From clean to diffusive mesoscopic systems: A semiclassical approach to the magnetic susceptibility

– We study disorder-induced spectral correlations and their effect on the magnetic susceptibility of mesoscopic quantum systems in the non-diffusive regime. By combining a diagrammatic perturbative approach with semiclassical techniques we perform impurity averaging for non-translational invariant systems. This allows us to study the crossover from clean to diffusive systems. As an application we consider the susceptibility of non-interacting electrons in a ballistic microstructure in the presence of weak disorder. We present numerical results for a square billiard and approximate analytic results for generic chaotic geometries. We show that for the elastic mean free path l larger than the system size, there are two distinct regimes of behaviour depending on the relative magnitudes of l and an inelastic scattering length. Phase-coherent, disordered conductors where the electron motion equals a random walk between impurities have traditionally been of interest in mesoscopic physics [1]. Random walks occur in the diffusive regime where the elastic mean free path l is much smaller than the system size L. On the other hand, the development of high-mobility semiconductor heterostructures, combined with advanced lithographic techniques, have allowed the confinement of electrons to two-dimensional microstructures of controllable, non-random geometry. They have been coined “ballistic” since l > L. Nevertheless, residual impurity scattering is nearly unavoidable even in these systems and it became clear that disorder can be strong enough to mix energy levels and effect the two-level correlation function, K(ε1, ε2), even in the “ballistic” regime [2, 3]. There it is necessary to consider both disorder averaging, 〈. . .〉d, and size (or energy) averaging, 〈. . .〉L. After such averaging the two-level correlation function may be divided into two separate terms [3], 〈 K(ε1, ε2) 〉 L = 〈〈ν(ε1)ν(ε2)〉d〉L − 〈〈ν(ε1)〉d〈ν(ε2)〉d〉L and K(ε1, ε2) = 〈〈ν(ε1)〉d〈ν(ε2)〉d〉L − ν̄, where ν denotes the single particle density of states and ν̄ = 〈〈ν(ε)〉d〉L its mean part. K is a measure of disorder-induced correlations of ν, while K is given by size-induced correlations. The orbital magnetism of isolated mesoscopic systems has been the subject of much theoretical interest, in particular because it is sensitive to spectral correlations: For a system with Typeset using EURO-LTEX 2 EUROPHYSICS LETTERS a fixed number of particles it is necessary to consider averaging under canonical conditions [4, 5, 6] resulting in a large contribution to the average magnetism. The corresponding susceptibility is given by [6] 〈χ(H)〉 = − (∆/2)∂/∂H 〈 δN(μ;H) 〉 . Here H is the magnetic field, ∆ is the mean level spacing and 〈 δN(μ;H) 〉 is the variance in the number of energy levels within an energy interval of width equal to the chemical potential, μ. This variance is related to K(ε1, ε2;H) by integration of the level energies ε1, ε2 over the energy interval. In the following we will label the contributions to the susceptibility, corresponding to 〈 K 〉 L and K, as 〈 χ(H) 〉 and 〈 χ(H) 〉 , respectively, so that 〈χ(H)〉 = 〈 χ(H) 〉