Cooling Electrons by Magnetic-Field Tuning of Andreev Reflection

A solid-state cooling principle based on magnetic-field-driven tunable suppression of Andreev reflection in superconductor/two-dimensional electron gas nanostructures is proposed. This cooling mechanism can lead to very large heat fluxes per channel up to ${10}^4$ times greater than currently achieved with superconducting tunnel junctions. This efficacy and its availability in a two-dimensional electron system make this method of particular relevance for the implementation of quantum nanostructures operating at cryogenic temperatures.

Figure 1

(a) Scheme of the proposed setup. Ferromagnetic strips ($F$) placed on top of the structure generate at each $S$-2DEG junction localized magnetic fields whose strength can be controlled by changing the orientation of magnetization ($\mathcal{M}$). (b) Idealized semiclassical picture of the Andreev reflection suppression mechanism: An electron ($e$) impinges on the localized magnetic field (yellow region) at the critical angle (${\theta }_{\mathrm{crit}}$) above which no Andreev reflection takes place.

Figure 2

(a) Heat current $J$ of a ballistic $S$-2DEG system extracted from the 2DEG side vs bias voltage $V_c$ applied across the junction, calculated for several angles $\varphi$ at $T=0.3{\Delta }_0/k_B$. $k_B$ is the Boltzmann constant. The dashed-dotted line represents $J$ for a $N$-2DEG junction at $\varphi =0°$. (b) $J$ vs $V_c$ calculated for several temperatures at $\varphi =0°$. (c) $J$ calculated at the optimal bias voltage across the junction vs $T$ at $\varphi =0°$. $J$ is maximized at $T=T_{\mathrm{opt}}\simeq 0.25{\Delta }_0/k_B$. (d) Efficiency ($\eta$) of the cooling process calculated at the optimal bias voltage vs $T$ at $\varphi =0°$. The junction normal-state resistance is $R_N(\varphi =0°)=1.07\times {10}^4\Omega$.

Figure 3

(a) Sketch of the system considered: The 2DEG is assumed to be in quasiequilibrium at temperature $T_e$ and to interact with the lattice phonons at temperature $T_L$ through piezoelectric coupling ($P_L$). The thermal load due to Joule dissipation ($P_J$) in the 2DEG as well as the heat flux ($J$) flowing through the system are indicated with additional arrows. (b) System equivalent electric circuit: The 2DEG is shown as a lumped resistor ($R_{\mathrm{load}}$), and $V$ is the total bias voltage applied across the symmetric structure.

Figure 4

(a) Electron temperature $T_e$ in the 2DEG of a $S\mathrm{\text{−}}2\mathrm{DEG}\mathrm{\text{−}}S$ refrigerator vs $V$ calculated for several angles $\varphi$ at $T_L=0.3\mathrm{K}$. (b) $T_e$ vs $V$ for different lattice temperatures $T_L$ at $\varphi =0°$. (c) $T_e$ vs $V$ for several charge densities of the 2DEG ($n_{2\mathrm{DEG}}$) at $\varphi =0°$ and $T_L=0.3\mathrm{K}$.