Steady-state spectra, current, and stability diagram of a quantum dot: A nonequilibrium variational cluster approach

We calculate steady-state properties of a strongly correlated quantum dot under voltage bias by means of nonequilibrium cluster perturbation theory and the nonequilibrium variational cluster approach, respectively. Results for the steady-state current are benchmarked against data from accurate matrix product state based time evolution. We show that for low to medium interaction strength, nonequilibrium cluster perturbation theory already yields good results, while for higher interaction strength the self-consistent feedback of the nonequilibrium variational cluster approach significantly enhances the accuracy. We report the current-voltage characteristics for different interaction strengths. Furthermore we investigate the nonequilibrium local density of states of the quantum dot and illustrate that within the variational approach a linear splitting and broadening of the Kondo resonance is predicted which depends on interaction strength. Calculations with applied gate voltage, away from particle-hole symmetry, reveal that the maximum current is reached at the crossover from the Kondo regime to the doubly occupied or empty quantum dot. Obtained stability diagrams compare very well to recent experimental data [A. V. Kretinin et al., Phys. Rev. B 84, 245316 (2011)].

Figure 1

Illustration of the SIAM as prepared for use within nCPT and nVCA. An interacting quantum dot (QD) is located in the middle of two noninteracting electronic leads. In order to employ the cluster approaches, the system is divided into three pieces, a left part, a right part, and a central region of length $L$, which includes the quantum dot as well as parts of left and right leads.

Figure 2

Steady-state current-voltage characteristics in the particle-hole symmetric case ($V_G=0$). Results are shown for small to large interaction strength: $U=4\Delta$ (top row), $U=8\Delta$ (second row), $U=12\Delta$ (third row), and $U=20\Delta$ (bottom row). The first column contains the full current-voltage characteristics as obtained with nCPT for $L=3,7,\text{and}11$. As benchmarks, accurate TEBD results (Ref. 68) and the linear response result $j_{\text{lin}}=2G_0{V}_B$ are depicted as well. In some bias regions, only an upper bound can be obtained by TEBD. This upper limit is depicted as a magenta bar. In the second column a close-up of the low bias region is presented. Here we compare nCPT ($L=3,7,\text{and}11$), nVCA${}_T$ ($L=3,7,\text{and}11$) with TEBD data as well as QMC results (Ref. 43). Note that the QMC data have been obtained in a wide-band limit. The third column contains results obtained for $L=7$ for various sets of nVCA variational parameters: nCPT, nVCA${}_T$, nVCA${}_{t^{\prime }}$, nVCA${}_{t,t^{\prime }}$, nVCA${}_{t_b}$. Legends are displayed once in the top row for the respective column.

Figure 3

Nonequilibrium local density of states of the quantum dot ${\rho }_f$ as a function of energy $\omega$ and bias voltage $V_B$ in the particle-hole symmetric case ($V_G=0$). Region A represents the experimentally most interesting region where no effects of the lead bands come into play and one observes a splitting of the equilibrium Kondo resonance. Region B is dominated by high-energy incoherent excitations which are expected to form continuous Hubbard bands in the thermodynamic limit. In region C, the lead density of states becomes relevant, causing a new low-energy resonance for $L\ge 7$. Data are depicted for $U=12\Delta$ and an artificial numerical broadening of $0^{+}=0.05$. nCPT results are shown in the first row for system sizes of $L=3$ (left), $L=7$ (middle), and $L=11$ (right). The second row displays the corresponding nVCA${}_T$ data. For reasons of visualization the color scale is cut off at ${\rho }_{\text{max}}=2$. Note that a horizontal line through the center of each plot at $V=0$ corresponds to the equilibrium density of states. The spectral function sum rule Eq. (A3) is fulfilled for each applied bias voltage. The $L=11$ results are noisy as fewer data points have been obtained.

Figure 4

Nonequilibrium local density of states of the interacting quantum dot. Data are shown for $U=12\Delta$ and an artificial numerical broadening of $0^{+}=0.05$. Results are obtained by nVCA${}_T$ with $L=3$.

Figure 5

Interaction-dependent splitting of the Kondo resonance at ${\omega }_K$ under bias. Data shown are for different interaction strengths $U=4,8,12,\text{and}20\Delta$ (solid lines). The resonance merges with the incoherent high-energy spectrum (Hubbard bands), located at ${\omega }_H\approx \pm \frac{U}{2}$ (dotted lines) at a certain bias voltage $V_m\approx 15,17,21,\text{and}28$ respectively. These values have been obtained by the linear fits (indicated as dashed lines). If the Kondo resonance would pin at the chemical potential of the leads ${\omega }_K$, the linear behavior indicated as ${\mu }_{L/R}$ (dash-dotted line) would follow.

Figure 6

Stability diagram and steady-state current as a function of bias voltage $V_B$ and gate voltage $V_G$. In the first column, the steady-state current is shown as a function of bias voltage for various gate voltages. The second column contains a density plot of the steady-state current in the full $V_B/V_G$ parameter space. In the third column a similar density plot is shown for the differential conductance $G=\frac{dI}{d{V}_B}$ (stability diagram). The four rows depict results obtained by nCPT for $U=4\Delta$, nVCA${}_T$ for $U=4\Delta$, nCPT for $U=20\Delta$, and nVCA${}_T$ for $U=20\Delta$. The squares in the two stability diagrams bottom right mark the experimentally most interesting region. In the nVCA${}_T$ result, the Kondo region (vertical line in the center) is reproduced extremely well, as are the Coulomb blockade regions ($\propto V_B$) above and below. For comparison to recent experimental data see Ref. 2 (Fig. 5 therein). Effects at higher bias voltage arise from the finite bandwidth used here and are typically not seen in experiment.

Figure 7

Dependence of the steady-state current on gate voltage $V_G$. The largest steady-state current is obtained in a parameter regime which crosses from the Kondo plateau to the doubly occupied or empty quantum dot. Data as obtained by nVCA${}_T$ ($L=3$) are presented for an interaction strength of $U/\Delta =12$.