Distinct reduction of Knight shift in superconducting state of Sr$_2$RuO$_4$ under uniaxial strain

Shortly after the discovery of superconductivity in Sr$_2$RuO$_4$, spin-triplet pairing was proposed and further corroborated by a constant Knight shift ($K$) across the transition temperature (T$_c$). However, a recent experiment observed a drop in $K$ at T$_c$ which becomes larger under uniaxial strain, ruling out several spin-triplet scenarios. Here we show that inter-orbital spin-triplet pairing features a d-vector that rotates when uniaxial strain is applied, leading to a larger drop in the spin polarization perpendicular to the strain direction, distinct from spin-singlet pairing. We propose that anisotropic spin polarization under strain will ultimately differentiate triplet vs.~singlet pairing.

Introduction -The discovery of superconductivity in Sr 2 RuO 4 [1] has had great attention over the past two decades.It has been considered the best solid-state system which exhibits a time-reversal symmetry breaking pwave spin-triplet pairing analog of the A-phase in 3 He.[2] The microscopic route to the spin-triplet pairing in 3 He is ferromagnetic fluctuations.[3] Since a sister compound, SrRuO 3 , is a ferromagnetic metal, the p + ip spin triplet pairing proposed by Rice and Sigrist [4] was a promising candidate.Earlier experiments of nuclear magnetic resonance (NMR) and µSR had corroborated this proposal, because no change in the NMR Knight shift [5] and a broken time reversal symmetry signal across T c in µSR [6] are consistent with the order parameter.However, a scanning magnetic imaging [7] measurement showed a null signal of the associated chiral super-current, which does not support the chiral p-wave spin-triplet pairing.Since then, the pairing symmetry of Sr 2 RuO 4 has remained a mystery with controversial experimental results.[8,9] Recently, Pustogow et al. [10] made an important breakthrough in determining the spin component of the order parameter, as they reported a 20-30% drop in the spin polarization M s below T c in unstrained samples in contrast to the earlier NMR reports.[5] When the sample is strained along the a-axis, the spin polarization along the b-axis drops almost 75%.This rules out the d-vector along the c-axis as in the chiral pairing proposal [4].Since the no-change Knight shift across T c has been the strong piece of evidence of a spin-triplet, this observation may rule out several spin-triplet pairings including the d-vector in the ab-plane and along the c-axis as listed in Ref. 10.
Here we show that inter-orbital spin-triplet pairings [11] exhibit significant reduction in the spin polarization under strain, and it becomes anisotropic relative to the strain direction.For inter-orbital spin-triplet pairings, the d-vector is locked in momentum space via spin-orbit coupling (SOC) as shown in Fig. 1.When the uniaxial strain is applied, the strength of the pairing is enhanced due to the van Hove singularity (vHS), which is true for both spin-singlet and -triplet.However, for an inter-orbital singlet spin-triplet, there is an important additional effect of uniaxial strain.It not only enhances the magnitude of the pairing, but also rotates the direction of the d-vector as shown in Fig. 1(b), be- cause the strain changes the composition of orbitals which then affects the d-vector direction.The d-vector rotation creates an anisotropy between spin polarizations parallel vs. perpendicular to the strain direction.When the strain is applied along the a-axis, the d-vector rotates towards the b-axis as shown by the red arrows in Fig. 1, leading to a larger drop in the spin-polarization along the b-axis than that of the unstrained case, as reported in Ref. 10.With the same strain condition, the a-axis polarization drop should be smaller.For a singlet, the two spin polarizations are the same.Thus we propose a NMR Knight shift experiment with the field along the a-axis under the a-axis strain, to be compared with the b-axis polarization.This will ultimately differentiate spin-triplet vs. -singlet pairings.
Below we formulate the proposed idea using a model which consists of a Kanamori interaction and a t 2g tight binding model with SOC.While the on-site Kanamori interaction restricts the pairing on a site (i.e., s-wave), it can arXiv:1912.02215v1[cond-mat.supr-con]4 Dec 2019 be generalized to further neighbour interactions leading to any even-parity inter-orbital-singlet spin-triplet pairing.
