High magnetic field ultrasound study of spin freezing in La$_{1.88}$Sr$_{0.12}$CuO$_4$

High-$T_{\rm{c}}$ cuprate superconductors host spin, charge and lattice instabilities. In particular, in the antiferromagnetic glass phase, over a large doping range, lanthanum based cuprates display a glass-like spin freezing with antiferromagnetic correlations. Previously, sound velocity anomalies in La$_{2-x}$Sr$_{x}$CuO$_4$ (LSCO) for hole doping $p\geq 0.145$ were reported and interpreted as arising from a coupling of the lattice to the magnetic glass [Frachet, Vinograd et al., Nat. Phys. 16, 1064-1068 (2020)]. Here we report both sound velocity and attenuation in LSCO $p=0.12$, i.e. at a doping level for which the spin freezing temperature is the highest. Using high magnetic fields and comparing with nuclear magnetic resonance (NMR) measurements, we confirm that the anomalies in the low temperature ultrasound properties of LSCO are produced by a coupling between the lattice and the spin glass. Moreover, we show that both sound velocity and attenuation can be simultaneously accounted for by a simple phenomenological model originally developed for canonical spin glasses. Our results point towards a strong competition between superconductivity and spin freezing, tuned by the magnetic field. A comparison of different acoustic modes suggests that the slow spin fluctuations have a nematic character.


I. INTRODUCTION
The coupling of electronic instabilities to the crystal lattice plays a significant role in shaping the phase diagram of some high-T c cuprate superconductors. The case of La-based cuprates is emblematic. Upon cooling, La 2−x Ba x CuO 4 (LBCO) and rare-earth doped (Nd, Eu) y -La 2−x−y Sr x CuO 4 ((Nd,Eu)-LSCO) evolve from a high-T tetragonal (HTT) to a mid-T orthorhombic (OMT) and finally to a low-T tetragonal (LTT) crystal structure. The LTT order pins stripe order, a combination of mutually commensurate spin and charge modulations, initially found in Nd-LSCO [1]. Within this context sound velocity and attenuation are particularly relevant quantities. Ultrasound measurements directly probe the lattice properties and they are sensitive to any strain dependent instability.
Among the La-based cuprate family La 2−x Sr x CuO 4 (LSCO) appears peculiar. First, the OMT-LTT structural phase transition does not occur, although LTTlike distortions exist locally [2][3][4]. Moreover, scattering evidence for charge ordering inside the pseudogap phase has remained elusive until recently [5][6][7][8]. In La 1.88 Sr 0.12 CuO 4 quasi-static charge modulation appears below T CDW = 70 ± 15 K with a maximal in-plane correlation length ξ (T c ) 30 Å, a value practically one order of magnitude smaller than in LBCO at the same doping.
In the same compound incommensurate antiferromagnetic (AFM) correlations are also found at low field for 0.02 ≤ p 0.135 [9]. The temperature at which these correlations appear static depends upon the probe frequency [9,10], revealing the glassy nature of the magnetic state. However, as in other La-based compounds close to p ≈ 0.12, one observes that the incommensurabilities of charge and spin density waves (respectively CDW and SDW) follow 2δ spin = δ charge , a relation reminiscent of charge-spin stripe ordering [6]. Close to the hole doping level p ≈ 0.12 elastic anomalies have been reported in both sound velocity and attenuation. Specifically, in single crystal studies and near the superconducting T c , a broad sound velocity minimum has been observed in different acoustic modes [11,12]. In a similar range of temperature, an attenuation maximum of longitudinal waves has been found in polycrystals [13,14]. Different interpretations have been proposed to explain this peculiar behaviour [11][12][13][15][16][17]. Recently, using NMR and sound velocity measurements in high magnetic field in LSCO for p ≥ 0.145, we have shown that the anomalous sound velocity appears to be caused by a coupling of the AFM glass to the lattice [18].
