Symmetry Field Breaking Effects in Sr 2 RuO 4
SSymmetry Field Breaking Effects in Sr RuO Pedro Contreras , † , Jos´e Florez and Rafael Almeida Centro de F´ısica Fundamental, Universidad de Los Andes, M´erida 5101, Venezuela Departamento de F´ısica, Universidad Distrital Francisco Jos´e de Caldas, Bogot´a , Colombia Departamento de Qu´ımica, Universidad de Los Andes, M´erida 5101, Venezuela † Current address: Departamento de Ciencias B´asicas, Facultad de Ingenier´ıa,Universidad Aut´onoma de Caribe, Barranquilla, Colombia. (Dated: December 18, 2018)In this work, after reviewing the theory of the elastic properties of Sr RuO ,an extension suitable to explain the sound speed experiments of Lupien et. al. [2]and Clifford et. al. [3] is carried out. It is found that the discontinuity in theelastic constant C gives unambiguous experimental evidence that the Sr RuO superconducting order parameter Ψ has two components and shows a broken time-reversal symmetry state. A detailed study of the elastic behavior is performed bymeans of a phenomenological theory employing the Ginzburg-Landau formalism. Keywords:
Elastic properties; unconventional superconductors; time reversalsymmetry; Ginzburg-Landau theory; sound speed.
PACS numbers: 74.20.De; 74.70.Rp, 74.70.Pq
I. INTRODUCTION
In a triplet superconductor the electrons in the Cooper pairs are bound with spins parallelrather than antiparallel to one another, i.e. they are bound in spin triplets [5, 7, 13]. Forthis kind of superconductors, the spins are lying on the basal plane, while the pair orbitalmomentum is directed along the z-direction and their order parameter Ψ is representedby a three-dimensional vector d ( k ). If Ψ is of the type k x ± i k y , there is a Cooper pairresidual 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 supercon- a r X i v : . [ c ond - m a t . s up r- c on ] D ec ducting transition temperature T c [8, 9], it has been proposed that Sr RuO is a tripletsuperconductor. These experiments showed that Pauli spin susceptibility of the conductionelectrons in the superconducting state remains unchanged respect to its value in the normalstate. Moreover, it has been reported [10] that Ψ breaks time reversal symmetry, whichconstitutes another key feature of unconventionality.The Sr RuO elastic constants C ij have been measured as the temperature T is loweredthrough T c . The results show a discontinuity in one of the elastic constants [2]. This impliesthat Ψ has two different components with the time reversal symmetry broken. Similarconclusions from a muon spin relaxation ( µ SR) experiment were reported by Luke et. al.[10]. Recently, experiments on the effects of uniaxial stress σ i , as a symmetry-breaking field,were performed by Clifford and collaborators [3], reporting that for Sr RuO 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 modelsto explain the changes occurring in the C ij at T c , which, as far as we know, has not beencarried out even in quite recent works [3]. Thus, the aim of our work is to extend anelasticity property phenomenological theory to show that Sr RuO is an unconventionalsuperconductor with a two-component Ψ [4, 11]. Here, let us mention that a differenttheory of Sr RuO elastic properties was presented by Sigrist [12]. However, unlike thispaper, Sigrist work does not take into account the splitting of T c due to σ i , and directlycalculates the jumps at zero stress, where the derivative of T with respect to σ i doesn’t exist.In this work, we first perform an analysis based on a Ψ that transforms as one of the twodimensional irreducible representations of the Sr RuO point group [4, 13]. Subsequently,we construct the Sr RuO superconducting phase diagram under an external σ i . Thisphase diagram is employed to develop a complete theory of the elastic behavior of Sr RuO , based on a two component Ginzburg-Landau ( GL ) model. This theory allows to properlycalculate the jumps in the components of the elastic compliances S ij . Finally, we proposethat there are significant advantages for using Sr RuO as a material for a detailed studyof symmetry-breaking effects in superconductivity described by a two-component Ψ. II. EHRENFEST RELATIONS FOR A UNIAXIAL STRESS σ i Provided that σ i does not split the phase transition [4], for applied σ i , Ehrenfest relationscan be derived in analogous manner to the case of applied hydrostatic pressure [11, 14], underthe condition that T c is known as a function of σ i . In order to