Nonlinear Fermi liquid transport through a quantum dot in asymmetric tunnel junctions

We study the nonlinear conductance through a quantum dot, specifically its dependence on the asymmetries in the tunnel couplings and bias voltages $V$, at low energies. Extending the microscopic Fermi liquid theory for the Anderson impurity model, we obtain an exact formula for the steady current $I$ up to terms of order $V^3$ in the presence of these asymmetries. The coefficients for the nonlinear terms are described in terms of a set of the Fermi liquid parameters: the phase shift, static susceptibilities, and three-body correlation functions of electrons in the quantum dots, defined with respect to the equilibrium ground state. We calculate these correlation functions, using the numerical renormalization group approach (NRG), over a wide range of impurity-electron filling that can be controlled by a gate voltage in real systems. The NRG results show that the order $V^2$ nonlinear current is enhanced significantly in the valence fluctuation regime. It is caused by the order $V$ energy shift of the impurity level, induced in the presence of the tunneling or bias asymmetry. Furthermore, in the valence fluctuation regime, we also find that the order $V^3$ nonlinear current exhibits a shoulder structure, for which the three-body correlations that evolve for large asymmetries play an essential role.

Figure 1

Schematic diagram of a quantum dot coupled to two leads via the tunnel couplings ${\mathrm{\Gamma }}_L$ and ${\mathrm{\Gamma }}_R$. The bias voltage $eV={\mu }_L-{\mu }_R$ is applied in a way such that ${\mu }_L=E_F+{\alpha }_{L}eV$ and ${\mu }_R=E_F-{\alpha }_{R}eV$, with ${\alpha }_L+{\alpha }_R=1$. The chemical potentials of the left and right leads are measured from the Fermi energy at equilibrium $E_F=0$. The tunnel coupling is smaller for the barrier with thicker width.

Figure 2

NRG results for the Fermi liquid parameters are plotted vs ${\epsilon }_d/U$, for $U/\left(\pi \mathrm{\Delta }\right)=1,2,3,4,5$. (a) $\langle n_{d\sigma }\rangle$, (b) ${sin}^2\delta$, (c) $R-1=-{\chi }_{\uparrow \downarrow }/{\chi }_{\uparrow \uparrow }$, and (d) $T_K/T^{*}$ for which $T^{*}\equiv 1/\left(4{\chi }_{\uparrow \uparrow }\right)$ depends on ${\epsilon }_d/U$ and $T_K$ is defined as the value of $T^{*}$ at half filling ${\epsilon }_d=-U/2$.

Figure 3

NRG results for the three-body correlation functions, ${\mathrm{\Theta }}_{\mathrm{I}}\equiv \frac{sin2\delta }{2\pi {\chi }_{\uparrow \uparrow }^2}{\chi }_{\uparrow \uparrow \uparrow }^{\left[3\right]}$ and ${\mathrm{\Theta }}_{\mathrm{II}}\equiv \frac{sin2\delta }{2\pi {\chi }_{\uparrow \uparrow }^2}{\chi }_{\uparrow \downarrow \downarrow }^{\left[3\right]}$, are plotted vs ${\epsilon }_d/U$ in (a) and (b), respectively, for $U/\left(\pi \mathrm{\Delta }\right)=1,2,3,4,5$.

Figure 4

NRG results for $sin2\delta$ are plotted vs ${\epsilon }_d/U$, for $U/\left(\pi \mathrm{\Delta }\right)=1,2,3,4,5$.

Figure 5

Effective impurity level ${\tilde{\epsilon }}_d^{\left(1\right)}$ vs $eV$ for a junction with tunnel asymmetries ${\mathrm{\Gamma }}_L/{\mathrm{\Gamma }}_R=1/9$, 1/3, 2/3, 1, 3/2, 3, 9 for chemical potentials ${\mu }_L=-{\mu }_R=eV/2$ (${\alpha }_{\mathrm{dif}}^{}=0$). The other parameters chosen to be $U/\left(\pi \mathrm{\Delta }\right)=4.0$ and ${\epsilon }_d=-U$, at which the FL parameters take values $z=0.75, R-1=0.52$, and ${\tilde{\epsilon }}_d^{\left(1\right)}=-0.91\mathrm{\Delta }$ at $eV=0$.

Figure 6

Behavior of $A^{\left(1\right)}\left(\omega \right)$ defined in Eq. (41) near the Fermi level for a junction with tunnel asymmetries of ${\mathrm{\Gamma }}_L/{\mathrm{\Gamma }}_R=1/9$ (dashed line), 1 (solid line), 9 (dotted line). Bias voltage is applied such that ${\mu }_L=-{\mu }_R=eV/2$ with $eV=0.7\mathrm{\Delta }$, choosing $U/\left(\pi \mathrm{\Delta }\right)=4$ and ${\epsilon }_d=-U$. The area between the curve for $A^{\left(1\right)}\left(\omega \right)$ and horizontal axis in the shaded region, which represents the bias window, determines order ${\left(eV\right)}^2$ nonlinear current with the formula Eq. (20).

