Anisotropic shear stress $\sigma_{xy}$ effects in the basal plane of Sr$_2$RuO$_4$

TL;DR

This paper revisits the effects of anisotropic shear stress on Sr$_2$RuO$_4$, clarifies elastic theoretical frameworks, and provides experimental evidence supporting a two-component superconducting order parameter with broken time-reversal symmetry.

Contribution

It offers a detailed analysis of shear stress effects in Sr$_2$RuO$_4$, clarifies elastic theory for sound speed experiments, and presents evidence for a two-component order parameter with broken time-reversal symmetry.

Findings

Discontinuity in elastic constant C$_{xyxy}$ indicates a two-component superconducting order parameter.

Elastic theoretical framework clarifies interpretation of sound speed and strain experiments.

Coupling of the $ ext{γ}$ band to in-plane shear stress supports anisotropic electron-phonon interaction model.

Abstract

In this short note, we repeat the calculations the jumps for the specific heat C$_{\sigma_{xy}}$, the elastic compliance S$_{xyxy}^{\sigma_{xy}}$ and the thermal expansion $\alpha_{\sigma_{xy}}$ due to a shear stress $\sigma_{xy}$ in the basal plane of $Sr_2RuO_4$. Henceforth we clarify some issues regarding the elastic theoretical framework suitable to explain the sound speed experiments of Lupien et al. (2001,2002), and partially the strain experiments of Hicks et al. (2014), and Steppke et al. (2016) in strontium ruthenate. We continue to propose that the discontinuity in the elastic constant C$_{xyxy}$ of this tetragonal crystal gives unambiguous experimental evidence that the superconducting order parameter $\Psi$ has two components with a broken time-reversal symmetry state, and that the $\gamma$ band couples the anisotropic electron-phonon interaction to the $[xy]$ in-plane shear stress according to Walker and collaborators [4] and [3]. Some importants words about the roll of the spin equal to one for the transversal phonons are added in the conclusion following Levine [34].

Figures

Equations

$$\Delta G_{{\sigma }_{xy}}$$

$$\Delta G_{{\sigma }_{xy}}={\alpha }_{+}{\eta }_{\xi }^2+{\alpha }_{-}{\eta }_{\mu }^2+\frac{1}{4}({\eta }_{\xi }^2+{\eta }_{\mu }^2{)}^2+(b_2+b_3)({\eta }_{\xi }^2-{\eta }_{\mu }^2{)}^2.$$

$$\Delta G_{\sigma_{xy}}=\alpha_{+}\eta^{2}\sin^{2}\chi+\alpha_{-}\eta^{2}\cos^{2}\chi+\frac{\eta^{4}}{4}\Big{[}b_{1}+(b_{2}+b_{3})\cos^{2}2\chi\Big{]}.$$

$$({\psi }_x,{\psi }_y)\approx \eta (e^{\frac{i\phi }{2}},e^{-\frac{i\phi }{2}}).$$

$$T_{c+}({\sigma }_{xy})$$

$$T_{c-}({\sigma }_{xy})$$

$$\frac{d{T}_{c+}}{d{\sigma }_{xy}}$$

$$\Delta C_{{\sigma }_{xy}}^{+}$$

$$\Delta {\alpha }_{{\sigma }_{xy}}^{+}$$

$$\Delta {\text{S}}_{xyxy}^{+}$$

$$\frac{d{T}_{c-}}{d{\sigma }_{xy}}$$

$$\Delta C_{{\sigma }_{xy}}^{-}$$

$$\Delta {\alpha }_{{\sigma }_{xy}}^{-}$$

$$\Delta {\text{S}}_{xyxy}^{-}$$

$$\Delta C_{{\sigma }_{xy}}^0$$

$$\Delta {\text{S}}_{xyxy}^0$$

$$\Delta {\alpha }_{{\sigma }_{xy}}^0$$

Full text

Keywords: Shear stress; time-reversal symmetry; Ehrenfest relations; elastic constants; thermal expansion; spin one phonons.

I Introduction

In Sr2RuO4 the electrons in the Cooper pairs are bound in spin triplets, where the spins are lying on the basal plane and the pair orbital momentum is directed along the z-direction. Henceforth, the order parameter $\Psi$ is represented by a vector ${\bf d}({\bf k})$ (of the type $k_{x}\pm i\>k_{y}$) mae94 ; mae1 ; mac1 ; ric2 .

