Heat transport in nonuniform superconductors

We calculate electronic energy transport in inhomogeneous superconductors using a fully self-consistent nonequilibrium quasiclassical Keldysh approach. We develop a general theory and apply it to a superconductor with an order parameter that forms domain walls of the type encountered in the Fulde-Ferrell-Larkin-Ovchinnikov state. The heat transport in the presence of a domain wall is inherently anisotropic and nonlocal. The bound states in the nonuniform region play a crucial role and control heat transport in several ways: (i) they modify the spectrum of quasiparticle states and result in Andreev reflection processes and (ii) they hybridize with the impurity band and produce a local transport environment with properties very different from those in a uniform superconductor. As a result of this interplay, heat transport becomes highly sensitive to temperature, magnetic field, and disorder. For strongly scattering impurities, we find that the transport across domain walls at low temperatures is considerably more efficient than in the uniform superconducting state.

Figure 1

A typical experimental setup for heat conductivity measurements involves establishing a steady-state current in the sample and measuring the effective local temperature $T_1$ and $T_2$. The heat conductance is defined as $\kappa =I_h\times 2L/(T_1-T_2)$. We set a two-dimensional superconductor in the $xy$ plane and for a given heat current $I_h$ along the $x$ axis we calculate the temperature difference between thermometers TH1 and 2 at $x=\pm L$. The sin line is a schematic representation of an FFLO modulation of the order parameter with shaded regions representing domain walls.

Figure 2

Local vs global equilibrium picture. The system is driven out of equilibrium by a steady uniform heat current $j_h$. The local equilibrium picture assumes that when heat flows, a local temperature $T\left(x\right)$ can be defined, and its gradient determines the magnitude of $j_h$. It is typically used in uniform-state problems, and we use it here to define boundary conditions for distribution functions away from the nonuniform (shaded) region of the order parameter. Our main approach, however, is to expand the propagators around some global equilibrium value of the temperature $T$: $g\left(x\right)=g_{\mathrm{eq}}\left(T\right)+g_c\left(x\right)$ where $g_c\left(x\right)$ determines current $j_h$.

Figure 3

Numerical integrations of Eqs. (15) and (20), in the shaded region, performed from $x=\mp L$ to $x=\pm L$ for right/left going $\left({\widehat{p}}_x\lessgtr 0\right)$ trajectories along $\widehat{p}$. We start the numerical integration at the white/black circles with the uniform Riccati amplitude given by Eqs. (30) and (31), see main text. The (half-) temperature bias $dT$ is the unknown that we numerically determine to satisfy Eq. (23).

Figure 4

(Top) Self-consistent OP profile $\mathrm{\Delta }\left(x\right)$ for a single DW (solid line). This solution is used to construct a non-self-consistent profile with $N_{\mathrm{DW}}$, case of 4 DWs with separation $X_{\mathrm{FFLO}}\approx 20\xi$ is shown by the dashed line. (Bottom) Local density of states (DoS) in Born (orange) and unitary (blue) limits for ${\ell }_{\mathrm{N}}=\pi \xi /0.3$. At the domain (solid lines), the peak at zero energy indicates the Andreev bound states (ABS).

Figure 5

Effective change in thermal length $d{L}^{\prime }$ (in units of $\xi$) across a single domain wall relative to the uniform case, as function of temperature. Numerical system size is $2L=16\pi \xi$. Different colors correspond to unitary limit (U, blue), Born (B, orange), and intermediate phase shift $\delta =\pi /4$ (I, green). Solid lines are for scattering rate $2\mathrm{\Gamma }{sin}^2\delta =1/{\tau }_N=0.6T_c$, dashed lines are for a cleaner case $1/{\tau }_N=0.2T_c$. In the unitary limit, $d{L}^{\prime }$ is nonmonotonous, and at low temperature $T<W_{\mathrm{imp}}$, the heat conductance through a domain wall is larger than in uniform case.

Figure 6

Energy dependence of the heat current kernel at $T=0.3T_c$ for transport across a domain wall (solid lines) and the uniform superconductor (dashed lines) for ${\ell }_{\mathrm{N}}=\pi \xi /0.3$. In Born or clean limit (orange lines), the ability to transport heat at low energy is suppressed by the presence of a domain: at low energy $K_{\mathrm{DW}}\left(\epsilon \right)<K_u\left(\epsilon \right)$. By contrast, in unitary limit (blue lines), the coupling between impurity band and Andreev bound states enhances energy transport: at low energy $K_{\mathrm{DW}}\left(\epsilon \right)>K_u\left(\epsilon \right)$.

