Elastocaloric determination of the phase diagram of Sr2RuO4

One of the main developments in unconventional superconductivity in the past two decades has been the discovery that most unconventional superconductors form phase diagrams that also contain other strongly correlated states. Many systems of interest are therefore close to more than one instability, and tuning between the resultant ordered phases is the subject of intense research1. In recent years, uniaxial pressure applied using piezoelectric-based devices has been shown to be a particularly versatile new method of tuning2,3, leading to experiments that have advanced our understanding of the fascinating unconventional superconductor Sr2RuO4 (refs. 4–9). Here we map out its phase diagram using high-precision measurements of the elastocaloric effect in what we believe to be the first such study including both the normal and the superconducting states. We observe a strong entropy quench on entering the superconducting state, in excellent agreement with a model calculation for pairing at the Van Hove point, and obtain a quantitative estimate of the entropy change associated with entry to a magnetic state that is observed in proximity to the superconductivity. The phase diagram is intriguing both for its similarity to those seen in other families of unconventional superconductors and for extra features unique, so far, to Sr2RuO4.


I. FREQUENCY, TEMPERATURE, AND STRAIN DEPENDENCE OF THE AC STRAIN AMPLITUDE
In our study of the elastocaloric effect we used the in situ determined displacement amplitude d exc to calculate the applied AC strain amplitude ∆ε. In the following we demonstrate that the applied displacement amplitude d exc displays only small variations as function of frequency, temperature, and strain in the parameter range of our measurements. Figure S1 displays the frequency response of the AC displacement amplitude and the corresponding phase shift at 4 K for an excitation amplitude of 0.5 V. d exc is in a wide range independent of frequency. Below about 70 Hz we observe an increased scattering of the data. At high-frequency, above 2 kHz, we find an abrupt increase in d exc (f ) indicating a nonlinear voltage-displacement response of the piezo-electric actuator. In this frequency range also the phase shift starts to deviate considerably from 90 • . We conclude that in our set-up, the response of the actuators is appropriate for measurement in the range between 70 Hz and approximately 2 kHz.
The AC displacement amplitude exhibits a small temperature dependence. It monotonically increases by approximately 6% between 1 and 8 K. Figure S2 shows data taken at a frequency of 1513 Hz with an AC voltage amplitude V exc = 0.5 V used to drive the piezo-electric actuator. The same frequency and AC voltage amplitude was used to investigate the dependence of d exc on applied tuning strain. Figure S3 displays data at three selected temperatures.
The general behavior of d exc (ε) does not change with temperature. At all temperature the maximum change in d exc (ε) is less than 11 % over the range of strain used in our experiments.

II. THEORETICAL CONSIDERATIONS
In section II A we provide a review on the relation between the elastocaloric effect and the uniaxial Grüneisen parameter Γ. In section II B we provide details of the calculation of Γ across the superconducting transition close to the strain-induced Van-Hove singularity of Sr 2 RuO 4 using a parametrisation of its γ-band; some results of this calculation are shown in the main text. Finally, in section II C we discuss the entropy quench across the superconducting transition within an effective theory valid close to the Van-Hove singularity, and we compare the scenario of a full gap versus a gap with nodes at the Van-Hove point in momentum space. We find that a nodal gap at the Van-Hove point is inconsistent with experimental signatures.

