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Revista mexicana de física

Print version ISSN 0035-001X

Rev. mex. fis. vol.62 n.5 México Oct. 2016

 

Research

Symmetry field breaking effects in Sr2RuO4

P. Contrerasa 

J. Florezb 

Rafael Almeidac 

aCentro de Física Fundamental, Universidad de Los Andes, Mérida 5101, Venezuela.

bDepartamento de Física, Universidad Distrital Francisco José de Caldas, Bogotá, Colombia.

cDepartamento de Química, Universidad de Los Andes, Mérida 5101, Venezuela

ABSTRACT

In this work, after reviewing the theory of the elastic properties of Sr2RuO4, an extension suitable to explain the sound speed experiments of Lupien et al. and Clifford et al. is carried out. It is found that the discontinuity in the elastic constant C66 gives unambiguous experimental evidence that the Sr2RuO4 superconducting order parameter Ψ has two components and shows a broken time-reversal symmetry state. A detailed study of the elastic behavior is performed by means of a phenomenological theory employing the Ginzburg-Landau formalism.

Keywords: Elastic properties; unconventional superconductors; time reversal symmetry; Ginzburg-Landau theory; sound speed

RESUMEN

En este trabajo, luego de realizar una revisión de la teoría de las propiedades elásticas del compuesto Sr2RuO4, se presenta una extensión que permite explicar los resultados de los experimentos, sobre la rapidez del sonido realizados por Lupien y colaboradores y Clifford y colaboradores . Se muestra que la discontinuidad observada en la constante elástica C66 constituye una evidencia experimental directa de que el parámetro de orden Ψ tiene dos componentes y rompe la simetría de inversión temporal. También se realiza un estudio detallado del comportamiento elástico usando una teoría fenomenológica basada en el formalismo de Ginzburg-Landau.

Palabras clave: Propiedades elásticas; superconductores no convencionales; simetría de inversión temporal; teoría de Ginzburg-Landau; rapidez del sonido

PACS: 74.20.De; 74.70.Rp; 74.70.Pq

1. Introduction

In a triplet superconductor the electrons in the Cooper pairs are bound with spins parallel rather than antiparallel to one another, i.e. they are bound in spin triplets5,7,13. For this kind of superconductors, the spins are lying on the basal plane, while the pair orbital momentum is directed along the z-direction and their order parameter Ψ is represented by a three-dimensional vector d(k). If Ψ is of the type kx ± i ky, there is a Cooper pair residual orbital magnetism, which gives place to an state of broken time reversal symmetry, edge currents in the surface of the superconductor, and a tiny magnetic field around non-magnetic impurities.

Based on the results of the Knight shift experiment performed through the superconducting transition temperature Tc8,9, it has been proposed that Sr2RuO4 is a triplet superconductor. These experiments showed that Pauli spin susceptibility of the conduction electrons in the superconducting state remains unchanged respect to its value in the normal state. Moreover, it has been reported10 that Ψ breaks time reversal symmetry, which constitutes another key feature of unconventionality.

The Sr2RuO4 elastic constants Cij have been measured as the temperature T is lowered through Tc. The results show a discontinuity in one of the elastic constants2 . This implies that Ψ has two different components with the time reversal symmetry broken. Similar conclusions from a muon spin relaxation (μSR) experiment were reported by Luke et al.10 . Recently, experiments on the effects of uniaxial stress σi, as a symmetry-breaking field were performed by Clifford and collaborators3 , reporting that for Sr2RuO4 the symmetry-breaking field can be controlled experimentally. Additionally, experiments by Lupien et al.2 showed the existence of small step in the transverse sound mode T[100].

This body of results evidences the need of extending or developing theoretical models to explain the changes occurring in Cij at Tc, which, as far as we know, has not been carried out even in quite recent works3 . Thus, the aim of our work is to extend an elasticity property phenomenological theory to show that Sr2RuO4 is an unconventional superconductor with a two-component Ψ 4,11. Here, let us mention that a different theory of Sr2RuO4 elastic properties was presented by Sigrist12 . However, unlike this paper, Sigrist work does not take into account the splitting of Tc due to σi, and directly calculates the jumps at zero stress, where the derivative of T with respect to σi doesn’t exist.

