yields an expression for electron mobility with a random distribution of dopants: $\mu_{RD}\propto $(a$_{B}^{*})^{1/2}$ n$^{-5/6}$, in reasonable agreement with the magnitude and concentration dependence of the low-temperature mobility in three dilute metals whose insulating parents are a wide-gap (SrTiO$_{3}$), a narrow-gap (PbTe) and a "zero"-gap (TlBiSSe) semiconductor.