Landauer Formula without Landauer’s Assumptions

The Landauer formula for dissipationless conductance lies at the heart of modern electronic transport, yet it remains without a clear microscopic basis. We analyze the Landauer formula microscopically, and give a straightforward quantum kinetic derivation for open systems. Some important experimental implications follow. These lie beyond the Landauer result as popularly received. In 1957 Rolf Landauer published a prescient interpretation of metallic resistivity [1]. It heralded one of the most dramatic predictions of modern condensed-matter physics: the perfect quantization, in steps of 2e/h, of electrical conductance in one-dimensional metallic channels [2]. Such quantization is quite independent of the material properties of the contact and of its leads. It is universal insofar as one may validly neglect the disruptive influences of inelastic dissipation within the transport process. Landauer argued that the current, not the applied electromotive voltage, should be understood as the active probe by which a device reveals its conductance. The observed carrier flux is understood as a kind of diffusive flow, tending to shift carriers from a “high”to a “low”-density reservoir (lead). In the mesoscopic realm, this flow between leads is conditioned by the intervening device channel, which presents a quantum tunnelling barrier to the non-interacting electrons making up the flux. The Landauer formula, then, has two cardinal tenets: (i) current is the flow of independent and degenerate electrons as they follow a nominal density gradient across reservoirs, and (ii) conductance is lossless transmission through an interposed quantum barrier. These underpin Landauer’s assumptions, namely that (a) transport ensues when a pair of leads connected to the device are set to different chemical potentials μL, μR; (b) the density mismatch due to μL − μR sustains the current; (c) μL − μR is the applied voltage across the device; (d) the Fermi energy is much larger than the thermal and electrical energies; and (e) there are no inelastic processes to dissipate the electrical energy gained by the electrons. Energy dissipation does not appear in the classic Landauer derivation [1, 3]. For a sample of mesoscopic dimensions, the model admits only elastic barrier scattering § To whom correspondence should be addressed ([email protected]) Landauer Formula without Landauer’s Assumptions 2 and excludes any role for inelastic processes within the active device and its interfaces. Yet it is dissipative inelastic scattering, and that alone, which ensures the energetic stability of resistive transport, and hence a steady state for conduction. Finite conductance and electrical energy loss are indivisible phenomena. The fundamental expression of their basic inseparability is the fluctuation-dissipation relation [4]. This establishes the equivalence of the mean-square fluctuation strength for the current and the conductance coefficient G in the the energy dissipation rate P = GV , where V is the applied voltage. There is a missing link between Landauer’s universal – and lossless – conductance formula, which has been critical in the development of mesoscopic science [3], and the dissipative inelastic processes that are absolutely vital to the microscopic origin of resistance. Repeated attempts have been made to obtain the Landauer formula from microscopic-like arguments [2, 5, 6]; see also Ciraci et al [7]. However, a convincing resolution has not yet materialized [8, 9]. The absence of so crucial a connection is a puzzling theoretical conundrum for Landauer’s approach to mesoscopics, which is otherwise so empirically compelling. In this letter we answer the question: How can the Landauer formula, in seemingly bypassing all inelastic processes, predict a finite – invariably dissipative – conductance that fulfils the fluctuation-dissipation theorem (FDT)? Below we offer a straightforward microscopic interpretation of Landauer’s result, for a mesoscopic contact open to the macroscopic environment. Our treatment differs from all earlier attempts by directly addressing the essential physics of dissipation. To obtain conductance quantization within an open contact, the explicit interplay of elastic and dissipative processes is necessary and sufficient. Neglect of either mechanism, in favour of the other, negates the formula’s microscopic basis. Both kinds of scattering are needed. We also show that the traditional Landauer assumptions of pseudo-diffusive current and lossless scattering are not required in a first-principles analysis of Landauer conductance. Our model relies solely upon orthodox quantum kinetics, as embodied in the microscopic Kubo-Greenwood (KG) formalism [10, 11]. The KG formulation automatically guarantees the FDT; it is not invoked as an additional hypothesis. Both dissipative and lossless scattering appear within the resulting fluctuation-dissipation relation, and both are assigned equal physical importance. First, we briefly recall the KG formula and the essential charge conservation built into it. Next we discuss the form of the KG relaxation time, which fixes the conductance. Finally, we show how the physical constraints on a one-dimensional open ballistic channel, connected to macroscopic leads, leads naturally to Landauer’s ideal quantized conductance. We go on to examine some of the measurable effects of device non-ideality on the Landauer conductance. The Kubo-Greenwood theory [10, 11] decribes the carriers’ full many-body density matrix. All of the transport and fluctuation properties are contained within it. Thus, the conductivity for the system appears as the trace of the current-current correlation function: