Thermal drag in electronic conductors

We study the electronic thermal drag in two different Coulomb-coupled systems, the first one composed of two Coulomb-blockaded metallic islands and the second one consisting of two parallel quantum wires. The two conductors of each system are electrically isolated and placed in the two circuits (the drive and the drag) of a four-electrode setup. The systems are biased, either by a temperature $\mathrm{\Delta }T$ or a voltage $V$ difference, on the drive circuit, while no biases are present on the drag circuit. In the case of a pair of metallic islands we use a master equation approach to determine the general properties of the dragged heat current $I_{\mathrm{drag}}^{\left(\mathrm{h}\right)}$, accounting also for cotunneling contributions and the presence of large biases. Analytic results are obtained in the sequential tunneling regime for small biases, finding, in particular, that $I_{\mathrm{drag}}^{\left(\mathrm{h}\right)}$ is quadratic in $\mathrm{\Delta }T$ or $V$ and nonmonotonic as a function of the interisland coupling. Finally, by replacing one of the electrodes in the drag circuit with a superconductor, we find that heat can be extracted from the other normal electrode. In the case of the two interacting quantum wires, using the Luttinger liquid theory and the bosonization technique, we derive an analytic expression for the thermal transresistivity ${\rho }_{12}^{\left(\mathrm{h}\right)}$, in the weak-coupling limit and at low temperatures. ${\rho }_{12}^{\left(\mathrm{h}\right)}$ turns out to be proportional to the electrical transresistivity, in such a way that their ratio (a kind of Wiedemann-Franz law) is proportional to $T^3$. We find that ${\rho }_{12}^{\left(\mathrm{h}\right)}$ is proportional to $T$ for low temperatures and decreases like $1/T$ for intermediate temperatures or like $1/T^3$ for high temperatures. We complete our analyses by performing numerical simulations that confirm the above results and allow us to access the strong-coupling regime.

Figure 1

Sketch of Coulomb-coupled systems, which consists of an upper (drive) biased circuit, and a lower (drag) unbiased circuit. The conductors are represented as gray rectangles and are attached to two leads each. The two conductors are coupled only through the Coulomb interaction. As indicated by the black arrows, the sign of charge and heat currents is positive when they enter an electrode.

Figure 2

Sketch of the first system under consideration composed of two capacitively coupled metallic islands labeled by 1 (in the drive circuit) and 2 (in the drag circuit). L1, L2, R1, and R2 label the four electrodes which are tunnel-coupled to the islands.

Figure 3

Energy scheme for the lower, drag circuit. Green rectangles represent the tunnel barriers. Gray areas represent the Fermi distribution functions of the leads, whose common equilibrium electrochemical potential, set to zero, is indicated by a dashed line. Thick horizontal black lines indicate the position of the chemical potential of the island for $n_{x1}=n_{x2}=1/2$. (a) The upper island is empty and the chemical potential is $\delta U_2(0,0)$: electrons can jump on the island. (b) The upper island is occupied and the chemical potential is $\delta U_2(1,0)$: electrons can jump out of the island.

Figure 4

Dragged heat currents plotted as functions of $n_{x2}$ for different values of $n_{x1}$. The results accounting for sequential tunneling only are plotted in black, while results including cotunneling contributions are plotted in red. ${\mathcal{R}}_{\mathrm{L}1}={\mathcal{R}}_{\mathrm{R}1}={\mathcal{R}}_{\mathrm{L}2}=5{\mathcal{R}}_{\mathrm{Q}},{\mathcal{R}}_{\mathrm{R}2}=10{\mathcal{R}}_{\mathrm{Q}},k_{\mathrm{B}}T=0.05E_C$, and $E_{\mathrm{I}}=0.4E_C$. (a) $\mathrm{\Delta }T=0$ and $V=0.08E_C/e$; (b) $V=0$ and $\mathrm{\Delta }T=0.08E_C/k_B$. The solid curves have been multiplied by a factor $1/20$ in panel (a) and by a factor $1/4$ in panel (b). The heat current is given in units of $I_0^{\left(\mathrm{h}\right)}=e^2/\left(4C^2\mathcal{R}\right)$.

Figure 5

Dragged heat current (sequential tunneling only) plotted as a function of $E_{\mathrm{I}}$ for the case $V=0.08E_C$ and $\mathrm{\Delta }T=0$ (red lines) the case $\mathrm{\Delta }T=0.08E_C$ and $V=0$ (black lines) for $n_{x1}=n_{x2}=1/2$. Thin black and red curves are plots of the analytic expressions Eqs. (15) and (16), respectively. The other parameters are chosen as follows: ${\mathcal{R}}_{\mathrm{L}1}={\mathcal{R}}_{\mathrm{R}1}={\mathcal{R}}_{\mathrm{L}2}=5{\mathcal{R}}_{\mathrm{Q}},{\mathcal{R}}_{\mathrm{R}2}=10{\mathcal{R}}_{\mathrm{Q}}$, and $k_{\mathrm{B}}T=0.05E_C$.

Figure 6

Heat currents $I_{\mathrm{R}1}^{\left(\mathrm{h}\right)}$ (solid black curve) and $I_{\mathrm{R}2}^{\left(\mathrm{h}\right)}$ (dashed red curve) plotted as a function of $E_{\mathrm{I}}$ for the case $\mathrm{\Delta }T=0$ and accounting for sequential tunneling only. The other parameters are chosen as follows: ${\mathcal{R}}_{\mathrm{L}1}={\mathcal{R}}_{\mathrm{L}2}={\mathcal{R}}_{\mathrm{R}1}={\mathcal{R}}_{\mathrm{R}2}=10{\mathcal{R}}_{\mathrm{Q}},n_{x1}=n_{x2}=1/2,V=0.08E_C/e$, and $k_{\mathrm{B}}T=0.05E_C$.