simplify the calculations, wemake use of the Voigt notation i = xx, yy, zz, yz, xz, 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 σ i , the elastic strain e i = − ( ∂G/∂σ i ) T arecontinuous functions of σ i and T. Therefore, at the transition line, ∆ e i ( T, σ j ) = 0 and∆ S ( T, σ j ) = 0. From this, for S and e i , the boundary conditions between the two phasesare∆ (cid:20)(cid:16) ∂S∂T (cid:17) σ j (cid:21) dT + ∆ (cid:20)(cid:16) ∂S∂σ j (cid:17) T (cid:21) dσ j = 0∆ (cid:20)(cid:16) ∂e i ∂T (cid:17) σ j (cid:21) dT + ∆ (cid:20)(cid:16) ∂e i ∂σ j (cid:17) T (cid:21) dσ i = 0 (1)By using the definitions of the thermal expansion α i = ( ∂e i /∂T ) σ , the specific heat atconstant stress, C σ = T ( ∂S/∂T ) σ , and the elastic compliances S ij = ( ∂e i /∂σ j ) T , togetherwith the Maxwell identity ( ∂S/∂σ i ) T = ( ∂e i /∂T ) σ i , the previous relations can be rewrittenas, ∆ C σ T + dσ i dT ∆( α i ) σ i = 0∆( α i ) σ j + dσ j dT ∆(S ij ) T = 0 . (2)From the first expression in eqn. (2), the relation for α i is found to be∆ α i = − ∆ C σ d ln T c ( σ i ) dσ i , (3)likewise, from the second expression of eqn. (2), the relation for S ij is obtained to be,∆S ij = − ∆ α i dT c ( σ j ) dσ j . (4)It is important to distinguish that the print letter S denotes the entropy, while the symbol S ij means the elastic compliances. In similar manner, the print letter C stands for the specificheat and the symbol C ij for the elastic stiffness. Let us also point out that in deriving theseexpressions, we used the fact that for a given thermodynamic quantity Q , its discontinuityalong the transition line points is obtained from ∆ Q = Q ( T c + 0 + ) − Q ( T c − + ), where 0 + is a positive infinitesimal quantity. Finally, by combining Eqs. (3) and (4), the variation inS ij is found to be: ∆S ij = ∆ C σ T c dT c ( σ i ) dσ i dT c ( σ j ) dσ j . (5)Before continuing, it is interesting to mention that besides of our previous works [4, 11],we are not aware of any other works that have derived Ehrenfest relations for the case whereapplied σ i produces a phase transition splitting. III. GINZBURG - LANDAU MODEL
In this section, a phenomenological model which takes into account the Sr RuO crys-tallographic point group D h is derived and employed. As we show, the analysis of G , usingan order parameter which belongs to any of the one dimensional representations of D h isnot able to describe the splitting of T c under an external stress field. In order to accountproperly for the splitting, superconductivity in Sr RuO must be described by a Ψ, trans-forming as one of the D h two dimensional irreducible representations, E g or E u , which atthis level of theoretical description render identical results [4, 11]. A. Superconducting free energy
In order to derive a suitable GL free energy G Γ , we first will suppose that the Sr RuO superconductivity is described by an order parameter ψ Γ , which transforms according to oneof the eight one-dimensional representations of D h : Γ = A g , A g , B g , B g , A u , A u , B u ,or B u . Let us notice that an analysis employing the D point group renders similar results.Here we will analyze the terms in G Γ linear in σ i and quadratic in ψ Γ : G Γ = G + α ( T ) | ψ Γ | + b | ψ Γ | +[ a ( σ xx + σ yy ) + c σ zz ] | ψ Γ | . (6)The terms proportional to σ xx , σ yy and σ zz in eqn. (6) give rise to discontinuities inthe elastic constants, evidenced from sound speed measurements [17]. On the other hand,discontinuities in the elastic compliance S and in the elastic constant C [27] arise from thelinear coupling with σ xy . However, due to symmetry, the later linear coupling does not existfor any Γ; therefore, S and C are expected to be continuous at T c for any of the one-dimensional irreducible representation that assumes a one-dimensional ψ Γ . Nevertheless,the results of Lupien et. al. experiments [2] showed a discontinuity in C . Hence, basedexclusively on sound speed measurements, we conclude that none of the one-dimensionalirreducible representations can provide an appropriate description of superconductivity inSr RuO . As far as we know, this conclusion has not been previously established in theliterature [3]. Let us mention that for any one-dimensional Γ, a detailed analysis of thecalculation of the jumps in C is presented in ref.