A. Elastocaloric effect and uniaxial Grüneisen parameter
In the case that the temperature T and uniaxial strain, say, along the x-axis ε xx can be controlled experimentally the entropy should be considered as a function S = S(T, ε xx ). Its differential is then given by where C εxx = T ∂S ∂T εxx is the specific heat at constant uniaxial strain ε xx . Under adiabatic conditions dS = 0, it follows where Γ is the Grüneisen parameter generalized to the case of uniaxial strain ε xx . Up to a factor of T , it quantifies the change of temperature upon adiabatically varying ε xx and thus describes an elastocaloric effect. The Grüneisen parameter defined in Eq. S2 is dimensionless. , and Young's modulus, 1 where σ xx is the corresponding stress component. The uniaxial Grüneisen parameter is then B. Grüneisen parameter arising from the γ-band of Sr 2 RuO 4 1. Tight-binding model for the γ-band We follow Refs. [1,2] and consider an experimentally determined tight-binding model of the γ-band of Sr 2 RuO 4 with the two-dimensional electron spectrum where µ is the chemical potential. The nearest-neighbour hopping amplitudes are t x and t y , and t ′ is the next-nearest-neighbour hopping amplitude within the crystallographic (a, b)plane with lattice constants a x and a y .
In the absence of strain the system possesses tetragonal symmetry with lattice constants a x = a y . In order to obtain the values for the hopping parameters at zero strain, we make use of tight-binding fits to the ARPES data by Burganov et al. [3] that yields the values t x = t y = t 0 = 0.119 eV, t ′ 0 = 0.392t 0 , and µ = 1.48t 0 . The value for t 0 and the chemical potential µ are directly taken from Ref. [3]. For the next-nearest neighbour hopping Burganov et al. give the value t ′ 0 = 0.41t 0 ; however, this value yields an energy of 19 meV at the point (k x , k y ) = (0, π) in the two-dimensional Brillouin zone that overestimates the value obtained experimentally. Instead of a global fit performed by Burganov et al. [3], we refitted their data with a local fit optimizing the behavior around this point in the Brillouin zone yielding a value for t ′ 0 that is 4% smaller. We note that the value for t ′ 0 of Burganov et al. [3] will not qualitatively change the behavior of the Grüneisen parameter but change its magnitude by approximately a factor of two due to an overestimate of the electronic energy.
In the presence of normal strains ε xx and ε yy , the amplitudes are modified, and for small strains we can approximate up to linear order with linear coefficients α and α ′ . The uniaxial strain along the y-direction, ε yy = −ν xy ε xx , is determined by ε xx with the Poisson ratio whose value at ambient pressure and 4 K is given by ν xy ≈ 0.508 [4]. The Van-Hove singularity is reached in the experiment for a strain of ε xx = −0.44%, see above. The tight-binding dispersion exhibits a Van-Hove point at (k x , k y ) = (0, π) for a choice of the parameters α = α ′ ≈ 7.604, where we used α = α ′ for simplicity. This value is consistent with the calculations of Barber et al. [5]. The parameter α will determine the overall magnitude of the Grüneisen parameter.
The corresponding density of states per spin is given by We have multiplied the integral by the area of the two-dimensional unit cell, a x a y , so that ν(ε) represent the number of states per energy and per unit cell. This density of states is independent of the lattice constants, as becomes manifest upon substituting the integration variables k x =k x /a x and k y =k y /a y . The strain dependence of ν(ε) thus arises from the hopping amplitudes in the dispersion of Eq. S4. ν(ϵ) and other densities of states that we use in the analysis of the Grüneisen parameter can be expressed in terms of elliptic integrals.
The evolution of the energy-dependent density of states with uniaxial strain is illustrated in Fig. S4. The Van-Hove singularity at zero uniaxial strain splits into two for finite ε xx . A Van-Hove singularity reaches the Fermi level for ε xx = −0.44%. The density of states at the Fermi level ν(0) as a function of uniaxial strain is shown in Fig. S5. A Van-Hove singularity at the Fermi level is realized for compressive strain ε xx ≈ −0.44% as well as tensile strain ε xx ≈ 0.34%. In between the density of states is minimal for ε xx ≈ −0.07%. Van-Hove singularity at the Fermi level is realized for compressive strain ε xx ≈ −0.44% as well as tensile strain ε xx ≈ 0.34%. As a result, a minimum arises at ε xx ≈ −0.07%.