Figure 7

Dragged heat current $I_{\mathrm{drag}}^{\left(\mathrm{h}\right)}$ plotted as a function of $V$ accounting for sequential tunneling only (solid black line) and including cotunneling contributions (dashed red line) for $\mathrm{\Delta }T=0$ and $n_{x1}=n_{x2}=0.478$. The other parameters are chosen as follows: ${\mathcal{R}}_{\mathrm{L}1}={\mathcal{R}}_{\mathrm{R}1}={\mathcal{R}}_{\mathrm{L}2}=2{\mathcal{R}}_{\mathrm{Q}},{\mathcal{R}}_{\mathrm{R}2}=4{\mathcal{R}}_{\mathrm{Q}},E_{\mathrm{I}}=0.1E_C$, and $k_{\mathrm{B}}T=0.01E_C$.

Figure 8

Heat (a) and charge (b) drag currents versus $n_{x2}$ for fixed $n_{x1}=1/2$ in the case where R2 is a superconducting electrode. The other parameters are chosen as follows: ${\mathcal{R}}_{\mathrm{L}1}={\mathcal{R}}_{\mathrm{R}1}={\mathcal{R}}_{\mathrm{R}2}=\mathcal{R}=5{\mathcal{R}}_{\mathrm{Q}},{\mathcal{R}}_{\mathrm{L}2}=200{\mathcal{R}}_{\mathrm{Q}},E_{\mathrm{I}}=0.3E_C,k_{B}T=0.05E_C,\mathrm{\Delta }=0.4E_C$. The charge current is given in units of $I_0^{\left(\mathrm{c}\right)}=e/\left(2C\mathcal{R}\right)$. Charge drag current (b) is obtained by applying a thermal bias $k_B\mathrm{\Delta }T=0.08E_C$, whereas the heat drag current (a) is obtained by applying a voltage bias $\mathrm{\Delta }V=0.08E_{\mathrm{C}}/e$.

Figure 9

Sketch of the energies in the presence of a superconducting electrode on R2, for $n_{x1}=1/2$. The red line represent the superconducting DOS, with a gap equal to $\mathrm{\Delta }$ centered at the equilibrium electrochemical potential of the electrodes (dashed thin line). Blue lines represent the two chemical potentials of the lower island, $\delta U_2(1,0)$ and $\delta U_2(0,0)$. Such chemical potentials, according to Eqs. (13) and (14), are symmetric with respect to the electrochemical potential of the electrodes when $n_{x2}=0.5$ [panel (a)], shift downwards [panel (b)] when $n_{x2}>0.5$, and shift upwards [panel (c)] when $n_{x2}<0.5$.

Figure 10

Sketch of the two Coulomb-coupled quantum wires of length $L$. The region where the interwire interaction is present is $L_{\mathrm{I}}$ (with $L_{\mathrm{I}}\ll L$) long.

Figure 11

Plot of ${\rho }_{12}^{\left(\mathrm{h}\right)}/{\rho }_{12}^{\left(0\right)}$ for various values of $g$ and fixed $T_1/T_0=0.2$ (top panel), and for various values of $T_1$ and fixed $g=1$ (bottom panel). We have defined ${\rho }_{12}^{\left(0\right)}\equiv m{U}^2\left(0\right)/\left(v_F{k}_B^2{T}_0\right)$.

Figure 12

3D plot of ${\rho }_{12}^{\left(\mathrm{h}\right)}/{\rho }_{12}^{\left(0\right)}$ as a function of $T/T_0$ and $T_1/T_0$ for $g=1$.

Figure 13

Diagram of the discrete system used with next-neighbor hopping ($\mathcal{J}$) and interaction ($\mathcal{J}\mathrm{\Delta }$) terms.

Figure 14

Schematic picture of the thermal state of the system for $t\le 0$ (top panel) and of the time evolution of the system for $t>0$ (bottom panel). In the upper panel, for each half of the wire the inverse temperature is shown. In the lower panel, the coupling region is colored in blue and the drive and drag currents are shown.

Figure 15

Plot of the stationary value of $I_{\mathrm{drag}}^{\left(\mathrm{h}\right)}$ as a function of $U$ for $\mathrm{\Delta }=0,{\beta }_L=0.5{\mathcal{J}}^{-1}$, and ${\beta }_R=0.75{\mathcal{J}}^{-1}$. The square points represent the data from the simulations, the blue solid line ($a{U}^2+b{U}^4$) fits all the points, while the red dotted line (parabola) misses the last two points (for $U\gtrsim 0.5\mathcal{J}$).

Figure 16

Plot of the time-dependent $I_{\mathrm{drag}}^{\left(\mathrm{h}\right)}\left(t\right)$ calculated for ${\beta }_L=0.5{\mathcal{J}}^{-1},{\beta }_R=0.75{\mathcal{J}}^{-1},\mathrm{\Delta }=0$, and $U=0.3\mathcal{J}$ (solid line) or $U=0.2\mathcal{J}$ (dashed line). The stationary-state values of $I_{\mathrm{drag}}^{\left(\mathrm{h}\right)}$ are marked in green.