Scaling theoriesof localisation in disordered materialspredict that the conductance and localisation lengths respectively vary as IE -E, I above and below the mobility edge. On the other hand some experiments have observed conductances varying as IE -E,ll 2 . In this paper it is shown that for systems having highly anisotropic effective mass tensors, localisation occurs for small values of disorder, which enables an analytic theory to be developed. This anisotropic theory gives an exponent of 1/2. The theory is applicable to ntype silicon provided that inter-valley scattering can be neglected, a result which accords with recent observations. The theory also shows in a natural way how the 1D and 2D limits of localisation theory can be taken by increasing the effective mass in one or two directions. The well known result that all states are localised in 2D is reproduced and can be ascribed to underlying symmetries in the system.
A new (2+1)-dimensional differential-difference equation is considered. With the aid of the nonlinearization of the Lax pair, the (2+1)-dimensional differential-difference equation is decomposed into a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems. Based on the theory of algebra curve, the continuous flow and discrete flow are straightened out in view of the introduced Abel-Jacobi coordinates.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.