Abstract
A generalization of the Vollhardt-Wölfle self-consistent localization theory is proposed to take into account spatial dispersion of the kinetic coefficients of a two-dimensional disordered system. It is shown that the main contribution to the singular part of the collision integral of the Bethe-Salpeter equation in the limit ω→0 is from the diffusion pole iω=(p+p′)2 D (|p+p′|,ω), which provides an anomalous increase in the probability of backscattering p→−p′. In this limit the dependence of the diffusion coefficient on q and ω exhibits localization behavior, D(q,ω)=−iωf(l Dq), where |f(z)|⩽(0)=d 2 (d is the localization length). According to the Berezinski\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \)-Gor’kov criterion, D(q,0)=0 for all q. Spatial dispersion of D(q,ω) is manifested on a scale q ∝ 1/l D, where l D is the frequency-dependent diffusion length. In the localization state l D≪l, where l is the electron mean free path; l D ∝ ω as ω→0, suggesting the suppression of spatial dispersion of the kinetic coefficients down to atomic scales. Under the same conditions σ(q,ω) exhibits a strong dependence on q on a scale q ∝ 1/d, i.e., the nonlocality range of the electrical conductivity is of the order of the localization length d. At the microscopic level these results corroborate the main conclusions of Suslov (Zh. Éksp. Teor. Fiz. 108, 1686 (1995) [JETP 81, 925 (1995)]), which were obtained to a certain degree phenomenologically in the limit ω→0. A major advance beyond the work of Suslov in the present study is the analysis of spatial dispersion of the kinetic coefficients at finite (rather than infinitely low) frequencies.
Similar content being viewed by others
References
P. W. Anderson, Phys. Rev. 109, 1492 (1958).
N. F. Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd ed., Clarendon Press, Oxford (1979).
Y. Nagaoka and H. Fukuyama (eds.), Anderson Localization, Springer-Verlag, Berlin-New York (1982).
P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
M. V. Sadovskii, Sverkhprovodimost’ (KIAE) 8, 337 (1995).
D. Vollhardt and P. Wolfle, Phys. Rev. B 22, 4666 (1980).
D. Vollhardt and P. Wolfle, Phys. Rev. Lett. 45, 842 (1980).
P. Wölfle and D. Vollhardt, in Anderson Localization, Y. Nagaoka and H. Fukuyama (eds.), Springer-Verlag, Berlin-New York (1982), p. 26.
F. J. Wegner, Z. Phys. B. 25, 327 (1976).
E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
D. Yoshioka, Y. Ono, and H. Fukuyama, J. Phys. Soc. Jpn. 50, 3419 (1981).
D. Yoshioka, in Anderson Localization, Y. Nagaoka and H. Fukuyama (eds.), Springer-Verlag, Berlin-New York (1982), p. 44.
A. Theumann and M. A. Pires Idiart, J. Phys. Condens. Matter 3, 3765 (1991).
É. Z. Kuchinskii and M. V. Sadovskii, Sverkhprovodimost’ (KIAE) 4, 2278 (1991).
I. M. Suslov, Zh. Éksp. Teor. Fiz. 108, 1686 (1995) [JETP 81, 925 (1995)].
V. L. Berezinskii and L. P. Gor’kov, Zh. Éksp. Teor. Fiz. 77, 2498 (1979) [Sov. Phys. JETP 50, 1209 (1979)].
T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982).
S. F. Edwards, Philos. Mag. 8, 1020 (1958).
D. N. Zubarev, “Modern methods of the statistical theory of nonequilibrium processes,” in Contemporary Problems in Mathematics [in Russian], Vol. 15, VINITI AN SSSR (1979).
V. P. Silin, Introduction to the Kinetic Theory of Gases [in Russian], Nauka, Moscow (1971).
W. Götze, Philos. Mag. 43, 219 (1981).
L. P. Gor’kov, A. I. Larkin, and D. E. Khmel’nitskii, JETP Lett. 30, 228 (1979).
A. A. Samarskii and A. V. Gulin, Numerical Methods [in Russian], Nauka, Moscow (1989).
Author information
Authors and Affiliations
Additional information
Zh. Éksp. Teor. Fiz. 111, 1787–1802 (May 1997)
Rights and permissions
About this article
Cite this article
Groshev, A.G., Novokshonov, S.G. Localization and space-time dispersion of the kinetic coefficients of a two-dimensional disordered system. J. Exp. Theor. Phys. 84, 978–985 (1997). https://doi.org/10.1134/1.558188
Received:
Issue Date:
DOI: https://doi.org/10.1134/1.558188