Thermopower oscillations in mesoscopic Andreev interferometers

Anomalously large thermopower of mesoscopic normal-metal/superconductor interferometers has been investigated by Chandrasekhar et al. It was shown that, depending on the geometry of the interferometer, the thermopower is either symmetric or antisymmetric periodic function of the magnetic flux. We develop a detailed theory of the observed thermoelectric phenomena in the framework of the nonequilibrium quasiclassical approach. In particular, we provide a possible explanation of the symmetric thermopower oscillations. This effect is attributed to the electron-hole symmetry violation that originates in the steady-state charge imbalance between different arms of the interferometer. Our theory can be tested by an additional control over the charge imbalance in a modified setup geometry. We also predict a sign reversal behavior of the thermopower with increasing temperature that is consistent with the experiments by Parsons et al.

Figure 1

The experimentally observed thermopower oscillations (solid line) in the (a) parallelogram and in the (b) house interferometer. The figure is adopted from Ref. 12.

Figure 2

The three-terminal junction formed by a contact attached to the main normal-metal wire $N$. The generalized rigid boundary conditions at the junction are given by Eqs. (27, 28) in the matrix notation. Green’s function $\check{g}$ is continuous along the wire $N$ but there is a jump in Green’s function across the tunnel barrier so that $\check{g}\ne \check{g}$ at the junction. The kinetic component of Eq. (28) is equivalent to Eqs. (30, 31).

Figure 3

The parallelogram interferometer. Diffusive normal-metal wire $N$ of the length $L=L_1+L_2$ connects the temperature reservoirs denoted as $T_1$ and $T_2$. The superconducting wire (dark line) is contacting $N$ at two junctions separated by a distance $d$. The $NS$ interfaces are characterized by the transparency parameters ${\alpha }_1$ and ${\alpha }_2$. Only antisymmetric dependence of the thermopower on the magnetic flux is allowed by symmetry. The effect requires ${\ell }_1\ne {\ell }_2$ or ${\alpha }_1\ne {\alpha }_2$.

Figure 4

(Color online) The function $Y\left(T/E_c,d/L,\pi /2\right)$ from Eq. (61) vs the ratio $T/E_c$ for the parallelogram interferometer with the symmetric arms ${\ell }_1={\ell }_2$. The thermopower is maximal for $T\sim 2E_c$, where $E_c=D/L^2$ is the Thouless energy associated with the normal-metal wire $N$. At high temperatures the thermopower decays according to the stretched exponential law [Eq. (60)] that is determined by the Thouless energy $E_c^{\prime }=D/d^2$ associated with the distance between the $NS$ junctions.

Figure 5

The house interferometer. The wires $N$ and $N^{\prime }$ are normal-metal wires. The dark wire is superconducting. The transparency parameters $\alpha$ and ${\tilde{\alpha }}_{1,2}$ are defined by the ratio between the transmission probability per channel and the effective barrier length. Both symmetric and antisymmetric dependences of the thermopower on the magnetic flux $\Phi$ is possible in this setup. The antisymmetric effect, however, requires $d_1\ne d_2$. A charge imbalance between the superconducting wire and the temperature reservoirs is necessary for the symmetric thermopower.

Figure 6

An example of interfering trajectories that contribute to the thermoelectric response of the house interferometer.

Figure 7

The function $\tilde{Y}\left(T/E_c\right)$ from Eqs. (87, 88) vs the ratio $T/E_c$ for the house interferometer with $d_1=d_2$. The function is maximal for $T\approx 4E_c$ and changes sign at $T\approx 24E_c$, where $E_c=D/L^2$ is the Thouless energy associated with the normal-metal wire $N$. The inset shows the temperature dependence of the thermopower in Eq. (87) under the assumption that the imbalance $\mu$ and the length $L$ are temperature independent. The thermopower is maximal at $T\approx E_c$ and decays as $T^{-2}$ for $T⪢24E_c$.