Heat switch and thermoelectric effects based on Cooper-pair splitting and elastic cotunneling

In this paper, we demonstrate that the hybrid normal-superconducting-normal (NSN) structure has potential for a multifunctional thermal device which could serve for heat flux control and cooling of microstructures. By adopting the scattering matrix approach, we theoretically investigate thermal and electrical effects emerging in such structures due to the Cooper pair splitting (CPS) and elastic cotunneling phenomena. We show that a finite superconductor can, in principle, mediate heat flow between normal leads, and we further clarify special cases when this seems contradictory to the second law of thermodynamics. Among other things, we demonstrate that the CPS phenomenon can appear even in the simple case of a ballistic NSN structure.

Figure 1

Different NSN configurations. (a) The superconductor is large. An incident electron (blue) reflects as a hole (green) with probability equal to unity, $R_{eh}^{LL}=1$. (b) The length of the superconductor is comparable with the coherence length $\xi$. Along with AR, the EC becomes efficient: An incident electron can either propagate as an electron or reflect as a hole, $R_{eh}^{LL}+T_{ee}^{LR}=1$. (c) Once normal scattering on the NS interface becomes possible, the CPS process enables electrons to be transmitted as holes. Due to the unitary condition, $R_{ee}^{LL}+R_{eh}^{LL}+T_{ee}^{LR}+T_{eh}^{LR}=1$. Red arrows indicate the selected direction of the thermal ($I_Q^{L\left(R\right)}$) and electrical ($I_e^{L\left(R\right)}$) currents in the left and right normal leads.

Figure 2

(a) Electron-to-hole ($T_{eh}^{LR}$) and (b) electron-to-electron ($T_{ee}^{LR}$) transmission probabilities and (c) difference between transmission probabilities of EC and CPS processes as functions of the superconductor length $L$ in a ballistic NSN structure ($k_0=\sqrt{2m\mathrm{\Delta }/{\hslash }^2}$) for ${\mu }_{\perp }=\{0.01\mathrm{\Delta },0.02\mathrm{\Delta },0.03\mathrm{\Delta }\}$ (orange, blue, and green lines, respectively). The plots represent the limiting case where $\varepsilon =0$ and ${\mu }_{\perp }\ll \mathrm{\Delta }$.

Figure 3

Working principle of the NSN heat switch utilizing double barrier scatterers with the energy-dependent transmission probability (X). Each barrier is perfectly transparent at its resonance energy, whereas particles with other energies are completely reflected. The positions of the resonances can be adjusted by the external gate voltages. (a) The resonance settings realizing ideal transmission. Symmetric configuration allows for unity electron-to-hole transmission probability at resonance, $T_{eh\text{(res)}}^{LR}=1$; dashed resonance curve marks perfect EC configuration, $T_{ee\text{(res)}}^{LR}=1$. Heat is transferred through the system. (b) An example of a cutoff configuration: particles do not propagate through the switch. Heat flow is absent. (c)–(e) Color plots of the heat conductance, $G_Q=I_Q^L/\delta \mathrm{\Theta }$, as function of the resonance energies at the left (${\varepsilon }_L$) and right (${\varepsilon }_R$) double barriers for $pL=\{\pi n,0.15\pi +\pi n,0.5\pi +\pi n\}$ ($n$ is integer). $E_F\gg \mathrm{\Delta }$, $L={\xi }_0=k_F\mathrm{\Delta }/\left(2E_F\right)$ and $\mathrm{\Theta }=0.1\mathrm{\Delta }$; the parameters of the X barriers are ${\mathrm{\Gamma }}_X=0.03\mathrm{\Delta }$, $T_{\text{(res)}}^X=1$.

Figure 4

(a) NXN refrigerator: The heat extraction from the left lead occurs due to the electron concentration difference at energies around the resonance of the X scatterer. (b) CPS refrigerator: The resonance configuration of the barriers corresponds to the perfect electron-to-hole transmission probability. Similar to the previous scheme, the cooling is based on the concentration difference of the quasiparticles in the normal leads. (c) The NXSXN configuration for which CPS is replaced by EC. The scheme is essentially reduced to the NXN one.

Figure 5

Heat currents $I_Q^{\mathrm{NN}}$ (blue; $T_{\left(\mathrm{res}\right)}^X=1$) and $I_Q^{\text{NSN}}$ (red, solid, and dashed lines correspond to, respectively, $T_{eh\left(\mathrm{res}\right)}^{LR}=1$ and $T_{eh\left(\mathrm{res}\right)}^{LR}=0.5$ (nonideal CPS)) as functions of the bias voltage $V_{\mathrm{B}}$. The dashed vertical line indicates where the solid lines are separated the most. The positiveness of the heat current directed from the left colder region to the right warmer one (${\mathrm{\Theta }}_R>{\mathrm{\Theta }}_L$) means that the heat flows against the temperature gradient. The parameters are $\delta \varepsilon =0.05\mathrm{\Delta }$, ${\mathrm{\Theta }}_L=0.1\mathrm{\Delta }$, ${\mathrm{\Theta }}_R=0.2\mathrm{\Delta }$, ${\mathrm{\Gamma }}_X={\mathrm{\Gamma }}_{\mathrm{XSX}}=0.003\mathrm{\Delta }$.