Normal state specific heat in the cuprate superconductors <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>La</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Sr</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi>CuO</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Bi</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Sr</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi><mml:mo>−</mml:mo><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>La</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi>CuO</mml:mi><mml:mrow><mml:mn>6</mml:mn><mml:mo>+</mml:mo><mml:mi>δ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math> near the critical point of the pseudogap phase
C Girod, D. LeBoeuf, A. Demuer, G. Seyfarth, Shusaku Imajo, Koichi Kindo, Yoshimitsu Kohama, M. Lizaire, Anaëlle Legros, Adrien Gourgout, H. Takagi, T. Kurosawa, Masahiro Oda, N. Momono, J. Chang, Shimpei Ono, G.-q. Zheng, C. Marcenat, Louis Taillefer, T. Klein
Abstract
The specific heat $C$ of the cuprate superconductors ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ and ${\mathrm{Bi}}_{2+y}{\mathrm{Sr}}_{2\ensuremath{-}x\ensuremath{-}y}{\mathrm{La}}_{x}{\mathrm{CuO}}_{6+\ensuremath{\delta}}$ 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 ${C}_{\mathrm{e}}$, the electronic contribution to the normal-state specific heat at $T\ensuremath{\rightarrow}0$. In ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ at $x=p=0.22$, 0.24 and 0.25, ${C}_{\mathrm{e}}/T=15\phantom{\rule{0.28em}{0ex}}\mathrm{to}\phantom{\rule{0.28em}{0ex}}16\phantom{\rule{0.28em}{0ex}}\mathrm{mJ}\phantom{\rule{0.16em}{0ex}}{\mathrm{mol}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}2}$ at $T=2\phantom{\rule{0.28em}{0ex}}\mathrm{K}$, 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 ${C}_{\mathrm{e}}$ at ${p}^{★}\ensuremath{\simeq}0.19$ as previously reported for samples in which superconductivity was destroyed by Zn impurities. Moreover, at those three dopings, we find a logarithmic growth as $T\ensuremath{\rightarrow}0$ such that ${C}_{\mathrm{e}}/T\ensuremath{\sim}B\phantom{\rule{0.16em}{0ex}}ln({T}_{0}/T)$. The peak versus $p$ and the logarithmic dependence versus $T$ are the two typical thermodynamic signatures of quantum criticality. In the very different cuprate ${\mathrm{Bi}}_{2+y}{\mathrm{Sr}}_{2\ensuremath{-}x\ensuremath{-}y}{\mathrm{La}}_{x}{\mathrm{CuO}}_{6+\ensuremath{\delta}}$, we again find that ${C}_{\mathrm{e}}/T\ensuremath{\sim}B\phantom{\rule{0.16em}{0ex}}ln({T}_{0}/T)$ at $p\ensuremath{\simeq}{p}^{★}$, strong evidence that this $ln(1/T)$ dependence of the electronic specific heat---first discovered in the cuprates ${\mathrm{La}}_{1.8\ensuremath{-}x}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ and ${\mathrm{La}}_{1.6\ensuremath{-}x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$---is a universal property of the pseudogap critical point.