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arXiv:1207.5641 · Physical Review B

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

Martin Nuss, Christoph Heil, Martin Ganahl, Michael Knap, Hans Gerd Evertz, Enrico Arrigoni, Wolfgang von der Linden

Published Dec 19, 2012• 20 citations
arXivHTMLPDF

Abstract

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)].

DOI
10.1103/physrevb.86.245119

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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)].
We study non-equilibrium properties of a strongly correlated quantum dot system, represented by the single impurity Andersonmodel, by means of the recently introduced non-equilibrium variational cluster approach [1]. We focus on the steady-state current density and non-equilibrium density of states in a parameter region of high interaction strength in the particle-hole symmetric case, which lies in the center of the Kondo regime. The bias dependent current density shows a gradually developing two peak structure with increasing interaction strength. It agrees with the linear response result in the low bias region and vanishes as expected at high bias due to a vanishing band overlap. The non-equilibrium density of states exhibits a linear splitting of the Kondo resonance with applied bias voltage. We compare results obtained via non-equilibrium variational cluster approach with non-equilibrium cluster perturbation theory.
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We investigate the dynamics of a strongly correlated quantum dot system in the mixed valence regime based on the hierarchical equations of motion (HEOM) approach. The transient and steady state transport properties after a quantum quench from equilibration by rapidly applying a bias voltage in a range of temperature below and above the Kondo temperature are described. We find that the time-dependent current exhibits a linear response behavior for weak bias voltage and outside of the linear response regime for larger bias voltage due to the transition of the voltage dependent quantum dot occupancies. The influence of the temperature, finite strongly correlated electron-electron interaction and energy level of the quantum dot on the nonlinear behavior and steady state values of current indicating the Kondo physics are explored in detail.
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Numerical renormalization group (NRG) is formulated for nonequilibrium steady-state by converting finite-lattice many-body eigenstates into scattering states. Extension of the full-density-matrix NRG for a biased Anderson impurity model, simplified by formulating with the original orbital basis as the Hamiltonian, enables detailed studies of the sub-Kondo spectral evolution in the zero-temperature limit, confirming the double-resonance structure at bias of the Kondo energy scale TKT_KTK​. The distribution shows distinct multi-scale spectral features at energy ωωω below the Kondo scale (ω≲TKω\lesssim T_Kω≲TK​) and near the bias (ω≳Vω\gtrsim Vω≳V), leading to the nonequilibrium temperature TlocT_{\rm loc}Tloc​ local to the Kondo dot scaling as kBTloc≈Vk_BT_{\rm loc}\approx VkB​Tloc​≈V for V≫TKV\gg T_KV≫TK​. The current-voltage relation in the low-temperature limit (T≪TKT\ll T_KT≪TK​) deviates from the unitary limit as the bias exceeds the Kondo scale (V/2≳TKV/2\gtrsim T_KV/2≳TK​) and reaches the current saturation regime.
We examine the recently proposed imaginary-time formulation for strongly correlated steady-state nonequilibrium for its range of validity and discuss significant improvements in the analytic continuation of the Matsubara voltage as well as the fermionic Matsubara frequency. The discretization error in the conventional Hirsch-Fye algorithm has been compensated in the Fourier transformation with reliable small frequency behavior of self-energy. Here we give detailed discussions for generalized spectral representation ansatz by including high-order vertex corrections and its numerical analytic continuation procedures. The differential conductance calculations agree accurately with available data from other nonequilibrium transport theories. It is verified that, at finite source-drain voltage, the Kondo resonance is destroyed at bias comparable to the Kondo temperature. Calculated scaling coefficients of differential conductance for zero-bias anomaly fall within the range of experimental estimates.