Vertical Line Nodes in the Superconducting Gap Structure of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

There is strong experimental evidence that the superconductor ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ has a chiral $p$-wave order parameter. This symmetry does not require that the associated gap has nodes, yet specific heat, ultrasound, and thermal conductivity measurements indicate the presence of nodes in the superconducting gap structure of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. Theoretical scenarios have been proposed to account for the existence of deep minima or accidental nodes (minima tuned to zero or below by material parameters) within a $p$-wave state. Other scenarios propose chiral $d$-wave and $f$-wave states, with horizontal and vertical line nodes, respectively. To elucidate the nodal structure of the gap, it is essential to know whether the lines of nodes (or minima) are vertical (parallel to the tetragonal $c$ axis) or horizontal (perpendicular to the $c$ axis). Here, we report thermal conductivity measurements on single crystals of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ down to 50 mK for currents parallel and perpendicular to the $c$ axis. We find that there is substantial quasiparticle transport in the $T=0$ limit for both current directions. A magnetic field $H$ immediately excites quasiparticles with velocities both in the basal plane and in the $c$ direction. Our data down to $T_c/30$ and down to $H_{c2}/100$ show no evidence that the nodes are in fact deep minima. Relative to the normal state, the thermal conductivity of the superconducting state is found to be very similar for the two current directions, from $H=0$ to $H=H_{c2}$. These findings show that the gap structure of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ consists of vertical line nodes. This rules out a chiral $d$-wave state. Given that the $c$-axis dispersion (warping) of the Fermi surface in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ varies strongly from sheet to sheet, the small $a-c$ anisotropy suggests that the line nodes are present on all three sheets of the Fermi surface. If imposed by symmetry, vertical line nodes would be inconsistent with a $p$-wave order parameter for ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. To reconcile the gap structure revealed by our data with a $p$-wave state, a mechanism must be found that produces accidental line nodes in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$.

Popular Summary

Superconductivity is a state of matter appearing at low temperature wherein an electric current can flow without any resistance whatsoever. The electrons in a superconductor spontaneously form pairs, and the quantum-mechanical wave function adopts a particular symmetry, dictated by the nature of the pairing interaction. In conventional superconductors, for example, the symmetry is called $s$ wave, which forms when the force that binds the electrons together is the same in all directions. The ruthenium oxide ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ is one of the rare superconductors whose symmetry is believed to be $p$ wave—a state for which paired electrons have parallel spins. One way to investigate the pairing is to measure how electrons carry heat in different directions of a crystal. We report on an investigation of heat conduction in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ that raises questions about how the electrons are actually paired in the superconductor.

We measured heat conduction in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ down to a temperature of 50 mK for currents parallel and perpendicular to the ${\mathrm{RuO}}_{-2}$ planes in the crystal. We detect unpaired electrons at temperatures near absolute zero along ‘‘vertical” lines (or nodes), perpendicular to the ${\mathrm{RuO}}_{-2}$ planes, implying a pronounced angular variation in the pairing interaction. This is typical of the $d$-wave state (where the electron pairing strength resembles a four-leaf clover) seen in some exotic superconductors, but not a $p$-wave state. However, other experiments also reveal $p$-wave behavior. One possible solution is that the electrons are paired in an $f$-wave state, a combination of $p$ wave and $d$ wave.

We hope that our findings will stimulate further investigations of this fascinating superconductor. In particular, measurements that are sensitive to the precise position of the nodes (do the leaves of the clover point along the crystal axis or along the diagonal?) and the phase (do all of the leaves have the same sign?) would elucidate the symmetry of the pairing interaction.

Figure 1

Residual linear term in the thermal conductivity, ${\kappa }_0/T$, as a function of impurity scattering rate $\mathrm{\Gamma }$, both normalized to unity at $\mathrm{\Gamma }={\mathrm{\Gamma }}_c$, the critical scattering rate needed to suppress superconductivity. The blue lines are theoretical calculations for a $d$-wave state [22] and a fully-gapped $p$-wave state [27], as indicated (left axis). In the clean limit ($\mathrm{\Gamma }\to 0$), ${\kappa }_0/T$ vanishes in the $p$-wave case, while it reaches a nonzero value in the $d$-wave case (open circle), whose value is given by Eq. (1), estimated at ${\kappa }_0/T=15.8\mathrm{mW}/{\mathrm{K}}^2\mathrm{cm}$ in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ (see text). The experimental values of ${\kappa }_0/T$ measured in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ are plotted as red symbols (right axis, circles, Ref. [21]; square, this work), taking $\hslash {\mathrm{\Gamma }}_c=k_B{T}_{c0}$. The black lines are the zero-energy density of states $N_0$, normalized by the normal-state value $N_F$ (left axis, solid, full-gap $p$-wave [28]; dashed, $d$-wave [23]). Black triangles show $N_0/N_F$ for a $p$-wave state with a deep gap minimum (${\mathrm{\Delta }}_{\min }\simeq {\mathrm{\Delta }}_{\max }/4$) [30].