Based on the results of the Knight-shift experiment performed through Tc ish1 ; duf1 , it has been proposed that Sr2RuO4 is a triplet superconductor, also it has been reported luk1 that $\Psi$ breaks time-reversal symmetry, which constitutes another key feature of unconventionality. The Sr2RuO4 elastic constants Cxyxy have been carefully measured as the temperature T is lowered through Tc, showing the existence of small step in the transverse sound mode $T[100]$ lup2 . This experimental result theoretically implies that $\Psi$ has two different components with a time-reversal symmetry broken state con2 . Similar conclusions from a muon-spin relaxation ($\mu$SR) experiment were reported luk1 . Recently, experiments on the effects of uniaxial strain $\epsilon_{xy}$, were performed hick , reporting that for Sr2RuO4 the symmetry-breaking field can be controlled experimentally.

Additionally, a most recently experiment tep1 found that the transition temperature Tc in the superconductor Sr2RuO4 rises dramatically under the application of a planar anisotropic strain, followed by a sudden drop beyond a larger strain. Furthermore, recent theoretical work suggests that those recent experiments tuned the Fermi surface topology efficiently by applying planar anisotropic strain emphasizing again, the point of view that in-plane effects (even by means of a more complicated renormalization group theory framework) also shows clear evidence of a symmetry broken stated in Sr2RuO4 . Furthermore, they reported a rapid initial increase in the superconducting transition temperature Tc, that the authors associated with the evolution of the Fermi surface toward a Lifshitz Fermi surface reconstruction under an increasing strain liu .

Here, we aim to clarify some particular concepts and methods following an elastic phenomenological (GL) approach lan1 ; walk ; lan2 ; tes1 ; aul1 ; mus ; jab1 . First, the differences between using stress or strain (which is the response of a system to applied stress) grechka to explain a time-reversal symmetry broken state. Second, there is a claim liu that only the $\gamma$ band responses to the strain sensitively, and we emphasize that this physical phenomenon is caused by the $\gamma$ band coupling the anisotropic electron-phonon interaction to the $[xy]$ plane wak1 ; con1 . Third, we do not expand our analysis to a Lifshitz reconstruction of the Fermi surface mainly because we do not have experimental evidences that show a topological Lifshitz transition in Sr2RuO4 even in its normal state, neither we have observed a generalized topological transition in Sr2RuO4 . In our opinion a 2D Fermi contour evolution under an applied external strain as the one for the $\gamma$ band in Sr2RuO4 needs further interpretations in terms of a topological electronic Lifshitz phase transition lif ; kag1 ; kag2 ; kag3 . We remember that $\sigma_{ik}=-p\;\delta_{ik}$ shows how pressure and stress are in general related, the stress $\sigma_{ik}$ becomes the delta function $\delta_{ik}$ if a volumetric pressure is applied to a sample lan4 .

These experimental results and theoretical interpretations need to clarify moderately in order to unify several theoretical criteria which try to explain the changes occurring in the Cxyxy elastic constant at Tc lup2 ; hick ; tep1 .

Thus, the aim of this short note is to clarify again that an elastic Ginzburg-Landau phenomenological approach partially demonstrates that Sr2RuO4 is an unconventional superconductor with a two-component order parameter $\Psi$ con1 ; con2 . We based our interpretation on a Sr2RuO4 $[T100]$ transversal response-impulse mode experimentally measured as the temperature T is lowered through Tc showing only an small step change lup2 (for that particular result please see bottom panel on Fig 5.7 and Fig 5.8 pp. 138-139 lup2 ). The result clearly shows a discontinuity in the $T[100]$ mode.

Here, let us mention that a different theory of Sr2RuO4 elastic properties was presented sig2 . However, the approach followed does not take into account the splitting of Tc due to shear $\sigma_{xy}$, and directly calculates the jumps at zero stress, where the derivative of T with respect to $\sigma_{xy}$ does not exist walk (see fig.1).

II Shear stress $\sigma_{xy}$ analysis

In this section, we make use of the $4^{th}$ rank tensor notation because the Voigt notation has a disadvantage; the stress and strain are treated differently. Voigt mapping only preserves the elastic stiffnesses. We also call the uniaxial shear stress as shear stress only because the effect observed is in the basal plane $[xy]$. As was stated previously con1 ; con2 when shear stress $\sigma_{xy}$ is applied to the basal plane of Sr2RuO4 , the crystal tetragonal symmetry is broken, and a second-order transition to a superconducting state occurs. Accordingly, for this case the analysis of the sound speed behavior at Tc also requires a systematic study of the two successive second order phase transitions (see fig.2). Hence, the Cxyxy discontinuity lup2 at Tc, can be explained in this context. Due to the absence of discontinuity in Sxyxy for any of the one-dimensional $\Gamma$ representations, the superconductivity in Sr2RuO4 must be described by the two dimensional irreducible representation E2u of the tetragonal point group D4h con2 .

