Wiedemann-Franz law in the underdoped cuprate superconductor ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$

The electrical and thermal Hall conductivities of the cuprate superconductor ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y, {\sigma }_{xy}$ and ${\kappa }_{xy}$, were measured in a magnetic field up to 35 T, at a hole concentration (doping) $p=0.11$. In the $T=0$ limit, we find that the Wiedemann-Franz law, ${\kappa }_{xy}/T=({\pi }^2/3){(k_{\mathrm{B}}/e)}^2{\sigma }_{xy}$, is satisfied for fields immediately above the vortex-melting field $H_{\mathrm{vs}}$. This rules out the existence of a vortex liquid at $T=0$ and it puts a clear constraint on the nature of the normal state in underdoped cuprates, in a region of the doping phase diagram where charge-density-wave order is known to exist. As the temperature is raised, the Lorenz ratio, $L_{\mathrm{xy}}={\kappa }_{\mathrm{xy}}/\left({\sigma }_{\mathrm{xy}}T\right)$, decreases rapidly, indicating that strong small-$q$ scattering processes are involved.

Figure 1

Magnetic-field-temperature phase diagram of YBCO at a hole concentration (doping) $p=0.11$, showing the upper critical field $H_{\mathrm{c}2}\left(T\right)$ (red squares), detected in the longitudinal thermal conductivity ${\kappa }_{\mathrm{xx}}$ (open squares, from [18]) and in the thermal Hall conductivity ${\kappa }_{\mathrm{xy}}$ (full squares, this work). The red dashed line is a guide to the eye, showing how $H_{\mathrm{c}2}\left(T\right)$ might extrapolate to zero at $T_{\mathrm{c}}$ [18]. The blue symbols mark $H_{\mathrm{vs}}\left(T\right)$, the field at which the vortex solid melts and above which the electrical resistance is no longer zero; the solid line is a fit to the theory of vortex-lattice melting [30]. In the limit of $T=0$, the fit extrapolates to $H_{\mathrm{vs}}\left(0\right)=24\pm 2\mathrm{T}$, so that $H_{\mathrm{vs}}\left(T\right)=H_{\mathrm{c}2}\left(T\right)$ at $T=0$.

Figure 2

(a) Longitudinal (${\rho }_{\mathrm{xx}}={\rho }_{\mathrm{a}}$) and (b) transverse (Hall; ${\rho }_{\mathrm{xy}}$) electrical resistivities of our $a$-axis YBCO sample with $p=0.11$ ($y=6.54$), plotted as a function of field up to $H=35\mathrm{T}$, well above the upper critical field $H_{\mathrm{c}2}\left(0\right)=24\mathrm{T}$, at various temperatures as indicated. These data are used to obtain the electrical Hall coefficient $R_{\mathrm{H}}$ and conductivity ${\sigma }_{\mathrm{xy}}$ in Fig. 3.

Figure 3

(a) Electrical Hall coefficient $R_{\mathrm{H}}$ as a function of magnetic field $H$, measured on the same sample of YBCO ($p=0.11$) on which ${\kappa }_{\mathrm{xy}}$ was measured, at various temperatures as indicated. The Fermi-surface reconstruction causes $R_{\mathrm{H}}$ at high field to go from a positive value at $T=80\mathrm{K}$ to a large negative value at $T=4\mathrm{K}$, evidence that a small high-mobility electron pocket emerges upon cooling [1, 2, 6, 15]. Note that $R_{\mathrm{H}}\left(H\right)$ at $T=4\mathrm{K}$ is constant above $H_{\mathrm{c}2}\left(0\right)=24\mathrm{T}$, showing that there is no flux flow due to vortices above that field. (b) Hall conductivity ${\sigma }_{\mathrm{xy}}$ vs $H$, for temperatures as indicated, obtained from measurements of ${\rho }_{\mathrm{xx}}$ and ${\rho }_{\mathrm{xy}}$ (Fig. 2). Note that above $H_{\mathrm{c}2}=24\pm 1\mathrm{T}$ (vertical gray band) the various isotherms of ${\sigma }_{\mathrm{xy}}$ collapse onto the same curve. The dashed blue line is a plot of the normal-state conductivity ${\sigma }_{\mathrm{xy}}^{\mathrm{N}}={\rho }_{\mathrm{xy}}/({\rho }_{\mathrm{xx}}^2+{\rho }_{\mathrm{xy}}^2)$, where ${\rho }_{\mathrm{xy}}=R_{\mathrm{H}}H$, with $R_{\mathrm{H}}=-11.5{\mathrm{mm}}^3/\mathrm{C}$, and ${\rho }_{\mathrm{xx}}=5\mu \mathrm{\Omega }\mathrm{cm}$, values appropriate for the high-field normal state at $T=10\mathrm{K}$ [Figs. 3 and 2].

Figure 4

Thermal Hall conductivity ${\kappa }_{\mathrm{xy}}$ for YBCO with $p=0.11$ ($y=6.54$), plotted as ${\kappa }_{\mathrm{xy}}/\left(TH\right)$ vs $H$ at different temperatures as indicated. The 68-K isotherm (brown) is multiplied by a factor of 5 to make it visible above the 54 K isotherm (black).

Figure 5

(a) Thermal Hall conductivity ${\kappa }_{\mathrm{xy}}$ of YBCO at $p=0.11$, plotted as ${\kappa }_{\mathrm{xy}}/T$ vs $H$, at $T=0.7\mathrm{K}$. The straight dashed black line is a linear fit to the rapid rise in $|{\kappa }_{\mathrm{xy}}|$ vs $H$. A lower bound on the upper critical field $H_{\mathrm{c}2}$ is the deviation from linearity at 23 T, while an upper bound is the minimum in ${\kappa }_{\mathrm{xy}}$ at 25 T, so that $H_{\mathrm{c}2}=24\pm 1\mathrm{T}$ (vertical gray band). The dashed blue line is the same as in Fig. 3, but multiplied by the constant $L_0={\pi }^2/3{(k_{\mathrm{B}}/e)}^2$, giving the measured value of $L_0{\sigma }_{\mathrm{xy}}$ at low temperature, above $H_{\mathrm{c}2}$. The fact that ${\kappa }_{\mathrm{xy}}/T=L_0{\sigma }_{\mathrm{xy}}$ for $H>H_{\mathrm{c}2}$, within error bars, shows that the Wiedemann-Franz law is satisfied—compelling evidence that superconductivity is entirely suppressed and the normal state is fully reached at $H_{\mathrm{c}2}=24\mathrm{T}$. Error bars on ${\sigma }_{\mathrm{xy}}$ and ${\kappa }_{\mathrm{xy}}$ are defined in Methods. (b) Lorenz ratio $L_{\mathrm{xy}}={\kappa }_{\mathrm{xy}}/\left({\sigma }_{\mathrm{xy}}T\right)$, plotted as $L_{\mathrm{xy}}/L_0$ vs $H$. The isotherm at $T=0.7\mathrm{K}$ is used for ${\kappa }_{\mathrm{xy}}/T$ [Fig. 5]; the isotherm at $T=10\mathrm{K}$ is used for ${\sigma }_{\mathrm{xy}}$ [Fig. 3]. $L_{\mathrm{xy}}$ saturates above $\sim 25\mathrm{T}$, to a value $L_{\mathrm{xy}}/L_0=1.1\pm 0.2$, showing that the Wiedemann-Franz law ($L_{\mathrm{xy}}=L_0$; dashed line) holds, within error bars, when $H>H_{\mathrm{c}2}$.

Figure 6

(a) Electrical and thermal Hall conductivities of YBCO as a function of temperature, measured on the same sample with $p=0.11$, plotted as $L_0{\sigma }_{\mathrm{xy}}$ (blue squares) and ${\kappa }_{\mathrm{xy}}/T$ (red circles), respectively, for $H=27\mathrm{T} >H_{\mathrm{c}2}\left(0\right)$. The ${\sigma }_{\mathrm{xy}}$ data are obtained from isotherms of ${\rho }_{\mathrm{xx}}$ and ${\rho }_{\mathrm{xy}}$ (Fig. 2); the ${\kappa }_{\mathrm{xy}}$ data are obtained from isotherms in Figs. 4 and 8. (b) Lorenz ratio $L_{\mathrm{xy}}={\kappa }_{\mathrm{xy}}/\left({\sigma }_{\mathrm{xy}}T\right)$, plotted as $L_{\mathrm{xy}}/L_0$ vs $T$ (green squares). The Wiedemann-Franz law ($L_{\mathrm{xy}}=L_0$; dashed line) is seen to hold in the limit of $T=0$ where scattering is elastic. With increasing $T$, however, $|{\kappa }_{\mathrm{xy}}/T|$ decreases rapidly even though $|L_0{\sigma }_{\mathrm{xy}}|$ remains constant, at least initially, reflecting the effect of inelastic scattering in the normal state of underdoped YBCO.

Figure 7

Thermal Hall conductivity of our two samples of YBCO with $p=0.11$ ($y=6.54$), at $T=8\mathrm{K}$, plotted as ${\kappa }_{\mathrm{xy}}/T$ vs $H$. In one sample, the heat current flows along the $a$ axis of the orthorhombic crystal structure (red, $J\parallel a$), while in the second it flows along the $b$ axis (blue, $J\parallel b$) (Methods). Within error bars ($\pm 20$ %), we see that both samples yield the same curve, so that ${\kappa }_{\mathrm{ab}}={\kappa }_{\mathrm{ba}}$, as expected from the Onsager reciprocity relation. The dashed line is a linear fit to the steepest part of $|{\kappa }_{\mathrm{xy}}|$ vs $H$. We define $H_{\mathrm{c}2}$ as the field above which $|{\kappa }_{\mathrm{xy}}|$ departs from that linear rise (vertical gray band), giving $H_{\mathrm{c}2}=25\pm 1\mathrm{T}$, as plotted in Fig. 1.

Figure 8

Thermal Hall conductivity ${\kappa }_{\mathrm{xy}}$ as a function of field at $T=10\mathrm{K}$ obtained from two different measurements of the transverse temperature difference $d{T}_{\mathrm{y}}$, using (1) a Cernox sensor (blue curve) and (2) a type-E thermocouple (red dots). In both cases, the longitudinal temperature difference $d{T}_{\mathrm{x}}$ is measured with two Cernox sensors. The two measurements of ${\kappa }_{\mathrm{xy}}$ show excellent agreement, confirming that our thermometry in high fields is reliable. The dashed line is a linear fit to the steepest part of $|{\kappa }_{\mathrm{xy}}|$ vs $H$. We define as $H_{\mathrm{c}2}$ the field above which $|{\kappa }_{\mathrm{xy}}|$ departs from that linear rise (vertical gray band), giving $H_{\mathrm{c}2}=25\pm 1\mathrm{T}$ (Fig. 1). The dotted line shows the linear behavior (${\kappa }_{\mathrm{xy}}\sim H$) expected of a metal when ${\kappa }_{\mathrm{xy}}$ becomes comparable to or smaller than ${\kappa }_{\mathrm{xx}}$.