In this work, we first perform an analysis based on a Ψ that transforms as one of the two dimensional irreducible representations of the Sr2RuO4 point group4,13. Subsequently, we construct the Sr2RuO4 superconducting phase diagram under an external σi . This phase diagram is employed to develop a complete theory of the elastic behavior of Sr2RuO4, based on a two component Ginzburg-Landau (GL) model. This allows to properly calculate the jumps in the components of the elastic compliances Sij. Finally, we propose that there are significant advantages for using Sr2RuO4 as a material for a detailed study of symmetry-breaking effects in superconductivity described by a two-component Ψ.

2.Ehrenfest relations for a uniaxial stress σi

Provided that σi does not split the phase transition , for applied σi, Ehrenfest relations can be derived in analogous manner to the case of applied hydrostatic pressure 11,14, under the condition that Tc is known as a function of σi. In order to simplify the calculations, we make use of the Voigt notation i = xx, yy, zz, yz, zx, xy.15

For a second order phase transition, the Gibbs free energy G derivatives respect to T, the entropy S = -(∂G/∂T)σ, and respect to σi, the elastic strain ei = -(G/σi)T are continuous functions of σi and T. Therefore, at the transition line, Δei(T,σj)=0 and ΔS(T, σj) = 0. From this, for S and ei, the boundary conditions between the two phases are

Δ[(ST)σj]dT+Δ[(Sσj)T]dσj=0Δ[(eiT)σj]dT+Δ[(eiσj)T]dσi=0 (1)

By using the definitions of the thermal expansion αi=(ei/T)σ, the specific heat at constant stress, Cσ = T(∂S/∂T)σ, and the elastic compliances Sij=(ei/σj)T, together with the Maxwell identity (S/σi)T=(ei/T)σi, the previous relations can be rewritten as,

ΔCσT+dσidTΔ(αi)σi=0Δ(αi)σj+dσjdTΔ(Sij)T=0. (2)

From the first expression in Eq. (2), the relation for αi is found to be

Δαi=-ΔCσdlnTc(σi)dσi, (3)

likewise, from the second expression of Eq. (2), the relation for Sij is obtained to be,

ΔSij=-ΔαidTc(σj)dσj. (4)

It is important to distinguish that the print letter S denotes the entropy, while the symbol Sij means the elastic compliances. In similar manner, the print letter C stands for the specific heat and the symbol Cij for the elastic stiffness. Let us also point out that in deriving these expressions, we used the fact that for a given thermodynamic quantity Q, its discontinuity along the transition line points is obtained from ΔQ = Q(Tc + 0+)-Q(Tc - 0+), where 0+ is a positive infinitesimal quantity. Finally, by combining Eqs. (3) and (4), the variation in Sij is found to be:

ΔSij=ΔCσTcdTc(σi)dσidTc(σj)dσj. (5)

Before continuing, it is interesting to mention that besides of our previous works4,11, we are not aware of any other works that have derived Ehrenfest relations for the case where applied σi produces a phase transition splitting.

3. Ginzburg-Landau model

In this section, a phenomenological model which takes into account the Sr2RuO4 crystallographic point group D4h is derived and employed. As we show, the analysis of G, using an order parameter which belongs to any of the one dimensional representations of D4h is not able to describe the splitting of Tc under an external stress field. In order to account properly for the splitting, superconductivity in must be described by a Ψ, transforming as one of the D4h two dimensional irreducible representations, E2g or E2u, which at this level of theoretical description render identical results4,11.

3.1. Superconducting free energy

In order to derive a suitable GL free energy GΓ, we first will suppose that the Sr2RuO4 superconductivity is described by an order parameter ψΓ, which transforms according to one of the eight one-dimensional representations of D4h: Γ = A1g, A2g, B1g, B2g, A1u, A2u, B1u, or B2u. Let us notice that an analysis employing the D4 point group renders similar results. Here we will analyze the terms in GΓ linear in σi and quadratic in ψΓ:

GΓ=G0+α(T)|ψΓ|2+b2|ψΓ|4+[a(σxx+σyy)+cσzz]|ψΓ|2. (6)

The terms proportional to σxx, σyy and σzz in Eq. (6) give rise to discontinuities in the elastic constants, evidenced from sound speed measurements17 . On the other hand, discontinuities in the elastic compliance S66 and in the elastic constant C66i arise from the linear coupling with σxy . However, due to symmetry, the later linear coupling does not exist for any Γ; therefore, S66 and C66 are expected to be continuous at Tc for any of the one-dimensional irreducible representation that assumes a one-dimensional ψΓ,. Nevertheless, the results of Lupien et al. experiments2 showed a discontinuity in C66. Hence, based exclusively on sound speed measurements, we conclude that none of the one-dimensional irreducible representations can provide an appropriate description of superconductivity in Sr2RuO4. As far as we know, this conclusion has not been previously established in the literature3 . Let us mention that for any one-dimensional Γ, a detailed analysis of the calculation of the jumps in C66 is presented in Ref. 11.

Due to the absence of discontinuity in S66 for any of the one-dimensional Γ, the superconductivity in Sr2RuO4 must be described by an order parameter ψE transforming as one of the two-dimensional representations E2g or E2u . The GL theory establishes that only the parameters of one of the irreducible representations becomes non-zero at Tc. Therefore, following the evidence provided in Refs. 5 and 19, we choose the E2u spin-triplet state as the correct representation for Sr2RuO4, and the speed measurements are analyzed in terms of the model ψE = (ψx, ψy), with ψx and ψy transforming as the components of a vector in the basal plane. The expression for G is determined by symmetry arguments based on the analysis of the second and fourth order invariants (real terms) of GΓ. To maintain gauge symmetry, only real and even products of Ψ can occur in the expansion of GΓ; thus, we find that all real invariants should be formed by second and fourth order products of ψ’sii. To obtain its expression, we use the fact that G is invariant with respect to a transformation by the generators c4z and c2x of D4h. Applying the generators to different second and fourth order combination of products of ψ’s, we find only one second order invariant |ψx|2 + |ψy|2 and three fourth order invariants, namely |ψx|2|ψy|2, |ψx|4 + |ψy|4, and ψx2ψy*2+ψx*2ψy2.

For the zero σi case, the expansion of G gives place to:

G=G0+α(T)|ψx|2+|ψy|2+b14|ψx|2+|ψy|22+b2|ψx|2|ψy|2+b32ψx2ψy*2+ψy2ψx*2, (7)

where α = α'(T - Tc0) and the coefficients b1, b2, and b3 are material-dependent real constants 20,21. These coefficients have to satisfy special conditions in order to maintain the free energy stability. The analysis of G is accomplished by considering two component (ψx, ψy) with the form:

(ψx,ψy)=(ηxeiφ/2,iηye-iφ/2); (8)

where ηx and ηy are both real and larger than zero. After substitution of ψx and ψy in Eq. (7), G becomes:

G=G0+α(T)(ηx2+ηy2)+b14(ηx2+ηy2)2+(b2-b3)ηx2ηy2+2b3ηx2ηy2sin2φ. (9)

For fixed values of the coefficients b1 and b2, if b3 > 0, G will reach a minimal value if the last term vanishes, i.e. if φ = 0. Moreover, if ηx and ηy have the form ηx = ηsin χ and ηx = ηcos χ, G becomes

G=G0+α(T)η2+b14η4-b̃4η4sin22χ, (10)

where b̃b3-b2. If b̃>0, G reaches its minimum value if sin22χ = 1, this condition is satisfied if χ = π/4; and therefore ηx = ηy. On the other hand, if b̃<0, then G becomes minimal if sin2 2χ= 0. In this case, either ηx = 0 or ηy = 0. Since for a superconducting state (ψx, ψy) ∼ (1, ± i), from the previous analysis, the lowest G state corresponds to b3 - b2 > 0. This thermodynamic state breaks time-reversal-symmetry; and hence, it is believed to be the state describing superconductivity in Sr2RuO44,5,7. In addition, it is found that for the phase transition to be of second order, it is required that bb1 + b2 - b3 > 0.

At this point it is important to understand why the state (ψx, ψy) ∼ (1, ± ) has been chosen for the analysis of σ6 and why it gives rise to the discontinuity in S6611. Minimization of Eq. (7) with respect to φ and χ, and employing Eq. (8) renders a set of solutions for the two-component order parameter which depend on the relation between the coefficients b1, b2, and b3 and also on the value of the phases φ and χ. Thus, for the E representation, solutions of the form,

ψ1=η(1,0)eiφ, (11)

are obtained, which are very similar to those found for the D4 one-dimensional irreducible representation. Therefore, these solutions are not able to account for the jump in C66. However, solutions with both components different than zero are also attained:

ψ2=22eiπ/4(1,1)η,ψ3=32(1,i)η. (12)

Each of these solutions corresponds to different relations for the bi. This is illustrated by Fig. 1, which shows the phase diagram, displaying the domains of ψ1, ψ2 and ψ3 as a function of b1, b2, and b3. Now, if the jump in C66 corresponds to a G minimum, the coupling term with σ6 must be taken to be different from zero. If the solution ψ2 is considered, the term containing σ6 becomes zero; therefore it is not acceptable. On the other hand, this requirement is satisfied by Ψ3, with the form (1, i)η. Hence, the GL analysis renders Ψ3 as the solution that breaks time reversal symmetry.

Figure 1 Superconducting state phase diagram for the two dimensional representation E of the tetragonal group D4 as unction of the material parameters b2 and b3 showing the domains which correspond to the order parameters ψ1, ψ2 and ψ3. Each domain corresponds to a different superconducting class. 

3.2. Coupling of the order parameter to an external stress

The transition to an unconventional superconducting state shows manifestations as the breakdown of symmetries, such as the crystal point group or the time reversal symmetry20,21. This loss of symmetry has measurable manifestations in observable phenomena, as the splitting of Tc under an elastic deformation. The coupling between the crystal lattice and the superconducting state is described Refs. 20 and 21. As explained there, close to Tc, a new term is added to G, which couples in second order Ψ with eij and in first order Ψ with σij. These couplings give place to discontinuities in Sijkl at Tc.

3.3 Analysis of the phase diagram

An expression for G accounting for a phenomenological coupling to C66 in the basal plane is given by

G=G0+α'(T-Tc0)(|ψx|2+|ψy|2)+b2|ψx|2|ψy|2+b14(|ψx|2+|ψy|2)2+b32(ψx2ψy*2+ψy2ψx*2)-12Sijσiσj+σiΛi+σidijEj. (13)

Here, Λi are the temperature-dependent αi, dij are the coupling terms between Ψ and Sij and Ej are the invariant elastic compliance tensor components, defined below. In order to determine these invariants describing the coupling of the order parameter to the stress tensor, we construct the tensor Ej with Voigt components E1 = |ψx|2, E2 = |ψy|2 and E6=ψx*ψy+ψxψy*; where E6 couples σ6 and Ψ. The tensor dij couples Ei with σj and has the same nonzero components as Sij. By applying symmetry considerations4, it is shown that the only non-vanishing independent components of dij are d11, d12 = d21, d31 = d32, and d66. Contributions to G that are quadratic in both, Ψ and σ6 were neglected. Such terms would have given an additional T dependence to the Sij17. However, given the large number of independent constants occurring in the associated sixth rank tensor, at this point, it is not clear whether or not the explicit inclusion of such terms would be productive.

Now, let us consider the case of uniaxial compression along the a axis (only with σ1< 0). If in Eq. (13), only quadratic terms in Ψ are kept, this equation can be written as

Gquad=α'[T-Tc+σ1]|ψx|2+α'[T-Tcyσ1]|ψy|2, (14)

here Tc+ (σ1) and Tcy (σ1) are given by

Tc+(σ1)=Tc0-σ1d11α',Tcy(σ1)=Tc0-σ1d12α'. (15)

In what follows, we assume that d11d12> 0, such that T c+> Tcy . Notice that this does not imply any lost in generality, assuming d11d12< 0, would render an identical model, simply by exchanging the x and y indices. Here, Tc+ is the higher of the two critical temperatures at which the initial transition occurs. As should be expected, just below Tc+, only ψ x is non zero. As T is further lowered, another phase transition happens at Tc-, which is different than Tcy. Below Tc-, the ψy is also different from zero (see Fig. (2)). Thus, in the presence of a non zero compressible σ1, Ψ has the form (ψx, ψy) ≈ ψ(1, ± I ϵ) where ϵ is real and equal to zero between Tc+ and Tc- (phase 1), and increases from ϵ = 0 to ϵ ≈ 1 as T becomes smaller than Tc- (phase 2), as illustrated in Figs. (1) and (2).