If there is a double transition, the derivative of Tc with respect to $\sigma_{xy}$ i.e. $dT_{c}/d\sigma_{xy}$ is different for each of the two transition lines (see the Tc-$\sigma_{xy}$ phase diagram in fig.2). At each of these transitions, the specific heat C${}_{\sigma_{xy}}$, the thermal expansion $\alpha_{\sigma_{xy}}$, and the elastic compliance S${}_{xyxy}^{\sigma_{xy}}$ show discontinuities con2 , the sum of them gives the correct expressions for the discontinuities at zero shear stress, where the Ehrenfest relations do not hold.

In the case of an applied shear stress $\sigma_{xy}$, the change for the in-plane Gibss free energy is given by

[TABLE]

where $d_{xyxy}$ term couples the stress $\sigma_{xy}$ to the order parameter, the thermal expansion coefficient $\alpha$ = $\alpha^{\prime}(T-T_{c0})$, and the minimization of $\Delta G_{\sigma_{xy}}$ is performed by substituting the general expression for $\Psi$ as was previously calculated con1 ; sig2 . Therefore $\Delta G_{\sigma_{xy}}$ becomes

[TABLE]

In the presence of $\sigma_{xy}$, the second order term determines the phase below Tc+, which is characterized by $\psi_{x}$ and by $\psi_{y}$ $=$ 0. As the temperature is lowered below Tc-, depending of the value of b3 a second component $\psi_{y}$ may appear. If at Tc- a second component occurs, the fourth order terms in eqn. (2) will be the dominant one. Thus for very low T’s, or for $\sigma_{xy}$ $\rightarrow$ 0, a time-reversal symmetry-breaking superconducting state may emerge. The analysis of eqn. (2) depends on the relation between the coefficients b2 and b3. It also depends on the values of the quantities $\eta_{x}$ and $\eta_{y}$, and of the phase $\varphi$. If b3 $<$ 0, and $\eta_{x}$ and $\eta_{y}$ are both nonzero, the state with minimum energy has a phase $\varphi$ = $\pi/2$. The transition temperature is obtained from eqn. (2), by performing the canonical transformations: $\eta_{x}$ = $\frac{1}{\sqrt{2}}(\eta_{\mu}+\eta_{\xi})$ and $\eta_{y}$ = $\frac{1}{\sqrt{2}}(\eta_{\mu}-\eta_{\xi})$. After their substitution, eqn. (2) becomes

[TABLE]

As it was before done, $\eta_{\xi}=\eta\hskip 2.84544pt\sin\chi$ and $\eta_{\mu}=\eta\hskip 2.84544pt\cos\chi$, eqn. (3) takes the form

[TABLE]

$\Delta G_{\sigma_{xy}}$ is minimized if $\cos 2\chi$ = 1, this is, if $\chi$ $=$ 0. Also, in order for the phase transition to be of second order, $b^{\prime}$, defined as $b^{\prime}\equiv b_{1}+b_{2}+b_{3}$, must be larger than zero. Therefore, if $\sigma_{xy}$ is non zero, the state with the lowest free energy corresponds to $b_{3}$ $<$ 0, phase $\varphi$ equal to $\pi/2$, and $\Psi$ of the form:

[TABLE]

In phase 1 of fig. (2), $\varphi$ = 0, and as T is lowered below Tc-, phase 2, $\varphi$ grows from 0 to approximately $\pi/2$. The two transition temperatures Tc+ and Tc- are obtained to be:

[TABLE]

The derivative of Tc+ with respect to $\sigma_{xy}$, and the discontinuity in $C^{+}_{\sigma_{xy}}$ at Tc+ are respectively:

[TABLE]

After applying the Ehrenfest relations lan1 the results for $\Delta\alpha_{\sigma_{xy}}$ and $\Delta\text{S}_{xyxy}$ at Tc+ are:

[TABLE]

For Tc-, the derivative of this transition temperature with respect to $\sigma_{xy}$, and the discontinuities in the specific heat, thermal expansion and elastic stiffness respectively are:

[TABLE]

[TABLE]

Since for the case of $\sigma_{xy}$, the derivative of Tc with respect to $\sigma_{xy}$ is not defined at zero stress point (see fig.1), the Ehrenfest relations do not hold at Tc0. Thus, the discontinuities occurring at Tc0, in the absence of $\sigma_{xy}$, are calculated by adding the expressions obtained for the discontinuities at Tc+ and Tc- in elastic stiffness fig. (3), elastic compliances and thermal expansion are given by:

[TABLE]

in this case, there is no discontinuity for the thermal expansion of Sr2RuO4 at zero stress $\alpha_{\sigma_{xy}}^{0}$ which is another physical feature we previously predicted  con2 (see fig.4). Some experimental studies on the changes in the thermal expansion coefficient $\alpha_{i}$ below $T_{c}$ in HTS reported  asa that an additive lattice jump was found to appear spontaneously at $T_{c}$ for a high Tc compound with one-component order parameter.