Figure 2

(a) In-plane ($a$-axis) thermal conductivity ${\kappa }_a(T)$ of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ at $H=0$ (open circles). The black line is a Fermi-liquid fit to the normal-state data, ${\kappa }_N/T=L_0/(a+b{T}^2)$, extended below $T_c$. The arrow marks the location of the superconducting transition temperature, $T_c=1.2\mathrm{K}$, defined as the temperature below which $\kappa /T$ deviates from its normal-state behavior. Note that the contribution of phonons to ${\kappa }_a$, ${\kappa }_p$, is negligible up to 3 K, so ${\kappa }_a\simeq {\kappa }_e$, which is the electronic contribution. The red dashed line is a calculation for a three-band model of a $p$-wave state with deep minima in the gap structure [17] (see text). It provides a good description of the data at high temperatures, but it fails below 0.3 K. (b) Zoom at low temperatures. Data taken in a magnetic field $H=0.25\mathrm{T}$ ($H\parallel a$) are also shown (blue dots). The solid black lines are a fit of the data to the form $\kappa /T={\kappa }_0/T+c{T}^n$.

Figure 3

Out-of-plane ($c$-axis) thermal conductivity of ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. (a) At $H=0$. The black line is a linear fit to the normal-state data. The arrow marks the location of $T_c=1.2\mathrm{K}$. In this direction, ${\kappa }_e\ll {\kappa }_p$, so the purely electronic term is obtained as ${\kappa }_c(T)/T$ in the $T=0$ limit. (b) Zoom of the data at low temperatures, at $H=0$ (red dots). The black line is a fit of the data to Eq. (3) below 0.35 K, with the phonon conductivity in the $T\to 0$ limit given by ${\kappa }_p=B{T}^{\alpha }$, with $\alpha =3.0$ and $B$ given by sound velocity and sample dimensions (see text). The other two lines are the same fit but with $\alpha =2.7$, to take into account the effect of specular reflection, and with $B$ either fixed (red line) or free (blue line) (see text). (c) Same data as in (b) (red, $H=0$), compared with data in a magnetic field $H=25\mathrm{mT}$ (blue) and $H=0.35\mathrm{T}$ (burgundy), with $H\parallel a$. (d) Increase in ${\kappa }_c/T$ with field $H\parallel a$, where ${\kappa }_c/T$ is either measured at $T=60\mathrm{mK}$ (open circles) or extrapolated to $T=0$, either linearly as in panel (c) (crosses) or through a fit as in panel (b) (full red dots). The red line is a guide to the eye.

Figure 4

Residual linear term ${\kappa }_0/T$ as a function of a magnetic field $H$ applied along the $a$ axis ($H\parallel a$). (a) For a current in the plane ($J\parallel a$). The data points (black dots) are ${\kappa }_{a0}/T$ obtained by fitting ${\kappa }_a/T$ vs $T$ as in Fig. 2. The black line is a constant fit to the data above $H_{c2}$ (negligible magnetoresistance for that current direction). It defines ${\kappa }_{aN}/T$ vs $H$, and it is consistent with the $H=0$ value obtained by extrapolating ${\kappa }_N/T$ above $T_c$ to $T\to 0$ [Fig. 2]. (b) Same as in (a) but for a current along the $c$ axis ($J\parallel c$). The data points (red dots) are ${\kappa }_{c0}/T$ obtained by fitting ${\kappa }_c/T$ vs $T$ as in Fig. 3. Above $H_{c2}$, ${\kappa }_{c0}/T$ decreases slightly due to magnetoresistance (see text). The red line is a fit of the data above $H_{c2}$ to ${\kappa }_N/T=a/(b+c{H}^2)$, which defines ${\kappa }_{\mathrm{cN}}/T$ vs $H$ for this current direction. The value at $H\to 0$ is ${\kappa }_{cN}/T=67\pm 7\mu \mathrm{W}/{\mathrm{K}}^2\mathrm{cm}$. (c) Field dependence of ${\kappa }_{a0}/T$ (black dots) and ${\kappa }_{c0}/T$ (red dots) normalized to their normal-state value, both plotted as $({\kappa }_0/T)/({\kappa }_N/T)$ vs $H/H_{c2}$, with $H_{c2}=1.25\mathrm{T}$. For simplicity, we define ${\kappa }_0/{\kappa }_N\equiv ({\kappa }_0/T)/({\kappa }_N/T)$. The error bars on ${\kappa }_0/{\kappa }_N$ come from the combined uncertainties in extrapolating $\kappa /T$ to $T=0$ to obtain ${\kappa }_0/T$ and in extending ${\kappa }_N/T$ below $H_{c2}$.

Figure 5

Residual linear term ${\kappa }_0/T$ as a function of magnetic field, for a field along the $c$ axis ($H\parallel c$) and a heat current along the $c$ axis ($J\parallel c$). The data are plotted as $({\kappa }_0/T)/({\kappa }_N/T)$ vs $H/H_{c2}$, with $H_{c2}=0.055\mathrm{T}$. For this field direction (in the longitudinal configuration with a tiny $H_{c2}$), the magnetoresistance in the normal state is negligible, so ${\kappa }_N/T$ is a constant below $H_{c2}$. The data points are ${\kappa }_{c0}/T$ obtained by fitting ${\kappa }_c/T$ vs $T$ as in Fig. 3 (red fit). The solid red line is a theoretical calculation for a single-band $d$-wave superconductor [59].