Interference-induced thermoelectric switching and heat rectification in quantum Hall junctions

Interference represents one of the most striking manifestations of quantum physics in low-dimensional systems. Despite evidence of quantum interference in charge transport having been known for a long time, signatures of interference-induced thermal properties have been reported only recently, paving the way for the phase-coherent manipulation of heat in mesoscopic devices. In this work we show that anomalous thermoelectric properties and efficient heat rectification can be achieved by exploiting the phase-coherent edge states of quantum Hall systems. By considering a tunneling geometry with multiple quantum point contacts, we demonstrate that the interference paths effectively break the electron-hole symmetry, allowing for a thermoelectric charge current flowing either from hot to cold or vice versa, depending on the details of the tunnel junction. Correspondingly, an interference-induced heat current is predicted, and we are able to explain these results in terms of an intuitive physical picture. Moreover, we show that heat rectification can be achieved by coupling two quantum Hall systems with different filling factors, and that this effect can be enhanced by exploiting the interference properties of the tunnel junction.

Figure 1

Scheme of two fractional quantum Hall systems with different filling factors ${\nu }_L$ and ${\nu }_R$ at temperatures $T_L>T_R$. Counterpropagating edge states are coupled by a tunneling region, schematically depicted in the middle. The zoom in the right panel shows the case of tunneling due to multiple quantum point contacts ($n=3$ in this case) equally spaced with distance $d$.

Figure 2

Tunneling charge current $J_C$ in units of $J_C^{\left(0\right)}=2\pi {\left|\lambda \right|}^{2}e\mu /{\omega }_{\mathrm{c}}^2$ for ${\nu }_L={\nu }_R=1$, as a function of $\eta ={\overline{k}}_{\mathrm{F}}d$. (a) Different curves represent different numbers of QPCs with temperature $T_R=20$ mK and $T_L=300$ mK. The $n=2$ curve displays oscillations of period $\pi$, modulated by a nonmonotonic envelope function that reaches its maximum for $\eta \sim 4.5\pi$. More complicated interference patterns arise for the $n=10$ curve. Note that the single QPC ($n=1$) does not break particle-hole symmetry, resulting in a null charge current. (b) Density plot of charge current for $n=2$ QPCs as a function of $\eta$ ($x$ axis) and $T_L/T_R$ ($y$ axis). $T_R$ is fixed and equal to 20 mK, while the ratio $T_L/T_R$ goes from 1 to 15 (corresponding to a maximum temperature of 300 mK). The figure shows an increasing or decreasing monotonic behavior of the charge current as a function of the ratio $T_L/T_R$ at fixed $\eta$, depending on the sign of $J_C$. The oscillating behavior described in panel (a) is also visible. In both panels the chemical potential is set to $\mu =10$ K.

Figure 3

Schematic representation of electron and hole contributions to the charge current in the presence of a thermal gradient (here $T_L=300$ mK, $T_R=20$ mK, and $\mu =10$ K). The electron contribution $g_n(E+\mu )n_L\left(E\right)\left[1-n_R\left(E\right)\right]$ is represented in red, while the hole one $g_n(E+\mu ){\overline{n}}_L\left(E\right)\left[1-{\overline{n}}_R\left(E\right)\right]$ is represented in blue [see Eq. (20)]. The insets show the transmission functions $g_n(E+\mu )$. (a) Single QPC: the transmission function is energy independent and electrons and holes contribute equally, thus giving $J_C=0$. (b)–(d) Two QPCs. (b) $\eta =4\pi$: despite the transmission function being energy dependent, it does not break particle-hole symmetry, so that $J_C=0$. (c) $\eta =4\pi -\pi /4$: particle-hole symmetry is broken and electron tunneling is enhanced with respect to hole tunneling, giving $J_C<0$. (d) $\eta =4\pi +\pi /4$: particle-hole symmetry is broken and hole tunneling is enhanced with respect to electron tunneling, giving $J_C>0$.

