Role of bias and tunneling asymmetries in nonlinear Fermi-liquid transport through an $\mathrm{SU}\left(N\right)$ quantum dot

We study how bias and tunneling asymmetries affect nonlinear current through a quantum dot with $N$ discrete levels in the Fermi liquid regime, using an exact low-energy expansion of the current derived up to terms of order $V^3$ with respect to the bias voltage. The expansion coefficients are described in terms of the phase shift, the linear susceptibilities, and the three-body correlation functions, defined with respect to the equilibrium ground state of the Anderson impurity model. In particular, the three-body correlations play an essential role in the order $V^3$ term, and their coupling to the nonlinear current depends crucially on the bias and tunnel asymmetries. The number of independent components of the three-body correlation functions increases with $N$ the internal degrees of the quantum dots, and it gives a variety in the low-energy transport. We calculate the correlation functions over a wide range of electron fillings of the Anderson impurity model with the $\mathrm{SU}\left(N\right)$ internal symmetry, using the numerical renormalization group. We find that the order $V^3$ nonlinear current through the $\mathrm{SU}\left(N\right)$ Kondo state, which occurs at electron fillings of 1 and $N-1$ for strong Coulomb interactions, significantly varies with the three-body contributions as tunnel asymmetries increase. Furthermore, in the valence fluctuation regime toward the empty or fully occupied impurity state, a sharp peak emerges in the coefficient of $V^3$ current in the case at which bias and tunneling asymmetries cooperatively enhance the charge transfer from one of the electrodes.

Figure 1

Fermi-liquid parameters of an SU(4) dot are plotted as a function of ${\xi }_d^{}/U$ for $U/\left(\pi \mathrm{\Delta }\right)=1/3, 2/3, 5/3, 10/3$, 5.0: (a) $\langle n_d\rangle$, (b) ${sin}^2\delta$, (c) rescaled Wilson ratio $\tilde{K}\equiv (N-1)(R-1)$, (d) renormalized factor $z$, (e) $T_K/T^{*}$, and (f) $sin2\delta$. Here $T_K$ is defined as the value of $T^{*}\equiv 1/\left(4{\chi }_{\sigma \sigma }^{}\right)$ at half filling ${\xi }_d^{}=0$.

Figure 2

Three-body correlation functions of an SU(4) dot plotted vs ${\xi }_d^{}/U$ for $U/\pi \mathrm{\Delta }=1/3, 2/3, 5/3, 10/3$, 5.0: (a) ${\mathrm{\Theta }}_{\mathrm{I}}$, (b) ${\tilde{\mathrm{\Theta }}}_{\mathrm{II}}$, and (c) ${\tilde{\mathrm{\Theta }}}_{\mathrm{III}}$. (d) Comparison of ${\mathrm{\Theta }}_{\mathrm{I}}, -{\tilde{\mathrm{\Theta }}}_{\mathrm{II}}$, and ${\tilde{\mathrm{\Theta }}}_{\mathrm{III}}$ at a large interaction $U/\left(\pi \mathrm{\Delta }\right)=5.0$. It indicates that ${\mathrm{\Theta }}_{\mathrm{I}}\simeq -{\tilde{\mathrm{\Theta }}}_{\mathrm{II}}\simeq {\tilde{\mathrm{\Theta }}}_{\mathrm{III}}$ over a wide range of $-U\lesssim {\xi }_d^{}\lesssim U$.

Figure 3

Fermi-liquid parameters of an SU(6) dot are plotted as a function of ${\xi }_d^{}/U$ for $U/\left(\pi \mathrm{\Delta }\right)=2/5$, 1.0, 2.0, 5.0. (a) $\langle n_d\rangle$, (b) ${sin}^2\delta$, (c) rescaled Wilson ratio $\tilde{K}\equiv (N-1)(R-1)$, (d) renormalized factor $z$, (e) $T_K/T^{*}$, and (f) $sin2\delta$. Here $T_K$ is defined as the value of $T^{*}\equiv 1/\left(4{\chi }_{\sigma \sigma }^{}\right)$ at half filling ${\xi }_d^{}=0$.

Figure 4

Three-body correlation functions of an SU(6) dot plotted vs ${\xi }_d/U$ for $U/\pi \mathrm{\Delta }=2/5$, 1.0, 2.0, 5.0: (a) ${\mathrm{\Theta }}_{\mathrm{I}}$, (b) ${\tilde{\mathrm{\Theta }}}_{\mathrm{II}}$, and (c) ${\tilde{\mathrm{\Theta }}}_{\mathrm{III}}$. (d) Comparison of ${\mathrm{\Theta }}_{\mathrm{I}}, -{\tilde{\mathrm{\Theta }}}_{\mathrm{II}}$, and ${\tilde{\mathrm{\Theta }}}_{\mathrm{III}}$ at a large interaction $U/\left(\pi \mathrm{\Delta }\right)=5.0$. It indicates that ${\mathrm{\Theta }}_{\mathrm{I}}\simeq -{\tilde{\mathrm{\Theta }}}_{\mathrm{II}}\simeq {\tilde{\mathrm{\Theta }}}_{\mathrm{III}}$ over a wide range of $-(N-1)/2U\lesssim {\xi }_d^{}\lesssim (N-1)/2$ for $N=6$.

Figure 5

Effects of tunnel asymmetry ${\gamma }_{\mathrm{dif}}^{}$ on order ${\left(eV\right)}^2$ nonlinear current: $C_V^{\left(2\right)}/{\gamma }_{\mathrm{dif}}^{}$ given by Eq. (4.3) for symmetrical bias voltages ${\alpha }_{\mathrm{dif}}^{}=0.0$ is plotted vs ${\xi }_d^{}/U$, varying interactions from weak to strong. (a) For SU(4) quantum dots with $U/\left(\pi \mathrm{\Delta }\right)=1/3, 2/3, 5/3, 10/3$, 5.0. (b) For SU(6) quantum dots with $U/\left(\pi \mathrm{\Delta }\right)=2/5$, 1.0, 2.0, 5.0.

Figure 6

Effects of bias-voltage asymmetry ${\alpha }_{\mathrm{dif}}^{}$ on order ${\left(eV\right)}^2$ nonlinear current: $C_V^{\left(2\right)}/{\alpha }_{\mathrm{dif}}^{}$ given by Eq. (4.4) for symmetric junctions with ${\gamma }_{\mathrm{dif}}^{}=0.0$ is plotted vs ${\xi }_d^{}/U$, varying interactions from weak to strong. (a) For SU(4) quantum dots with $U/\left(\pi \mathrm{\Delta }\right)=1/3, 2/3, 5/3, 10/3$, 5.0. (b) For SU(6) quantum dots with $U/\left(\pi \mathrm{\Delta }\right)=2/5$, 1.0, 2.0, 5.0.

Figure 7

$C_V^{\left(2\right)}$ for a junction with large tunnel asymmetry ${\gamma }_{\mathrm{dif}}^{}=0.8$ is plotted vs ${\xi }_d^{}/U$, varying bias-voltage asymmetries, as ${\alpha }_{\mathrm{dif}}=-1.0$ ($•$), $-0.5$ ($\times$), 0.0 ($\blacksquare$), 0.5 ($\square$), and 1.0 ($▵$). (Left panels) For SU(4) quantum dots and (right panels) for SU(6) ones, with (top panels) a strong interaction $U/\left(\pi \mathrm{\Delta }\right)=5.0$ and with (bottom panels) weak interactions $U/\left(\pi \mathrm{\Delta }\right)=1/3$ for $\mathrm{SU}(4$) and $U/\left(\pi \mathrm{\Delta }\right)=2/5$ for $\mathrm{SU}(6$).

Figure 8

$C_V^{\left(2\right)}$ for a large bias-voltage asymmetry ${\alpha }_{\mathrm{dif}}^{}=1.0$ is plotted vs ${\xi }_d^{}$, varying tunnel asymmetries, as ${\gamma }_{\mathrm{dif}}^{}=-0.8$ ($\circ$), $-0.5$ ($\times$), $-0.2$ ($\diamond$), 0.0 ($\blacksquare$), 0.2 ($▿$), 0.5 ($\square$), and 0.8 ($▵$). (Left panels) For SU(4) quantum dots and (right panels) for SU(6) ones, with (top panels) a strong interaction $U/\left(\pi \mathrm{\Delta }\right)=5.0$ and (bottom panels) weak interactions $U/\left(\pi \mathrm{\Delta }\right)=1/3$ for $\mathrm{SU}(4$) and $U/\left(\pi \mathrm{\Delta }\right)=2/5$ for $\mathrm{SU}(6$).

Figure 9

Coefficient $C_V^{\left(3\right)}$ is plotted vs ${\xi }_d/U$ for the case where both tunnel couplings and bias voltages are symmetric, ${\gamma }_{\mathrm{dif}}^{}=0.0$ and ${\alpha }_{\mathrm{dif}}^{}=0.0$. Upper panels show the total value of $C_V^{\left(3\right)}=({\pi }^2/64)(W_V+{\mathrm{\Theta }}_V)$ together with the two-body $W_V$ and three-body ${\mathrm{\Theta }}_V$ components (a) for SU(4) and (b) for SU(6) quantum dots choosing a large interaction $U/\left(\pi \mathrm{\Delta }\right)=5.0$. Bottom panels show $U$ dependence of $C_V^{\left(3\right)}$: (c) $U/\left(\pi \mathrm{\Delta }\right)=1/3, 2/3, 5/3, 10/3$, 5.0 for SU(4) and (d) $U/\left(\pi \mathrm{\Delta }\right)=2/5$, 1.0, 2.0,5.0 for SU(6).

