Metal-insulator transition in anisotropic systems

We study the three-dimensional Anderson model of localization with anisotropic hopping, i.e., weakly coupled chains and weakly coupled planes. In our extensive numerical study we identify and characterize the metal-insulator transition by means of the transfer-matrix method and energy level statistics. Using high accuracy data for large system sizes we estimate the critical exponent as ν = 1.6 ± 0.3. This is in agreement with its value in the isotropic case and in other models of the orthogonal universality class. Previous studies of Anderson localization [1] in threedimensional (3D) disordered systems with anisotropic hopping using the transfer-matrix method (TMM) [2, 3,4], multifractal analysis (MFA) [5] and energy-level statistics (ELS) [6] show that an MIT exists even for very strong anisotropy. In Refs. [7,8], we studied critical properties of this second-order phase transition with high accuracy. Here we shall demonstrate the significance of irrelevant scaling exponents for an accurate determination of the critical disorder Wc and the critical exponent ν. Previous highly accurate TMM studies for isotropic systems of the orthogonal universality class reported ν = 1.54± 0.08 [9], ν = 1.58± 0.06 [10], ν = 1.61± 0.07, and ν = 1.54± 0.03 [11], whereas for anisotropic systems of weakly coupled planes ν = 1.3±0.1 and ν = 1.3±0.3 was found [3]. We emphasize that this variation in theoretical values has its counterpart in the experiments where a large variation of ν has been reported with values ranging from 0.5 [12] over 1.0 [13], 1.3 [14], up to 1.6 [15]. Possibly this experimental “exponent puzzle” [14] is due to other effects such as electron-electron interaction [15] or sample inhomogeneities [14,16,17]. A further important aspect of anisotropic hopping besides the question of universality is the connection to experiments which use uniaxial stress, tuning disordered Si:P or Si:B systems across the MIT [12,13,14,15]. Applying stress reduces the distance between the atomic orbitals, the electronic motion becomes alleviated, and the system changes from insulating to metallic. Thus, although the explicit dependence of hopping strength on stress is material specific and in general not known, it is reasonable to relate uniaxial stress in a disordered system to an anisotropic Anderson model with increased hopping between neighboring planes. We use the standard Anderson Hamiltonian [1]