Thermodynamics arising from the γ-band
In order to account for the behavior of the Grüneisen parameter across the superconducting transition close to the Van-Hove singularity of Sr 2 RuO 4 , we consider the entropy per unit cell, where E(ε) = √ ε 2 + ∆ 2 with the superconducting gap ∆, and the Fermi function f (E) = (e E/(k B T ) + 1) −1 . The factor of 2 in front of the integral of Eq. S7 accounts for the spin degree of freedom. The superconducting gap ∆ is determined by the gap equation with a cutoff provided by the frequency ω 0 , and the coupling constant g > 0 with units of energy. Our choice for the cutoff frequency ℏω 0 /k B = 400 K was informed by inelastic neutron scattering data on Sr 2 RuO 4 [6]. In order to reproduce the critical temperature T c ≈ 3.5 K as observed at the strain-induced Van-Hove singularity of Sr 2 RuO 4 , we need to choose g/k B ≈ 714 K. The gap ∆ = ∆(T, ε xx ) itself will not only depend on temperature T but also on strain ε xx , see Figure S6.
With the help of the entropy S7 the Grüneisen parameter can be evaluated using Eq. S2 by taking derivatives with respect to T and strain ε xx . The specific heat is given by . For the strain derivative we obtain where we introduced the auxiliary function In the normal phase ∆ = 0 sufficiently far away from the Van Hove singularity where the density of states ν(ε) only smoothly varies in the range |ε| ≲ k B T around the Fermi energy, standard Fermi liquid behaviour is obtained. The entropy then depends linearly on and the Grüneisen parameter reduces to a constant that quantifies the strain dependence of the density of states at the Fermi level ν(0). From this expression follows that the Grüneisen parameter in the Fermi-liquid limit will change sign at maxima as well as minima of ν(0) as a function of uniaxial strain ε xx . The pronounced  singularity above the critical temperature, entropy accumulates resulting in a maximum of S(ε xx ) and thus a sign change of ∂S/∂ε xx | T and Γ. As the superconducting phase is entered upon lowering the temperature, a large part of the entropy is quenched so that the maximum in S(ε xx ) is converted into a minimum. This leads to a saddle point in the strain dependence of the entropy S(ε xx ) close to the maximum of the critical temperature T c .

Influence of inter-layer hopping on the Van Hove singularity
A finite inter-plane hopping of electrons will eventually cut off the logarithmic divergence of the Van Hove singularity in the density of states and, as a consequence, weaken and broaden the associated thermodynamic signatures. Here, we estimate the temperature and strain scale at which this cut-off is expected to happen.
We start the discussion with an extended tight-binding Hamiltonian following Ref. [7] that is based on the three t 2g Ru orbitals, Here, s = ±1 represents the spin degree of freedom, η is the spin-orbit coupling, and the dispersions are parameterised as We used already dimensionless wavevectors, a i k i → k i with i = x, y, z and k i ∈ (−π, π]. This Hamiltonian gives rise to the α, β, and γ bands. The values of the hopping parameters can be found in Ref. [7] where they were obtained by a fit to ab initio calculations. We already accounted for a strain dependence ε xx of the hopping parameterst 1 andt 2 with a strength α and α ′ , similar to Eq. S5, and neglected for simplicity the dependence of all other hopping parameters on strain. Note that the ab initio parameters of Ref. [7] overestimate the values of the hopping parameters. For example, the valuet 1 = 0.3568 eV [7] should be compared to t 0 = 0.119 eV of section II B 1. Using the ab initio hopping parameters of Ref. [7] and the Poisson ratio ν xy = 0.508 [4] setting α ′ = α for simplicity, the γ band dispersion ε γ ( ⃗ k) of Eq. S14 vanishes at (k x , k y , k z ) = (0, π, 0) for the critical strain ε VH xx = −0.44% for a value of the parameter α = 15.62. These values based on ab initio calculations will be used for the following estimates.
First, let us recall the behavior of the γ band of the two-dimensional model of section II B 1. Close to the point (k x , k y ) = (0, π) its dispersion possesses a saddle point with positive coefficients c 1 , c 2 , c 3 , and δε xx = ε xx − ε VH xx is the distance to the critical strain. This leads to a logarithmic singularity in the two-dimensional density of states The dispersion of the γ band resulting from the full three-dimensional model of Eq. S14 possesses close to (k x , k y , 0) = (0, π, k z ) the modified form We kept here only the first order Fourier components of k z except for the last term with coefficient c 6 that involves a second order 2k z Fourier component. We also neglected contri- The resulting density of states for δε xx = 0 is approximately constant in the range |ε| < c 6 with two cusps, i.e., typical non-analyticities for a three-dimensional density of states at the two edges ε = ±c 6 , see Fig. S10(a).
The regularization of the logarithmic singularity in the density of states by the inter-layer hopping leads to a temperature scale T 3d ∼ c 6 /k B and a strain scale ε 3d xx ∼ c 6 /c 3 . Using the above estimates we obtain the values T 3d ≈ 5.8 K and δε 3d xx ∼ 3.8 × 10 −5 , i.e., approximately 10 −2 of the absolute value of the critical strain. The latter is consistent with comparing T 3d and the energy of the γ band at (k x , k y , k z ) = (0, π, 0) and zero strain that is 53 meV or 615 K using the ab initio parameters (compared to just over 10 meV measured experimentally).
Both scales are reflected for example in the dependence of the specific heat coefficient of the normal state. In the low temperature limit, on the one hand, the specific heat coefficient is proportional to the density of states. In order to resolve the inter-layer regularization a strain resolution of order δε 3d xx would be required that is far beyond the experimental resolution. On the other hand, at the critical strain the specific heat coefficient diverges logarithmically as a function of decreasing temperature for the two-dimensional model. For the three dimensional model with inter-layer hopping this divergence is cut off, and the specific heat coefficient saturates but only below a temperature of order 0.1T 3d ∼ 0.6 K, see We can conclude for Sr 2 RuO 4 that its thermodynamics in the normal state is only barely affected by the inter-layer hopping. First, its signatures as a function of strain occur on a scale beyond the strain resolution of the experiment. Second, its signatures as a function of temperature are most pronounced at the critical strain but are negligible for temperatures T > T c,max ≈ 3.5 K ≈ 0.6T 3d . For lower temperatures the Van Hove singularity seems to be rather cut off by the superconducting order parameter masking the effect of the inter-layer hopping on the Van Hove singularity in density of states. Finally, note that the ab initio parameters overestimate the scales so that our estimates about the influence of inter-layer hopping are on the conservative side.