Figure 10

Effects of tunnel asymmetry ${\gamma }_{\mathrm{dif}}^{}$ on order ${\left(eV\right)}^3$ nonlinear current: (top panels) $C_V^{\left(3\right)}$ and the components (upper middle panels) $W_V$ and (lower middle panels) ${\mathrm{\Theta }}_V$ defined in Eqs. (5.6) and (5.7) for symmetrical bias voltages ${\alpha }_{\mathrm{dif}}^{}=0.0$ are plotted vs ${\xi }_d^{}/U$, varying tunnel asymmetries, as for ${\gamma }_{\mathrm{dif}}=0.0$ ($\blacksquare$), 0.2 ($▿$), 0.5 ($\square$), 0.58 ($\times$), 0.8 ($▵$). Interaction strength is chosen to be $U/\left(\pi \mathrm{\Delta }\right)=5.0$ for (a)–(f): (left panels) for SU(4) quantum dots and (right panels) for SU(6) quantum dots. (Bottom panels) $U$ dependence of $C_V^{\left(3\right)}$, calculated for a fixed large tunnel asymmetry ${\gamma }_{\mathrm{dif}}^{}=0.8$, (g) $U/\left(\pi \mathrm{\Delta }\right)=1/3$ ($\blacksquare$), $2/3$ ($\diamond$), $5/3$ ($\circ$), $10/3$ ($▵$), 5.0 ($\times$) for SU(4) and (h) $U/\left(\pi \mathrm{\Delta }\right)=2/5$ ($\blacksquare$), 1.0 ($\diamond$), 2.0 ($\circ$), 5.0 ($▵$) for SU(6).

Figure 11

Behavior of $C_V^{\left(3\right)}$ for two different tunnel couplings, (top panels) ${\gamma }_{\mathrm{dif}}^{}=0.0$ and (bottom panels) ${\gamma }_{\mathrm{dif}}^{}=0.8$, is plotted as a function of ${\xi }_d^{}$, varying bias asymmetries ${\alpha }_{\mathrm{dif}}^{}=-1.0$ ($•$), $-0.8$ ($\circ$), $-0.5$ ($\times$), $-0.2$ ($\diamond$), 0.0 ($\blacksquare$), 0.2 ($▿$), 0.5 ($\square$), 0.8 ($▴$), and 1.0 ($▵$). Interaction strength is fixed at $U/\left(\pi \mathrm{\Delta }\right)=5.0$ for both (left panels) SU(4) and (right panels) SU(6) quantum dots.

Figure 12

Behavior of $C_V^{\left(3\right)}$ at large bias asymmetry ${\alpha }_{\mathrm{dif}}^{}=1$ is plotted vs ${\xi }_d^{}$ for (left panels) SU(4) and (right panels) SU(6) quantum dots. Top panels: Tunnel asymmetry is varied as ${\gamma }_{\mathrm{dif}}^{}=-0.8$ ($\circ$), $-0.5$ ($\times$), $-0.2$ ($\diamond$), 0.0 ($\blacksquare$), 0.2 ($▿$), 0.5 ($\square$), and 0.8 ($▵$). In addition, two-body part $W_V$ ($\square$) and three-body part ${\mathrm{\Theta }}_V$ ($\circ$) are plotted together with $C_V^{\left(3\right)}$ ($▴$) for two large opposite tunnel asymmetries: (upper middle panels) ${\gamma }_{\mathrm{dif}}=0.8$, and (lower middle panels) ${\gamma }_{\mathrm{dif}}=-0.8$. Interaction strength is chosen to be $U/\left(\pi \mathrm{\Delta }\right)=5.0$ in (a)–(f). Bottom panels show the $U$ dependence of $C_V^{\left(3\right)}$ for ${\gamma }_{\mathrm{dif}}=0.8$: (g) $U/\left(\pi \mathrm{\Delta }\right)=1/3, 2/3, 5/3, 10/3$, 5.0 for SU(4) and (h) $U/\left(\pi \mathrm{\Delta }\right)=2/5$, 1.0, 2.0, 5.0 for SU(6).

Figure 13

Correlation functions emerging on the right-hand side of Eqs. (C1) and (C2) are plotted vs ${\xi }_d/U$ for $N=4$. [(a) and (b)] the diagonal component ${\left(\pi \mathrm{\Delta }\right)}^2{\chi }_{\sigma \sigma \sigma }^{\left[3\right]}$. (c) ${\left(\pi \mathrm{\Delta }\right)}^2\frac{\partial {\chi }_{\sigma \sigma }}{\partial {\epsilon }_d}$. (d) ${\left(\pi \mathrm{\Delta }\right)}^2(N-1)\frac{\partial {\chi }_{\sigma {\sigma }^{\prime }}}{\partial {\epsilon }_d}$. Panel (b) represents an enlarged view of (a). In each of these panels, interaction strength is chosen to be $U/\left(\pi \mathrm{\Delta }\right)=1/3, 2/3, 5/3, 10/3$, 5.0.