In this study, we strengthen this interpretation with high magnetic field measurements of sound velocity and attenuation in LSCO p = 0.12. Comparing ultrasound attenuation with NMR measurements on crystals from the same batch, we reinforce the link between the slowing down of magnetic fluctuations and the ultrasound anomalies observed in the (c 11 − c 12 )/2 and c 11 elastic constants. Moreover, we show that the ultrasound prop-erties of the (c 11 −c 12 )/2 mode can be semi-quantitatively reproduced by a phenomenological dynamical susceptibility model initially developed for canonical spin glasses. Finally, by comparing different acoustic modes, we find that the spin freezing produces an enhanced susceptibility in the B 1g channel, which is associated with nematicity in cuprates. This paper is organized as follows. In section II, we describe the sample studied and the experimental technique. Then, in section III we report the experimental sound velocity and attenuation measurements. We present a phenomenological model of ultrasound in spin glasses and use it to analyze the ultrasound data in section IV. Then, in section V, we discuss the magnetic field effect on the ultrasound properties, the differences between the acoustic modes studied and the symmetry of the AFM fluctuations inferred from our measurements. We summarize our conclusions in section VI.

II. METHODS
A high quality LSCO single crystal was grown by the traveling solvent floating zone method. From this crystal, three samples were cut along different crystallographic directions in order to probe different elastic constants. Typical samples dimensions are 2 × 2 × 2 mm 3 . The hole doping p = 0.122 ± 0.002 has been determined by measuring T st = 252 K, the temperature of the HTT-OMT structural phase transition by sound velocity, as described in Ref. [18]. The different samples share a similar T st and thus a similar doping. The superconducting transition temperature T c = 29 ± 3 K has been determined by sound velocity, in-plane resistivity, and magnetic susceptibility measurements.
A standard pulse-echo technique with phase comparison was used to measure variations of sound velocity, ∆v/v, and sound attenuation, ∆α [19]. Ultrasound was generated and detected using commercial LiNbO 3 transducers glued onto parallel, clean and polished surfaces of the samples. The excitation frequency, ω, ranged from 50 to 300 MHz. For high symmetry propagation direction, the sound velocity variation of a given acoustic mode can be converted to the associated elastic constant change using ∆c ii /c ii = 2∆v/v.
Zero-field and static-field experiments were performed at the LNCMI Grenoble using 20 T superconducting and 28 T resistive magnets. Field cooled conditions were used. Pulsed-fields experiments up to 60 T were carried out at the LNCMI Toulouse. In all cases, the field was applied along the crystallographic c-axis. No magnetic field is applied, the curves are arbitrarily superimposed at T =70 K. T min refers to the minimum in the c11 elastic constant that coincides with the superconducting Tc in zero magnetic field. Tα indicates the spin freezing temperature at the µeV ultrasound energy scale, defined by an attenuation peak (see Fig. 2 Table I: Properties of the different elastic constants measured: direction of propagation ( k) and polarisation ( u) of the acoustic wave, strain ( ij ) and associated symmetry. The indices of elastic constants are expressed in the Voigt notation. Crystallographic directions are those of the HTT phase (D 4h point group).

A. Sound velocity in zero magnetic field
We begin with a zero magnetic field study of different elastic constants in LSCO p = 0.12 as shown in Fig.  1. The description of the different modes studied is reported in table I. The c 44 acoustic mode follows the classical variation expected in solids: upon cooling the sound velocity increases continuously and eventually saturates at low temperature [20]. This behaviour contrasts with the c 33 elastic constant which shows a downward jump at T c . This mean-field anomaly at the superconducting transition is expected for a longitudinal mode and is related to the specific heat jump through the Ehrenfest with ∆C p (T c ) the specific heat jump at T c and V mol the molar volume. The amplitude of the anomaly, ∆v/v(T c ) 0.2 × 10 −3 , is consistent with literature values on samples with similar doping levels [21,22]. For T ≥ T c , the temperature dependence of c 11 and (c 11 − c 12 )/2 elastic constants is anomalous. In both of these modes, the normal state sound velocity decreases upon cooling, until the temperature hits T c where it shows an upturn. Consequently, the sound velocity in these modes has a minimum at T min T c . Fig. 1 shows that the anomalous lattice softening appears only in acoustic modes having a B 1g strain component, namely c 11 and (c 11 − c 12 )/2. Note that so far we have not been able to measure the B 2g mode (c 66 ) for T < T st .