Influence of disorder on the Van Hove singularity
The Van Hove singularity might in principle also be smeared out by elastic scattering off impurities. This becomes relevant once the elastic scattering rate ℏτ −1 0 exceeds the scale for inter-layer coupling, which corresponds to several Kelvin [8]. We are, however, confident that the scattering rate must be significantly below this value. Sr 2 RuO 4 is known to be sensitive to pair-breaking due to non-magnetic impurities [9]. Hence, the very observation of a superconducting transition implies that ℏτ −1 0 ≪ k B T c and elastic scattering events will not smear out the Van Hove singularity in the temperature regime of our measurements.
C. Entropy quench for a superconducting gap with nodes at the Van Hove singularity In section II B it was assumed that the superconductor is fully gapped at the Van Hove point in momentum space. In this case, the calculated entropy, that is enhanced at the Van Hove singularity, gets strongly quenched upon entering the superconducting phase, see Fig. S9. This behavior is consistent with the experimental observations as presented in the main text. In the present section, we demonstrate that a superconducting gap with nodes at the Van Hove points is not able to reproduce such a substantial entropy quench.

Effective low-energy theory for the Van Hove singularity and nodal superconductivity
For this purpose, we consider an effective theory that is only valid close to the Van Hove singularity. Expanding the dispersion of Eq. S4 close to the Van Hove singularity at k y = π/a y , it acquires in lowest order the following form where r = −µ + 2(t y − t x ) + 4t ′ tunes the distance to the Van Hove point, the mass m = tx−2t ′ . We will supplement the effective theory with a hard energy cutoff (q 2 x + q 2 y )/(2m) ≤ ε 0 where ε 0 is on the order of the hopping; we will use ε 0 = 1 2maxay .
We assume that the superconducting gap depends in general analytically on momentum, ∆ q = ∆γ q , and we consider the following three scenarios: The scenario (1) corresponds to a hard gap at the Van Hove singularity, and this should reproduce the asymptotic behavior of section II B. The scenarios (2) and (3)  The superconductor possesses the dispersion E q = ε 2 q + ∆ 2 q . For a momentumdependent gap, it is convenient to consider in this section the density of states defined as follows where the integral over wavevectors is evaluated with the hard energy cutoff ε 0 ≥ (q 2 x + q 2 y )/(2m). It is clear that ν(E) is symmetric in r because the case r < 0 can be mapped onto r > 0 by interchanging q x and q y in the definition of ν(E). In the following, we therefore focus on positive r > 0.
The density of states is illustrated in Fig. S11 for the three gap scenarios of Eq. S26. The fully gapped scenario (1) produces a hard gap in the density of states E gap = ∆. The nodal scenario (2) also produces a small hard gap whose size however depends on both the tuning parameter and the cutoff, E gap = ∆|r| √

