Tunneling through nanosystems: Combining broadening with many-particle states

We suggest an approach for transport through finite systems based on the Liouville equation. By working in a basis of many-particle states for the finite system, Coulomb interactions are taken fully into account and correlated transitions by up to two different contact states are included. This latter extends standard rate equation models by including level-broadening effects. The main result of the paper is a general expression for the elements of the density matrix of the finite size system, which can be applied whenever the eigenstates and the couplings to the leads are known. The approach works for arbitrary bias and for temperatures above the Kondo temperature. We apply the approach to standard models and good agreement with other methods in their respective regime of validity is found.

Figure 1

(Color online) The time-dependent current calculated with our method (full line) and with the time-dependent Green function method from Ref. 21 (dashed line) as response to a steplike modulation of the bias with step height ${\mu }_L$. The coupling is ${\Gamma }_L={\Gamma }_R=\Gamma ∕2$, the temperature $k_{B}T=0.05\Gamma$, and the half-width of the band is $W=30\Gamma$.

Figure 2

(Color online) Stationary current through the double quantum dot structure for different values of the interdot Coulomb repulsion $U$. The triangles are from a nonequilibrium Green function calculation, and the dotted line is the result by Stoof and Nazarov (Ref. 22) valid in high-bias limit for $U\to \infty$. The levels of the dot are placed symmetrically around the zero-bias with $E_{\alpha }-E_{\beta }=\Gamma$. We use the interdot tunneling coupling $\Omega =\Gamma$, ${\Gamma }_L={\Gamma }_R=\Gamma ∕2$, the temperature $k_{B}T=0.1\Gamma$, and the half-width of the band $W=20\Gamma$.

Figure 3

(Color online) Zero-bias conductance as a function of level position for different temperatures. All parameters are according to the experimental data shown in Fig. 2 of Ref. 23.

Figure 4

(Color online) Differential conductance for finite bias. Parameters as in Fig. 3.

Figure 5

(Color online) Current versus bias for $E_1=0$ and $T=0.8\mathrm{K}$. Parameters as in Fig. 3. The dashed horizontal lines correspond to the rate equation model by Gurvitz and Prager (Ref. 7), where we added the respective formulas.