Finally, for T 15 K or so, a rapid stiffening is observed in c 11 upon cooling. Indeed, the sound velocity in the T = 0 limit exceeds largely what would be expected from an extrapolation of the high temperature bare elastic constant (e.g. following the c 44 elastic constant). A similar upturn is found in c 33 and (c 11 − c 12 )/2 upon cooling for T 15 K, although much weaker than in c 11 .

B. Sound velocity and attenuation in applied magnetic field
In Fig. 2 and Fig. 3 we investigate how the anomalous sound velocity, and the corresponding sound attenuation, evolve as a function of temperature at different magnetic fields, in the c 11 and (c 11 − c 12 )/2 modes respectively. In both these modes no signature of the vortex lattice is observed, as discussed in Appendix A.
The anomalous features of the zero field sound velocity in the c 11 and (c 11 − c 12 )/2 acoustic modes are enhanced by a magnetic field: both the amplitudes of the lattice softening (for T ≥ T min ) and stiffening (T ≤ T min ) increase with increasing magnetic field. For both acoustic modes an attenuation peak is found at T α ≤ T min . The amplitude of this attenuation peak and T α increase monotonically with increasing field.
The magnetic field dependencies of T α and T min from c 11 measurements are shown in the phase diagram of  given magnetic field similar T α and T min . In contrast with T α , T min has a non-monotonic field dependence: it decreases for 0 ≤ B ≤ 2 T and increases for higher fields. The initial decrease is caused by the lattice coupling to the superconducting order parameter as further detailed in Appendix B. However, for B ≥ 5 T or so, the two temperature scales have similar field dependence, indicating that they are coupled and caused by the same phenomenon.

C. Comparison with NMR 1/T1
In Fig. 4 we compare the ultrasound attenuation, ∆α, with the 139 La NMR spin-lattice relaxation rate, 1/T 1 , both measured in LSCO p = 0.12 samples from the same batch and in a magnetic field B = 28 T. The comparison is striking, both quantities display remarkably similar temperature dependencies and show a maximum at comparable temperatures.
The peak in 1/T 1 is a classical signature of spin freezing in superconducting LSCO [23][24][25][26]. This peak is understood within the so-called Bloembergen-Purcell-Pound (BPP) model [23,26,27] originating from a diverging correlation time τ (T ). Upon cooling, spin fluctuations are gradually slowing down. At the temperature where the condition ω NMR τ = 1 is fulfilled (ω NMR being the NMR frequency), 1/T 1 is maximum. This temperature defines the freezing temperature T f at the NMR timescale. Correspondingly, Fig. 4 reveals that the ultrasound attenuation ∆α is governed by a similar correlation time. At the temperature where the condition ω US τ = 1 is met, with ω US the ultrasound frequency, a peak in the ultrasound attenuation is observed. The good agreement between T f and T α is provided by the fact that ω NMR ≈ ω US ≈ 10 8 Hz. Finally, note that the small difference observed between ∆α and 1/T 1 could arise from a small variation in doping level between the two samples, but also from a disparity in the way these probes couple to the magnetic moments. This is discussed in the next section.

IV. MODELLING
A broad sound velocity minimum at T min [28][29][30] and an attenuation peak at T α ≤ T min [28,31] are common characteristics of -insulating or metallic -canonical spin glasses around the spin freezing temperature. The sound velocity and attenuation of a cobalt fluorophosphorate spin glass shown in Fig. 5(a) exemplify those In the following, we focus on the transverse (c 11 −c 12 )/2 acoustic mode shown in Fig. 3 and demonstrate that it can be semi-quantitatively reproduced by a phenomenological model developed for spin glasses. The strong increase observed in c 11 (T ) at low temperature is not explained by this model and will be discussed later.
