Heat transport through quantum Hall edge states: Tunneling versus capacitive coupling to reservoirs

We study the heat transport along an edge state of a two-dimensional electron gas in the quantum Hall regime, in contact to two reservoirs at different temperatures. We consider two exactly solvable models for the edge state coupled to the reservoirs. The first one corresponds to filling $\nu =1$ and tunneling coupling to the reservoirs. The second one corresponds to integer or fractional filling of the sequence $\nu =1/m$ (with $m$ odd), and capacitive coupling to the reservoirs. In both cases, we solve the problem by means of nonequilibrium Green function formalism. We show that heat propagates chirally along the edge in the two setups. We identify two temperature regimes, defined by $\Delta$, the mean level spacing of the edge. At low temperatures, $T<\Delta$, finite size effects play an important role in heat transport, for both types of contacts. The nature of the contacts manifests itself in different power laws for the thermal conductance as a function of the temperature. For capacitive couplings, a highly nonuniversal behavior takes place, through a prefactor that depends on the length of the edge as well as on the coupling strengths and the filling fraction. For larger temperatures, $T>\Delta$, finite-size effects become irrelevant, but the heat transport strongly depends on the strength of the edge-reservoir interactions, in both cases. The thermal conductance for tunneling coupling grows linearly with $T$, whereas for the capacitive case, it saturates to a value that depends on the coupling strengths and the filling factors of the edge and the contacts.

Figure 1

Sketch of the studied setup of a fractional (Laughlin) quantum Hall fluid in contact with a source and a drain and with a thermometer. The fractional quantum Hall edge state is represented by a ring. Two reservoirs, drain and source, with temperatures $T_1<T_2$, are connected to the ring at positions $x_1$ and $x_2$, either through contacts that allow tunneling of particles with couplings strengths $w_1$, $w_2$, or through capacitive couplings with strengths $V_1$, $V_2$ where only an energy current can flow through the leads. A third reservoir is weakly connected at $x_3$ in order to sense the local temperature $T_3$. For tunneling contacts, the only exactly solvable case corresponds to filling $\nu =1$. For capacitive coupling, any filling $\nu$ can be exactly treated.

Figure 2

(Top) Transmission function ${\mathcal{T}}_{12}\mathrm{t}$ as function of $\omega$ for a tunneling coupling between the edge and two reservoirs with amplitudes $w_1=w_2=0.1$. (Bottom) Transmission function ${\mathcal{T}}_{12}\mathrm{c}$ as function of $\omega$ for a capacitive coupling between the edge and two reservoirs with amplitudes $V_1=V_2=0.2$. The remaining parameters, common to the two cases, are $v_F=1$ and $L=200$, $x_1=0$ and $x_2=100$. All energies are expressed in natural units ($\hslash =1$).

Figure 3

Behavior of the thermal conductance as a function of the coupling strength within the regime $k_{B}T\gg \Delta$. (Top) Tunneling thermal conductance $G_{\mathrm{th}}^t$, having set $w_1=w_2=w$, for different temperatures: $T=0.04$ (red, solid), $T=0.05$ (green, dashed), $T=0.06$ (blue, dot-dashed). (Bottom) Capacitive thermal conductance $G_{\mathrm{th}}^c$ as a function of $\sqrt{\nu {\nu }^{\prime }}V$ for different temperatures: $T=0.04$ (red, solid), $0.05$ (green, dashed), and $0.06$ (blue, dot-dashed). We have set $V_1=V_2=V$ and ${\nu }_1={\nu }_2={\nu }^{\prime }$, and $\nu$ is the filling factor of the ring. All energies are expressed in natural units ($\hslash =1$).

Figure 4

Low-temperature behavior of the tunneling thermal conductance through reservoirs 1 and 2 as a function of temperature, for different values of the chemical potential of the reservoirs: $\mu =0$ corresponding to resonance (red, solid), and two off-resonant values $\mu =\pi /\left(5L\right)$ (green, dashed), $\pi /L$ (blue, dot-dashed). The couplings are $w_1=w_2=0.1$, and the ring length is $L=400$. The arrows indicate $T=\gamma ,\Delta$. A zoom of the linear regime for very low temperatures ($T\sim \gamma$) in the resonant case is shown in the inset.

Figure 5

Map plot of the tunneling thermal conductance through reservoirs 1 and 2, as function of the temperature and the chemical potential of the reservoirs. The couplings are $w_1=w_2=0.1$, and the ring length $L=400$.

Figure 6

Capacitive thermal conductance as a function of the temperature for different ring lengths: $L=100$ (red, solid), 200 (green, dashed), and 400 (blue, dot-dashed). The couplings are $V_1=V_2=1$.

Figure 7

Local temperature along the edge, as a function of the position of the thermometer, for different values of the couplings with the reservoirs. For the tunneling case (a), we have set $w_1=w_2=0.4$ (red, solid), $0.6$ (green, dashed), $0.8$ (blue, dotted-dashed) and $\mu =0$. For the capacitive case (b), the values are $V_1=V_2=1$ (red, solid), 2 (green, dashed), and 3 (blue, dotted-dashed). The remaining parameters are $x_1=0$, $x_2=200$, $L=400$, $T_1=0.009$, and $T_2=0.011$. A filling factor $\nu =1$ is considered. The temperature difference between the two arms of the ring $(T_u-T_d)$ as a function of $\nu$ is shown in the inset, for two values of the couplings: $V_1=V_2=1,3.$