[11].Due to the absence of discontinuity in S for any of the one-dimensional Γ, the super-conductivity in Sr RuO must be described by an order parameter ψ E transforming as oneof the two-dimensional representations E g or E u [4]. The GL theory establishes that onlythe parameters of one of the irreducible representations becomes non-zero at T c . Therefore,following the evidence provided in ref. ([5, 19]), we choose the E u spin-triplet state as thecorrect representation for Sr RuO , and the speed measurements are analyzed in terms ofthe model ψ E = ( ψ x , ψ y ), with ψ x and ψ y transforming as the components of a vector inthe basal plane. The expression for G is determined by symmetry arguments based on theanalysis of the second and fourth order invariants (real terms) of G Γ . To maintain gaugesymmetry, only real and even products of Ψ can occur in the expansion of G Γ ; thus, we findthat all real invariants should be formed by second and fourth order products of ψ ‘s [28]. Toobtain its expression , we use the fact that G is invariant with respect to a transformationby the generators c z and c x of D h . Applying the generators to different second and fourthorder combination of products of ψ ‘s, we find only one second order invariant | ψ x | + | ψ y | and three fourth order invariants, namely | ψ x | | ψ y | , | ψ x | + | ψ y | , and ψ x ψ ∗ y + ψ ∗ x ψ y .For the zero σ i case, the expansion of G gives place to: G = G + α ( T )( | ψ x | + | ψ y | ) + b | ψ x | + | ψ y | ) + b | ψ x | | ψ y | + b ψ x ψ ∗ y + ψ y ψ ∗ x ) , (7)where α = α (cid:48) ( T − T c ) and the coefficients b , b , and b are material-dependent realconstants [20, 21]. These coefficients have to satisfy special conditions in order to maintainthe free energy stability. The analysis of G is accomplished by considering two component( ψ x , ψ y ) with the form: ( ψ x , ψ y ) = ( η x e iϕ/ , i η y e − iϕ/ ); (8)where η x and η y are both real and larger than zero. After substitution of ψ x and ψ y inequation (7) , G becomes: G = G + α ( T )( η x + η y ) + b η x + η y ) +( b − b ) η x η y + 2 b η x η y sin ϕ. (9)For fixed values of the coefficients b and b , if b > G will reach a minimal value ifthe last term vanishes, i.e. if ϕ = 0. Moreover, if η x and η y have the form η x = η sin χ and η y = η cos χ , G becomes G = G + α ( T ) η + b η − ˜ b η sin χ, (10)where ˜ b ≡ b − b . If ˜ b > G reaches its minimum value if sin χ = 1, this condition issatisfied if χ = π/
4; and therefore η x = η y . On the other hand, if ˜ b <
0, then G becomesminimal if sin χ = 0. In this case, either η x = 0 or η y = 0. Since for a superconductingstate ( ψ x , ψ y ) ∼ (1 , ± i ), from the previous analysis, the lowest G state corresponds to b − b >
0. This thermodynamic state breaks time-reversal-symmetry; and hence, it is believed tobe the state describing superconductivity in Sr RuO [4, 5, 7]. In addition, it is found thatfor the phase transition to be of second order, it is required that b ≡ b + b − b > ψ x , ψ y ) ∼ (1 , ± i(cid:15) ) has beenchosen for the analysis of σ and why it gives rise to the discontinuity in S [11]. Minimiza-tion of eqn. (7) with respect to ϕ and χ , and employing eqn. (8) renders a set of solutions forthe two-component order parameter which depend on the relation between the coefficientsb , b , and b and also on the value of the phases ϕ and χ . Thus, for the E representation,solutions of the form, ψ = η (1 , e i ϕ , (11)are obtained, which are very similar to those found for the D one-dimensional irreduciblerepresentation. Therefore, these solutions are not able to account for the jump in C .However, solutions with both components different than zero are also attained: ψ = √ e i π/ (1 , η, ψ = √
32 (1 , i ) η. (12) FIG. 1: Superconducting state phase diagram for the two dimensional representation E of thetetragonal group D as function of the material parameters b and b showing the domains whichcorrespond to the order parameters ψ , ψ , and ψ . Each domain corresponds to a differentsuperconducting class. Each of these solutions corresponds to different relations for the b i . This is illustratedby Fig. (1), which shows the phase diagram, displaying the domains of ψ , ψ and ψ as afunction of b , b and b . Now, if the jump in C corresponds to a G minimum, the couplingterm with σ must be taken to be different from zero. If the solution ψ is considered, theterm containing σ becomes zero; therefore it is not acceptable. On the other hand, thisrequirement is satisfied by Ψ , with the form (1 , i ) η . Hence, the GL analysis renders Ψ asthe solution that breaks time reversal symmetry. B. Coupling of the order parameter to an external stress
