Abstract
In one electron picture, the electronic states in a crystalline solid are extended Bloch states. This is a direct consequence of the Bloch-Floquet’s theorem. These quantum states are labelled by the wave-vector, к, and the band index, n. For Bloch states these are good quantum numbers. In a disordered solid, on the other hand, with the breakdown of long-ranged order, the wave-vector, к, is no longer a good quantum number. Of course, the states may still be extended, but cannot be characterized by к. The seminal paper by Anderson [1] stated that if the disorder is strong enough, all the electronic states become localized and at T = 0°o leads to a metal-insulator transition.He considered a tight-binding Hamiltonian with only on-site disorder characterized by a probability density of width W and a crystalline band-width of, B. Anderson showed that when the dimensionless parameter W/B is greater than a critical value (W/B) c all states in the band of the electron in the disordered lattice are localized.
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References
P.W. Anderson, Phys. Rev. 109, 1492 (1958)
N.F. Mott, Adv. Phys. 16, 49 (1967)
A.F. Ioffe and A.R. Regel, Prog. Semicond. 4, 237 (1960)
N.F. Mott, Phil. Mag. 19, 835 (1969)
E. Abrahams, P.W. Anderson, D.C. Licciardello, and T.V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979)
D.J. Thouless, Phys. Rep. 13, 93 (1974)
V. Chaudhry and A. Mookerjee, Pramana, (1975)
A. McKinnon and B. Kramer, Phys. Rev. Lett. 147, 1546 (1981)
A. McKinnon and B. Kramer, Z. Phys. B 53, 1 (1985)
R.E. Borland, Proc. Roy. Soc. A 274, 529 (1963)
M. Azbel Ya., Phys. Rev. B 28, 4106 (1983)
J.B. Pendry, J. Phys. C: Solid State Phys. 20, 733 (1987)
S.J. Bending and M.R. Beasley, Phys. Rev. B 35, 2548 (1987)
M. Vaito and M.R. Beasley, Phys. Rev. B 35, 9926 (1987)
D. Popovic, A.B. Fowler, and S. Washburn, Phys. Rev. Lett. 67, 2870 (1991)
N. Kumar, J. Phys. C: Solid State Phys. 16, L745 (1983)
N. Kumar and A.M. Jayannavar, Phys. Rev. B 32, 3345 (1985)
K.A. Muttalib, J. Picard and A.D. Stone, Phys. Rev. Lett. 59, 2473 (1987)
T.J. Godin and R. Haydock, Phys. Rev. B 38, 5237 (1988)
T.J. Godin and R. Haydock, Europhys. Lett. 14(2), 137 (1991)
S. Kirkpatrick and T.P. Eggarter, Phys. Rev. B 6, 3598 (1972)
N. Tit, N. Kumar, J.W. Halley, and H. Shore, Phys. Rev. B 47, 15988 (1993)
R.E. Peierls, Z. Phys. 80, 763 (1933)
J. Pimental and S.L.A. de Queiroz, J. Phys. A: Math. Phys. 22, L345 (1983)
T. Odagaki and K.C. Chang, Phys. Rev. B 30, 1612 (1984)
L.J. Root, J.D. Bauer, and J.L. Skinner, Phys. Rev. B 37, 5518 (1988)
K.C. Chang and T. Odagaki, J. Phys. A: Math. Phys. 20, L1027 (1987)
J. Stein and U. Krey, Z. Phys. B 37, 13 (1980)
R. Raghavan and D.C. Mattis, Phys. Rev. B 23, 4791 (1981)
R. Raghavan, Phys. Rev. 29, 748 (1984)
J.T. Edwards and D.J. Thouless, J. Phys. C: Solid State Phys. 5, 807
D.C. Licciardelo and D.J. Thouless, J. Phys. C: Solid State Phys. 8, 4157 (1975)
Th. Koslowski and W. von Neissen, Phys. Rev. B 42, 10342 (1990)
J.P.G. Taylor and A. McKinnon, J. Phys. C: Solid State Phys. 1, 9963 (1989)
C.M. Soukoulis and G.S. Grest, Phys. Rev. B 44, 4685 (1991)
H.M. Pastawski, J.F. Weisz, and S. Albornoz, Phys. Rev. B 28, 6896 (1983)
Shapir Y., Aharony A. and Harris A.B., Phys. Rev. Lett., 49 486 (1982)
Meir Y., Aharony A. and Harris A.B., Phys. Rev. Lett., 56 976 (1986)
Meir Y., Aharony A. and Harris A.B., Europhys. Lett., 10 275 (1989)
J.L. Gammel Padé Approximants and their Applications (Acadmeic Press, New York, 1973)
R. Haydock, Solid State Physics (Aacdemic Press, New York, 35, 216, 1980)
R. Haydock, Philos. Mag. B 43, 203 (1981)
R. Haydock and R. Te, Phys. Rev. B 49, 10845 (1994)
R. Haydock and R. Te, Phys. Rev. B 57 296 (1997)
A. Aharony and D. Stauffer, Inroduction to Percolation Theory (Taylor and Francis, London, 1991)
D.E. Khmelnitskii, Physica 126 B + C 235 (1984)
R. Haydock, Philos. Mag. 53, 554 (1986)
J.L. Picard and G. Sharma, J. Phys. C: Solid State Phys. 14, L127, L617 (1981)
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Mookerjee, A., Saha-Dasgupta, T., Dasgupta, I. (2009). Quantum Transmittance Through Random Media. In: Chakrabarti, B., Bardhan, K., Sen, A. (eds) Quantum and Semi-classical Percolation and Breakdown in Disordered Solids. Lecture Notes in Physics, vol 762. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85428-9_3
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