Transport calculations based on density functional theory, Friedel's sum rule, and the Kondo effect

Friedel's sum rule provides an explicit expression for a conductance functional $\mathcal{G}\left[n\right]$, valid for the single-impurity Anderson model at zero temperature. The functional is special because it does not depend on the interaction strength $U$. As a consequence, the Landauer conductance for the Kohn-Sham (KS) particles of density functional theory (DFT) coincides with the true conductance of the interacting system. The argument breaks down at temperatures above the Kondo scale, near integer filling, $n_{\mathrm{d}\sigma }\approx 1/2$ for spins $\sigma =\uparrow \downarrow$. Here, the true conductance is strongly suppressed by the Coulomb blockade, while the KS conductance still indicates resonant transport. Conclusions of our analysis are corroborated by DFT studies with numerically exact exchange-correlation functionals reconstructed from calculations employing the density matrix renormalization group.

Figure 1

Ground-state density per spin of a QD (at $x=0$) coupled to a noninteracting reservoir with $M=100$ sites for growing on-QD interaction $U=0.0,0.6,4.0$. Parameters: ${\varepsilon }_{\mathrm{d}}=0.2,V=0.3$, bandwidth of conduction electrons: $2t=2$.

Figure 2

XC correlation potential corresponding to the evolution of the density shown in Fig. 1. The on-dot potential is denoted $v_0^{\mathrm{KS}}={\varepsilon }_{\mathrm{d}}^{\mathrm{KS}}$.

Figure 3

Local spectral function of the KS system, ${\mathcal{A}}_d^{\mathrm{KS}}=2\pi {\rho }_{\mathrm{imp}}$, on the QD for interactions $U=0,2$. Parameters: ${\varepsilon }_{\mathrm{d}}=0.1$, $V=0.15...$, ${\varepsilon }_{\mathrm{F}}=0$.

Figure 4

Comparison of conductances calculated via the Friedel sum rule (FSR) and directly from the Landauer formula for KS particles (10). Two parameter combinations, $V=0.3,U=1.8,2.1$ are shown. Inset: Detuned KS couplings $V^{\mathrm{KS}}$ that together with the belonging XC potential reproduce the evolution of the dot occupation and hence the conductance seen in main plot.