We use the phenomenological dynamical susceptibility model developed by Doussineau et al. [28,32]. Sound velocity and attenuation are expressed in terms of a complex elastic constant, c(ω, T ): With c 0 the bare elastic constant, g the spin-phonon coupling constant and ω the ultrasound measurement frequency. Ultrasound quantities are deduced through: ∆α(dB/cm) = ω v 10 log (10) m (∆c/c) Here χ 4 (ω, T ) is a dynamical susceptibility defined as: χ 4 (ω = 0, T ) is a static susceptibility and τ 4 (T ) is the correlation time of the spin fluctuations. In our case, since S = 1/2, the magneto-acoustic coupling arises from the Waller mechanism (also called exchangestriction mechanism), i.e. a modulation of the exchange interaction by the strain [19]. Consequently, the associated susceptibility is quadrupolar and the correlation time is involved in a four-spin correlation function. In contrast, the 1/T 1 NMR relaxation rate is governed by a correlation time τ 2 (T ) which is involved in a two-spin correlation function. This can produce slight differences between ∆α and 1/T 1 in Fig. 4 [33]. We use the following expressions for τ 4 (T ) and χ 4 (ω = 0, T ): C curie controls the amplitude of the lattice softening for T ≥ T min , E 0 is an energy scale that governs T min and T α , χ 0 is the constant term of the susceptibility and finally τ ∞ is the correlation time of spin fluctuations for T E 0 . Note that Eq. 6 and Eq. 7 are motivated by an analysis of 139 La NMR 1/T 1 [23,26] and ac-susceptibility measurements in the AFM glass of LSCO [34,35] respectively. As inferred from various experiments [36], the value of τ ∞ is fixed to exp(−30) 10 −13 s. Moreover, as usually done in spin glasses [28,37], and especially in the AFM glass phase of LSCO [26,36], we consider that τ 4 (T ) is inhomogeneous using a gaussian-distribution of E 0 with full width at half maximum 2∆E 0 . Within this framework it is possible to fit simultaneously ∆v/v and ∆α, and to extract both E 0 and g 2 C curie as a function of magnetic field. A representative example is shown on Fig. 5(b): the model reproduces most of the salient features seen in the two ultrasound quantities.
The evolution of the fitting parameters is shown in Fig. 5(c, d). Up to B = 60 Ti.e. well above our T → 0 extrapolation of the vortex melting field B v on Fig. 2(c) -E 0 and g 2 C curie increase continuously. The former increase is related to the non saturating values of the temperature scales T α and T min . The latter is explained by the continuous increase of the amplitudes of the lattice softening and attenuation peak up to 60 T (see Fig. 3).
The NMR 1/T 1 data at B = 28 T shown in Fig. 4 can be fitted with the BPP formula using Eq. 6 for τ 2 (T ) and a gaussian distribution of activation energy E 0 [26,38]. This parametrization of 1/T 1 data yields an activation energy in fair agreement with E 0 inferred from ultrasound data (see Fig. 5(c)). It has been suggested previously that the activation energy is analogous to the spinstiffness 2πρ s [24,38,39]. The value of E 0 ≈ 200 K found here for B = 20 T is comparable to what is obtained in Nd-LSCO x = 0.12 in zero magnetic field [24,39,40]. It is an order of magnitude smaller than the spin stiffness of the antiferromagnetic parent compound La 2 CuO 4 where 2πρ s ≈ J [41].
Finally, in the paramagnetic state of a classical Néel AFM C curie ∝ µ 2 , where µ is the magnetic moment.
Since the dynamical susceptibility model is purely phenomenological, we cannot extract microscopic information. As such, the increase of g 2 C curie with magnetic field (see Fig. 5(d)) could originate from an enhanced µ [42,43] or from an increased magnetic volume [44].