Figure 2 Temperature behavior of the two component order parameter (ψx, ψy) for the case of a nonzero uniaxial stress below Tc. Notice that only the BCS component ψx (Tc+) ecomes non zero for temperatures between Tc+ and Tc-. The second unconventional component ψyTc-) only appears below Tc-

The next step is finding Tc-. To achieve this goal, the equilibrium value of the non zero component of ψx, ψx2 = -2αx/b1 is replaced in Eq. (13) and Tc- follows from

Tc+-Tc-=-[d11-d122α'][b̃+bb̃]σ1. (16)

To obtain Eq. (16), it is assumed that σ12σ1 and only linear terms in σ1 are kept. The phase diagram for this system is shown in Fig. (3).

Figure 3 Phase diagram showing the upper and lower superconducting transition temperatures, Tc++ and Tc-, respectively, as functions of the compressible stress -σi along the a axis. 

4. Calculation of the discontinuities

As discussed before, an external uniaxial stress acting on the Sr2RuO4 basal plane breaks the tetragonal symmetry of the crystal. As a consequence of this, when a second order transition to the superconducting state occurs, it splits into two transitions. For the case of applied σ1, the analysis of the behavior of the sound speed at Tc requires a systematic study of these second-order phase transitions. Moreover, thermodynamic quantities, such as dTc/dσi, Cσ, and ασ, which are needed in order to calculate the components Sijσ are accompanied by a discontinuity at each of the second order phase transitions.

As depicted in Fig. (3), for a given σ1 ≠ 0 as T is lowered below Tc+, a first discontinuity for a thermodynamical quantity Q is observed at the first line of transition temperatures. This discontinuity along the transition line, corresponding to the higher transition temperatures, T = Tc+ (σi) is given by ∆Q+ = Q(Tc+ + 0+) − Q(T c+ − 0+), where 0+ is a positive infinitesimal number. If T is further dropped below T c-, a second discontinuity arises, and the lower line of transition temperatures appears. The discontinuity along this line, a T = Tc- (σi), is defined by ∆Q = Q(Tc− + 0+) − Q(Tc− − 0+)18. The sum of these two discontinuities

ΔQ(Tc0,σ=0)=ΔQ++ΔQ-, (17)

gives the correct expressions for the discontinuities at Tc0, for the case with σi = 0, where the Ehrenfest relations do not hold directly 4. As an example of these discontinuities, the two jumps in Cσ under an external σi are sketched in Fig. (4).

Figure 4 Schematic dependence of the specific heat on the temperature, for the case of an uniaxial stress splitting the Sr2RuO4 transition Temperature. Notice the two jumps in the heat capacity near the transition temperatures Tc+ and Tc-

4.1. Jumps due to a uniaxial stress σ1

The free energy, Eq. (13), for the cases where both σ1 and σ6 are nonzero is:

G=G0+αx|ψx|2+αy|ψy|2+σ6d66(ψxψy*+ψx*ψy)+b14(|ψx|2+|ψy|2)2+b2|ψx|2|ψy|2+b32(ψx2ψy*2+ψy2ψx*2). (18)

Here αx = α'(TTc0 )+σ1d11 and αy = α'(TTc0 )+ σ1d12. If only σ1 is applied, this equation becomes:

ΔG=αx|ψx|2+αy|ψy|2+b14(|ψx|2+|ψy|2)2+b2|ψx|2|ψy|2+b32(ψx2ψy*2+ψy2ψx*2), (19)

Where ∆G = GG0(T). The nature of the superconducting state that follows from Eq. (19), depends on the values of the coefficients b1, b2, and b3. The analysis from Eq. (19) of the superconducting part of G is performed by using, as was done previously, an expression for Ψ given by Eq. (8).