Gap equation and phase diagram
The gap ∆ fulfils the gap equation with the coupling constant g, another cutoff E 0 ≪ ε 0 , and the auxiliary density of states is given by It turns out that the auxiliary density of states ν γ (E) for ∆ = 0 in the gap scenario (2) vanishes at the Van Hove point r = 0 as ν γ (E) ∝ E 2 log 1/E for E → 0. At zero energy, E = 0, and finite r it assumes a finite value ν γ (0) ∝ r 2 log 1/|r| for small r. As a result, the critical temperature T c (r) possesses a minimum at r = 0 and increases for increasing distance to the Van Hove point. As this behavior is inconsistent with the experimental signatures, we exclude this scenario (2) in the following.
For the numerical evaluation, we choose for the cutoffs ε 0 = 0.1 eV and E 0 = 0.01 eV.
Moreover, we demand that the critical temperature T c,max = 3.5 K, i.e., k B T c,max /ε 0 ≈ 0.003 at the Van Hove singularity, r = 0, in order to connect with experiment. The coupling constant g is chosen accordingly. The resulting phase diagrams for the scenarios (1) and (3) in the (r, T ) plane are shown in Fig. S12.  (1) and (3) with a common T c,max = 3.5 K. Note the different scales for the tuning parameter r on the x-axis. The gap scenario (2) possesses a critical temperature T c (r) that is minimal at r = 0 and, as a consequence, it is not considered further.

Entropy quench across the superconducting transition
Here, we discuss the behavior of entropy close to the top of both superconducting domes in Fig. S12. The entropy per unit cell and per spin is given by It is symmetric S(T, r) = S(T, −r) and we focus thus on positive r ≥ 0. The entropy as a function of r is evaluated for a few temperatures in Fig. S13.
The panels show the entropy divided by temperature because in the Fermi liquid limit this ratio is proportional to the density of states of the normal state, see also Eq. S12.
This limiting behavior is expected for T > T c (r) and r ≫ k B T . As k B T c,max ≈ 0.003 ε 0 this amounts to r/ε 0 ≫ 0.003 for T ≈ T c,max . Due to the different scales for the tuning parameter on the x-axis, this limit is mostly fulfilled in Fig. S13 for scenario (3) (provided that T > T c (r)) but not yet for scenario (1).
As the superconducting dome is entered as a function of decreasing r, the entropy for the fully gapped scenario (1) exhibits a characteristic kink where the entropy changes slope  Figure S13. Entropy divided by temperature close to the top of the superconducting dome for scenarios (1) and (3) as a function of the tuning parameter r. The red dashed line corresponds to S = π 2 3 k 2 B T ν(0)| ∆=0 and is proportional to the density of states in the normal state, see also the discussion in the context of Eq. S12.
∂ r S even close to the top of the dome for T = 0.95 T c,max . As a consequence, the maximum in S(r) at r = 0 for T > T c,max is rapidly converted into a minimum for T < T c,max . The behaviour for the nodal gap scenario (3) is distinctly different. As its density of states at the Van Hove point r = 0 is only capped by superconductivity but remains gapless, see Fig. S11, the entropy remains basically unquenched at r = 0 upon entering the superconducting dome.
A small quench only appears for finite r at T < T c (r) when the density of states develops a soft gap. This small quench is however not able to change the sign of the slope ∂ r S at least not down to temperatures of T = 0.6 T c,max .
Close to the Van Hove singularity, r = 0, the strain sensitivity is mostly attributed to the strain dependence of the tuning parameter, r = r(ε xx ). As a consequence, the slope ∂ r S(r) is basically proportional to the strain derivative ∂ εxx S. A sign change of ∂ r S is thus directly reflected in a sign change of the elastocaloric effect. The behavior for the fully gapped scenario (1) is, as expected, consistent with the calculation of section II B. It is also consistent with the experimentally observed signatures whereas the nodal gap scenario is not. So we conclude that the conversion of the maximum in entropy S(ε xx ) into a minimum upon entering the superconducting dome at its top as derived from measurements of the elastocaloric effect on Sr 2 RuO 4 strongly suggests a superconducting order with a full gap at