Since the phase diagram was determined as a function of $\sigma_{xy}$, rather than as a function of the shear strain $\epsilon_{xy}$, (see fig. 1), in this work, as in refs. con2 ; con1 , we make use of the 6 $\times$ 6 elastic compliance matrix S, and also of the full range tensor notation. However, the sound speed measurements  lup2 ; lup1 are best interpreted in terms of the elastic stiffness tensor C, with matrix elements Cijkl, which is the inverse of S nei1 . However, the strain is easier to measure than the stress because fiber optics yields dynamic fidelity of a fraction of a nanostrain grechka .

Therefore, it is important to be able to obtain the discontinuities in the elastic stiffness matrix in terms of the elastic compliance matrix for the shear stress case. Thus, close to the transition line we follow walk : $\text{C}(T_{c}+0^{+})$ = $\text{C}(T_{c}-0^{+})+\Delta\;\text{C}$ and $\text{S}(T_{c}+0^{+})$ = $\text{S}(T_{c}-0^{+})+\Delta\text{S}$, where $0^{+}$ is positive and infinitesimal. By making use of the fact that $\text{C}(T_{c}+0^{+})\;\text{S}(T_{c}+0^{+})$ = $\hat{1}$, where $\hat{1}$ is the unit matrix, it is shown that, to first order, the discontinuities satisfy, $\Delta\text{C}$ $\approx$ $-\text{C}\>\Delta\text{S}\>\text{C}$. In this manner, it is found that, for instance at Tc+, the expressions that define the jumps for the discontinuities in elastic stiffness and compliances, due to an external stress, have either a positive or a negative value. In this way, $\Delta\text{S}_{xyxy}$ are negative; while $\Delta\text{C}_{xyxy}$ are positive.

III Remarks

The most noteworthy outcome of this short note is that the observation of a discontinuity in the elastic constant Cxyxy lup1 ; lup2 is an evidence that the order parameter $\Psi$ in Sr2RuO4 has two components as the theoretical $GL$ analysis predicts. Also the theoretical indicator that the sum of the jumps $\Delta\alpha_{\sigma_{xy}}^{+}+\Delta\alpha_{\sigma_{xy}}^{-}=0$ for the in-plane thermal expansion coefficient con2 (see fig.4). Hence, the use of Sr2RuO4 as a material in detailed studies of superconductivity symmetry-breaking effects has significant advantages because is described by a two-component order parameter. Nevertheless, determining from Sr2RuO4 experimental measurements the magnitude of the parameters in the Ginzburg-Landau model is complicated con2 .

In the experimental work of the sound velocity measurements lup2 ; lup1 in Sr2RuO4 a discontinuity in the behavior for Cxyxy below Tc0, without a significant change in the sound speed slope as T goes below 1 Kelvin was understood as a signature of an unconventional transition to a superconducting phase lup2 ; con2 ; sig2 . Thus, this set of previous results, and other recent results hick ; tep1 ; liu ; acha , considers Sr2RuO4 as a strong candidate for a detailed experimental investigation of the effects of a symmetry-breaking field by means of strain or stress experimental measurements. We also suggest that an in-plane thermal expansion measurement at zero uniaxial shear stress might further clarify any previous interpretation.

However, according to Levine lev , transversal phonons may have spin equal to one for certain point group symmetries. It could be that it occurs to the D4h tetragonal group of strontium ruthenate, this could be the key to understand superconductivity and the time-reversal symmetry broken state in Sr2RuO4 . The possible fact that phonons in the jump observed in the sound transversal mode $T[100]$ and $T[110]$ lup1 ; lup2 have spin equal to one. As a consequence it could be that these symmetry breaking strain or stress fields effects are due the transversal spin waves in certain point group symmetries in metallic solids. Some words about the roll of the magnetic interaction should be added.

Acknowledgments

P. Contreras express his gratitude to Professors C. Lupien and L. Taillefer at Universite de Sherbrooke from for stimulating discussions regarding their experimental results several years ago at the University of Toronto. We also acknowledge Dr. Vladimir Grechka from Marathon Oil for clarifying certain aspects regarding the use of the shear strain for experimental purposes.