Transport in one-dimensional wires: the role of reservoirs

Using the formalism of Ford-Kac-Mazur we study nonequilibrium transport in fermionic systems connected to general particle and thermal reservoirs. Several exact results on electrical and thermal transport in one-dimensional disordered non-interacting fermionic wires are obtained. Landauer’s formula is shown to follow for a special choice of reservoirs. For general reservoirs the currents in the system depend on the spectral properties of the reservoirs and Landauer’s formula needs modification. Some experimental implications are discussed. PACS numbers: 05.60.-k, 72.10.Bg, 73.63.Nm, 05.40.-a Typeset using REVTEX 1 There is considerable current interest in the problem of transport through various nanoscale devices both from fundamental and from applied points of view. In this connection, Kubo’s transport formulas have to a large extent been superseded by different formalisms in the spirit of Bardeen’s tunneling model [1]. The Landauer formula [2] and the Keldysh technique [3], C algebraic formulas [4] and generalized scattering theory ideas [5] have been developed, allowing one to study systems in steady state arbitrarily far from the linear region where Kubo’s formulas are applicable. While these new formalisms are being developed, there is considerable experimental activity involving resistive elements, such as quantum dots, STM tips and single walled nanotubes, often coming up with unexpected physics. As an example in the experiment of Ref( [6,7]), one finds oscillations of current as a function of the applied voltage for nanotubes as much as a thousand atoms long, these are of quantum origin and are apparently related to the physics of interference between electron waves traversing the leads and the samples several times, resembling Fabry Perot oscillations in optical systems. Such a behaviour is characteristic of ballistic transport between semi permeable barriers, and as such is unexpected from the most straightforward applications of say Landauer’s formula. To this menagerie of techniques, we add a new and promising member in this paper. We adapt the formalism developed by Ford, Kac, and Mazur [8] to model quantum mechanical particle and heat reservoirs, and study charge and heat transport in disordered noninteracting fermionic systems. This method, originally devised to study Brownian motion in coupled oscillators can be modified to treat Fermi systems as we show here. It is very direct to interpret and seems more straightforward to apply than other methods of treating open quantum systems such as the Caldeira-Leggett [9], Keldysh [10] and scattering theory [5]. We obtain exact formal expressions for currents and local densities in the nonequilibrium steady state. The analysis here closely follows the one in [11] for the case of classical oscillator chains. The most popular alternative to Kubos formulas is the Landauer formula, proposed in 1957 [2]. Since then several derivations of the Landauer formula have been given [12,13,14,15,16,17,18] and this has led to a good understanding of the formula and espe2 cially the important differences between the two-probe and four-probe measurements. A large number of experiments are interpreted on the basis of Landauer’s formula successfully. The quantum of conductance e/h has been understood as a contact resistance which arises due to the squeezing of the reservoir degrees of freedom into a single channel [19,20]. While a physically careful statement of the conditions for validity of the Landauer formula can be found in Ref( [21]), we believe that a detailed mathematical theory of the role of reservoirs and the nature of the coupling between the wires and reservoirs does not exist. The role of the idealized reservoirs has been to serve as perfect sources and sinks of thermal electrons which travel into and out of the resistive system. This clearly will not be satisfied in all experimental conditions and it is necessary to have a better microscopic understanding of reservoirs and contacts. There has been some work [3,4,5,20,22] where a microscopic modeling of reservoirs has been done. But, to our knowledge, there has been no explicit demonstration of the conditions under which Landauer’s formula becomes valid. We show that for a special type of reservoir, Landauer’s results follow exactly while for general reservoirs they need to be modified. The set-up: We wish to study conduction in a disordered fermionic system connected to heat and particle reservoirs through ideal one-dimensional leads [see Fig. 1]. For simplicity we begin by considering the case where the system and leads are both one-dimensional while the reservoirs are taken to be quite general. We consider both the reservoirs and the system to be noninteracting and describable by the tight binding model. We will use the following notation: the indices l,m will denote points on the system or leads, greek indices λ, ν or λ, μ will denote points on the left or right reservoirs respectively, finally p, q will be used to denote points anywhere. Thus cl (l = 1, 2...N) denotes lattice fermionic operators on the (system + lead), cλ (λ = 1, 2...M) denotes operators on the left reservoir and cλ′ (λ ′ = 1, 2...M) denotes operators on the right reservoir. The cps’ satisfy the usual anticommutation relations {cp, cq} = 0; {c † p, c † q} = 0; {c † p, cq} = δpq. Out of the N = Ns + 2Nl, sites the first and last Nl sites refer to the left and right leads respectively while the middle Ns sites refer to the system. The Hamiltonian for the entire system is given by: 3 H = H + V + Vinteraction (1) H = [− N−1 ∑ l=1 (c†l cl+1 + c † l+1cl) + N ∑