The transition to an unconventional superconducting state shows manifestations as thebreakdown of symmetries, such as the crystal point group or the time reversal symmetry[20, 21]. This loss of symmetry has measurable manifestations in observable phenomena, asthe splitting of T c under an elastic deformation. The coupling between the crystal latticeand the superconducting state is described Refs. [20, 21]. As explained there, close to T c , anew term is added to G , which couples in second order Ψ with e ij and in first order Ψ with σ ij . These couplings give place to discontinuities in S ijkl at T c . C. Analysis of the phase diagram
An expression for G accounting for a phenomenological coupling to C in the Sr RuO basal plane is given by G = G + α (cid:48) ( T − T c )( | ψ x | + | ψ y | ) + b | ψ x | | ψ y | + b | ψ x | + | ψ y | ) + b ψ x ψ ∗ y + ψ y ψ ∗ x ) −
12 S ij σ i σ j + σ i Λ i + σ i d ij E j . (13)Here, Λ i are the temperature-dependent α i , d ij are the coupling terms between Ψ andS ij and E j are the invariant elastic compliance tensor components, defined below. In orderto determine these invariants describing the coupling of the order parameter to the stresstensor, we construct the tensor E j with Voigt components E = | ψ x | , E = | ψ y | and E = ψ ∗ x ψ y + ψ x ψ ∗ y ; where E couples σ and Ψ. The tensor d ij couples E i with σ j and has thesame nonzero components as S ij . By applying symmetry considerations [4], it is shown thatthe only non-vanishing independent components of d ij are d , d = d , d = d , and d .Contributions to G that are quadratic in both, Ψ and σ were neglected. Such terms wouldhave given an additional T dependence to the S ij [17]. However, given the large number ofindependent constants occurring in the associated sixth rank tensor, at this point, it is notclear 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 σ < G quad = α (cid:48) [ T − T c + ( σ )] | ψ x | + α (cid:48) [ T − T cy ( σ )] | ψ y | , (14)here T c + ( σ ) and T cy ( σ ) are given by T c + ( σ ) = T c − σ d α (cid:48) , T cy ( σ ) = T c − σ d α (cid:48) . (15)In what follows, we assume that d − d >
0, such that T c + > T cy . Notice that thisdoes not imply any lost in generality, assuming d − d <
0, would render an identicalmodel, simply by exchanging the x and y indices. Here, T c + is the higher of the two criticaltemperatures at which the initial transition occurs. As should be expected, just below T c + ,only ψ x is non zero. As T is further lowered, another phase transition happens at T c − , which FIG. 2: Temperature behavior of the two component order parameter ( ψ x , ψ y ) for the case ofa nonzero uniaxial stress below T c . Notice that only the BCS component ψ x ( T c + ) becomes nonzero for temperatures between T c + and T c − . The second unconventional component ψ y ( T c − ) onlyappears below T c − . is different than T cy . Below T c − , the ψ y is also different from zero (see fig. (2)). Thus, inthe presence of a non zero compressible σ , Ψ has the form ( ψ x , ψ y ) ≈ ψ (1 , ± i (cid:15) ), where (cid:15) isreal and equal to zero between T c + and T c − (phase 1), and increases from (cid:15) = 0 to (cid:15) ≈ c − (phase 2), as illustrated in figs. (1) and (2).The next step is finding T c − . To achieve this goal, the equilibrium value of the non zerocomponent of ψ x , ψ x = − α x /b is replaced in eqn. (13) and T c − follows from T c + − T c − = − (cid:104) d − d α (cid:48) (cid:105) (cid:104) ˜ b + b ˜ b (cid:105) σ . (16)To obtain eqn. (16), it is assumed that σ (cid:28) σ and only linear terms in σ are kept.The phase diagram for this system is shown in fig. (3). IV. CALCULATION OF THE DISCONTINUITIES
As discussed before, an external uniaxial stress acting on the Sr RuO basal planebreaks the tetragonal symmetry of the crystal. As a consequence of this, when a secondorder transition to the superconducting state occurs, it splits into two transitions. For thecase of applied σ , the analysis of the behavior of the sound speed at T c requires a systematicstudy of these second-order phase transitions. Moreover, thermodynamic quantities, suchas dT c /dσ i , C σ , and α σ , which are needed in order to calculate the components S σij are0 FIG. 3: Phase diagram showing the upper and lower superconducting transition temperatures, T c + and T c − , respectively, as functions of the compressible stress − σ i along the a axis. accompanied