The transition between the two-dimensional and three-dimensional weak-localisation correction to the Boltzmann conductivity

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Abstract
The weak-localisation correction to the Boltzmann conductivity is discussed for a two-dimensional system emerging from the electric quantum limit and eventually behaving in a three-dimensional manner. The form of the correction for a system with arbitrary subband wavefunctions is found when two sub-bands with well separated energy minima are occupied. This produces a correction similar to the two-dimensional result. A criterion for the breakdown of this regime is derived. Within the framework of an idealised model the weak localisation is calculated across the whole two- to three-dimensional range. The calculation shows how the form of the correction changes from ln (L/l) in two dimensions to (1/l-1/L) in three dimensions.