At Tc+ and in the presence of σ1, the second order terms in Eq. (19) dominate and Ψ has a single component ψx; whereas at Tc- a second component ψy appears. Thus, at very low T, the fourth order terms dominate the Eq. (19) behavior. Each of these two-component domains has the form of ψ2 given by Eq. (12). In this case, G can be written in terms of ηx and ηy as

ΔG=αxηx2+αyηy2+b14(ηx2+ηy2)2+(b2-b3)ηx2ηy2+2b3ηx2ηy2sin2φ. (20)

The analysis of Eq. (20) depends on the relation between the coefficients b1, b2, and b3. Assuming that b3 > 0, and ηx and η y are both different from zero, and following the procedure described after Eq. (9) one arrives to

(ψx,ψy)(1,±iϵ), (21)

where ϵ is real and grows from ϵ = 0 to ϵ ≈ 1 as T is reduced below Tc-, while Eq. (20) becomes

ΔG=αxηx2+αyηy2+b14(ηx2+ηy2)2-(b3-b2)ηx2ηy2. (22)

To calculate the jumps at Tc+, we use αx = α'(TTc+) and αy = α'(TTcy), and assume that Tc+> Tcy . For the interval Tc+> T > Tc−, the equilibrium value for Ψ satisfies αx > 0 and αy = 0, i.e. ηx > 0 and ηy = 0, with ηx2=-2αx/b1, obtaining that T c+ and its derivative with respect to σ1 are respectively,

Tc+(σ1)=Tc0-σ1α'd11,dTc+dσ1=-d11α'. (23)

The specific heat discontinuity at Tc+, relative to its normal state value, is calculated by using:

ΔCσ1=-T2ΔG2T|T=Tc+, (24)

and renders the result

ΔCσ1+=-2Tc+α'2b1. (25)

A schematic depiction of the Cσ discontinuities below this transition temperature is exhibited in Fig. (4). At Tc+, the discontinuity in ασ is calculated by applying the Ehrenfest relation of Eq. (3), yielding:

Δα1+=-2α'd11b1. (26)

The discontinuities in Sij are obtained by using Eqs. (4) and (5), rendering the result,

ΔSi'j'+=-2di'1dj'1b1. (27)

In the previous expression a prime on an index (as in i' or j') indicates a Voigt index taking only the values 1,2, or 3. Thus, from Eq. (27) the change in S11 at Tc+ can be calculated to be ΔS11σ1=-2d112b1.

To find the discontinuities at Tc-, the invariant (ηx2+ηy2)2 in Eq. (22) is expanded, after which G takes the form,

ΔG=αxηx2+b14ηx4+[αy+b12+b2-b3ηx2]ηy2+b14ηy4. (28)

In this expression, the second order term in ηy is renormalized by the square of ηx. The second transition temperature is determined from the zero of the total prefactor of ηy2, obtaining that Tc- and its derivative with respect to σ1 are:

Tc-(σ1)=Tc-σ12α'[d11+d12-bb~d12-d11],dTc-dσ1=-12α'[d11+d12-bb~d12-d11]. (29)

Below Tc- the superconducting free energy, Eq. (28) has to be minimized respect to both components of Ψ. After doing so, ηx and ηy for this temperature range are found to be

ηx2=-12bb~[b-b~αy+b+b~αx],ηy2=-12bb~[b-b~αx+b-b~αy]. (30)

This analysis shows that the second superconducting phase is different in symmetry, and that time reversal symmetry is broken. The change in Cσ1 at Tc-, with respect to its value in the normal phase, ΔCσ1-,N, is found to be, ΔCσ1-,N = -2Tc-α'1/b. The specific heat variation at Tc- is,

ΔCσ1-=ΔCσ1-,N-ΔCσ1+, (31)

which results in

ΔCσ1-=-2Tc-α'2b̃bb1. (32)

The size of these jumps is complicated to infer, because it depends on the material parameters b1, b2, and b3, and on the coupling constants d11 and d12.

With the help of the Ehrenfest relation, Eq. (3), the discontinuity in αi at Tc- is obtained to be

Δαi'-=-α'b̃bb1(di'+-bb̃di'-), (33)

and after employing Eqs. (4) and (5), the discontinuity in Si',j' at Tc- is shown to be

ΔSi'j'-=-b̃2bb1di'+-bb̃di'-dj'+-bb̃dj'-. (34)

Here di'±=di'1±di'2. The discontinuities occurring at T c0, in the absence of uniaxial stress, can be obtained by adding the discontinuities occurring at Tc+ and Tc-, yielding:

ΔCσ10=-2Tc0α'2b,Δαi'0=-α'di'+b,ΔSi'j'0=-12(di'+dj'+b+di'-dj'-b~). (35)

Before continuing, it is important to emphasize that at at zero stress, the derivative of Tc with respect to σi is not defined; therefore, there is no reason to expect any of the Ehrenfest relations to hold 4,11.