V. DISCUSSION
Let us summarize our results so far. (i) The (c 11 − c 12 )/2 and c 11 modes show a softening for T ≥ T min and a hardening for T ≤ T min . Those features are enhanced by magnetic field and survive when superconductivity is strongly suppressed by the field. Consequently, neither feature is caused by superconductivity. We attribute this broad sound velocity minimum to the freezing of the AFM glass. (ii) The striking similarity of the ultrasound attenuation with the NMR relaxation rate 1/T 1 shows that the AFM glass is also causing the anomalous attenuation peak in high magnetic field. (iii) The behaviour of the (c 11 −c 12 )/2 elastic constant found in LSCO p = 0.12 in high magnetic field is remarkably similar to what is found in canonical spin glasses. A dynamical susceptibility model, developed in the context of spin glasses, reproduces all features of the anomalous ultrasound properties in the (c 11 − c 12 )/2 mode.
The similar decrease of T min and T c with magnetic field B ≤ 14 T in LSCO at p ≈ 0.14 has previously motivated a scenario in which a competing lattice instability -that produces a lattice softening for T > T c -is quenched by the onset of superconductivity that induces a hardening for T < T c [12]. While we observe the same behaviour in LSCO p = 0.12 for B ≤ 2T (see Appendix B for more details), this scenario does not hold at higher field where we observe an increase of T min . All measurements reported here in LSCO p = 0.12 support the interpretation that the ultrasound anomalies are caused by the AFM glass phase via spin-phonon coupling [18].
In the following we discuss some implications of the aforementioned results. In particular we comment on the magnetic field effect on the ultrasound properties, the relation of this study with previous elastic experiments and the symmetry of the AFM quasi-static fluctuations.
A. The special coupling with B1g strain In canonical spin glasses such as cobalt fluorophosphorate, the magnetic moments are frozen in a random manner (note however that metallic spin glass CuMn show short ranged (∼ 20 Å) SDW correlations [45]). Consequently, longitudinal and transverse acoustic modes couple similarly to the spins in such systems (see Ref. [32]). The magnetic moments of LSCO have similar dynamical properties as canonical spin glasses: they gradually freeze as the system is cooled down, such that the onset temper-ature depends on the probe frequency [9,10]. However, the moments in LSCO arrange in a pattern displaying incommensurate AFM character, and Bragg peaks indicating correlation lengths as high as ∼ 200 Å in LSCO x = 0.12 are observed in neutron diffraction experiments [42,[46][47][48]. Consequently in LSCO the coupling between the frozen spins and the lattice varies dramatically from one mode to another, as shown in Fig. 1. The anomalous softening for T ≥ T min is observed only in modes transforming according to the B 1g irreducible representation (see Table I and Fig. 1). Note however, that we cannot exclude a similar coupling of the AFM glass to B 2g mode. Nonetheless this suggests a special role of the B 1g mode.
Within the framework of the dynamical susceptibility model, the lattice softening in the B 1g mode is caused by the growth of a Curie-like susceptibility χ 4 (ω = 0, T ). Eq. 2 is reminiscent of the elastic constant c = d 2 F/d 2 calculated using a Landau free energy F containing a bilinear coupling F c = g Q [19], with a strain and Q an order parameter. Indeed, within such a model, the softening is directly related to the increasing mean-field susceptibility of Q, ∆v/v ∝ −g 2 χ Q . For this bilinear coupling to exist, both and Q must transform according to the same irreducible representation. In this context, our result would suggest that the order parameter (and the fluctuations) associated with the AFM glass has a B 1g , i.e. nematic, character.
Although conjectural in the absence of a measurement of the B 2g mode, this interpretation of the ultrasound data is evocative of the B 1g susceptibility observed by symmetry-resolved Raman scattering in LSCO at x = 0.10 [49]. It is consistent with evidence of charge and spin stripe orders in this compound [6,48,50,51]. Nematicity can indeed result from fluctuating stripes [52]. We note that, at this doping, the observed B 1g susceptibility develops well below the pseudogap temperature T . Indeed, our ultrasound experiment does not detect any significant B 1g susceptibility in the vicinity of T ≈ 130 K [53]. The lack of B 1g susceptibility at the pseudogap temperature is also reported in symmetry-resolved electronic Raman scattering experiments in Bi 2 Sr 2 CaCu 2 O 8+δ [54]. The onset temperature of our detection of B 1g susceptibility is actually comparable to the CDW onset temperature T CDW = 70 ± 15 K [5][6][7]. This suggests that, in LSCO p = 0.12, charge-stripe order triggers slow magnetic fluctuations [55][56][57] with nematic character.

