Incompleteness of the Landauer formula for electronic transport

We show that the Landauer formula for the conductance of a nanoscale system is incomplete because it does not take into account many-body effects which cannot be treated as contributions to the single-particle transmission probabilities. We show that the physical origin of these effects is related to the viscous nature of the electron liquid and we develop a perturbative formalism, based on the time-dependent current-density-functional theory, for calculating the corrections to the resistance in terms of the “Kohn-Sham current distribution” and the exchange-correlation kernel. The difficulties that still remain in calculating the latter are critically discussed.

Figure 1

Schematic of a quantum system in a multiterminal configuration. The central region is connected to leads of noninteracting electrons, in turn connected adiabatically to reservoirs of electrons.

Figure 2

Diagrams for the proper current-current response function. The solid dots represent (particle) current vertices. The solid lines represent fully dressed single-particle propagators and the dotted lines Coulomb scattering processes.

Figure 3

A model two-terminal device in which the density changes only in one dimension. 1 and 2 are the lead regions.

Figure 4

Qualitative behavior of the viscosity of a homogeneous electron gas as a function of frequency. For frequencies smaller than $1/\tau$, with $\tau$ the quasiparticle relaxation time, $\eta$ approaches the dc limit ${\eta }_0$. ${\eta }_0$ vs density is shown in inset (a) for $T=300\text{K}$. The calculation is done with the Abrikosov-Khalatnikov formula (Ref. 40). For frequencies larger than $1/\tau$, but still much smaller than the Fermi frequency $E_F/\hslash$, $\eta$ tends to a different constant ${\eta }_{\infty }$. The behavior of ${\eta }_{\infty }$ vs density is shown in inset (b) in various approximations at zero temperature. CV is from Ref. 25 and QV is from Ref. 41. The atomic unit of viscosity is $\hslash /a_B^3=\simeq 7\times {10}^{-3}$ P ($a_B$ is the Bohr radius) and $n{a}_B^3=\frac{3}{4\pi r_s^3}$.