Thermopower of few-electron quantum dots with Kondo correlations

The thermopower of few-electron quantum dots is crucially influenced by on-dot electron-electron interactions, particularly in the presence of Kondo correlations. In this paper, we present a comprehensive picture which elucidates the underlying relations between the thermopower and the spectral density function of two-level quantum dots. The effects of various electronic states, including the Kondo states originating from both spin and orbital degrees of freedom, are clearly unraveled. Such a physical picture is affirmed by accurate numerical data obtained with a hierarchical equations of motion approach. Our findings and understandings provide an effective and viable way to control the thermoelectric properties of strongly correlated quantum dot systems.

Figure 1

Thermopower of a single-level QD calculated with two schemes: (i) $S_{\mathrm{def}}=\left(\frac{V}{\Delta T}\right){|}_{I=0}$ is computed by searching for the voltage $V$ that cancels the thermal current induced by a fixed $\Delta T={10}^{-2}\Delta$. (ii) $S_{\mathrm{linear}}=-L_T/G$ is obtained by computing the transport coefficients $L_T$ and $G$. The HEOM truncation level is $L=4$. The parameters of the QD are (in units of $\Delta$) $U=15,W=30$, and $T=0.2$.

Figure 2

(a) $G$ and (b) $S$ vs ${\epsilon }_1$ for a single-level QD of $U=15\Delta$ and $W=30\Delta$. (c) $G$ and (b) $S$ vs ${\epsilon }_1$ for a two-level QD with the parameters of (in units of $\Delta$) $\delta \epsilon ={\epsilon }_2-{\epsilon }_1=1,U=20$, and $W=80$. The equilibrium chemical potentials of leads $\mu$ are set as zero energy.

Figure 3

$S_{\mathrm{linear}}$ and $S_{\mathrm{Landauer}}$ of a two-level QD calculated by the HEOM approach at the truncation level of (a) $L=2$, (b) $L=3$, and (c) $L=4$. The parameters adopted are (in units of $\Delta$) $\delta \epsilon =1,U=20,W=80$, and $T=0.5$.

Figure 4

Spectral function of a single-level QD calculated with the HEOM approach at different truncation levels. The inset magnifies the Kondo spectral peak around $\omega =0$. The parameters of the QD are (in units of $\Delta$) ${\epsilon }_1=-5,U=15,W=30$, and $T=0.2$.

Figure 5

(a) $A_1\left(\omega \right)$ and (b) $A_2\left(\omega \right)$ of a two-level QD calculated at different truncation levels $L$. The parameters adopted are (in units of $\Delta$) ${\epsilon }_1=-5.5,\delta \epsilon =1,U=20,W=80$, and $T=0.5$.

Figure 6

Calculated $A_i\left(\omega \right) \left(i=1,2\right)$ of a two-level QD in the (a) $N=1$, (b) $N=2$, and (c) $N=3$ regimes. The parameters adopted are (in units of $\Delta$) $\delta \epsilon =1,U=20$, and $T=0.5$.

Figure 7

The line shapes of $-f^{\prime }\left(\omega \right)$ at various $T$. The boundaries of the thermal activation window are indicated by dotted vertical lines.

Figure 8

Schematic diagram of the dot spectral function of a two-level QD in the $N=1$ region. The spectral peaks originating from the various electronic states are depicted. The left and right columns correspond to temperatures $T_{\mathrm{L}}$ and $T_{\mathrm{H}}$, respectively. Each row represents a scenario corresponding to a certain range of parameters $(\delta \epsilon ,U)$, as illustrated in the inset in (b). The thermal activation window and the resultant sign of $S$ are also shown in each panel.

Figure 9

Calculated $A\left(\omega \right)$ of a two-level QD in the $N=1$ regime $\left({\epsilon }_1=-5.5\Delta \right)$ with $(\delta \epsilon ,U)=(\Delta ,20\Delta )$ at (a) $T_{\mathrm{L}}=0.5\Delta$ and (b) $T_{\mathrm{H}}=2.5\Delta$. The various spectral peaks are marked by arrows, and the thermal activation window is indicated by dashed lines. The inset in (a) magnifies $A\left(\omega \right)$ around $\mu =0$ at $T_{\mathrm{L}}$, while the inset in (b) depicts the calculated $S$ vs ${\epsilon }_1$.

Figure 10

Calculated $A\left(\omega \right)$ of a two-level QD in the $N=1$ regime $\left({\epsilon }_1=-6\Delta \right)$ with $(\delta \epsilon ,U)=(2\Delta ,20\Delta )$ at (a) $T_{\mathrm{L}}=0.5\Delta$ and (b) $T_{\mathrm{H}}=2.5\Delta$. The inset in (a) magnifies $A\left(\omega \right)$ around $\mu =0$ at $T_{\mathrm{L}}$, while the inset in (b) depicts the calculated $S$ vs ${\epsilon }_1$.

Figure 11

Calculated $A\left(\omega \right)$ of a two-level QD in the $N=1$ regime $\left({\epsilon }_1=-6\Delta \right)$ with $(\delta \epsilon ,U)=(\Delta ,15\Delta )$ at (a) $T_{\mathrm{L}}=0.5\Delta$ and (b) $T_{\mathrm{H}}=2.5\Delta$. The inset in (a) magnifies $A\left(\omega \right)$ around $\mu =0$ at $T_{\mathrm{L}}$, while the inset in (b) depicts the calculated $S$ vs ${\epsilon }_1$.

Figure 12

Calculated $A\left(\omega \right)$ of a two-level QD in the $N=1$ regime $\left({\epsilon }_1=-6\Delta \right)$ with $(\delta \epsilon ,U)=(2\Delta ,15\Delta )$ at (a) $T_{\mathrm{L}}=0.5\Delta$ and (b) $T_{\mathrm{H}}=2.5\Delta$. The inset in (a) magnifies $A\left(\omega \right)$ around $\mu =0$ at $T_{\mathrm{L}}$, while the inset in (b) depicts the calculated $S$ vs ${\epsilon }_1$.

Figure 13

Calculated $A_s\left(\mu \right)$ as a function of $T/T_{\mathrm{K}}$. The QD parameters are (in units of $\Delta$) $U=-2{\epsilon }_1=1.6$ and $W=20$.

Figure 14

${(G/G_0)}^{-1/s}$ vs ${(T/T_{\mathrm{K}})}^2$. The parameters adopted are ${\epsilon }_1=-U/2$ and $W=20\Delta$. The scaling constant is $s=0.217$.

Figure 15

Comparison between HEOM (scattered symbols) and NRG (lines) calculated zero-bias conductance of a two-level QD. The NRG data are extracted from Fig. 1 of Ref. [73], which uses a constant hybridization function; the HEOM approach adopts a Lorentzian form of a hybridization function with a finite bandwidth $W=100\Delta$. The dimensionless gate voltage $N_g\equiv \frac{1}{2}-{\epsilon }_1/U$. The QD parameters are $\delta \epsilon =0$ and $U=20\Delta$.