B. The effect of the magnetic field
Previous neutron scattering and µSR experiments have shown that the magnetism of LSCO at p ≈ 0.12 is enhanced by a magnetic field [42,43,[58][59][60][61], and this effect has been ascribed to a competition between superconducting and AFM order parameters. In line with this interpretation, we observe that the ultrasound signatures of the AFM glass are strengthened by a magnetic field (see Fig. 2, 3 and 5). The magnetic field dependence  [20]. (c11 − c12)/2 can be fully reproduced by the dynamical susceptibility model involving χ4(ω, T ) (dashed blue line). For T → 0, the difference between the sound velocity of the (c11 − c12)/2 mode and its background tends to zero. This contrasts with the behaviour of c11. For T → 0, the sound velocity of the c11 mode is larger than the background sound velocity. This means that in addition to the contribution from χ4(ω, T ) (dashed blue line) that causes the minimum in c11, another component contributes at low T . As discussed in the text, a contribution proportional to M 2 (dashed orange line) could explain the rapid increase of c11 at low temperature. Figure 7: Comparison between the magnetic field dependence of the superlattice Bragg peak intensity of incommensurate AFM seen by neutron diffraction, ∆Ineutron (blue circles, left scale), and the sound velocity, ∆v/v, in the c11 and c33 acoustic modes (up and down red triangles respectively, right scale). The neutron diffraction intensity is reproduced from Ref. [42]. The sound velocity in the c11 mode is divided by a factor 4.5. The sound velocity measurements presented here are taken at T ≤ 4 K in field-cooled conditions. The dashed line is a guide to the eye. of the ultrasound properties does not saturate up to 60 T and the magnetic field induced softening appears at temperatures as high as T ≈ 50 K (see Fig. 3). These observations are puzzling since at this doping T c ≈ 29 K and the extrapolation of the vortex melting line leads to B v (T −→ 0) ≈ 20 T. This raises important questions on the effect of magnetic fields on the magnetic freezing and the possible resilience of superconducting fluctuations in high field.
We note that this behaviour is reminiscent of the magnetoresistance producing an upturn in the resistivity of superconducting LSCO in high fields. This magnetoresistance is observed up to T 100 K at the doping level p = 0.12 [62]. The spin freezing has been previously discussed as a cause of the resistivity upturn in Labased cuprates [63][64][65][66][67]. Consequently, it is possible that the large magnetoresistance observed in LSCO p = 0.12 above T c is related to the field-induced gradual slowing down of magnetic fluctuations observed here.
C. Differences between c11 and (c11 − c12)/2 Finally, we discuss the differences between the c 11 and (c 11 − c 12 )/2 modes. As discussed above, the strength of the magneto-acoustic coupling is largest in the (c 11 − c 12 )/2 mode, where the largest softening is observed (see Fig. 1). The second difference between the response in these two modes is the field-enhanced hardening that is seen in c 11 at low temperature. The situation is schematically depicted in Fig. 6. In the (c 11 − c 12 )/2 mode, the difference between the measured sound velocity in the T → 0 limit and the background velocity is negligibly small. On the other hand this difference is significant in the c 11 mode, with the measured sound velocity being larger than the background velocity. This behaviour echoes the results from previous studies performed on polycrystals. Earlier ultrasound studies suggested that this behaviour can be caused by LTT distortions [13,68]. On the other hand, anelastic experiments implied a coupling between strain and AFM glass domain wall motion [69].Those studies are discussed in greater details in Appendix C.
