Deconstructing Kubo formula usage: Exact conductance of a mesoscopic system from weak to strong disorder limit

Abstract

In both research and textbook literature one often finds two "different" Kubo formulas for the zerotemperature conductance of a non-interacting Fermi system. They contain a trace of the product of velocity operators and single-particle (retarded and advanced) Green operators: Tr(vxĜ rv xĜ a ) or Tr(vx ImĜvxImĜ). The study investigates the relationship between these expressions, as well as the requirements of current conservation, through exact evaluation of such quantum-mechanical traces for a nanoscale (containing 1000 atoms) mesoscopic disordered conductor. The traces are computed in the semiclassical regime (where disorder is weak) and, more importantly, in the nonperturbative transport regime (including the region around localization-delocalization transition) where concept of mean free path ceases to exist. Since quantum interference effects for such strong disorder are not amenable to diagrammatic or nonlinear σ-model techniques, the evolution of different Green function terms with disorder strength provides novel insight into the development of an Anderson localized phase.PACS numbers: 72.10. Bg, 72.15.Rn, 05.60.Gg At first sight, the title of this paper might sound perplexing. What else can be said about Kubo formula 1 after almost a half of a (last) century of explorations in practice, as well as through numerous re-derivations in both research 2 and textbook 3,4 literature? Kubo linear response theory (KLRT) represents the first full quantum-mechanical transport formalism. It connects irreversible processes in nonequilibrium to the thermal fluctuations in equilibrium [fluctuation-dissipation theorem (FDT)]. Therefore, the study of transport is limited to the nonequilibrium states close to equilibrium. Nevertheless, the computation of linear kinetic coefficients is greatly facilitated since final expressions deal with equilibrium expectation values of relevant physical quantities (which are much simpler than the corresponding nonequilibrium ones 5 ). It originated 6 from the Einstein relation for the diffusion constant and mobility of a particle performing a random walk.Until the scaling theory of localization, 7 and ensuing computation of the lowest order quantum correction, weak localization 8 (WL), to the Drude conductivity, it almost appeared that microscopic and complicated Kubo formulation of quantum transport merely served to justify the intuitive Bloch-Boltzmann semiclassical approach 3 to transport in weakly disordered (k F ℓ ≫ 1, k F is the Fermi wave vector and ℓ is the mean free path) conductors. Furthermore, the advent of mesoscopic physics 9 has led to reexamination of major transport ideas-in particular, we learned how to apply properly KLRT to finite-size systems. Thus, the equivalence was established 2 between the rigorous Kubo formalism and heuristically founded Landauer-Büttiker 10 scattering approach to linear response transport of non-interacting quasiparticles.11 This has emerged as an important tool in for studying mesoscopic transport phenomena, where system size and interface...

Efficient linear-scaling quantum transport calculations on graphics processing units and applications on electron transport in graphene

“…Conceptually, one unambiguous way to define the length of a simulated sample is to connect it with two semi-infinite leads along the transport direction, which affect the effective Hamiltonian of the sample by adding the "self energies" arising from the interactions between the sample and the leads. This inevitably leads to the "mesoscopic Kubo-Greenwood formula" [46][47][48], or equivalently, the NEGF method [1].…”

Obtaining localization properties efficiently using the Kubo-Greenwood formalism

“…[10][11][12][13][14][15][16][17] The main difficulty of using the KG method in the localized regime is a lack of a length scale, which is crucial for characterizing the scaling behavior of conductance. This makes the original KG formula not very suitable for studying mesoscopic transport, while a 'mesoscopic KG formula' [18][19][20][21] was found to be equivalent to the RGF formalism. However, it has been realized recently that a definition of length is possible by recasting the KG formula into a time-dependent Einstein formula, 8,9,[22][23][24][25][26][27] in which the conductivity is expressed as a time-derivative of the mean square displacement (MSD).…”

Resistivity saturation in semi-conducting polyacetylene molecular wires

“…Within the Kubo formalism, the formula for the optical transmission along the x direction (when neglecting vertex corrections [71][72][73][74][75]) is given by [76] T…”