Conductivity exponent puzzle in doped semiconductors
In plain words
Adding phosphorus to silicon switches it from insulator to metal at a critical concentration, and the conductivity then rises as a power of the distance from that point. Measured powers differ between kinds of samples and from theory for non-interacting electrons, and the reason is disputed.
Precise statement
In uncompensated Si:P, Si:B and Ge:Ga, $\sigma(T = 0) \sim (n/n_c - 1)^{s}$ with $s \sim 0.5$ reported in several studies, while compensated samples give $s \sim 1$. The non-interacting 3D orthogonal-class value is $s = \nu \sim 1.57$ (Wegner scaling $s = \nu(d - 2)$), and the Chayes-Chayes-Fisher-Spencer bound (rigorous form of the Harris criterion) requires $\nu \ge 2/d$, which $s \sim 0.5 = \nu$ would violate. Explain the discrepancy (extrapolation of $\sigma(T)$ to $T = 0$, Coulomb interaction, local magnetic moments, hybridization of impurity and conduction bands, or dopant inhomogeneity) and give the true s for uncompensated material.
What would settle it
Measurements on dopant-controlled samples down to $T << 10\,\mathrm{mK}$ with a scaling analysis that includes finite-temperature corrections, compared with a controlled theory of the interacting transition.
Status in the literature
Claimed resolutions (Stupp et al., Physical Review Letters 71, 2634, 1993; Carnio, Hine and Romer, Physical Review B 99, 081201, 2019, linking the puzzle to hybridization of conduction and impurity bands) are not generally accepted.