Local temperatures of strongly-correlated quantum dots out of equilibrium

Probes that measure the local thermal properties of systems out of equilibrium are emerging as new tools in the study of nanoscale systems. One can then measure the temperature of a probe that is weakly coupled to a bias-driven system. By tuning the probe temperature so that the expectation value of some observable of the system is minimally perturbed, one obtains a parameter that measures its degree of local statistical excitation, and hence its local heating. However, one anticipates that different observables may lead to different temperatures and thus different local heating expectations. We propose an experimentally realizable protocol to measure such local temperatures and apply it to bias-driven quantum dots. By means of a highly accurate open quantum system approach, we show theoretically that the measured temperature is quite insensitive both to the choice of observable and to the probe-system coupling. In particular, even with observables that are distinct both physically and in their degree of locality, such as the local magnetic susceptibility of the quantum dot and the global spin-polarized current measured at the leads, the resulting local temperatures are quantitatively similar for quantum dots ranging from noninteracting to Kondo-correlated regimes, and are close to those obtained with the traditional “local equilibrium” definition.

Figure 1

Schematic diagram of the proposed protocol for the measurement of local temperature in a QD connected to two leads ($L$ and $R$). A weakly coupled probe with tunable ${\mu }_p$ and $T_p$ is used. The expectation value of an observable, $O_p$, is monitored while $T_p$ is varied.

Figure 2

(a) Local temperature $T^{*}$ of a noninteracting QD determined by different protocols versus background $T$. The dashed line is a guide to eyes. (b) Variation of the local magnetic susceptibility ${\chi }^m$ and total spin current $I_L^m$ with $T_p$ at a background $T=0.133\Delta$. The parameters adopted are (in units of $\Delta$): ${\epsilon }_d=-2.67, U=0, {\Delta }_L=2{\Delta }_R=0.67, {\mu }_R=-{\mu }_L=0.53$, and $W=40$.

Figure 3

(a) and (b) depict the energy distribution of electric and heat currents flowing into $\alpha$ lead ($\alpha =L,R$), respectively. (c) and (d) plot the energy distribution of electric and heat currents flowing into the probe when the local equilibrium condition $I_p=J_p^{\mathrm{H}}=0$ is satisfied, respectively. The parameters adopted are (in units of $\Delta$): ${\epsilon }_d=-2.67, U=0, {\Delta }_L=2{\Delta }_R=100{\Delta }_p=0.67, {\mu }_R=-{\mu }_L=0.53, W=40$, and $T_L=T_R=0.133$.

Figure 4

(a) Local temperature $T^{*}$ of an interacting QD determined by different protocols versus background $T$. The dashed line is a guide to eyes. The inset shows the Kondo spectral peak of the dot at various $T$. (b) Variation of system observables ${\chi }^m$ and $I_L^m$ with $T_p$, at a background $T=0.1\Delta$ lower than the Kondo temperature $T_K=0.82\Delta$. The parameters adopted are (in units of $\Delta$): ${\epsilon }_d=-1.2, U=2.4, {\Delta }_L={\Delta }_R=0.5, {\mu }_R=-{\mu }_L=0.25$, and $W=5$.

Figure 5

Local temperature $T^{*}$ of a noninteracting QD determined by different protocols versus the system-probe coupling strength ${\Delta }_p$. The background temperature is fixed at $T=0.133\Delta$. The parameters adopted are (in units of $\Delta$) ${\epsilon }_d=-2.67, U=0, {\Delta }_L=2{\Delta }_R=0.67, {\mu }_R=-{\mu }_L=0.53$, and $W=40$.

Figure 6

$T^{*}$ of a noninteracting QD vs the temperature of left lead $T_L$. The parameters adopted are (in units of $\Delta$) ${\epsilon }_d=-3.33, U=0, {\Delta }_L=5{\Delta }_R=0.83, {\mu }_R={\mu }_L=0, \Delta T=T_R-T_L=0.17$, and $W=50$.