Influence of vortices and phase fluctuations on thermoelectric transport properties of superconductors in a magnetic field

We study heat transport and thermoelectric effects in two-dimensional superconductors in a magnetic field. These are modeled as granular Josephson-junction arrays, forming either regular or random lattices. We employ two different models for the dynamics: relaxational model-A dynamics or resistively and capacitively shunted Josephson junction dynamics. We derive expressions for the heat current in these models, which are then used in numerical simulations to calculate the heat conductivity and Nernst coefficient for different temperatures and magnetic fields. At low temperatures and zero magnetic field the heat conductivity in the RCSJ model is calculated analytically from a spin wave approximation, and is seen to have an anomalous logarithmic dependence on the system size, and also to diverge in the completely overdamped limit $C\to 0$. From our simulations we find at low magnetic fields that the Nernst signal displays a characteristic “tilted hill” profile similar to experiments and a nonmonotonic temperature dependence of the heat conductivity. We also investigate the effects of granularity and randomness, which become important for higher magnetic fields. In this regime geometric frustration strongly influences the results in both regular and random systems and leads to highly nontrivial magnetic field dependencies of the studied transport coefficients.

Figure 1

Equivalent circuit model for Langevin dynamics. The cross denotes the Josephson junction with critical current $I_{\mathit{ij}}^C$.

Figure 2

Equivalent circuit model for RCSJ dynamics.

Figure 3

Part of a random lattice with the length $d_{\mathit{ij}}^{\perp }$ and the vector ${\mathbf{r}}_{\mathit{ji}}$ defined. The dark (green) lines make up the direct Delaunay lattice and the light (orange) structure is the dual Voronoi lattice.

Figure 4

On each plaquette $I$ and $J$, defined with respect to the direct lattice points $i$ and $j$, there are loop currents ${\mu }_I$ and ${\mu }_J$. The difference between these loop currents ${\mu }_{\mathit{ij}}={\mu }_I-{\mu }_J$ is the transverse part of the total current on the link from $i$ to $j$.

Figure 5

Heat conductivity $\kappa$ vs temperature $T$ for an $L\times L$ square lattice with underdamped RCSJ dynamics ($Q=10$). The different curves correspond to different system sizes $L$. The inset shows the logarithmic dependence on system size $L$ at low $T$ ($T=0.1$). The circles are simulation data and the smooth red curve is the analytic result obtained from a linearized model.

Figure 6

Heat conductivity $\kappa$ vs temperature $T$ for an $L\times L$ square lattice with RSJ dynamics. Here the temperature-independent spin wave background of $\kappa$ has been subtracted. The different curves correspond to different system sizes $L$. Inset: The same curves but scaled with $1/\ln (L/0.7)$. The collapse is very good over the entire temperature range, except from close to the transition temperature $T_{\mathrm{BKT}}\simeq 0.9$.

Figure 7

Nernst signal $e_N$, heat conductivity $\kappa$, and electrical resistivity $\rho$ vs temperature $T$ at different fillings $f$ for a $20\times 20$ square lattice with Langevin dynamics.

Figure 8

Nernst signal $e_N$, heat conductivity $\kappa$, and electrical resistivity $\rho$ vs temperature $T$ at different fillings $f$ for a $20\times 20$ square lattice with RCSJ dynamics ($Q=0.5$). In the $\kappa$ plot, the temperature-independent spin wave background has been subtracted.

Figure 9

Plot of $\kappa /e_N$ and ${\alpha }_{\mathit{xy}}T$ vs temperature $T$ for RCSJ dynamics ($Q=0.5$) on a square $20\times 20$ lattice at filling $f=0.02$. Both of these measure the heat transport per vortex $(J_x^Q/J_x^v{\Phi }_0)$. Inset: The $f$ dependence of these two quantities at low $T$ ($T=0.1$). The two smooth green curves are fits to the form $a+b\ln f$ (light green: $a=0.11$, $b=-0.19$; dark green: $a=-0.14$, $b=-0.22$).

Figure 10

Off-diagonal component of the thermoelectric tensor ${\alpha }_{\mathit{xy}}=e_N/\rho$ vs temperature $T$ for RCSJ dynamics ($Q=0.5$) on a square $20\times 20$ lattice at filling $f=0.01$. At low $T$ the curve follows the power law $~1/T$. In the high temperature region the falloff is much faster, about $~1/T^8$.

Figure 11

Nernst signal $e_N$, heat conductivity $\kappa$, and electrical resistivity $\rho$ vs filling $f$ at different temperatures for a $20\times 20$ square lattice with Langevin dynamics.

Figure 12

Nernst signal $e_N$, heat conductivity $\kappa$, and electrical resistivity $\rho$ vs filling $f$ at different temperatures for a $20\times 20$ square lattice with RSJ dynamics.

Figure 13

Contour plot of the Nernst signal $e_N$ for a $20\times 20$ lattice with Langevin dynamics in the temperature-filling plane. The dashed white line joins the maxima of $e_N$.

Figure 14

Nernst signal $e_N$, heat conductivity $\kappa$ and electrical resistivity $\rho$ vs filling $f$ at $T=1$ for random lattices of size $20\times 20$ with $d_{\min }=0.8$, using Langevin dynamics. Results shown are for two models with different critical current distributions $I_{\mathit{ij}}^c=I^c$ (red or gray) and $I_{\mathit{ij}}^c~d_{\mathit{ij}}^{\perp }$ (black). Each curve is an average over eight disorder realizations.

Figure 15

Nernst signal $e_N$, heat conductivity $\kappa$, and electrical resistivity $\rho$ vs filling $f$ at $T=1$ for random lattices of size $20\times 20$ with different degrees of disorder set by the parameter $d_{\min }$ using Langevin dynamics. Each curve is an average over 16 disorder realizations. The lower panel shows examples of Voronoi lattices obtained from random packings of grains with different $d_{\min }$.

Figure 16

Contour plot of the Nernst signal $e_N$ in the filling–grain-size standard deviation plane for random lattices with Langevin dynamics at $T=1$.