Pseudogap in ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\delta }$ is not bounded by a line of phase transitions: Thermodynamic evidence

We discuss a recent resonant ultrasound spectroscopy (RUS) study of ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6+\delta }$, which infers a line of phase transitions bounding the pseudogap phase and argue that this scenario is not supported by thermodynamic evidence. We show that the anomalies in RUS, heat capacity, and thermal expansion at the superconducting transition temperatures agree well, but there are large discrepancies between RUS and thermodynamic measurements at $T^{*}$ where the pseudogap phase transitions are purported to occur. Moreover, the frequency and temperature dependence of the RUS data for the crystal with $\delta =0.98$, interpreted in terms of critical slowing down near an electronic phase transition, is five orders of magnitude smaller than what is expected. For this crystal the RUS data near $T^{*}$ are more consistent with nonequilibrium effects, such as oxygen relaxation.

Figure 1

Adaptation of Figs. 2 and 2 from Shekhter et al. [15] showing (a) the relative change in RUS mode frequency $\delta f/f$ for their ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6.98}$ (YBCO) crystal superimposed on the electronic specific heat coefficient $\gamma \equiv C_V/T$ for YBCO at full oxygenation [1, 2]. The right-hand scale has been corrected from units of 10${}^{-4}$ to 10${}^{-2}$, consistent with Figs. 1 and 1 of the original article and as confirmed by the authors [17]. $T^{*}$ marks where a change in slope of $\delta f/f$ is observed along with a peak in the resonance width $\Gamma$, which is shown in (b). No feature is observed in $\gamma \left(T\right)$ at $T^{*}$ even though the relative scale for $\gamma$ has been greatly amplified so that the jump $\delta \gamma$ at $T_c$ is 90$\times$ the size of the jump in $\delta f/f$ at $T_c$.

Figure 2

An Arrhenius plot of the data in Fig. 4(b) of Ref. [15] for the ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{6.98}$ crystal, where ${\tau }^{-1}=2\pi f$, $f$ is the RUS frequency, and $T$ is the temperature of the peak in RUS width $\Gamma$. Also included are the data reported by Almond et al. from ultrasonic anelastic relaxation studies. Both show an activation energy of about 60 meV.