Peltier effects in Andreev interferometers

The superconducting proximity effect is known to modify transport properties of hybrid normal-superconducting structures. In addition to changing electrical and thermal transport separately, it alters the thermoelectric effects. Changes to one off-diagonal element $L_{12}$ of the thermoelectric matrix $L$ have previously been studied via the thermopower, but the remaining coefficient $L_{21}$, which is responsible for the Peltier effect, has received less attention. We discuss symmetry relations between $L_{21}$ and $L_{12}$ in addition to the Onsager reciprocity, and calculate Peltier coefficients for a specific structure. Similar to the thermopower, for finite phase differences of the superconducting order parameter, the proximity effect creates a Peltier effect significantly larger than the one present in purely normal-metal structures. This results from the fact that a nonequilibrium supercurrent carries energy.

Figure 1

Example of a four-probe structure considered in the text: five normal-metal wires connected to each other and to four terminals, of which two are superconducting $\left(S\right)$ and two normal $\left(N\right)$. We take the lengths $l$, cross-sectional areas $A$, and conductivities $\sigma$ of the wires to be $l∕l_0=(1.5,1,1.2,1,0.8)$ and $A\sigma ∕A_0{\sigma }_0=(0.8,1,0.8,1,1)$. Here, $l_0$, $A_0$, and ${\sigma }_0$ are some characteristic values controlling the energy scale $E_T=\hslash D∕{(l_3+l_4+l_5)}^2$ of the proximity effect. The system is chosen so as to bring out effects that depend on the magnitude of geometrical asymmetry. In the numerics, all wires are assumed to be quasi-one-dimensional, $l⪢\sqrt{A}$.

Figure 2

(Color online) Elements $L_{\alpha \beta }^{ij}$ for $i,j=1,2$ and $\alpha ,\beta =L,T$ for the asymmetric interferometer in Fig. 1, in units of $1∕R_2$. The order parameters in the superconducting terminals have $\arg {\Delta }_3-\arg {\Delta }_4=0.54\pi$ and $\mid {\Delta }_3\mid =\mid {\Delta }_4\mid =40E_T$. The thick blue solid line is $L^{11}$, the dash-dotted red line $L^{22}$, the black dotted line $L^{12}$, and the magenta dashed line $L^{21}$. Note the symmetry ${\tilde{L}}_{\alpha \beta }^{ij}\left(E\right)={(-1)}^{1-{\delta }_{\alpha \beta }}{\tilde{L}}_{\beta \alpha }^{ji}\left(E\right)$. Approximations found by solving Eqs. (4) to first order in $j_S$ and $\mathcal{T}$ are shown as thin solid lines—in general, they are indistinguishable from the exact numerical results.

Figure 3

(Color online) Peltier coefficients ${\Pi }_{NN}$ and ${\Pi }_{NS}$ for the same parameters as in Fig. 2. Approximations (14a, 14b) are shown as thin lines—deviation from the exact result is due to neglecting $\mathcal{T}$.

Figure 4

(Color online) (a) Temperature of terminal 1 in Fig. 1 for $\phi =\arg {\Delta }_3-\arg {\Delta }_4=\pm 0.54\pi$ (dashed and solid lines) and current configurations corresponding to ${\Pi }_{NN}$ and ${\Pi }_{NS}$ (red and blue lines) as a function of $I_c$. (b) $T_1$ for $e{R}_1{I}_c∕E_T=0.15$ as a function of $\phi$. The results are calculated assuming other terminals are at the temperature $T=2E_T∕k_B$. Deviation of $T_1$ from $T$ originates from Joule heating and the oscillation of the proximity-Peltier effect.