3.1. Superconducting free energy

In order to derive a suitable GL free energy G^Γ, we first will suppose that the Sr₂RuO₄ superconductivity is described by an order parameter ψ^Γ, which transforms according to one of the eight one-dimensional representations of D_4h: Γ = A_1g, A_2g, B_1g, B_2g, A_1u, A_2u, B_1u, or B_2u. Let us notice that an analysis employing the D₄ point group renders similar results. Here we will analyze the terms in G^Γ linear in σ_i and quadratic in ψ^Γ:

The terms proportional to σ_xx, σ_yy and σ_zz 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 σ_xy . However, due to symmetry, the later linear coupling does not exist for any Γ; therefore, S₆₆ and C₆₆ 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. 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 ψ^E 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 T_c. Therefore, following the evidence provided in Refs. 5 and 19, we choose the E_2u spin-triplet state as the correct representation for Sr₂RuO₄, and the speed measurements are analyzed in terms of the model ψ^E = (ψ_x, ψ_y), with ψ_x and ψ_y 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 G^Γ. To maintain gauge symmetry, only real and even products of Ψ can occur in the expansion of G^Γ; 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 c_4z and c_2x of D_4h. Applying the generators to different second and fourth order combination of products of ψ’s, we find only one second order invariant |ψ_x|² + |ψ_y|² and three fourth order invariants, namely |ψ_x|²|ψ_y|², |ψ_x|⁴ + |ψ_y|⁴, and ψx2ψy*2+ψx*2ψy2.

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

where α = α'(T - T_c0) 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 (ψ_x, ψ_y) with the form:

where η_x and η_y are both real and larger than zero. After substitution of ψ_x and ψ_y in Eq. (7), G becomes:

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 η_x and η_y have the form η_x = ηsin χ and η_x = ηcos χ, G becomes

where b̃≡b3-b2. If b̃>0, G reaches its minimum value if sin22χ = 1, this condition is satisfied if χ = π/4; and therefore η_x = η_y. On the other hand, if b̃<0, then G becomes minimal if sin2 2χ= 0. In this case, either η_x = 0 or η_y = 0. Since for a superconducting state (ψ_x, ψ_y) ∼ (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 Sr₂RuO₄^4,5,7. In addition, it is found that for the phase transition to be of second order, it is required that b ≡ b₁ + b₂ - b₃ > 0.

At this point it is important to understand why the state (ψ_x, ψ_y) ∼ (1, ± iϵ) 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,

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:

Each of these solutions corresponds to different relations for the b_i. 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 symmetry^20,21. This loss of symmetry has measurable manifestations in observable phenomena, as the splitting of T_c 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 T_c, a new 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.

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

Here, Λ_i are the temperature-dependent α_i, d_ij are the coupling terms between Ψ and S_ij and E_j 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 E_j with Voigt components E₁ = |ψ_x|², E₂ = |ψ_y|² and E6=ψx*ψy+ψxψy*; where E₆ couples σ₆ and Ψ. The tensor d_ij couples E_i with σ_j and has the same nonzero components as S_ij. By applying symmetry considerations⁴, it is shown that the only non-vanishing independent components of d_ij 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 S_ij¹⁷. 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

here T_c+ (σ₁) and T_cy (σ₁) are given by

In what follows, we assume that d₁₁ −d₁₂> 0, such that T _c+> T_cy . Notice that this does not imply any lost in generality, assuming d₁₁ − d₁₂< 0, would render an identical model, simply by exchanging the x and y indices. Here, T_c+ is the higher of the two critical temperatures 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 is different than T_cy. Below T_c-, the ψ_y is also different from zero (see Fig. (2)). Thus, in the presence of a non zero compressible σ₁, Ψ has the form (ψ_x, ψ_y) ≈ ψ(1, ± I ϵ) where ϵ is real and equal to zero between T_c+ and T_c- (phase 1), and increases from ϵ = 0 to ϵ ≈ 1 as T becomes smaller than T_c- (phase 2), as illustrated in Figs. (1) and (2).

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

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

To obtain Eq. (16), it is assumed that σ12≪σ1 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, T_c++ and T_c-, respectively, as functions of the compressible stress -σ_i along the a axis. 

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:

Here α_x = α^'(T −T_c0 )+σ₁d₁₁ and α_y = α^'(T −T_c0 )+ σ₁d₁₂. If only σ₁ is applied, this equation becomes:

Where ∆G = G − G₀(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 T_c+ and in the presence of σ₁, the second order terms in Eq. (19) dominate and Ψ has a single component ψ_x; whereas at T_c- a second component ψ_y 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 η_x and η_y as

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

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

To calculate the jumps at T_c+, we use α_x = α'(T −T_c+) and α_y = α'(T − T_cy), and assume that T_c+> T_cy . For the interval T_c+> T > T_c−, the equilibrium value for Ψ satisfies α_x > 0 and α_y = 0, i.e. η_x > 0 and η_y = 0, with ηx2=-2αx/b1, obtaining that T _c+ and its derivative with respect to σ₁ are respectively,

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

and renders the result

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

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

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 T_c+ can be calculated to be ΔS11σ1=-2 d112b1.

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

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 ηy2, obtaining that T_c- and its derivative with respect to σ₁ are:

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

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

which results in

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 α_i at T_c- is obtained to be

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

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 T_c+ and T_c-, yielding:

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 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 T_c 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 T_c, can be explained in this context.

If there is a double transition, the derivative of T_c with respect to σ₆ i.e. dT_c/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 T_c −σ₆ 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

Here α = α'(T − T_c0), 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

In the presence of σ₆, the second order term determines the phase below T_c+, which is characterized by ψ_x and by ψ_y = 0. As the temperature is lowered below T_c- , depending of the value of b₃ a second component ψ _y may appear. If at T_c- 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 η_y, and of the phase φ. If b₃< 0, and η_x and η_y 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

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

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:

In phase 1 of Fig. (3), φ = 0, and as T is lowered below T_c-, 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 T_c+ and T_c- are obtained to be:

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

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

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

Since for the case of σ₆, the derivative of T_c with respect to σ₆ is not defined at zero stress point, the Ehrenfest relations do not hold at T_c0. Thus, the discontinuities occurring at T_c0, in the absence of σ₆, are calculated by adding the expressions obtained for the discontinuities at T_c+ and T_c-,

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 S_ij. However, the sound speed measurements are best interpreted in terms of the elastic stiffness matrix C, with matrix elements C_ij, 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(T_c +0⁺) = C(T_c − 0⁺) + ∆ C and S(T_c + 0⁺) = S(T_c − 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 T_c+, ΔC11+≈(2(Cj1 dj1)2/b1). From this expression it is clear that ΔC11+ must be greater than zero. In general, at T_c+, T_c-, and T_c0, 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.