Microscopic Hamiltonian -Sr 2 RuO 4 is a multi-orbital system with non-negligible SOC.The orbital degrees of freedom allow four distinct pairings which satisfy the antisymmetric fermion wave function requirement, i.e., ∆(k) = − ∆T (−k).They are four types; (i) even-parity intra-orbital (or inter-orbital-triplet) spin-singlet (φ ν ), (ii) odd-parity inter-orbital-singlet spin-singlet (iii) odd-parity intra-orbital (or inter-orbital-triplet) spin-triplet, and (iv) even-parity inter-orbital-singlet spin-triplet ( Dν ), where ν represents intra-and inter-orbitals among t 2g .[11] A generic Hamiltonian H = H kin + H SOC + H int consisting of a tight binding model, SOC, and Kanamori interaction is considered.The tight binding and SOC terms are used to reproduce the Fermi surface (FS) reported in Ref. 12, and are listed in the Supplemental Material (SM).The underlying FS of three bands, α, β, and γ is reported earlier [13][14][15], and was further refined in Ref. 12 shown as the solid lines in Fig. 1.The interaction term is given by and can be expressed using the pairing order parameters: which are given by with l = x, y, z. λν and ˆ ν are 3 × 3 symmetric and antisymmetric matrices in the orbital basis under the exchange of the t 2g orbitals for three different inter-orbital matrices denoted with ν = X (between xz and xy orbitals), Y (yz and xy), Z (xz and yz), respectively.Their expressions are given in SM.The orbital-singlet spin-triplet channel has an attractive interaction for 3J > U , and the direction of d-vector denoted by ν is determined by the SOC.[11] The importance of the SOC in Sr 2 RuO 4 was also addressed earlier [11,[16][17][18][19], and recently re-emphasized.[12] Pairing gap under strain -Since the inter-orbital-singlet pairing corresponds to the inter-band-singlet pairing in the band basis, we consider the possibility of finite momentum pairing, i.e., FFLO state, because two electrons in different bands are at different energies when they have two opposite momenta k and −k.Using a self-consistent mean field theory, we find the zero-momentum q = 0 state is always the lowest state despite the pairing between different bands.However, the pairing amplitude appears to be extremely small.They are thousands of times smaller than the t 2g bandwidth, even though the attractive interaction is reasonably large.We set 3J − U = 0.401 for the current results and the mean field theory in general overestimates the gap size.In addition to the spin-triplet order parameters Dν , spin-singlets φν are also finite which are induced by the SOC.[11] Since the pairing occurs between different bands with different Fermi momenta, there exists a finite gap below the Fermi level in addition to the induced spinsinglet gap near the FS.The quasiparticle dispersion shown in SM represents strongly anisotropic gaps, which are very small in size, both near and below the Fermi level.This suggests that when the bandwidth is renormalized by electronic correlations, and becomes narrower, the inter-orbital singlet spin-triplet is further favoured.A recent dynamical mean field theory reported a strong mass renormalization of the bands [20], which would enhance the inter-orbital spintriplet pairing.
To study the uniaxial strain effects, we change the ratio of the hopping integrals along the a-and b-axes such that t jx = (1 − δ)t j and t jy = (1 + δ)t j for j = 1, 2, 3.A uniaxial strain along the a-axis corresponds to δ < 0. The change of different order parameters as a function of δ is shown in Fig. 2. The pairing gaps are roughly quadratic in δ as expected from the even parity pairing.When the γ band touches the vHS around δ = ±0.0475, the pairing amplitude is peaked.Since mean field theory causes the gap to be proportional to the transition temperature, T c is also peaked as reported in Refs.21 and 22.The overall gap size is minuscule in comparison to the energy scale of the kinetic and potential terms as discussed above.
