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arXiv:cond-mat/0110621 · Journal of the Physical Society of Japan

Perturbation Study of the Conductance through an Interacting Region Connected to Multi-Mode Leads

Yoshihide Tanaka, Akira Oguri, Hiroumi Ishii

Published Jan 1, 2002• 7 citations
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Abstract

We study the effects of electron correlation on transport through an interacting region connected to multi-mode leads based on the perturbation expansion with respect to the inter-electron interaction. At zero temperature the conductance defined in the Kubo formalism can be written in terms of a single-particle Green's function at the Fermi energy, and it can be mapped onto a transmission coefficient of the free quasiparticles described by an effective Hamiltonian. We apply this formulation to a two-dimensional Hubbard model of finite size connected to two noninteracting leads. We calculate the conductance in the electron-hole symmetric case using the order $U^2$ self-energy. The conductance shows several maximums in the $U$ dependence in some parameter regions of $t_y/t_x$, where $t_x$ ($t_y$) is the hopping matrix element in the $x$- ($y$-) directions. This is caused by the resonance occurring in some of the subbands, and is related with the $U$ dependence of the eigenvalues of the effective Hamiltonian.

DOI
10.1143/jpsj.71.211

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We study the effects of electron correlation on transport through a small interacting system connected to reservoirs using an effective Hamiltonian which describes the free quasiparticles of a Fermi liquid. The effective Hamiltonian is defined microscopically with the value of the self-energy at $\ensuremath{\omega}=0.$ Specifically, we apply the method to a Hubbard chain of finite size N (=1,2,3,…),(=1,2,3,\dots{}),(=1,2,3,…), and calculate the self-energy within the second order in U in the electron-hole-symmetric case. When couplings between the chain and the reservoirs on the left and right are small, the conductance for even N decreases with increasing N, showing a tendency toward a Mott-Hubbard insulator. This is caused by the off-diagonal element of the self-energy, and this behavior is qualitatively different from that in the special case examined in previous work. We also study the effects of the asymmetry in the two couplings. While a perfect transmission due to the Kondo resonance occurs for any odd N in the symmetric coupling, the conductance for odd N decreases with increasing N in the case of asymmetric coupling.
The dc conductance gN{g}_{N}gN​ through a finite Hubbard chain of size N (=1,2,3,…)(=1,2,3,\dots{})(=1,2,3,…) connected to reservoirs is studied at T=0T=0T=0 in an electron-hole symmetric case. We calculate a spatial dependence of the self-energy analytically at $\ensuremath{\omega}=0$ within the second order in U, and obtain an inter-site Green's function GN1{G}_{N1}GN1​, from which gN{g}_{N}gN​ can be determined, via the Dyson equation. The results depend strongly on whether N is even or odd. For odd N, a perfect transmission occurs, and ${g}_{N}\ensuremath{\equiv}{2e}^{2}/h$ independent of the values of U. This may be attributed to a Kondo resonance appearing at $\ensuremath{\omega}=0$. For even N, gN{g}_{N}gN​ decreases with increasing N, and converges to a finite constant which is a smooth decreasing function of U. These behaviors are essentially owing to the presence of the reservoirs, which makes a quasiparticle description valid for low-energy states at $\ensuremath{\omega}\ensuremath{\ll}\ensuremath{\Elzxh}{v}_{F}/(\mathrm{Na})$; where vF{v}_{F}vF​ is the Fermi velocity and a is the lattice constant.
A numerical analysis for evaluating the transport coefficient through a small interacting system is examined using the quantum Monte Carlo (QMC) method. To formulate the analysis, we consider a finite interacting system on a one-dimensional lattice that is connected to noninteracting leads, and investigate the expression of the dc conductance making use of the Kubo formula with the perturbation theory in the Coulomb interaction. At T=0T=0T=0 the dc conductance is written in terms of the transmission coefficient, which can be obtained numerically by extrapolating the QMC data for the Matsubara Green's function as $i{\ensuremath{\varepsilon}}_{n}\ensuremath{\rightarrow}{i0}^{+}$. We apply the analysis to the dc conductance through a region with a few resonance states.
We study the relation between the dc conductance and the transmission through an interacting region based on the Kubo formalism using the perturbation analysis in the Coulomb interaction developed by Yamada-Yosida and Shiba. We find that the contributions of the vertex correction to the dc conductance disappear at T=0 if the currents are measured in the noninteracting leads. Consequently, the dc conductance is written in a Landauer-type form using the transmission coefficient for single-particle-like excitation at the Fermi energy. The results are generalized to a system with a number of scattering channels, and may be regarded as an extension of the relation derived by Fisher-Lee.
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