Figure 7

Inverse local impurity scattering length ${\ell }_N/{\ell }_{\mathrm{imp}}(x,\widehat{p},\epsilon )\approx {\ell }_N/{\ell }_{\mathrm{imp}}(\epsilon ,x)$ (weakly dependent on momentum directions), as a function of energy, for ${\ell }_{\mathrm{N}}=\pi \xi /0.3$. In Born limit (orange), the mean-free path is large in the bulk (dashed lines) and becomes small at the domain wall (solid lines). For unitary scattering (blue), on the right, this behavior is reversed: the zero-energy peak in the DOS results in suppression of scattering rate at the domain wall and longer mean-free path.

Figure 8

Effective thermal length $d{L}^{\prime }$ (normalized by $\xi$) across $N_{\mathrm{DW}}$ domain walls, for low temperature $T/T_c=0.05$ (large symbols, solid lines) and intermediate temperature $T/T_c=0.5$ (small symbols, dashed lines). The scattering rate $1/{\tau }_N=0.3T_c$ is used for various impurity strengths: Born (B), unitary (U), and intermediate $\delta =\pi /4$ (I). This is an “independent domain walls” regime where the heat conductivity contributions from each domain add up, as is clear from linear dependency $c_1{N}_{\mathrm{DW}}+c_2$ shown by lines. At low temperature, the unitary and intermediate strength disorder has a negative slope consistent with the single-domain result in Fig. 5, coming from low-energy states' transport. At intermediate temperature we have a suppression of heat flow due to independent Andreev reflection processes, with positive slope and linear increase in the thermal length $d{L}^{\prime }$ with $N_{\mathrm{DW}}$. In the Born limit at low temperature, the dependence is more complicated due to the large extent of bound states and a more intricate impurity band energy dependence for $T\sim W_{\mathrm{imp}}$.

Figure 9

Effect on thermal conductance of Andreev reflections from a set of $N_{\mathrm{DW}}$ domain walls (clean limit). Heat transport through more than ten consecutive domains is dominated by extended bound states along nodal directions on the Fermi surface. Phase space of those states and their contribution to the heat transport grow with temperature. The fitting line through numerical points is explained in the text.

Figure 10

Effect of the Zeeman field splitting $h=\mu H/T_c$ on thermal transport across a single domain wall. Unitary limit with scattering rate $1/{\tau }_N=0.3T_c$. The bound states, shifted by $h=0.5$ contribute to a reduction of the thermal length, $d{L}^{\prime }$, in a wide range of temperatures. When the Zeeman shift is very large $h=1$, the contributions from bound and continuum states mix up leading to a very nonmonotonic temperature dependence.

Figure 11

Uniform thermal conductivity as a function of temperature. At low temperature, $T\lesssim 0.3T_c \left(\epsilon \lesssim 0.6T_c\right)$, the thermal conductivity in Born limit (green) is higher than that in the Unitary limit (blue), indicating that it is dominated by the large mean-free path of quasiparticles. Solid and dashed lines correspond to mean-free paths ${\ell }_N=\pi \xi /0.3$ and ${\ell }_N=\pi \xi /0.2$, respectively.

Figure 12

Spectral and transport properties of a uniform $d$-wave superconductor. Angle-resolved DoS, $N(\widehat{p},\epsilon )/N_F$ (top row), mean-free path ${\ell }_e(\widehat{p},\epsilon )/{\ell }_N$ (middle row), and impurity scattering length ${\ell }_{\mathrm{imp}}\left(\epsilon \right)/{\ell }_N$ (bottom row) are plotted in Born and unitary limits, for the normal state mean-free path ${\ell }_N\approx 10\xi$, where ${\ell }_{\mathrm{imp}}=v_F/2\mathrm{Im}\left[{\mathrm{\Sigma }}_{\mathrm{imp}}\right]$. Different curves represent different momentum directions spanning the $d$-wave clover from a node to antinode (solid blue to red dashed lines), as shown in inset. In unitary limit, the low-energy impurity band in DoS is large, and the mean-free path is reduced by enhanced impurity scattering. By contrast, in the Born limit, the impurity band is exponentially small and the mean-free path of nodal quasiparticles is longer.

Figure 13

Thermal transport properties of a clean superconductor across a single domain wall (solid lines) compared against a uniform superconductor (dashed lines). (a) Local DoS for the momenta directions shown in (d) at the domain wall $N(\widehat{p},\epsilon ,x)$ with part of the spectral weight (shaded area) moved from continuum states into zero-energy bound states, which form a very sharp peak not resolved on this scale. (b) The Andreev reflection length scale ${\ell }_{\mathrm{\Delta }}(\epsilon ,\widehat{p},x)$. In a uniform superconductor, it is infinite for above-gap energies $1/{\ell }_{\mathrm{\Delta }}\left(\left|\epsilon \right|>\left|\mathrm{\Delta }\right(\widehat{p}\left)\right|\right)=0$, while at the domain wall it is finite for all energies and even changes sign. (c) Kernel of the heat current $K(\widehat{p},\epsilon )$ for four momentum directions and integrated over the Fermi surface. With the domain wall the kernel $K(\epsilon ,\widehat{p})<1$ is suppressed due to Andreev reflection.