Rotation of the d-vector under uniaxial strain -For spintriplet pairing, the d-vector represents the direction along which the spin projection of the condensed pair has eigenvalue zero [3].When SOC is finite, the mean field solutions find the pinning of the d-vector depending on the interorbital composition via SOC.For the pairing between xz and xy orbitals, the d-vector points along the x-direction, yz and xy along the y-direction, and xz and yz along the z-axis.The x-,y-, and z-axes are the same as the crystallographic axes of a, b, and c, as Sr 2 RuO 4 is a tetragonal lattice.The d-vector changes in momentum space as shown in Fig. 1(a), as the orbital composition changes along the FS.The red arrows represent the d-vector directions.The shorter the length of arrow, the bigger the z-component of the d-vector.There is a finite d-vector at every momentum point, and on average it is finite in all directions leading to a reduction of the spin polarization in all directions.
In the absence of a strain, due to the tetragonal symmetry, there is a π 2 rotational symmetry between dx and dy .This leads to the same reduction of the spin polarization along the a-and b-axes (and any other directions related to the symmetry of the tetragonal lattice).However, when the uniaxial strain is applied, the orbital composition changes mainly around X and Y regions of the Brillouin Zone (BZ) as shown by the underlying Fermi surface in Fig. 1(b).Most importantly, the yz orbital contribution to all bands increases, causing the d-vector at every momentum to rotate towards the b-axis, with the most change occurring around the diagonal direction of the BZ.This will then affect the magnitude of the spin polarization in the superconducting state, and generates a directional dependence, which we show below.
Spin polarization under strain -The magnetic susceptibility χ jj measured by the NMR Knight shift is given by M j /B j where M is the magnetization, B is an external magnetic field, and j = x, y, z.We compute the spin polarization at a site i, M i = g l L i + g s S i using a Zeeman coupling H Zeeman = i (g l L i + g s S i ) • B. The g-factors g l and g s are renormalized by SOC in solids.While the Knight shift with strong SOC should take into account total angular momentum [23], Sr 2 RuO 4 has intermediate SOC strength.Therefore, we present the angular momentum and spin polarizations separately.
The results are shown in Fig. 3 where the main plot shows the spin polarization along the x-and y-directions as the strain changes, and the inset plot shows the angular momentum polarization.Here we assume that the g-factor merely changes under the strain and temperature, and plot the ratio of the polarizations between the strained values, and the normal state unstrained cases.
As expected from the d-vector rotation under the a-axis strain, we find a greater drop in the magnetization from the normal to superconducting state in the y-direction compared with the x-direction for both the spin and orbital magnetizations, with a difference of about 20% in the spin magnetization of the superconducting state.The magnetization in the x-direction also drops due to the strain bringing the sample deeper into the superconducting state.
Extending to three-dimensional bands -Sr 2 RuO 4 has a layered structure, and one expects to see more k z dispersion of the bands originating from xz and yz orbitals due to their shape, and less dispersion from the xy orbital.The momentum dependent t 2g -orbital projection of the wavefunction for the α, β and γ bands on the three-dimensional FS was reported [19], which is consistent with the three dimensional (3D) tight binding model constructed in Ref. 24.
The β and γ bands still have significant overlap of xy and one dimensional (1D) orbitals, even though detailed composition depends on k z as shown in Ref. 19, while the α band is mainly made of 1D orbitals.Thus the above analysis done in the two-dimensional (2D) system can be simply generalized to a layered three-dimensional system.The qualitative uniaxial strain effect, i.e., the relative directional dependence of the spin polarization under a uniaxial strain, is independent of the details of c-axis hopping parameters, even though the quantitative drop may depend on the strength of the hopping parameters.Using the tight binding parameters in Ref. 24, we found the d-vector directions are similar to the 2D case.The angle φ represents the tilting from the ab-plane, which is about 17 − 19 • depending on k z .A clear rotation of the averaged d-vector is shown as a red arrow in a top corner in Fig. 4, and the main conclusion of the d-vector rotation can be generalized to the 3D model including the layer coupling.Discussion and summary -In multi-orbital systems, orbital degrees of freedom extend the types of superconducting pairings.Even-parity spin-triplet pairings are allowed when the pairing occurs between different orbitals with the antisymmetric fermionic wavefunction condition, i.e., inter-orbital singlets.In the band basis, this maps to interband spin-triplet pairing, and intra-band spin-singlet pair- ing when SOC is finite.
