Normal State Specific Heat In The Cuprate Superconductors La2-Xsrxcuo4 And Bi2+Ysr2-X-Ylaxcuo6+Delta Near The Critical Point Of The Pseudogap Phase

The specific heat C of the cuprate superconductors La2-xSrxCuO4 and Bi2+ySr2-x-yLaxCuO6+delta was measured at low temperatures (down to 0.5 K) for dopings p close to p(star), the critical doping for the onset of the pseudogap phase. A magnetic field up to 35 T was applied to suppress superconductivity, giving direct access to the normal state at low temperatures, and enabling a determination of C-e, the electronic contribution to the normal-state specific heat at T -> 0. In La2-xSrxCuO4 at x = p = 0.22, 0.24 and 0.25, C-e/T = 15 to 16 mJ mol(-1) K-2 at T = 2 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) similar or equal to 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 -> 0 such that C-e/T similar to B ln(T-0/T). The peak versus p and the logarithmic dependence versus T are the two typical thermodynamic signatures of quantum criticality. In the very different cuprate Bi2+ySr2-x-yLaxCuO6+delta, we again find that C-e/T similar to B ln(T-0/T) at p similar or equal to p(star), strong evidence that this ln(1/T) dependence of the electronic specific heat-first discovered in the cuprates La1.8-xEu0.2SrxCuO4 and La1.6-xNd0.4SrxCuO4-is a universal property of the pseudogap critical point.