Figure 7

(a) NRG results of $C_V^{\left(2\right)}$ vs ${\epsilon }_d/U$ for different tunnel asymmetries ${\mathrm{\Gamma }}_L/{\mathrm{\Gamma }}_R=1/9(\circ ), 1/3(\times ), 2/3(\diamond ), 1\left(\blacksquare \right), 3/2\left(\nabla \right), 3(\square ), 9\left(▴\right)$ for $U/\left(\pi \mathrm{\Delta }\right)=4.0$, keeping $\mathrm{\Delta }={\mathrm{\Gamma }}_L+{\mathrm{\Gamma }}_R$ unchanged. (b) $C_V^{\left(2\right)}$ for several values of interactions $U/\left(\pi \mathrm{\Delta }\right)=1$, 2, 3, 4, 5, for fixed tunnel couplings ${\mathrm{\Gamma }}_L=9{\mathrm{\Gamma }}_R$. For both (a) and (b), bias voltages are chosen to be symmetric ${\alpha }_L={\alpha }_R=1/2$, i.e., ${\alpha }_{\mathrm{dif}}^{}=0$.

Figure 8

NRG results of $C_V^{\left(2\right)}$ for symmetric tunnel couplings with ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R$ are plotted for different degrees of bias asymmetries ${\alpha }_{\mathrm{dif}}=-0.8(\circ ), -0.5(\times ), -0.2(\diamond ), 0.0\left(\blacksquare \right), 0.2\left(\nabla \right), 0.5(\square ), 0.8\left(▴\right)$. Coulomb interactions are chosen to be (a) $U=4\pi \mathrm{\Delta }$ and (b) $U=\pi \mathrm{\Delta }$.

Figure 9

NRG results of $C_V^{\left(2\right)}$ for junctions with large imbalanced couplings ${\mathrm{\Gamma }}_L=9{\mathrm{\Gamma }}_R$ are plotted for different degrees of bias asymmetries ${\alpha }_{\mathrm{dif}}=-0.8(\circ ), -0.5(\times ), -0.2(\diamond ), 0.0\left(\blacksquare \right), 0.2\left(\nabla \right), 0.5(\square ), 0.8\left(▴\right)$. Coulomb interactions are chosen to be (a) $U=4\pi \mathrm{\Delta }$ and (b) $U=\pi \mathrm{\Delta }$.

Figure 10

NRG results of $C_V^{\left(2\right)}$ for a maximized bias asymmetry ${\alpha }_L=1$ and ${\alpha }_R=0$ (i.e., ${\mu }_L=eV$ and ${\mu }_R=0$) for different degrees of tunnel asymmetries ${\mathrm{\Gamma }}_L/{\mathrm{\Gamma }}_R=1/9(\circ ), 1/3(\times ), 2/3(\diamond ), 1\left(\blacksquare \right), 3/2\left(\nabla \right), 3(\square ), 9\left(▴\right)$, for a fixed Coulomb interaction $U/\left(\pi \mathrm{\Delta }\right)=4$. Coulomb interactions are chosen to be (a) $U=4\pi \mathrm{\Delta }$ and (b) $U=\pi \mathrm{\Delta }$.

Figure 11

NRG results of the coefficient for order ${\left(eV\right)}^3$ nonlinear current $C_V^{\left(3\right)}=({\pi }^2/64)(W_V+{\mathrm{\Theta }}_V)$ are plotted vs ${\epsilon }_d/U$ for a symmetric bias ${\alpha }_L={\alpha }_R=1/2$, choosing $U/\left(\pi \mathrm{\Delta }\right)=4$. In this case ${\alpha }_{\mathrm{dif}}^{}=0$; both the two-body $W_V$ and three-body ${\mathrm{\Theta }}_V$ contributions become independent of tunnel asymmetries $({\mathrm{\Gamma }}_L-{\mathrm{\Gamma }}_R)/\mathrm{\Delta }$ from the definition summarized in Table 1.

Figure 12

$C_V^{\left(3\right)}$ for ${\alpha }_{\mathrm{dif}}^{}=-0.8(\circ ), -0.5(\times ), -0.2(\diamond ), 0.0\left(\blacksquare \right), 0.2(▿), 0.5(\square ), 0.8\left(▴\right)$ are plotted for three different tunnel couplings (a) ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R$, (b) ${\mathrm{\Gamma }}_L/{\mathrm{\Gamma }}_R=9$, and (c) ${\mathrm{\Gamma }}_L/{\mathrm{\Gamma }}_R=1/9$, choosing the interaction to be $U/\left(\pi \mathrm{\Delta }\right)=4$.

Figure 13

$C_V^{\left(3\right)}$ for a maximized bias asymmetry ${\alpha }_{\mathrm{dif}}^{}=1$ (${\mu }_L=eV$ and ${\mu }_R=0$). (a) Different degrees of tunnel asymmetries are examined, taking ${\mathrm{\Gamma }}_L/{\mathrm{\Gamma }}_R=1/9(\circ ), 1/3(\times ), 2/3(\diamond ), 1\left(\blacksquare \right), 3/2\left(\nabla \right), 3(\square ), 9\left(▴\right)$. (b) and (c): $W_V$ and ${\mathrm{\Theta }}_V$ are plotted together with $C_V^{\left(3\right)}$ for (b) ${\mathrm{\Gamma }}_L=9{\mathrm{\Gamma }}_R$ and (c) ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R/9$. The Coulomb interaction is chosen to be $U/\left(\pi \mathrm{\Delta }\right)=4$.

Figure 14

$C_V^{\left(3\right)}$ for ${\alpha }_{\mathrm{dif}}^{}=1$ and ${\mathrm{\Gamma }}_L/{\mathrm{\Gamma }}_R=9$ is plotted for different values of Coulomb interactions $U/\left(\pi \mathrm{\Delta }\right)=1$, 2, 3, 4, 5. Panel (b) shows an enlarged view of the shaded region in (a), describing an evolution of the shoulder structure in the valence fluctuation regime.