Normal state specific heat in the cuprates La${}_{2-x}$Sr${}_x$CuO${}_4$ and Bi${}_{2+y}$Sr${}_{2-x-y}$La${}_x$CuO${}_{6+\hat{I}´}$ near the critical point of the pseudogap phase

Abstract

The specific heat $C$ of the cuprate superconductors La${}_{2-x}$Sr${}_x$CuO${}_4$ and Bi${}_{2+y}$Sr${}_{2-x-y}$La${}_x$CuO${}_{6+\hat{I}´}$ was measured at low temperature (down to $0.5\mathrm{K}$), for dopings $p$ close to $p^{\star }$, the critical doping for the onset of the pseudogap phase. A magnetic field up to $35\mathrm{T}$ was applied to suppress superconductivity, giving direct access to the normal state at low temperature, and enabling a determination of $C_e$, the electronic contribution to the normal-state specific heat, at $T\to 0$. In La${}_{2-x}$Sr${}_x$CuO${}_4$ at $x=p=0.22$, $0.24$ and $0.25$, $C_e/T=15-16{\mathrm{mJmol}}^{-1}{\mathrm{K}}^{-2}$ at $T=2\mathrm{K}$, values that are twice as large as those measured at higher doping ($p>0.3$) and lower doping ($p<0.15$). This confirms the presence of a broad peak in the doping dependence of $C_e$ at $p^{\star }\simeq 0.19$, as previously reported for samples in which superconductivity was destroyed by Zn impurities. Moreover, at those three dopings, we find a logarithmic growth as $T\to 0$, such that $C_e/T\sim \mathrm{B}\ln (T_0/T)$. The peak vs $p$ and the logarithmic dependence vs $T$ are the two typical thermodynamic signatures of quantum criticality. In the very different cuprate Bi${}_{2+y}$Sr${}_{2-x-y}$La${}_x$CuO${}_{6+\hat{I}´}$, we again find that $C_e/T\sim \mathrm{B}\ln (T_0/T$) at $p\simeq p^{\star }$, strong evidence that this $\ln (1/T)$ dependence - first discovered in the cuprates La${}_{1.8-x}$Eu${}_{0.2}$Sr${}_x$CuO${}_4$ and La${}_{1.6-x}$Nd${}_{0.4}$Sr${}_x$CuO${}_4$ - is a universal property of the pseudogap critical point. All four materials display similar values of the $\mathrm{B}$ coefficient, indicating that they all belong to the same universality class.

8 pages, 6 figures