Thermoelectric transport through strongly correlated quantum dots

The thermoelectric properties of strongly correlated quantum dots, described by a single-level Anderson model coupled to conduction-electron leads, is investigated using Wilson’s numerical renormalization-group method. We calculate the electronic contribution, $K_{\text{e}}$, to the thermal conductance, the thermopower, $S$, and the electrical conductance, $G$, of a quantum dot as a function of both temperature, $T$, and gate voltage, $v_g$, for strong, intermediate, and weak Coulomb correlations, $U$, on the dot. For strong correlations and in the Kondo regime, we find that the thermopower exhibits two sign changes, at temperatures $T_1\left(v_g\right)$ and $T_2\left(v_g\right)$ with $T_1<T_2$. We find that $T_1>T_p\left(v_g\right)\approx T_K\left(v_g\right)$, where $T_p\left(v_g\right)$ is the position of the Kondo-induced peak in the thermopower, $T_K\left(v_g\right)$ is the Kondo scale, and $T_2=O\left(\Gamma \right)$, where $\Gamma$ is the level width. The loci of $T_1\left(v_g\right)$ and $T_2\left(v_g\right)$ merge at a critical gate voltage, $v_g=v_g^c\left(U/\Gamma \right)$ beyond which no sign change occurs at finite gate voltage (measured relative to midvalley). We determine $v_g^c$ for different $U/\Gamma$ finding that $v_g^c$ coincides, in each case, with entry into the mixed-valence regime. No sign change is found outside the Kondo regime, or, for weak correlations $U/\Gamma \lesssim 1$, making such a sign change in $S\left(T\right)$ a particularly sensitive signature of strong correlations and Kondo physics. The relevance of this to recent thermopower measurements of Kondo correlated quantum dots is discussed. The results for quantum dots are compared also to the relevant transport coefficients of dilute magnetic impurities in nonmagnetic metals: the electronic contribution, ${\kappa }_{\text{e}}$, to the thermal conductivity, the thermopower, $S$, and the impurity contribution to the electrical resistivity, $\rho$. In the mixed-valence and empty-orbital regimes, we find, as a function of temperature, two peaks in $K_{\text{e}}$ as compared to a single peak in ${\kappa }_{\text{e}}$, and similarly, $G\left(T\right)$ exhibits a finite-temperature peak on entering the mixed-valence regime whereas such a pronounced peak is absent in $\rho \left(T\right)$ even far into the empty-orbital regime. We compare and contrast the figure of merit, power factor, and the extent of violation of the Wiedemann-Franz law in quantum dots and dilute magnetic impurities. The extent of temperature scaling in the thermopower and thermal conductance of quantum dots in the Kondo regime is discussed.

Figure 1

(Color online) Temperature dependence of the occupancy for different values of the gate voltages $v_g>0$ in the Kondo (solid lines), mixed-valence (dashed lines), and empty-orbital (dotted lines) regimes. We define these regimes by $\left|n_d\left(T=0\right)-1\right|\lesssim 0.25$, $\left|n_d\left(T=0\right)-0.50\right|\lesssim 0.25$, and $\left|n_d\left(T=0\right)\right|\lesssim 0.25$, respectively.

Figure 2

(Color online) Dependence on gate voltage, $v_g=\left({\epsilon }_d+U/2\right)/\Gamma$, of, (a), the occupancy of the quantum dot, $n_d$, and, (b), the double occupancy, $D=⟨n_{d\uparrow }{n}_{d\downarrow }⟩$, at several temperatures.

Figure 3

(Color online) (a)–(c) The normalized electrical conductance, $G/G_0$, where $G_0=2e^2/h$, (d)–(f), the thermopower, $S$, in units of $k_B/e$, and, (g)–(i), the normalized electronic contribution to the thermal conductance, $K_{\text{e}}/G_0$, multiplied by a factor ${10}^2$ for clarity of presentation, as a function of $T/\Gamma$, in the strongly correlated regime $U/\Gamma =8$ for a range of gate voltages, $v_g=\left({\epsilon }_d+U/2\right)/\Gamma$, in the Kondo (first column), mixed-valence (second column), and empty-orbital (third column) regimes. The range of $v_g$ is indicated in the top panels for each regime and the increment used was 0.25. Arrows indicate the evolution of the transport quantities with increasing $v_g$. The inset of (d) compares the Kondo scale $T_K$ in the Kondo regime with the peak position of the low-temperature peak, $T_p$, in $S$ below the first sign change at $T_1$ as a function of $v_g>0$. In the Kondo regime $T_K\approx T_p$ and for gate voltages approaching the mixed-valence regime, the two scales deviate, as expected. The inset for $K_{\text{e}}$ in (g) and (h) shows the crossing point at $T/\Gamma \approx 0.6$ in more detail (vertical dotted line) and the evolution of the two-peaked structure for gate voltages approaching the mixed-valence regime.