Here we propose an alternative mechanism that could cause the low temperature stiffening in the c 11 mode. We start by noticing that a stiffening is also observed in the c 33 mode for T T α (see Fig. 1 ). As shown in Fig. 7, the low temperature increase of the sound velocity in both the c 11 and c 33 modes has a field dependence that scales with the increase in µ 2 , the ordered moment squared inferred from neutron diffraction experiment, as discussed previously [13,17]. This scaling can be explained by invoking a biquadratic coupling F c = λ 2 M 2 with M the magnetization, and λ a coupling constant. Note that F c is symmetry-allowed for all elastic constants of Table I. This coupling produces ∆v/v ∝ λM 2 and can naturally account for the experimental observations if we assume that the coupling constant λ is larger for longitudinal modes (c 11 and c 33 ) than it is for transverse modes (c 44 and (c 11 − c 12 )/2).

VI. SUMMARY
In summary, we studied sound velocity and attenuation in La 1.88 Sr 0.12 CuO 4 in high magnetic field. The behaviour of the c 11 and (c 11 − c 12 )/2 elastic constants is highly anomalous. By comparing the anomalies with 139 La NMR 1/T 1 we confirm that they originate from the AFM glass phase via a magneto-acoustic coupling. A semi-quantitative analysis of this contribution is made based on a phenomenological model of spin glass systems. Our ultrasound data points toward a strong competition between spin freezing and superconductivity in high magnetic field. A symmetry analysis reveals that the slowing down of spin fluctuations could be associated with a growing nematic susceptibility.  Fig. 5 and −Tα × ln(ωτ∞) (diamonds, right scale) determined directly from the data shown in Fig. 2(b), ω = 2π × 110 MHz and τ∞ ≈ 10 −13 s. The energy scale −Tα × ln(ωτ∞) corresponds to an experimental determination of E0 solely based on Tα, using the condition ωτ4(Tα) = 1 in Eq. 6 (see text). The rapid drop of −Tα × ln(ωτ∞) at low B is most likely due to the impact of superconductivity on the spin dynamics. Error bars on this quantity are smaller than the size of the symbols.
at p ≈ 0.14 leads to the same observation [18]. As field increases the superconducting contribution to the sound velocity becomes weaker and the spin freezing contribution larger. For B ≥ 5 T the superconducting contribution is dwarfed by the contribution from the magnetic slowing down. This explains why the difference between T min and T α = T f is large and strongly field dependent for B < 5 T, while smaller and constant for higher fields (see Fig. 2(c)).
The temperature scale T α is insensitive of a direct contribution from superconductivity. While in conventional superconductors, the opening of the superconducting gap causes an attenuation drop, there is no corresponding behaviour in LSCO p ≈ 0.12. However ∆α(T ) can be indirectly impacted by the onset of superconductivity at low field because the latter modifies the spin dynamics that controls ∆α(T ). This is best illustrated by the zero field ∆α(T ) that shows a remarkable kink at T c and then a maximum at T α 9.5 K (see Fig. 2(b)). Within the dynamical susceptibility model, the ultrasound attenuation is mostly governed by the energy scale E 0 entering τ 4 as indicated by Eq. 4. The kink anomaly at T c in ∆α(T ) in zero field can be interpreted as a decrease of E 0 for T < T c caused by the onset of superconductivity.
Because the dynamical susceptibility model does not take into account the impact of superconductivity on spin dynamics, we use an alternative scheme in order to extract E 0 for B < 5 T. At T = T α , the condition ωτ 4 (T ) = 1 is met. Solving Eq. 6 for E 0 at T = T α hence yields E 0 = −T α / ln(ωτ ∞ ) [38]. In Fig. B.1 we compare this T α derived E 0 with E 0 of Fig. 5(c) obtained with the parametrization of the ultrasound data. Good agreement is found between the two estimations of E 0 . As seen in Fig. B.1, −T α / ln(ωτ ∞ ) decreases rapidly at low fields, dropping from E 0 ∼ 150K for B = 5 T to E 0 = 92 K for B = 0. This rapid drop reflects the competition between spin freezing and superconductivity.