by a discontinuity at each of the second order phase transitions.As depicted in fig. (3), for a given σ (cid:54) = 0 as T is lowered below T c + , a first discontinuity fora thermodynamical quantity Q is observed at the first line of transition temperatures. Thisdiscontinuity along the transition line, corresponding to the higher transition temperatures, T = T c + ( σ i ) is given by ∆ Q + = Q ( T c + + 0 + ) − Q ( T c + − + ), where 0 + is a positiveinfinitesimal number. If T is further dropped below T c − , a second discontinuity arises, andthe lower line of transition temperatures appears. The discontinuity along this line, at T= T c − ( σ i ), is defined by ∆ Q − = Q ( T c − + 0 + ) − Q ( T c − − + ) [18]. The sum of these twodiscontinuities ∆ Q ( T c , σ = 0) = ∆ Q + + ∆ Q − , (17)gives the correct expressions for the discontinuities at T c , for the case with σ i = 0, wherethe Ehrenfest relations do not hold directly [4]. As an example of these discontinuities, thetwo jumps in C σ under an external σ i are sketched in fig. (4). A. Jumps due to a uniaxial stress σ The free energy, eqn. (13), for the cases where both σ and σ are nonzero is:1 FIG. 4: Schematic dependence of the specific heat on the temperature, for the case of an uniaxialstress splitting the Sr RuO transition Temperature. Notice the two jumps in the heat capacitynear the transition temperatures T c + and T c − . G = G + α x | ψ x | + α y | ψ y | + σ d ( ψ x ψ ∗ y + ψ ∗ x ψ y ) + b | ψ x | + | ψ y | ) + b | ψ x | | ψ y | + b ψ x ψ ∗ y + ψ y ψ ∗ x ) . (18)Here α x = α (cid:48) ( T − T c ) + σ d and α y = α (cid:48) ( T − T c ) + σ d . If only σ is applied, thisequation becomes: ∆ G = α x | ψ x | + α y | ψ y | + b | ψ x | + | ψ y | ) + b | ψ x | | ψ y | + b ψ x ψ ∗ y + ψ y ψ ∗ x ) , (19)where ∆ G = G − G ( T ). The nature of the superconducting state that follows from eqn. (19),depends on the values of the coefficients b , b , and b . The analysis from eqn. (19) of thesuperconducting part of G is performed by using, as was done previously, an expression forΨ given by eqn. (8).At T c + and in the presence of σ , the second order terms in eqn. (19) dominate and Ψ hasa single component ψ x ; whereas at T c − a second component ψ y appears. Thus, at very lowT, the fourth order terms dominate the eqn. (19) behavior. Each of these two-componentdomains has the form of ψ given by eqn. (12). In this case, G can be written in terms of η x and η y as2∆ G = α x η x + α y η y + b η x + η y ) +( b − b ) η x η y + 2 b η x η y sin ϕ. (20)The analysis of eqn. (20) depends on the relation between the coefficients b , b , andb . Assuming that b >
0, and η x and η y are both different from zero, and following theprocedure described after eqn. (9) one arrives to( ψ x , ψ y ) ≈ (1 , ± i(cid:15) ) , (21)where (cid:15) is real and grows from (cid:15) = 0 to (cid:15) ≈ c − , while eqn. (20)becomes ∆ G = α x η x + α y η y + b η x + η y ) − ( b − b ) η x η y . (22)To calculate the jumps at T c + , we use α x = α (cid:48) ( T − T c + ) and α y = α (cid:48) ( T − T cy ), and assumethat T c + > T cy . For the interval T c + > T > T c − , the equilibrium value for Ψ satisfies α x > α y = 0, i.e. η x > η y = 0, with η x = − α x /b , obtaining that T c + and its derivativewith respect to σ are respectively, T c + ( σ ) = T c − σ α (cid:48) d ,d T c + d σ = − d α (cid:48) . (23)The specific heat discontinuity at T c + , relative to its normal state value, is calculated byusing: ∆ C σ = − T ∂ ∆ G∂ T | T = T c + , (24)and renders the result ∆ C + σ = − T c + α (cid:48) b . (25)A schematic depiction of the C σ discontinuities below this transition temperature is ex-hibited in fig. (4). At T c + , the discontinuity in α σ is calculated by applying the Ehrenfestrelation of eqn. (3), yielding: ∆ α +1 = − α (cid:48) d b . (26)3The discontinuities in S ij are obtained by using eqns. (4) and (5), rendering the result,∆ S + i (cid:48) j (cid:48) = − d i (cid:48) d j (cid:48) b . (27)In the previous expression a prime on an index (as in i (cid:48) or j (cid:48) ) indicates a Voigt indextaking only the values 1,2, or 3. Thus, from eqn. (27) the change in S at T c + can becalculated to be ∆ S σ = − d b .To find the discontinuities at T c − , the term ( η x + η y ) in eqn. (22) is expanded, afterwhich G takes the form,∆ G = α x η x + b η x + (cid:104) α y + ( b b − b ) η x (cid:105) η y + b η y . (28)In this expression, the second order term in η y is renormalized by the square of η