Normal state specific heat in the cuprate superconductors La2-xSrxCuO4 and Bi2+ySr2-x-yLaxCuO6+δ near the critical point of the pseudogap phase

The specific heat C of the cuprate superconductors La2-xSrxCuO4 and Bi2+ySr2-x-yLaxCuO6+δ was measured at low temperatures (down to 0.5 K) for dopings p close to p∗, 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 Ce, 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, Ce/T=15to16mJmol-1K-2 at T=2K, 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 Ce at p∗≃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 Ce/T∼Bln(T0/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+δ, we again find that Ce/T∼Bln(T0/T) at p≃p∗, 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.