Transport DFT Lab
Density functional theory replaces interacting electrons by a fictitious non-interacting Kohn–Sham system that has the same density. Out of equilibrium that is not enough: the standard recipe (Kohn–Sham levels plus the Landauer formula) misses Coulomb blockade and gets thermoelectric coefficients wrong. i-DFT and iq-DFT fix this by making the current and the heat current basic variables too, which brings in an exchange–correlation bias and an exchange–correlation thermal gradient. Here you can compare both approaches with the exact answer on a single interacting quantum dot.
What you are seeing
- The problem. In a Kohn–Sham calculation the dot is described by a single level at vs = v + vHxc[n], and the current is obtained by plugging that level into the Landauer formula with the real bias. For weak interactions this works, but a strongly interacting dot has two addition energies, v and v+U. The standard approach (blue) smears them into one broad plateau and misses the Coulomb-blockade peaks.
- The xc bias. In i-DFT the pair (density, current) is in one-to-one correspondence with the pair (gate, bias). The Kohn–Sham electrons therefore feel a different bias, Vs = V + Vxc[n, I]. Look at the diagram: in the blockade valleys the xc bias eats most of the applied bias and the Kohn–Sham current is suppressed exactly as in the interacting system.
- Steps and pinning. The functionals have sharp steps (derivative discontinuities) where adding an electron costs an extra U. The self-consistent solution gets pinned to those steps, which is how a non-interacting system reproduces Coulomb blockade in the density, the current–voltage curve and the thermocurrent.
- Temperature gradient. Heat the left lead and a thermocurrent flows even at zero bias. The same functionals, with one effective temperature per lead, describe it.
- Onsager matrix. Close to equilibrium the charge and heat currents respond linearly to the bias and to the temperature difference. Conductance G, Seebeck coefficient S, thermal conductance κ and figure of merit ZT all follow from a 2×2 Onsager matrix.
- The standard approach fails again. Using the Kohn–Sham Onsager matrix directly (blue) gives a single broad peak for G and a Seebeck coefficient with the wrong structure, even with an accurate density.
- The xc kernel. iq-DFT adds, besides the xc bias, an xc thermal gradient Ψxc. Their derivatives with respect to the charge and heat currents form a 2×2 xc kernel, and a Dyson-like equation L = Ls + LsFxcL turns the Kohn–Sham matrix into the interacting one. With the analytic functionals used here, iq-DFT (amber) follows the exact result across the Coulomb-blockade peaks. The small remaining differences come only from the approximate density.
Model: single-impurity Anderson model in the wide-band limit, symmetric coupling γL = γR = γ/2, μL,R = ±eV/2, TL,R = T(1 ± ΔT/2T). Reference: many-body model with two Lorentzian peaks at v and v+U weighted by 1−n/2 and n/2, accurate in the Coulomb-blockade regime (kBT ≳ γ). i-DFT: analytic Coulomb-blockade functionals obtained by reverse engineering, with an effective temperature that accounts for the broadening. iq-DFT: single-site-model functionals and analytic xc kernel. Energies in units of γ; currents in units of γ/ħ; G in units of G0 = 2e2/h; S in units of kB/e; κ in units of 2kBγ/h. References: G. Stefanucci & S. Kurth, Nano Lett. 15, 8020 (2015); N. Sobrino, R. D'Agosta & S. Kurth, Phys. Rev. B 100, 195142 (2019); N. Sobrino, F. Eich, G. Stefanucci, R. D'Agosta & S. Kurth, Phys. Rev. B 104, 125115 (2021). Reference implementation: github.com/Nahualcsc/iq-DFT.