Thermal rectification with interacting electronic channels: Exploiting degeneracy, quantum superpositions, and interference

This paper explores different mechanisms that induce thermal rectification in the nanoscale. The presence of interacting energy channels combined with simple asymmetries is sufficient for promoting the desired behavior. We use simple quantum dot configurations, identifying the basic properties that enhance rectification for each case: the size of a quantum dot state space (which suggests the use of scaled up systems with many interacting channels), tunneling asymmetries due to coherent tunneling in a double quantum dot, or quantum interference in a triangular triple quantum dot. An efficient and tunable thermal diode is proposed using a channel capacitively coupled to a mesoscopic switch.

Figure 1

Scheme of the different quantum dot configurations discussed in the paper. (a) A single quantum dot is coupled to two terminals with asymmetric tunneling rates ${\mathrm{\Gamma }}_l$. There are two possible states the electrons can tunnel to. (b) A double quantum dot in series. Hybridization of the quantum dot orbitals due to coherent tunneling, ${\tau }_{\mathrm{LR}}$, introduces L-R asymmetric and energy-dependent tunneling rates even if the left and right barriers are identical. (c) A triple quantum dot in a triangular configuration introduces directionality due to the interference of trajectories coming from the left lead.

Figure 2

Two-level quantum dot coupled to two terminals. Left-right asymmetric and level-dependent tunneling lead to four different tunneling rates, ${\mathrm{\Gamma }}_{li}$. The latest can be due to an energy splitting $\mathrm{\Delta }E$ in energy-dependent barriers, or to an additional degree of freedom (e.g., spin).

Figure 3

Rectification of a single quantum dot whose spin states are split by a magnetic field: ${\varepsilon }_1=\varepsilon -\mathrm{\Delta }E/2$, ${\varepsilon }_2=\varepsilon +\mathrm{\Delta }E/2$, with $\mathrm{\Delta }E=g\mu B_{\mathrm{z}}$. (a) Heat current for $\mathrm{\Delta }T=T/2$, $V=0$, ${\mathrm{\Gamma }}_{\mathrm{L}}=2{\mathrm{\Gamma }}_{\mathrm{R}}=0.2k_{\text{B}}T$, with $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{L}}{\mathrm{\Gamma }}_{\mathrm{R}}/{\mathrm{\Gamma }}_{\mathrm{\Sigma }}$. (b) Forward and backward currents for $B_{\mathrm{z}}=0$. (c) Rectification coefficient with (d) cuts at zero and finite magnetic field. The inset in (b) zooms in the difference of the two currents around $\varepsilon =3k_{\text{B}}T$. Note that rectification is present even if $B_{\mathrm{z}}=0$.

Figure 4

Interacting channels in parallel realized in (a) two capacitively coupled quantum dots, and (b) a system of self-assembled quantum dots. (c) Rectification coefficient for a system of $N$ capacitively coupled quantum dots in parallel, with $\mathrm{\Delta }T=T/2$.

Figure 5

Double quantum dot in series. Electron delocalization due to coherent interdot tunneling, ${\tau }_{\mathrm{LR}}$, leads to hybridization of the quantum dot states. The relative weight of the orbitals in each quantum dot affects the tunneling rates.

Figure 6

Rectification of a double quantum dot whose energy levels, ${\varepsilon }_{\mathrm{L}}$ and ${\varepsilon }_{\mathrm{R}}$ are tuned by gate voltages. (a) Heat current for $\mathrm{\Delta }T=T/2$, $V=0$, ${\tau }_{\mathrm{L}\mathrm{R}}=k_{\text{B}}T$, ${\mathrm{\Gamma }}_{\mathrm{L}}={\mathrm{\Gamma }}_{\mathrm{R}}=0.2k_{\text{B}}T$, with $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{L}}{\mathrm{\Gamma }}_{\mathrm{R}}/{\mathrm{\Gamma }}_{\mathrm{\Sigma }}$. White-dashed lines mark the zeros of the eigenenergies $E_{\pm }$. (b) Forward and backward currents for ${\varepsilon }_{\mathrm{R}}-{\varepsilon }_{\mathrm{L}}=k_{\text{B}}T$. (c) Rectification coefficient with (d) cuts at different level splittings. It vanishes when the system is symmetric (${\varepsilon }_{\mathrm{L}}={\varepsilon }_{\mathrm{R}}$), and close to the conditions $E_{\pm }\approx 0$.

