Notes on Anderson localization

Lagendijk, van Tiggelen, and Wiersma reply: In his response to our story on the history of Anderson localization, Theodore Castner classifies our statements regarding both the scaling of the conductivity with temperature and the exponent puzzle as “misleading.” He also says we “ignored” important contributions, in particular his own proposition to explain a critical exponent 0.5 in weakly compensated semiconductors by the so-called ion-impurity scattering mechanism. ¹ That mechanism would lead to a Drude electronic conductivity proportional to the Fermi wavenumber.

It is true that the controversy over critical exponents around the mobility edge and the ongoing debate about the Mott minimum conductivity at the mobility edge marked the history of Anderson localization. They should be mentioned—as we did—by any review on the subject. Given length constraints, it was impossible for us to go into more detail and to discuss recent speculations, including claims on the Mott minimum conductivity. ² We found it important to state that an exponent around 0.5 observed in uncompensated semiconductors would violate the lower limit of 2⁄3 set by the scaling theory and would thus require a scenario other than the one proposed by Philip Anderson and coworkers. Another view on compensated semiconductors is that charge carriers tunnel between impurity states, much like what is proposed in the Anderson model with diagonal disorder. That approach would lead to an exponent of 1.5.

We are very much aware that the extrapolation to zero temperature of the conductivity was a struggle in the early work at Karlsruhe University and at Bell Labs (Castner’s references 1–3). In 1996 Issai Shlimak and coworkers claimed that with a better justified extrapolation toward zero temperatures, even uncompensated germanium:arsenic, germanium:antimony, and silicon:phosphorus would exhibit a critical exponent of order one. ³

For a more extended historical overview with due credit, see our website http://www.andersonlocalization.com , which has been in operation since 2008.