Figure 4

Tunneling heat current $J_Q$ in units of $J_Q^{\left(0\right)}=2\pi {\left|\lambda \right|}^2{\mu }^2/{\omega }_{\mathrm{c}}^2$ for ${\nu }_L={\nu }_R=1$. The chemical potential is set to $\mu =10$ K. (a) $J_Q$ as a function of $\eta ={\overline{k}}_{\mathrm{F}}d$. Different curves represent different numbers of QPCs with temperature $T_R=20$ mK and $T_L=300$ mK. As for the charge current, oscillations of period $\pi$ are visible in the $n=2$ signal. Here heat current oscillates around a nonvanishing mean value equal to half the value of the single QPC heat current. Note that the modulating function of the oscillating part of the signal changes its sign at $\eta \sim 4.5\pi$, displaying a sort of beat. More complicated interference patterns arises for the $n=10$ curve. (b) Density plot of heat current for $n=2$ QPCs as a function of $\eta$ ($x$ axis) and $T_L/T_R$ ($y$ axis). $T_R$ is fixed and equal to 20 mK, while the ratio $T_L/T_R$ goes from 1 to 15 (corresponding to a maximum temperature of 300 mK). The heat current is an increasing monotonic function of the ratio $T_L/T_R$ for all values of $\eta$. The oscillating behavior described in panel (a) is also visible.

Figure 5

Schematic representation of electron and hole contributions to the heat current (in units of $\mu$) in the presence of a thermal gradient (here $T_L=300$ mK, $T_R=20$ mK, and $\mu =10$ K). The electron contribution $E{g}_n(E+\mu )n_L\left(E\right)\left[1-n_R\left(E\right)\right]$ is represented in red, while the hole one $(-E)g_n(E+\mu ){\overline{n}}_L\left(E\right)\left[1-{\overline{n}}_R\left(E\right)\right]$ is represented in blue [see Eq. (21)]. In contrast to the charge current, the total heat current is given by the sum (and not the difference) of electron and hole contributions; see Eq. (21). The insets show the transmission functions $g_n(E+\mu )$. (a) Single QPC: the transmission function $g_1\left(E\right)=1$, giving the maximum heat current; see Fig. 4. (b)–(d) Two QPCs. The heat current shows a sequence of maxima and minima (compare with Fig. 4). (b) $\eta =2\pi -\pi /2$: both electrons and holes have low transmission, giving a minimum of $J_Q$. (c) $\eta =2\pi$: both electrons and holes have high transmission, giving a maximum of $J_Q$. (d) $\eta =2\pi +\pi /2$: both electrons and holes have low transmission, giving a minimum of $J_Q$.

Figure 6

Charge (a) and heat (b) currents for ${\nu }_L=1$ and ${\nu }_R=1/3$ as a function of $\eta ={\overline{k}}_{\mathrm{F}}d$ at fixed temperatures $T_R=20$ mK and $T_L=300$ mK. Units of $J_C$ or $J_Q$ are respectively $J_C^{\left(1\right)}=2\pi {\left|\lambda \right|}^{2}e{\mu }^3/{\omega }_{\mathrm{c}}^4\text{and} J_Q^{\left(1\right)}=2\pi {\left|\lambda \right|}^2{\mu }^4/{\omega }_{\mathrm{c}}^4$. Different curves refer to different numbers of QPCs. The chemical potential is set to $\mu =10$ K.

Figure 7

(a) Heat rectification coefficient $r_Q$ for multiple QPC geometry as a function of $T_{\mathrm{hot}}/T_{\mathrm{cold}}$ for $\eta =3\pi$. Different curves refer to different numbers of QPCs. In the case of a single QPC (solid line) $r_Q$ saturates to 2.43 as one can see from Eq. (23). (b) Density plot of the heat rectification coefficient as a function of $\eta$ ($x$ axis) and $T_{\mathrm{hot}}/T_{\mathrm{cold}}$ ($y$ axis) for $n=10$ QPCs. The ratio $T_{\mathrm{hot}}/T_{\mathrm{cold}}$ goes from 1 to 15, with fixed $T_{\mathrm{cold}}=20$ mK. Larger rectification effect occurs when $\eta =k\pi$ and is linked to interference patterns. For each value of $k$ the rectification coefficient is a nonmonotonic function of $T_{\mathrm{hot}}/T_{\mathrm{cold}}$, displaying a maximum whose position changes with $k$. For $\eta =3\pi$ and $T_{\mathrm{hot}}/T_{\mathrm{cold}}\sim 12$ one can reach a maximum value of $r_Q\sim 9.6$.