Figure 4

(Color online) Dependence of the temperatures $T_1$ (filled symbols, solid lines) and $T_2$ (open symbols, dashed lines) at which $S\left(T\right)$ changes sign as a function of $v_g\ge 0.25$ for $U/\Gamma =3,6,8$ (left panel: quantum dot, right panel: magnetic impurity). The critical gate voltage, $v_g^c$, beyond which no sign change occurs in $S\left(T\right)$ at finite $v_g$ is indicated in the legend for each case.

Figure 5

(Color online) Comparison of transport properties for quantum dots (solid lines) and magnetic impurities (dashed lines) in the strongly correlated regime $U/\Gamma =6$. (a)–(c): normalized electrical conductance $G\left(T\right)/G_0$ (quantum dot) and normalized resistivities $\rho /{\rho }_0$ (impurity), where ${\rho }_0$ is defined in Eq. (16). (d)–(f): thermopower $S\left(T\right)$ for quantum dots and impurities [inset of (d) shows the low-temperature Kondo peak in the thermopower in more detail]. (g)–(i): electronic contribution to the thermal conductance $c_1{K}_{\text{e}}/G_0$ (quantum dot) and thermal conductivity $c_2{\kappa }_{\text{e}}{\rho }_0$ (impurity), rescaled by $G_0$ and ${\rho }_0$, respectively, so that the same unit $\left(K\right)$ applies to both cases. The numerical factors $c_1={10}^2$ and $c_2=5\times {10}^{-4}$ are included for clarity of presentation. The inset in (g) for $K_{\text{e}}\left(T\right)/G_0$ shows the crossing point in $K_{\text{e}}\left(T\right)/G_0$ at $T/\Gamma \approx 0.6$ in more detail and the evolution of the second peak in the thermal conductance of the quantum dot as the mixed-valence regime is approached (with inclusion of four additional gate voltages $v_g=1.50,\dots ,2.25$ in the Kondo regime). The range of gate voltages is $v_g=0.25,\dots ,1.25$ (Kondo), $v_g=2.25,\dots ,3.75$ (mixed valence), and $v_g=4.0,\dots ,5.75$ (empty orbital).

Figure 6

(Color online) Temperature dependence of, (a)–(c), the “figure of merit,” $Z{T}_0$, (d)–(f), the “power factor,” $P{F}_0$, and, (g)–(i), the Lorenz number ratio $L\left(T\right)/L_0$ for a quantum dot (solid lines, positive $y$ axis) and for magnetic impurities (dashed lines, shown on the negative $y$ axis for clarity). Results are for $U/\Gamma =8$ and the same range of dimensionless gate voltages as in Fig. 3. For quantum dots we define $Z{T}_0=G{S}^{2}T/K_{\text{e}}$, $P{F}_0=S^{2}G/G_0$, and $L\left(T\right)/L_0=K_{\text{e}}/GT$ with $L_0={\pi }^2{k}_B^2/3e^2$. The corresponding quantities for magnetic impurities are defined by $Z{T}_0=\sigma S^{2}T/{\kappa }_{\text{e}}$, $P{F}_0=S^2\sigma /{\sigma }_0$, and $L\left(T\right)/L_0=\left({\kappa }_{\text{e}}/\sigma T\right)$ where $\sigma =1/\rho$ and ${\sigma }_0=1/{\rho }_0$. Arrows indicate the evolution of the transport quantities with increasing $v_g>0$. Insets in (a) and (d) for the Kondo regime, show the low-temperature peak in $Z{T}_0$ and $P{F}_0$ in the vicinity of $T_K\left(v_g\right)\gtrsim 2.6\times {10}^{-3}\Gamma$ (Table ). The inset in (g) for the Lorenz number shows the deviations from the Wiedemann-Franz law in the region around $T_K\left(v_g\right)$. In this inset, $L\left(T\right)/L_0$ is shown on the positive $y$ axis for both impurity and quantum-dot cases.

