Thermal and electrical currents in nanoscale electronic interferometers

We theoretically study thermal transport in an electronic interferometer comprising a parallel circuit of two quantum dots, each of which has a tunable single electronic state which are connected to two leads at different temperature. As a result of quantum interference, the heat current through one of the dots is in the opposite direction to the temperature gradient. An excess heat current flows through the other dot. Although locally, heat flows from cold to hot, globally the second law of thermodynamics is not violated because the entropy current associated with heat transfer through the whole device is still positive. The temperature gradient also induces a circulating electrical current, which makes the interferometer magnetically polarized.

Figure 1

(Color online) A quantum interferometer based on two quantum dots. Both dots are tunnel-coupled to the left and right leads. The tunneling amplitudes between the dots and the leads are denoted by ${\Gamma }_1$ and ${\Gamma }_2$. Each lead is described by an equilibrium Fermi-Dirac distribution with temperature $T_L$ and $T_R$ and electrochemical potential ${\mu }_L$ and ${\mu }_R$. The energy level position in each dot is measured as ${\epsilon }_1$ and ${\epsilon }_2$ relative to the Fermi energy in the leads.

Figure 2

Thermopower of the interferometer as a function of temperature for different level positions of the dots $({\epsilon }_1∕\Gamma ,{\epsilon }_2∕\Gamma )$. The tunneling amplitudes are taken as $\Gamma ={\Gamma }_1={\Gamma }_2$. For both energy levels of dots being above (below) the Fermi energy, the sign of the thermopower is negative (positive). The charge and heat are carried mainly through the electron (hole) channels. When the energy level of one dot is lying below the Fermi energy and that of the other is lying above the Fermi energy, the sign of the thermopower is changed as the temperature varies. Therefore, the main propagating channels for charge and heat determine the sign of the thermopower. The magnitude of the thermopower is of the order of $k_B∕e=86.17\mu \mathrm{V}∕\mathrm{K}$.

Figure 3

(a) Local electric currents induced by a temperature gradient as a function of temperature. The dot energy levels are taken as $({\epsilon }_1∕\Gamma ,{\epsilon }_2∕\Gamma )=(-0.9,0.3)$ for the tunneling amplitudes, $\Gamma ={\Gamma }_1={\Gamma }_2$. In contrast to the thermopower (compare Fig. 2), the level positions taken in this plot do not affect the physics of the local currents but only change its amplitude and sign. Even when the total electric transport current is zero $(I=0)$, the nonvanishing local electric currents indicates the existence of the circulating electric current which makes the interferometer magnetically polarized. The magnetic polarization current is then defined as the local current: $I_M\equiv I_1=-I_2$ for $I=0$. (b) The ratio of the magnetic polarization currents to the temperature gradient, $\Delta T$, as a function of temperature. The magnitude of the magnetic polarization currents is of the order of $nA$ for a temperature gradient of order 1 K.

Figure 4

(Color online) The ratio of the magnetic polarization current $I_M\equiv I_1=-I_2$ to the temperature gradient $\Delta T$ for $I=0$ as a function of the energy level positions of each dot $({\epsilon }_1∕\Gamma ,{\epsilon }_2∕\Gamma )$ for $k_{B}T=5.0\times {10}^{-2}\Gamma$. The tunneling amplitudes are taken as $\Gamma ={\Gamma }_1={\Gamma }_2$. Note that by varying the energy level of each dot the interferometer has a different magnetic state according to the direction of the magnetic polarization current.

Figure 5

(Color online) The ratio of the circulating heat current to the temperature gradient $\Delta T$ as a function of the energy level positions of each dot $({\epsilon }_1∕\Gamma ,{\epsilon }_2∕\Gamma )$ for $I=0$. The parameters are taken as $k_{B}T=5.0\times {10}^{-2}\Gamma$ and $\Gamma ={\Gamma }_1={\Gamma }_2$. The existence of the circulating heat current indicates that while the local heat current through one dot can be greater than the total transport heat current through the interferometer due to the quantum interference, the local heat current through the other dot flows against the temperature gradient. The current conservation requires that the local excess heat current circulate on the closed path in the interferometer.