4.2Jumps due to a shear stress σ6

When a shear stress σ6 is applied to the basal plane of Sr2RuO4, the crystal tetragonal symmetry is broken, and a second 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. Very important to mention that the C66 discontinuity observed by Lupien 2 at Tc, can be explained in this context.

If there is a double transition, the derivative of Tc with respect to σ6 i.e. dTc/dσ6 is different for each of the two transition lines. At each of these transitions, Cσ6, ασ6, and Sijσ6 show discontinuities. As discussed before, the sum of them gives the correct expressions for the discontinuities at zero shear stress, where the Ehrenfest relations do not hold.

The Tcσ6 phase diagram will be similar to that obtained for σ1; therefore, the diagram in Fig. (3) also qualitatively holds here. In the case of an applied σ6, ΔG is given by

ΔG=α(|ψx|2+|ψy|2)+σ6d66(ψxψy*+ψx*ψy)+b14(|ψx|2+|ψy|2)2+b2|ψx|2|ψy|2+b32(ψx2ψy*2+ψy2ψx*2). (36)

Here α = α'(TTc0), and the minimization of ΔG is performed as in the σ1 case, i.e. by substituting the general expression for Ψ given in Eq. (8). After doing so, ΔG becomes

ΔG=α(ηx2+ηy2)+2ηxηyσ6sinφd16+b14ηx2+ηy22+(b2-b3)ηx2ηy2+2b3ηx2ηy2sin2φ. (37)

In the presence of σ6, the second order term determines the phase below Tc+, which is characterized by ψx and by ψy = 0. As the temperature is lowered below Tc- , depending of the value of b3 a second component ψ y may appear. If at Tc- a second component occurs, the fourth order terms in Eq. (37) will be the dominant one. Thus for very low T’s, or for σ6 → 0, a time-reversal symmetry-breaking superconducting state emerges. The analysis of Eq. (37) depends on the relation between the coefficients b2 and b 3. It also depends on the values of the quantities η x and ηy, and of the phase φ. If b3< 0, and ηx and ηy are both nonzero, the state with minimum energy has a phase φ = π/2. The transition temperature is obtained from Eq. (37), by performing the canonical transformations: ηx=(1/2)(ημ+ηξ) and ηy=(1/2)(ημ-ηξ). After their substitution, Eq. (37) becomes

ΔG=α+ηξ2+α-ημ2+14(ηξ2+ημ2)2+(b2+b3)(ηξ2-ημ2)2. (38)

If, as was done before, ηξ = η sinχ and ηµ = η cosχ, Eq. (38) takes the form

ΔG=α+η2sin2χ+α-η2cos2χ+η44[b1+b2+b3cos22χ]. (39)

ΔG is minimized if cos2χ = 1, this is, if χ = 0. Also, in order for the phase transition to be of second order, b', defined as b' ≡ b1 + b2 + b3, must be larger than zero. Therefore, if σ6 is non zero, the state with the lowest free energy corresponds to b3< 0, phase φ equal to π/2, and Ψ of the form:

(ψx,ψy)η(eiφ2,e-iφ2). (40)

In phase 1 of Fig. (3), φ = 0, and as T is lowered below Tc-, phase 2, φ grows from 0 to approximately π/2. Again, following an analysis similar to that carried out for σ1, the two transition temperatures Tc+ and Tc- are obtained to be:

Tc+(σ6)=Tc0-σ6α'd66,Tc-(σ6)=Tc0+b2b3α'σ6d66. (41)

The derivative of Tc+ with respect to σ6, and the discontinuity in Cσ6+ at Tc+ are respectively found to be:

dTc+dσ6=-d66α',ΔCσ6+=-2Tc+α'2b'. (42)

After applying the Ehrenfest relations, Eqs. (4) and (5), the results for Δασ6 and ΔS66 at Tc+ are:

Δασ6+=-2α'd66b',ΔS66+=-2d662b'. (43)

For Tc-, the derivative of this transition temperature with respect to σ6, and the discontinuities in the specific heat, thermal expansion and elastic stiffness respectively are:

dTc-dσ6=bd662b3α',ΔCσ6-=-4Tc-α'2b3bb', (44)

Δασ6-=2α'd66b',ΔS66-=-d662bb'b3. (45)

Since for the case of σ6, the derivative of Tc with respect to σ6 is not defined at zero stress point, the Ehrenfest relations do not hold at Tc0. Thus, the discontinuities occurring at Tc0, in the absence of σ6, are calculated by adding the expressions obtained for the discontinuities at Tc+ and Tc-,

ΔCσ60=-2Tc0α'2b,ΔS660=-d662b3,Δασ60=0. (46)

Notice that in this case, there is no discontinuity for ασ60.