Figure 7

Rectification of a double quantum dot in the open-circuit configuration. (a) Heat current and (b) rectification coefficient as functions of the position of the levels, ${\varepsilon }_1$ and ${\varepsilon }_2$. The same parameters as in Fig. 6 are considered.

Figure 8

Orbitals in a triple quantum dot. When the levels in dots A and B are degenerate, a dark state is formed with forbidden tunneling to the level in dot C. Therefore, it is uncoupled from the right lead.

Figure 9

Rectification of a triple quantum dot whose energy levels, ${\varepsilon }_{\text{AB}}$ and ${\varepsilon }_{\text{C}}$ are tuned by gate voltages. (a) Forward and (b) backward heat currents for $\mathrm{\Delta }T=T$, $V=0$, $\tau =k_{\text{B}}T$, ${\mathrm{\Gamma }}_{\mathrm{L}}={\mathrm{\Gamma }}_{\mathrm{R}}=0.2k_{\text{B}}T$, with $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{L}}{\mathrm{\Gamma }}_{\mathrm{R}}/{\mathrm{\Gamma }}_{\mathrm{\Sigma }}$. (c) Forward and backward currents for ${\varepsilon }_{\text{AB}}-{\varepsilon }_{\text{C}}=0$. (d) Cut along ${\varepsilon }_{\text{AB}}-{\varepsilon }_{\text{C}}=0$ of the rectification coefficient plotted in (e) for different level positions. When ${\varepsilon }_{\text{AB}}<0$, the occupation of the dark state suppresses the backward current and high rectification coefficients are attained. White-dashed lines in (a) mark the zeros of the eigenenergies $E_i$.

Figure 10

Effect of temperature in a triple quantum dot. (a) Heat currents and (b) rectification coefficient for increasing temperature gradients. We assume ${\varepsilon }_{\text{AB}}={\varepsilon }_{\text{C}}V=0$, $\tau =2k_{\text{B}}T$, ${\mathrm{\Gamma }}_{\mathrm{L}}={\mathrm{\Gamma }}_{\mathrm{R}}=0.04k_{\text{B}}T$, with $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{L}}{\mathrm{\Gamma }}_{\mathrm{R}}/{\mathrm{\Gamma }}_{\mathrm{\Sigma }}$. Forward (backward) currents are plotted with dashed (solid) lines.

Figure 11

Two capacitively coupled quantum dots, one of which carries an electron transport, with the other one only supporting fluctuations by being coupled to only one lead. If the capacitive coupling is strong, the occupation of the later acts as a switch by preventing a second electron to tunnel into the conducting dot.

Figure 12

Rectification of a system of capacitively quantum dots, one of which is tunneled coupled to one lead only. (a) Forward heat current for $\mathrm{\Delta }T=T$, $V=0$, ${\mathrm{\Gamma }}_{li}=0.2k_{\text{B}}T$, except for ${\mathrm{\Gamma }}_{\mathrm{R}2}=0$, with $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{L}}{\mathrm{\Gamma }}_{\mathrm{R}}/{\mathrm{\Gamma }}_{\mathrm{\Sigma }}$. (b) Forward and backward currents along the maximum at ${\varepsilon }_1=-0.36k_{\text{B}}T$. (d) Cut along the same condition of the rectification coefficient plotted in (c) for different level positions. When ${\varepsilon }_2<0$, the occupation of the coupled dot suppresses the backward current, resulting in high rectification coefficients.

Figure 13

Rectification of a two-state quantum dot in open-circuit. (a) Heat current and (b) rectification coefficient as functions of the position of the level and an applied magnetic field. The same parameters as in Fig. 3 are considered.