Universal Bound to the Amplitude of the Vortex Nernst Signal in Superconductors

A liquid of superconducting vortices generates a transverse thermoelectric response. This Nernst signal has a tail deep in the normal state due to superconducting fluctuations. Here, we present a study of the Nernst effect in two-dimensional heterostructures of Nb-doped strontium titanate (STO) and in amorphous MoGe. The Nernst signal generated by ephemeral Cooper pairs above the critical temperature has the magnitude expected by theory in STO. On the other hand, the peak amplitude of the vortex Nernst signal below $T_c$ is comparable in both and in numerous other superconductors despite the large distribution of the critical temperature and the critical magnetic fields. In four superconductors belonging to different families, the maximum Nernst signal corresponds to an entropy per vortex per layer of $\approx k_{B}ln2$.

Figure 1

The Nernst effect in two-dimensional Nb-doped strontium titanate: (a) Schematic view of the heterostructure. (b) Sketch of the two-thermometers–one-heater setup used in these measurements. (c) Resistivity ${\rho }_{xx}$ as a function of temperature. The midpoint resistive transition at $T_c=0.341\mathrm{K}$ shifts to lower temperatures with increasing magnetic field. The inset shows the correlated evolution of this midpoint and the Nernst peak with temperature and magnetic field. (d)–(j) The Nernst signal $N$ and ${\rho }_{xx}$ vs temperature at different magnetic fields, both the Nernst peak and the resistive transition vanish at $B=0.1\mathrm{T}$.

Figure 2

The Nernst response in the normal state due to superconducting fluctuations: (a) The Nernst coefficient as a function of temperature in two-dimensional Nb:STO across the critical temperature at $B=5\mathrm{mT}$. (b) The off-diagonal component of the thermoelectric tensor, ${\alpha }_{xy}=(N/\rho )a_0$ as a function of reduced temperature, $\epsilon =(T-T_c)/T_c$. The dashed line represents ${\epsilon }^{-1.3}$, the solid line what is expected by Eq. (1). The inset compares the magnitude of normal state ${\alpha }_{xy}$ (at $T=1.5T_c$) in different superconductors [19, 20, 21] as a function of their upper critical field, $H_{c2}$. The dashed line represents the magnitude expected by Eq. (1) and a coherence length given by $H_{c2}(0)$.

Figure 3

Nernst effect in amorphous MoGe: (a) Evolution of the resistive superconducting transition in amorphous films of MoGe with magnetic field. (b) Nernst effect $N$ in the same sample. The color code for magnetic fields is identical to the one used in the upper panel. The inset schematically depicts the structure of the sample consisting of alternating superconducting and insulating layers.

Figure 4

Peak Nernst signal in different superconductors: (a) The Nernst signal as a function of temperature in STO and MoGe (present work) compared with data on an amorphous film of ${\mathrm{InO}}_x$ [26], on ${\mathrm{FeSe}}_{0.6}{\mathrm{Te}}_{0.4}$ [28], on $\kappa \text{−}(\mathrm{ET}{)}_2\mathrm{Cu}[\mathrm{N}(\mathrm{CN}{)}_2]\mathrm{Br}$ [29], on ${\mathrm{La}}_{1.92}{\mathrm{Sr}}_{0.08}{\mathrm{CuO}}_4$ [30] and on ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ [6]. For each system, the critical temperature is indicated together with the magnetic field at which the observed peak was the largest. In all systems, this magnetic field is the one at which the peak is largest, except in Bi2212 [6], for which the data were restricted to 12 T. The field and temperature dependence of the Nernst signal are shown as color plots for (b) Nb:STO, (c) MoGe, and (d) ${\mathrm{FeSe}}_{0.6}{\mathrm{Te}}_{0.4}$ [28]. Note the similarity in the peak amplitude in contrast to the large difference in the field and temperature scales.