Since the phase diagram was determined as a function of σ6, rather than as a function of the strain, (see Fig. (3)), in this work, as in Refs. 4 and 11, we make use of the 6 × 6 elastic compliance matrix S, whose matrix elements are Sij. However, the sound speed measurements are best interpreted in terms of the elastic stiffness matrix C, with matrix elements Cij, which is the inverse of S 23. Therefore, it is important to be able to obtain the discontinuities in the elastic stiffness matrix in terms of the elastic compliance matrix. Thus, close to the transition line, C(Tc +0+) = C(Tc − 0+) + ∆ C and S(Tc + 0+) = S(Tc − 0+) + ∆S, where 0+ is positive and infinitesimal. By making use of the fact that C(Tc+0+)STc+0+=1^, where 1^ is the unit matrix, it is shown that, to first order, the discontinuities satisfy, ΔC ≈ - C ΔS C. In this manner, it is found that, for instance at Tc+, ΔC11+(2(Cj1dj1)2/b1). From this expression it is clear that ΔC11+ must be greater than zero. In general, at Tc+, Tc-, and Tc0, 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, ∆S11, ∆S22, ∆S33, and ∆S66 are all negative; while, the stiffness components ∆S11, ∆S22, ∆S33, and ∆S66 are all positive.

5. Final remarks

Since for Sr2RuO4, the symmetry-breaking field, due to σi, is under experimental control, states of zero symmetry-breaking stress and of σi single direction can be a- chieved1,2,3. Hence, it has significant advantages the use of as a material in detailed studies of superconductivity symmetry-breaking effects, described by a two-component order parameter. Nevertheless, determining from experimental measurements the magnitude of the parameters in the Ginzburg-Landau model is complicated, because the number of independent parameters occurring for the case of tetragonal symmetry is greater than for the case of hexagonal symmetry (i.e. UPt3) 24,25,26. Thus for Sr2RuO4, three linearly independent parameters b1, b2, and b3, are required to specify the fourth order terms in Ψ occurring in Eq. (1); whereas only two independent parameters, b1 and b2, are required for UPt3. For Sr2RuO4, two independent ratios can be formed from the three independent bi parameters, and these two independent ratios could be determined, for example, by experimentally determining the ratios ΔCσ+/ΔCσ- in the presence of the σ1 and σ6.

Measurements results for the Sr2RuO4 elastic constants below Tc are presented in Ref. 2. There, it is concluded that the quantities C44 and C11C12 follow the same behavior as those of the BCS superconducting transition, which is evidenced by a change in slope below T c0. On the other hand, a discontinuity is observed for C66 below Tc0, without a significant change in the sound speed slope as T goes below 1 Kelvin. It has been previously stated 2,11 that this kind of C66 changes can be understood as a signature of an unconventional transition to a superconducting phase. Thus, this set of results and others, as those of Clifford et al. 3, lead to consider Sr2RuO4 as an excellent candidate for a detailed experimental investigation of the effects of a symmetry-breaking field in unconventional superconductors.

Acknowledgments

We thank Prof. Michael Walker from the University of Toronto, Prof. Kirill Samokhin from Brock University, and Prof. Christian Lupien from Université de Sherbrooke for stimulating discussions. We are also grateful to the Referee for his comments and Prof. Moisey I. Kaganov. This research was supported by the Grant CDCHTA-ULA number C-1908-14-05-B.

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i the Voigt notation for C66 means Cxyxy where 6 = xy.15

iiThe invariance under the gauge symmetry U(1) means that the quantities ψi must transform according to the rule ψxeiΦψx and ψyeiΦψy.

Received: May 28, 2015; Accepted: May 05, 2016

Current address: Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Autónoma de Caribe, Barranquilla, Colombia.

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