Notes on Anderson localization

Abstract

Ad Lagendijk, Bart van Tiggelen, and Diederik Wiersma, in their article “Fifty years of Anderson localization” (PHYSICS TODAY, August 2009, page 24), discuss the experimental studies in semiconductors such as weakly compensated phosphorus-doped silicon. However, the authors don’t accurately depict the situation, and they ignore important work.Lagendijk and coauthors note that in 1982 a Bell Labs group11. M. A. Paalanen, T. F. Rosenbaum, G. A. Thomas, R. N. Bhatt, Phys. Rev. Lett. 48, 1284 (1982). https://doi.org/10.1103/PhysRevLett.48.1284 found that for charge-carrier densities n above a critical value nc in weakly compensated Si:P, the conductivity, extrapolated to zero temperature, scaled with reduced density with an exponent s of approximately 0.5; for compensated semiconductors (also amorphous alloys), experiments yielded s of approximately 1, which agrees with the scaling theory. As the authors describe, that finding led to the “exponent puzzle.” But the zero-compensation case includes only off-diagonal order in contrast to the 1958 paper by Philip Anderson. The different disorder cases are characterized by different scaling exponents.Considerable controversy ensued in 1993–99. H. Stupp at Karlsruhe University and coauthors22. H. Stupp, M. Hornung, M. Lakner, O. Madel, H. von Löhneysen, Phys. Rev. Lett. 71, 2634 (1993); https://doi.org/10.1103/PhysRevLett.71.2634 T. Castner, Phys Rev. Lett. 73, 3600 (1994) https://doi.org/10.1103/PhysRevLett.73.3600 claimed an exponent of 1.3 for Si:P, but with nc 6% lower than the Bell group. I showed that for n between 3.52 × 1018 cm−3 and 3.69 × 1018 cm−3, the data were a better fit to Mott variable-range hopping; the finding suggests that these samples were insulating as T → 0. A 6% decrease in nc increased s from 0.5 to 1.3, which demonstrates the very strong coupling between s and nc. Lagendijk and coauthors state, without giving references, “In 1999, researchers argued that an exponent of 1 is recovered in the experiments on silicon if the conductivity is correctly extrapolated to zero temperature.” That statement is misleading. In 1999 two groups33. S. Bogdanovich, M. P. Sarachik, R. N. Bhatt, Phys. Rev. Lett. 82, 137 (1999); https://doi.org/10.1103/PhysRevLett.82.137S. Waffenschmidt, C. Pfleiderer, H. von Löhneysen, Phys. Rev. Lett. 83, 3005 (1999). https://doi.org/10.1103/PhysRevLett.83.3005 reported measurements of σ as a function of uniaxial stress on Si:B and Si:P. Both groups observed a substantial increase in s from near 0.5 to between 1.2 and 1.5 close to nc. However, compressive uniaxial stress introduces inhomogeneity from sample bending, which increases s.44. T. G. Castner, Phys. Rev. Lett. 87, 129701 (2001). https://doi.org/10.1103/PhysRevLett.87.129701 The features of the 1999 data were similar to the Bell Si:P data, but the Bell group didn’t analyze the tail portion of its data very close to nc where the stress inhomogeneity became dominant and σ is small.Are features like weak localization or carrier interactions relevant or essential to explain the scaling component of approximately 0.5 for the weakly compensated case? Although there may still be disagreement, some researchers believe the answer is no. Two different calculations, both featuring the two-component model, yield σ ∝ kF, the Fermi wavevector, and kF ∝ (EF − Ec)1/2 for noninteracting carriers. That explanation is consistent with a Boltzmann–Drude conductivity.55. J. C. Phillips, Phys Rev. B 45, 5863 (1992); https://doi.org/10.1103/PhysRevB.45.5863 T. G. Castner, Phys. Rev. Lett. 84, 1539 (2000). https://doi.org/10.1103/PhysRevLett.84.1539 Since kF = 2π/λdB, with λdB the de Broglie wavelength, the calculations demonstrate a second scaling length besides the ubiquitous correlation length ξ(n). Those explanations were ignored by Lagendijk and coauthors.REFERENCESSection:ChooseTop of pageREFERENCES <<1. M. A. Paalanen, T. F. Rosenbaum, G. A. Thomas, R. N. Bhatt, Phys. Rev. Lett. 48, 1284 (1982). https://doi.org/10.1103/PhysRevLett.48.1284 , Google ScholarCrossref, CAS2. H. Stupp, M. Hornung, M. Lakner, O. Madel, H. von Löhneysen, Phys. Rev. Lett. 71, 2634 (1993); https://doi.org/10.1103/PhysRevLett.71.2634 , Google ScholarCrossref, CAST. Castner, Phys Rev. Lett. 73, 3600 (1994) https://doi.org/10.1103/PhysRevLett.73.3600 , , Google ScholarCrossref, CAS3. S. Bogdanovich, M. P. Sarachik, R. N. Bhatt, Phys. Rev. Lett. 82, 137 (1999); https://doi.org/10.1103/PhysRevLett.82.137, Google ScholarCrossref, CASS. Waffenschmidt, C. Pfleiderer, H. von Löhneysen, Phys. Rev. Lett. 83, 3005 (1999). https://doi.org/10.1103/PhysRevLett.83.3005, , Google ScholarCrossref, CAS4. T. G. Castner, Phys. Rev. Lett. 87, 129701 (2001). https://doi.org/10.1103/PhysRevLett.87.129701 , Google ScholarCrossref, CAS5. J. C. Phillips, Phys Rev. B 45, 5863 (1992); https://doi.org/10.1103/PhysRevB.45.5863 , Google ScholarCrossref, CAST. G. Castner, Phys. Rev. Lett. 84, 1539 (2000). https://doi.org/10.1103/PhysRevLett.84.1539 , , Google ScholarCrossref, CAS© 2012 American Institute of Physics.