x .The second transition temperature is determined from the zero of the total prefactor of η y ,obtaining that T c − and its derivative with respect to σ are: T c − ( σ ) = T c − σ α (cid:48) (cid:104) d + d − b ˜ b ( d − d ) (cid:105) ,d T c − d σ = − α (cid:48) (cid:104) d + d − b ˜ b ( d − d ) (cid:105) . (29)Below T c − the superconducting free energy, eqn. (28) has to be minimized respect to bothcomponents of Ψ. After doing so, η x and η y for this temperature range are found to be η x = − b ˜ b (cid:104) ( b − ˜ b ) α y + ( b + ˜ b ) α x (cid:105) ,η y = − b ˜ b (cid:104) ( b − ˜ b ) α x + ( b − ˜ b ) α y (cid:105) . (30)This analysis shows that the second superconducting phase is different in symmetry, andthat time reversal symmetry is broken. The change in C σ at T c − , with respect to its valuein the normal phase, ∆ C − ,Nσ , is found to be, ∆ C − ,Nσ = − T c − α (cid:48) /b . The specific heatvariation at T c − is, ∆ C − σ = ∆ C − ,Nσ − ∆ C + σ , (31)which results in ∆ C − σ = − T c − α (cid:48) ˜ bb b . (32)The size of these jumps is complicated to infer, because it depends on the materialparameters b , b and b , and on the coupling constants d and d .4With the help of the Ehrenfest relation, eqn. (3), the discontinuity in α i at T c − is obtainedto be ∆ α − i (cid:48) = − α (cid:48) ˜ bb b (cid:16) d i (cid:48) + − b ˜ b d i (cid:48) − (cid:17) , (33)and after employing eqns. (4) and (5), the discontinuity in S i (cid:48) ,j (cid:48) at T c − is shown to be∆S − i (cid:48) j (cid:48) = − ˜ b b b ( d i (cid:48) + − b ˜ b d i (cid:48) − ) ( d j (cid:48) + − b ˜ b d j (cid:48) − ) . (34)Here d i (cid:48) ± = d i (cid:48) ± d i (cid:48) . The discontinuities occurring at T c , in the absence of uniaxialstress, can be obtained by adding the discontinuities occurring at T c + and T c − , yielding:∆ C σ = − T c α (cid:48) b , ∆ α i (cid:48) = − α (cid:48) d i (cid:48) + b , ∆S i (cid:48) j (cid:48) = − (cid:18) d i (cid:48) + d j (cid:48) + b + d i (cid:48) − d j (cid:48) − ˜ b (cid:19) . (35)Before continuing, it is important to emphasize that at at zero stress, the derivative of T c with respect to σ i is not defined; therefore, there is no reason to expect any of the Ehrenfestrelations to hold. B. Jumps due to a shear stress σ When a shear stress σ is applied to the basal plane of Sr RuO , the crystal tetrago-nal 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 T c also requires asystematic study of the two successive second order phase transitions. Hence, the C dis-continuity observed by Lupien et. al. [2] at T c , can be explained in this context.If there is a double transition, the derivative of T c with respect to σ i.e. dT c /dσ isdifferent for each of the two transition lines. At each of these transitions, C σ , α σ , and S σ ij show discontinuities. As discussed before, the sum of them gives the correct expressions forthe discontinuities at zero shear stress, where the Ehrenfest relations do not hold.The T c - σ phase diagram will be similar to that obtained for σ ; therefore, the diagram5in fig.(3) also qualitatively holds here. In the case of an applied σ , ∆ G is given by∆ G = α ( | ψ x | + | ψ y | ) + σ d ( ψ x ψ ∗ y + ψ ∗ x ψ y ) + b | ψ x | + | ψ y | ) + b | ψ x | | ψ y | + b ψ x ψ ∗ y + ψ y ψ ∗ x ) . (36)Here α = α (cid:48) ( T − T c ), and the minimization of ∆ G is performed as in the σ case, i.e. bysubstituting the general expression for Ψ given in eqn. (8). After doing so, ∆ G becomes∆ G = α ( η x + η y ) + 2 η x η y σ sin ϕ d + b η x + η y ) +( b − b ) η x η y + 2 b η x η y sin ϕ. (37)In the presence of σ , the second order term determines the phase below T c + , which ischaracterized by ψ x and by ψ y = 0. As the temperature is lowered below T c − , depending ofthe value of b a second component ψ y may appear. If at T c − a second component occurs,the fourth order terms in eqn. (37) will be the dominant one. Thus for very low T’s, or for σ →
0, a time-reversal symmetry-breaking superconducting state may emerge. The analysisof eqn. (37) depends on the relation between the coefficients b and b . It also depends onthe values of the quantities η x and η y , and of the phase ϕ . If b <
0, and η x and η y are bothnonzero, the state with minimum energy has a phase ϕ = π/