Figure 7

(Color online) (a) $S\left(T\right)/T$ for a quantum dot, scaled by its limiting low-temperature absolute value $\pi \gamma \text{cot}\left(\pi n_d/2\right)/\left|e\right|$ versus $\gamma T\sim T/T_K$ for $U/\Gamma =8$ and a range of gate voltages in the Kondo regime. Here, $\gamma \sim 1/T_K$ (Ref. 59). (b) The thermal conductance $K_{\text{e}}\left(T\right)$, scaled by $\alpha T$, versus $T/T_K^{\theta }$, for parameters as in (a), with $\alpha$ defined in Eq. (24) and $T_K^{\theta }$ a Kondo scale defined in Eq. (25). The electrical conductance $G\left(T\right)/G\left(0\right)$ versus $T/T_K^{\theta }$ at $v_g=0.25$ is also shown. The Kondo scale defined by $G\left(T=T_K\right)=G\left(0\right)/2$ is related to $T_K^{\theta }$ by $T_K\approx 1.9T_K^{\theta }$.

Figure 8

(Color online) Gate-voltage dependence of electrical and thermal transport properties of quantum dots [left panels (a)–(c)] and magnetic impurities [right panels (d)–(f)] for $U/\Gamma =8$ and at several representative temperatures $T/\Gamma$ (labeling the individual curves in the figures). Vertical dashed lines in the top panels delineate the Kondo regime from mixed-valence and empty-orbital regimes. (a): The normalized conductance $G\left(T\right)/G_0$, (d): the normalized resistivity $\rho \left(T\right)/{\rho }_0$ for the impurity case, and, (b), the rescaled thermal conductance $K_{\text{e}}\left(T\right)/G_0$, [(e): the rescaled thermal conductivity ${\kappa }_{\text{e}}\left(T\right){\rho }_0$ for the impurity case]. The bottom panel, (c), shows the dimensionless ratio $\left(K_{\text{e}}/T\right)/\left(L_0{G}_0\right)$ for the quantum dot [panel (f) shows the analogous dimensionless ratio for magnetic impurities ${\left({\kappa }_{\text{e}}/T\right)}^{-1}/\left({\rho }_0/L_0\right)$]. Deviations from $G\left(T\right)/G_0$ in (a) reflects deviations from the Wiedemann-Franz law [similarly, deviations of ${\left({\kappa }_{\text{e}}/T\right)}^{-1}/\left({\rho }_0/L_0\right)$ in (f) from $\rho \left(T\right)/{\rho }_0$ in (d) indicate deviations from the Wiedemann-Franz law for the impurity case]. For clarity not all temperatures are shown in (b) and (e).

Figure 9

(Color online) Gate-voltage dependence of the thermopower for the strong correlation case $U/\Gamma =8$ at several characteristic temperatures $T/\Gamma$ for, (a), the quantum dot case, and, (b), the impurity case. The vertical dashed lines delineate the Kondo regime (around $v_g=0$) from the surrounding mixed-valence and empty-orbital regimes.

Figure 10

(Color online) Gate-voltage dependence of the thermopower for $U/\Gamma =8$ at temperatures $T/\Gamma$ corresponding to the experimental ones $T_L$ from Ref. 20 using ${\Gamma }_{\text{exp}}=0.35\text{meV}$. Since the experimental gate voltage $V_E\sim -v_g$ we plotted $S$ versus $-v_g$ to facilitate comparing with the experimental data.

Figure 11

(Color online) Temperature dependence of, (a)–(c), the normalized electrical conductance $G/G_0$, (d)–(f), the thermopower, $S$, and, (g)–(i), the normalized thermal conductance, $K_{\text{e}}/G_0$, multiplied by a factor ${10}^2$ for clarity of presentation, as a function of $T/\Gamma$, in the moderately correlated regime $U/\Gamma =3$ and a range of dimensionless gate voltages, $v_g=\left({\epsilon }_d+U/2\right)/\Gamma >0$, in the Kondo (first column), mixed-valence (second column), and empty-orbital (third column) regimes. The range of $v_g$ is indicated in the top panels for each regime and the increment used was 0.25. Arrows indicate the evolution of the transport quantities with increasing $v_g$.

Figure 12

(Color online) Temperature dependence of, (a), the normalized electrical conductance, $G/G_0$, (b), the thermopower, $S$, and, (c), the normalized thermal conductance, $K_{\text{e}}/G_0$, multiplied by a factor ${10}^2$ for clarity of presentation, as a function of $T/\Gamma$, in the weakly correlated regime $U/\Gamma =1$ and a range of dimensionless gate voltages, $v_g=\left({\epsilon }_d+U/2\right)/\Gamma$. The range of $v_g$ is indicated in the top panel and the increment used was 0.25. Arrows indicate the evolution of the transport quantities with increasing $v_g$.