Given that spin-singlet and -triplets are mixed by SOC, the distinction between spin-singlet and triplets is not as sharp as pairings without SOC.However, one can still define a dominant pairing based on a primary attractive channel, and we suggest that the Hund's coupling offers the dominant inter-orbital spin-triplet pairings, and SOC induces intra-orbital spin-singlets in all three bands.On the other hand, since inversion is intact, odd-and even-parity pairings do not couple linearly, and all pairings presented here are even-parity.Odd-parity intra-orbital (and interorbital triplet) spin-triplet pairings are also possible when the ferromagnetic interaction is extended to further nearest neighbour sites [25], even though the impact of the enhanced density of states via the vHS is drastically reduced due to the odd-parity form factor sin(nk i ) where n is an integer representing nearest and further nearest neighbour distances with i = x, y, z.Similar to the current investigation, the directional dependence of the d-vector under the strain for odd-parity order parameters, is worthwhile to pursue in the future.
Another consequence of SOC is a complex order parameter.The SOC induced spin-singlet components always appear as a pure imaginary value iφ, while the spin-triplet is a purely real.The relative phase of π 2 between the two is determined by the atomic SOC.[11] While the order parameter is complex (D + iφ), it does not break time-reversal symmetry, as the time-reversal operator maps the order parameter to itself with an overall sign difference.Despite this, the complex order parameter may be important to understand the µSR [6], Josephson junction [26], and Kerr rotation [27] results, since the order parameter near impurities may change its relative phase π 2 leading to non-trivial effects.This is an open topic for future study.
Last, but not least, a limitation of the mean field theory needs some attention.The Hund's coupling should be larger than U/3 to attain an attractive interaction for spin-triplet pairings within the mean field theory [11].This is larger than most used values of Hund's coupling in 3d transition metals, where J is about 20-30% of U [28].A recent experimental study supports ferromagnetic fluctuations [29], and computational studies going beyond mean field theory support spin-triplet interactions originating from Hund's coupling at low energy scales [30,31].Thus despite the simplicity, the mean field theory captures the essence of the pairing symmetry as well as the unique response to the uniaxial strain, which is the main message of the current study.
In summary, we showed that an inter-orbital spin-triplet pairing with SOC leads to anisotropic Knight shift response under the uniaxial strain, different from spin-singlet pairing.When the strain is applied along the a-axis, interorbital pairing involving d yz and d xy is further enhanced leading to a rotation of the d-vector towards the b-axis.As a consequence of the d-vector rotation, the Knight shift becomes anisotropic relative to the strain axis.It has more drop magnetization when the magnetic field is perpendicular to the strain and less when the field is parallel to the strain.Such anisotropy is not expected in the spinsinglet, thus we propose the Knight shift measurement with the field along the a-axis, which can be compared with the data presented in Ref. 10.This will ultimately determine a long-standing debate of a possible spin-triplet pairing in Sr 2 RuO 4 .This idea can also be extended to other multiorbital systems with significant Hund's coupling and SOC.To express the spin-triplet and spin-singlet order parameters with orbital degrees of freedom, it is useful to define 3 × 3 matrices in the orbital basis.λν and ˆ ν in Eqs. ( 3) and ( 4) of the main text are defined by Given that the inter-orbital pairing occurs among t 2g orbitals, which form different bands with different Fermi momenta, the pairing gap is finite not only near the Fermi surface, but also below the Fermi level.For example, by finding the energy eigenvalues of the mean field Hamiltonian using the tight binding parameters listed above, as well as self consistent solutions for the various gap parameters, the quasiparticle dispersion is plotted in When a particle band crosses its own hole band at the Fermi level, a gap will form due to one of the induced intra-band spin-singlets, such as those labelled by the green circles in Fig. A.5(b).Therefore, all pairing at the Fermi level will be due to the induced intra-band spin-singlets.The gap on the Fermi surface is finite at every momentum, even though it is strongly anisotropic because of different orbital composition in each band.This is a combination of the spin-triplet and the induced spin-singlet order parameters.One may wonder if a finite momentum pairing, i.e., FFLO state occurs.We found that zero-momentum pairing between different bands has lower energy than an FFLO state for all parameters that we have considered.The gap parameters are increased by a factor 20× to make the gaps more visible.b) Magnification of (a) to show the pairing occurring away from the Fermi energy due to the inter-orbital nature of the triplet parameters, labelled with red circles.Two of the red circles are labelled with the parameter that is primarily responsible for the pairing at that location, while a third is left unlabelled, where there will be mixed contribution due to the mixed orbital character of the bands around this location.Green circles show two locations of the gaps at the Fermi energy due to the induced singlets, labelled with the singlet parameter primarily responsible for the pairing at that location.The dispersion is shown in the unstrained case, and therefore the π 2 rotational symmetry means that the pairings due to D Y y and φyz along Γ to Y are identical to D X x and φxz along Γ to Y, respectively.