2. The transition temperatureis obtained from eqn. (37), by performing the canonical transformations: η x = √ ( η µ + η ξ )and η y = √ ( η µ − η ξ ). After their substitution, eqn. (37) becomes∆ G = α + η ξ + α − η µ + 14 ( η ξ + η µ ) + ( b + b )( η ξ − η µ ) . (38)If, as was done before, η ξ = η sin χ and η µ = η cos χ , eqn. (38) takes the form∆ G = α + η sin χ + α − η cos χ + η (cid:104) b + ( b + b ) cos χ (cid:105) . (39)∆ G is minimized if cos 2 χ = 1, this is, if χ = 0. Also, in order for the phase transitionto be of second order, b (cid:48) , defined as b (cid:48) ≡ b + b + b , must be larger than zero. Therefore,if σ is non zero, the state with the lowest free energy corresponds to b <
0, phase ϕ equalto π/
2, and Ψ of the form: ( ψ x , ψ y ) ≈ η ( e iϕ , e − iϕ ) . (40)6In phase 1 of fig. (3), ϕ = 0, and as T is lowered below T c − , phase 2, ϕ grows from 0 toapproximately π/
2. Again, following an analysis similar to that carried out for σ , the twotransition temperatures T c + and T c − are obtained to be: T c + ( σ ) = T c − σ α (cid:48) d ,T c − ( σ ) = T c + b b α (cid:48) σ d . (41)The derivative of T c + with respect to σ , and the discontinuity in C + σ at T c + are respectivelyfound to be: d T c + d σ = − d α (cid:48) , ∆ C + σ = − T c + α (cid:48) b (cid:48) . (42)After applying the Ehrenfest relations, eqns. (4) and (5), the results for ∆ α σ and ∆S atT c + are: ∆ α + σ = − α (cid:48) d b (cid:48) , ∆S +66 = − d b (cid:48) . (43)For T c − , the derivative of this transition temperature with respect to σ , and the disconti-nuities in the specific heat, thermal expansion and elastic stiffness respectively are: d T c − d σ = b d b α (cid:48) , ∆ C − σ = − T c − α (cid:48) b b b (cid:48) , (44)∆ α − σ = 2 α (cid:48) d b (cid:48) , ∆S − = − d bb (cid:48) b . (45)Since for the case of σ , the derivative of T c with respect to σ is not defined at zero stresspoint, the Ehrenfest relations do not hold at T c . Thus, the discontinuities occurring at T c ,in the absence of σ , are calculated by adding the expressions obtained for the discontinuitiesat T c + and T c − ,7∆ C σ = − T c α (cid:48) b , ∆S = − d b , ∆ α σ = 0 . (46)Notice that in this case, there is no discontinuity for α σ .Since the phase diagram was determined as a function of σ , rather than as a functionof the strain, (see fig. (3)), in this work, as in refs. [4, 11], we make use of the 6 × ij . However, the sound speedmeasurements are best interpreted in terms of the elastic stiffness matrix C, with matrixelements C ij , which is the inverse of S [23]. Therefore, it is important to be able to obtain thediscontinuities in the elastic stiffness matrix in terms of the elastic compliance matrix. Thus,close to the transition line, C( T c + 0 + ) = C( T c − + ) + ∆ C and S( T c + 0 + ) = S( T c − + ) + ∆S,where 0 + is positive and infinitesimal. By making use of the fact that C( T c + 0 + ) S( T c + 0 + )= ˆ1, where ˆ1 is the unit matrix, it is shown that, to first order, the discontinuities satisfy,∆C ≈ − C ∆S C. In this manner, it is found that, for instance at T c + , ∆C +11 ≈ j d j ) b .From this expression it is clear that ∆C +11 must be greater than zero. In general, at T c + ,T c − , and T c , the expressions that define the jumps for the discontinuities in elastic stiffnessand compliances, due to an external stress, have either a positive or a negative value. In thisway, ∆S , ∆S , ∆S , and ∆S are all negative; while, the stiffness components ∆C ,∆C , ∆C , and ∆C are all positive. V. FINAL REMARKS
Since for Sr RuO , the symmetry-breaking field, due to σ i , is under experimental con-trol, states of zero symmetry-breaking stress and of σ i single direction can be achieved [1–3].Hence, it has significant advantages the use of Sr RuO as a material in detailed studies ofsuperconductivity symmetry-breaking effects, described by a two-component order param-eter. Nevertheless, determining from Sr RuO experimental measurements the magnitudeof the parameters in the Ginzburg-Landau model is complicated, because the number ofindependent parameters occurring for the case of tetragonal symmetry is greater than forthe case of hexagonal symmetry (i.e. UP t ) [24–26]. Thus for Sr RuO , three linearly8independent parameters, b , b , and b , are required to specify the fourth order terms in Ψoccurring in eqn. (7) ; whereas only two independent parameters, b and b , are requiredfor UPt . For Sr RuO , two independent ratios can be formed from the three indepen-dent b i parameters, and these two independent ratios could be determined, for example, byexperimentally determining the ratios ∆ C + σ / ∆ C − σ in the presence of the σ and σ [4, 11].Measurements results for the Sr RuO elastic constants below T c are presented in Ref.