Spin-Triplet Superconductivity with Line Nodes in Sr₂RuO₄

Abstract

Several possible odd-parity states are listed up group-theoretically and examined in light of recent experiments on Sr2RuO4. Those include some of the f -wave pairing states, d(k) ∝ zkxky(kx + iky) andẑ(k 2 x − k 2 y )(kx + iky) and otherẑ(kx + iky) cos ckz (c is the c-axis lattice constant) as most plausible candidates. These are time-reversal symmetry broken states and have line nodes running either vertically (the former two) or horizontally (the latter), consistent with experiments. Characterizations of these states and other possibilities are given.KEYWORDS: spin triplet superconductivity, line node of the energy gap, time-reversal symmetry breakingThe possibility of the spin triplet pairing in Sr 2 RuO 4 is pointed out 1) soon after the discovery of the superconductivity.2) Experimental evidence for the spin triplet pairing is accumulated. The temperature-independent Knight shift for the external magnetic field parallel to the conducting plane (the basal plane) is thought to be the evidence for the spin triplet pairing with the d-vector aligned to the z-axis.3) The µSR experiment 4) indicates that the magnetic field is spontaneously induced and the time reversal symmetry seems to be broken in the superconducting state. These experiments were explained by the p-wave spin triplet state with the order parameter,where ∆ 0 is a constant. This state is analogous to the Anderson-Brinkman-Morel (ABM) state, which is identified as the A phase in superfluid 3 He.7) The AMB state in superfluid 3 He has point nodes of the energy gap at k x = k y = 0 on the spherical Fermi surface. However, since the Fermi surface of Sr 2 RuO 4 consists of three cylindrical surfaces, the above p-wave state has a finite energy gap on the Fermi surface. The previous experiments of specific heat 8) C(T ) and NMR relaxation rateshowed that about a half of the density of states remains at T = 0 in the superconducting state. In order to explain this residual density of states, the non-unitary states and the orbital dependent superconductivity are proposed. 10, 11)However, it is reported in very recent experiments that the residual density of states are very small in the better samples, i.e. C(T ) ∝ T 2 and T −1 1∝ T 3 at low temperatures.12, 13) These temperature dependences are interpreted as a consequence of the line nodes of the energy gap. The residual density of states observed in the previous experiments may be due to the impurity effects on the anisotropic superconductivity with line nodes.An anisotropic energy gap model caused by the finite range interaction in the two-dimensional plane is proposed to explain the line-node-like temperature dependences of the specific heat and relaxation rate in Sr 2 RuO 4 .14) The order parameter in this case is given bywhere a is the lattice constant in the basal plane of the tetragonal crystal D 4h . This anisotropic energy gap should give the exponential temperature dependence as exp(−α∆/T ) at temperatures below the smallest energy gap ∆ in clean samples, where α is a constant of the order 1. As ...

Universal Heat Transport inSr2RuO4

“…This value is close to typical values for several nodal gap structures. For the 2D f -wave gap with four vertical line nodes ∆ = ∆ 0 cos(2φ) [30], a is 4/π [4], µ = 2, ∆ 0 = 2.14k B T c in the weak coupling limit and we obtain µ∆ 0 /a = 0.43 meV. For the state with horizontal line node, indicated in recent directional experiments [25][26][27], the universal thermal conductivity was not considered theoretically.…”

“…Early theoretical prediction of the fully gapped p-wave chiral state [20] did not find support in recent experiments, suggesting the SC gap with line nodes [21][22][23][24], likely running parallel to the conducting plane [25][26][27]. Several new theoretical models have been proposed [28][29][30][31] to explain the experimentally observed "triplet-chiral-nodal" SC order parameter. …”

Edge states and determination of pairing symmetry in superconductingSr2RuO4

“…Ref. [17] also proposed the pwave with horizontal lines of nodes at certain k z . Its pairing potential can be obtained from Eq.…”

“…a small value of k max y . In this situation, only the states with k y ≈ 0 and hence E(k y ) ≈ 0 are probed; thus all pairing potentials, except the f -wave (17), would show a ZBCP. However, such a ZBCP, in contrast to nonchiral cuprates [25] and organic superconductors [34], does not identify the pairing symmetry uniquely.…”

“…However, the power-law temperature dependences found in specific heat [9], nuclear relaxation rate [10], penetration depth [11], thermal conductivity [12], and ultrasonic attenuation [13,14] indicate nodes in the energy gap. In response to these experiments, the following alternative order parameters were proposed [15]: anisotropic p-wave [16], p-wave with horizontal lines of nodes [17], and fwaves with vertical [17,18] or horizontal lines of nodes [19]. It was also proposed that the α and β bands of Sr 2 RuO 4 are either not superconducting [20] or have horizontal lines of nodes [21].…”