Figure 1 .
Figure 1.The red arrows at representative momenta show the dvector on the Fermi surface (FS) obtained by the self-consistent mean field theory presented in the main text for (a) unstrained and (b) uniaxial-strain along a-axis.The d-vector rotation occurs the most in the diagonal direction of the Brillouin zone.The length of each arrow represents the in-plane component; the shorter the arrow, the bigger the c-axis component.The blue colour on the FS denotes the size of gap.The red arrow at the bottom corner of each panel represents the averaged d-vector direction projected onto the ab-plane denoted by θ; θ = 45 and 67.2 degrees in (a) and (b) respectively.

Figure 2 .
Figure 2. The size of the gap |D| as a function of the strain δ showing a roughly quadratic behavior in δ, and has a maximum at the vHS as expected.The induced spin-singlet on each orbitals φ are also shown.φ xz/yz shows the expected asymmetry with respect to ±δ.The gap denoted by the blue color intensity on the Fermi surface in Fig. 1 is dominated by the spin-singlets.See the main text for the gap distribution under the Fermi surface.

Figure 3 .
Figure 3.The spin-magnetization S in the normal (n) and superconducting (s) states as a function of the strain δ normalized to the unstrained, normal state value, where Sx 0n = Sy 0n, for a field magnitude |B| about half of the zero strain |D| value.The change in the y-direction from the normal to superconducting state of a strained sample is greater than the change in the x-direction due to the rotation of the d-vector.The orbital contribution to the magnetization (inset) shows a slight change in the normal state under the strain.The reductions in Lx and Ly are similar in the superconducting state.

Figure 4 .
Figure 4.The d-vectors on the 3D γ-band FS are shown at various momentum points for (a) no strain and (b) strain along the a-axis.The average d-vector indicated by the red arrow at the top corner shows the rotation of the d-vector denoted by (θ, φ) towards b-axis and slightly c-axis under the a-axis uniaxial strain.Similar to the 2D case, most of the rotation of d-vector occurs near the diagonal direction of the BZ.

Fig. A. 5 .
When a particle β-band intersects a hole γ-band at a finite energy above or below the Fermi level, a finite gap of |D ν | is clearly present, as shown by the red circles in Fig. A.5(b).

Figure
Figure A.5. a) Bogoliubov quasiparticle dispersion, featuring bands of mixed particle and hole character, obtained by finding the energy eigenvalues of the Hamiltonian listed in the main text solved in the mean field approximation with φ and D values obtained by self-consistent mean field theory for 3J − U = 0.401 along the path shown on the inset Fermi surface.The gap parameters are increased by a factor 20× to make the gaps more visible.b) Magnification of (a) to show the pairing occurring away from the Fermi energy due to the inter-orbital nature of the triplet parameters, labelled with red circles.Two of the red circles are labelled with the parameter that is primarily responsible for the pairing at that location, while a third is left unlabelled, where there will be mixed contribution due to the mixed orbital character of the bands around this location.Green circles show two locations of the gaps at the Fermi energy due to the induced singlets, labelled with the singlet parameter primarily responsible for the pairing at that location.The dispersion is shown in the unstrained case, and therefore the π This work was supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant 06089-2016, and the Center for Quantum Materials at the University of Toronto.