Localization in weakly coupled planes and weakly coupled wires

Physical Review B - Condensed Matter and Materials Physics

Volumn 56, Issue 19, 1997, Pages 12221-12231

  Zambetaki, I ^a   Li, Qiming ^a   Economou, E ^a   Soukoulis, C ^a  

^a IOWA STATE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001471986     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.56.12221     Document Type: Article

References (35)

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25. P. W. Anderson, 23, 4828 (1981)] that a well-behaved quantity that can serve as a single scaling parameter and is also additive with respect to the length (Formula presented) is the ratio (Formula presented) where (Formula presented) is a slowly varying function of (Formula presented) It was found numerically that (Formula presented) varies between 1.13 and 1 as (Formula presented) varies from infinity to much less than unity. Thus (Formula presented) can be taken to be 1 in Eq. (1). The above derivation of Eq. (1) is based also upon the double inequality (Formula presented), (Formula presented) where (Formula presented) is the cross section of the bar of length (Formula presented) and mean free path (Formula presented). Thus Eq. (1) is certainly not valid in the ballistic regime where (Formula presented) or (Formula presented). However, it is valid in the weak scattering (but not ballistic) regime (Formula presented) and (Formula presented) (in this regime (Formula presented), (Formula presented), and thus the double inequality (Formula presented) is easily satisfied). It is worthwhile to point out that numerical calculations show that Eq. (1) works even near the critical regime (where (Formula presented) in spite of the breakdown of the inequality (Formula presented) In the weak disorder limit (Formula presented), the dimensionless conductance (Formula presented) can be expanded and gives that (Formula presented). By using the data of Fig. 22, which obey Eq. (6), we obtain that (Formula presented) and therefore (Formula presented), where (Formula presented) is the correlation length. The coefficient (Formula presented) is approximately a constant of order (Formula presented) away from the critical point. Notice that (Formula presented) and (Formula presented) for (Formula presented). For the infinite-size system (Formula presented), (Formula presented), and therefore (Formula presented). In the weak disorder limit (Formula presented) and therefore (Formula presented). The ratio of the localization lengths can be obtained by using the length rescaling idea. The conductances in all the directions are the same if the dimension of the system is proportional to the localization length in that direction. This implies the expression (Formula presented) and since (Formula presented), we obtain that (Formula presented).
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