[2]. There, it is concluded that the quantities C and C - C follow the same behavior asthose of the BCS superconducting transition, which is evidenced by a change in slope belowT c . On the other hand, a discontinuity is observed for C below T c , without a significantchange in the sound speed slope as T goes below 1 Kelvin. It has been previously stated[2, 11] that this kind of C changes can be understood as a signature of an unconventionaltransition to a superconducting phase. Thus, this set of results and others, as those of Clif-ford et. al [3], lead to consider Sr RuO as an excellent candidate for a detailed experimentalinvestigation of the effects of a symmetry-breaking field in unconventional superconductors. Acknowledgments
We thank Prof. Michael Walker from the University of Toronto, Prof. Kirill Samokhinfrom Brock University, and Prof. Christian Lupien from Universit de Sherbrooke for stimu-lating discussions. We are also grateful to the Referee for his comments. This research wassupported by the Grant CDCHTA-ULA number C-1908-14-05-B. [1] C. Lupien, W. A. MacFarlane, Cyril Proust, Louis Taillefer, Z. Q. Mao, and Y. Maeno, Phys.Rev. Lett. , 5986 (2001).[2] C. Lupien, Ph. D. Thesis, University of Toronto, 2002.[3] W. Clifford, et al. Science , 283 (2014).[4] M. Walker and P. Contreras, Phys. Rev. B , 214508 (2002).[5] Y. Maeno, T. M. Rice, and M. Sigrist, Physics Today , No. 1, 42 (2001).[6] T. M. Rice and M. Sigrist, J. Phys. Condens. Matter , L643 (1995).[7] A. Mackenzie, and Y. Maeno, Rev. of Modern Physics , 657 (2003). [8] K. Ishida, H. Mukuda, Y. Kitaoka, K. Asayama, Z. Q. Mao, Y. Mori, Y. Maeno, Nature(London) , 658 (1998).[9] J. Duffy, S. Hayden, Y. Maeno, Z. Mao, J. Kulda, and G. McIntyre, Phys. Rev. Lett. , 5412(2000).[10] G. M. Luke, Y. Fudamoto, K. M. Kojima, M. I. Larkin, J. Merrin, B. Nachumi, Y. J. Uemura,Y. Maeno, Z. Q. Mao, H. Nakamura, and M. Sigrist, Nature , 558 (1998).[11] P. Contreras, PhD Thesis, University of Toronto, (2006) 96.[12] M. Sigrist Prog. Theor. Phys. , 917 (2002).[13] T. M. Rice and M. Sigrist, J. Phys. Condens. Matter , L643 (1995).[14] L. D. Landau and E. M. Lifshitz, Statistical Physics , Butterwoth-Heinemann (Oxford, 1980)[15] A. Auld,
Acoustic fields and waves in solids (Wiley ans Sons, New York, 1973).[16] L. D. Landau and E. M. Lifshitz,
Elasticity Theory , Pergamon Press (London, 1970)[17] L. R. Testardi, Phys. Rev. B , 95 (1971).[18] M. Walker, Phys. Rev. B (1980) 1338.[19] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Jujita, J. G. Bednorz, and f. Lichten-berg, Nature (London) , 532 (1994).[20] M. Sigrist and K. Ueda, Rev. Mod. Phys. , 239 (1991).[21] V. P. Mineev and K. V. Samokhin, Introduction to Unconventional Superconductivity (Gordonand Breach, Amsterdam, 1999).[22] E. Bauer G. et. al, Phys. Rev. Lett. , 027003 (2004).[23] F. Ney Physical Properties of Crystals (Oxford, London, 1979).[24] D. W. Hess, T. A. Tokuyasu and J. A. Sauls, J. Phys.: Condens. Matter , 8135 (1989).[25] B. Lussier, L. Taillefer, W. J. L. Buyers, T. E. Mason, and T. Petersen Phys. Rev. B ,R6873 (1996).[26] R. Joynt, and L. Taillefer, Rev. of Modern Physics , 235 (2002).[27] the Voigt notation for C means C xyxy where 6 = xy [15][28] The invariance under the gauge symmetry U (1) means that the quantities ψ i must transformaccording to the rule ψ x → e i Φ ψ x and ψψ
Elasticity Theory , Pergamon Press (London, 1970)[17] L. R. Testardi, Phys. Rev. B , 95 (1971).[18] M. Walker, Phys. Rev. B (1980) 1338.[19] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Jujita, J. G. Bednorz, and f. Lichten-berg, Nature (London) , 532 (1994).[20] M. Sigrist and K. Ueda, Rev. Mod. Phys. , 239 (1991).[21] V. P. Mineev and K. V. Samokhin, Introduction to Unconventional Superconductivity (Gordonand Breach, Amsterdam, 1999).[22] E. Bauer G. et. al, Phys. Rev. Lett. , 027003 (2004).[23] F. Ney Physical Properties of Crystals (Oxford, London, 1979).[24] D. W. Hess, T. A. Tokuyasu and J. A. Sauls, J. Phys.: Condens. Matter , 8135 (1989).[25] B. Lussier, L. Taillefer, W. J. L. Buyers, T. E. Mason, and T. Petersen Phys. Rev. B ,R6873 (1996).[26] R. Joynt, and L. Taillefer, Rev. of Modern Physics , 235 (2002).[27] the Voigt notation for C means C xyxy where 6 = xy [15][28] The invariance under the gauge symmetry U (1) means that the quantities ψ i must transformaccording to the rule ψ x → e i Φ ψ x and ψψ y → e i Φ ψψ