Heat current across a capacitively coupled double quantum dot

We study the heat current through two capacitively coupled quantum dots coupled in series with two conducting leads at different temperatures $T_L$ and $T_R$ in the spinless case (valid for a high applied magnetic field). Our results are also valid for the heat current through a single quantum dot with strongly ferromagnetic leads pointing in opposite directions (so that the electrons with given spin at the dot can jump only to one lead) or through a quantum dot with two degenerate levels with destructive quantum interference and high magnetic field. Although the charge current is always zero, the heat current is finite when the interdot Coulomb repulsion $U$ is taken into account due to many-body effects. We study the thermal conductance as a function of temperature and the dependence of the thermal current with the couplings to the leads, $T_L-T_R$, energy levels of the dots and $U$, including conditions for which an orbital Kondo regime takes place. When the energy levels of the dots are different, the device has rectifying properties for the thermal current. We find that the ratio between the thermal current resulting from a thermal bias $T_L>T_R$ and the one from $T_L<T_R$ is maximized for particular values of the energy levels, one above and the other below the Fermi level.

Figure 1

Sketch of the system analyzed in this work in which two capacitively quantum dots are attached to two conducting leads of spinless electrons and at different temperatures and chemical potentials. It also describes two additional models as explained in the main text.

Figure 2

Schematic picture for the transport of heat in the presence of interactions. Although Eq. (2) is defined for spinless electrons, the sketch considers also spin, in order to also represent the corresponding configurations of the single-dot spinfull model with polarized reservoirs discussed at the end of Sec. 2a.

Figure 3

Thermal conductance as a function of the temperature for the symmetric configuration $E_L=E_R=-U/2$ and different values of $U$, calculated with PT (see Sec. 3b1). The dotted line indicates the weak-coupling prediction Eq. (26) for $U/\mathrm{\Gamma }=7$. The thin line corresponds to the quantum of thermal conductance [Eq. (1)]. (Inset) Low-temperature part of the plots for $U/\mathrm{\Gamma }=4,7$, with the temperature expressed in units of $T_K$. The limit $U\to \infty$, calculated with RPT is also shown by the dash-dot-dot line.

Figure 4

Low-temperature behavior of the thermal conductance in the limit $U\to \infty$ for $E_L=E_R=-4\mathrm{\Gamma }$, as a function of $T/T_K$, calculated with RPT (see Sec. 3b2) and NCA (see Sec. 3b3). The straight line corresponds to the quantum bound [Eq. (1)]. (Inset) Thermal conductance as a function of $T$ calculated with NCA.

Figure 5

Thermal current as a function of dot-lead couplings For $T_L=2T_R, U=k_B{T}_R/10$ and $E_L=E_R=-U/2$.

Figure 6

Thermal current as a function of $T_L=\mathrm{\Delta }T$ for $T_R=0, E_L=E_R=-U/2$, and several values of $U$. The dotted line corresponds to Eq. (20) with $U=7\mathrm{\Gamma }$.

Figure 7

Thermal current as a function of $\mathrm{\Delta }T$ for $U\to \infty$ and $E=-4\mathrm{\Gamma }$. Inset) Thermal current calculated with NCA for a wide range of temperatures.

Figure 8

Thermal current as a function of dot energies $E$ for $k_B\mathrm{\Delta }T=50\mathrm{\Gamma }$ and $U\to \infty$ (solid line). The dashed line corresponds to $A/E^2$ where $A$ is a constant.

Figure 9

Thermal current as a function of $U$ for different $\mathrm{\Delta }T=T_L, T_R=0$, and $E_L=E_R=-U/2$. Results have been multiplied by a factor in some case, in order to present them in the same scale.

Figure 10

Same as Fig. 9 for $\mathrm{\Delta }T=T_L=10\mathrm{\Gamma }$. Dashed line corresponds to Eq. (20).

Figure 11

Thermal currents given by Eq. (19) as a function of $\mathrm{\Delta }T$ for $T_R=\mathrm{\Gamma }, {\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=\mathrm{\Gamma }, U=20\mathrm{\Gamma }$, and full line $E_L=-14.3\mathrm{\Gamma }, E_R=+5\mathrm{\Gamma }$ (level nearest to the Fermi energy next to the cold lead), dashed line $E_L=+5\mathrm{\Gamma }, E_R=-14.3\mathrm{\Gamma }$ (level nearest to the Fermi energy next to the hot lead).

Figure 12

Rectification coefficient given by Eq. (28) calculated in the atomic limit as a function of both energy levels and $\mathrm{\Delta }T=20\mathrm{\Gamma }, T_R=\mathrm{\Gamma }$, and $U=20\mathrm{\Gamma }$.

Figure 13

(Top) Rectification coefficient calculated with NCA in the $U\to \infty$ limit as a function of both energy levels and $\mathrm{\Delta }T=2.5\mathrm{\Gamma }$ and $T_R=\mathrm{\Gamma }$. (Bottom) Thermal currents as a function of $\mathrm{\Delta }T$ for $T_R=0, {\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=\mathrm{\Gamma }, U=\infty$ and full line $E_L=-10\mathrm{\Gamma }, E_R=+1.5\mathrm{\Gamma }$ (level nearest to the Fermi energy next to the cold lead), dashed line $E_L=+1.5\mathrm{\Gamma }, E_R=-10\mathrm{\Gamma }$ (level nearest to the Fermi energy next to the hot lead).