Coherent $c$-axis transport in the underdoped cuprate superconductor YBa${}_2$Cu${}_3$O${}_y$

The electrical resistivity ${\rho }_c$ of the underdoped cuprate superconductor YBa${}_2$Cu${}_3$O${}_y$ was measured perpendicular to the CuO${}_2$ planes on ultrahigh quality single crystals in magnetic fields large enough to suppress superconductivity. The incoherent insulating-like behavior of ${\rho }_c$ at high temperature, characteristic of all underdoped cuprates, is found to cross over to a coherent regime of metallic behavior at low temperature. This crossover coincides with the emergence of the small electron pocket detected in the Fermi surface of YBa${}_2$Cu${}_3$O${}_y$ via quantum oscillations, the Hall and Seebeck coefficients, and with the detection of a unidirectional modulation of the charge density as seen by high-field nuclear magnetic resonance measurements. The low coherence temperature is quantitatively consistent with the small hopping integral $t_{\perp }$ inferred from the splitting of the quantum oscillation frequencies. We conclude that the Fermi-surface reconstruction in YBa${}_2$Cu${}_3$O${}_y$ at dopings from $p=0.08$ to at least $p=0.15$, attributed to stripe order, produces a metallic state with three-dimensional coherence deep in the underdoped regime.

Figure 1

Electrical resistivity ${\rho }_c$ of YBa${}_2$Cu${}_3$O${}_y$ for a current $I$ and a magnetic field $B$ along the $c$ axis ($I\parallel B\parallel c$). Three underdoped samples were measured at different temperatures below $T_c$ (as indicated) in pulsed magnetic fields up to 60 T. The doping level of each sample is (a) $p$ = 0.097, (b) $p$ = 0.109, and (c) $p$ = 0.120. Insets: Same data between 10 and 50 K with a fit of each isotherm (thin solid lines) using a two-band model above the superconducting transition (see Sec. 5).

Figure 2

Electrical resistivity ${\rho }_c$ of YBa${}_2$Cu${}_3$O${}_y$ (a) $p=0.097$, (b) $p=0.109$, and (c) $p=0.120$ plotted as a function of temperature for different values of the magnetic field. Dashed lines are a guide to the eye. The increase of the in-field $c$-axis resistivity down to about 4 K is in part due to the strong magnetoresistance which develops at temperatures below 60 K.

Figure 3

Field dependence of (a) the Nernst coefficient (Ref. 12), (b) the Hall coefficient (Ref. 11), and (c) the $c$-axis resistivity [Fig. 1(c)] of YBa${}_2$Cu${}_3$O${}_y$ at $p=0.12$ and $T=10$ K. Above a threshold field of about 30 T (indicated by the right vertical dashed red line), the Nernst coefficient $\nu$ [panel (a); green curve] saturates to its negative quasiparticle value, as demonstrated by its derivative [panel (a); red curve] which goes to zero as $B\approx 30$ T. This saturation shows that the positive contribution to the Nernst coefficient from superconducting fluctuations has become negligible above 30 T. Above this field, the Hall coefficient is almost flat [dashed blue line in panel (b)] and the two-band model fit to the $c$-axis resistivity [dashed magenta line in panel (c)] merges with the data. We conclude that at 10 K and above 30 T the flux-flow contribution to the transport properties is negligible, and the large magnetoresistance at high field is purely a property of the normal state at this doping level.

Figure 4

Temperature dependence of the $c$-axis resistivity of YBa${}_2$Cu${}_3$O${}_y$ ($p=0.109$) measured at zero magnetic field (black solid line) and at $B=55$ T (orange symbols). Blue solid line is a $1/T$ fit to the high temperature data up to 300 K. The saturation of the in-field resistivity at low temperature contrasts with the insulating-like behavior seen at high temperature. The green open squares correspond to the resistivity from which the magnetoresistance has been subtracted using a two-band model to extrapolate the normal-state data of Fig. 1 to $B=0$ (see Sec. 5).

Figure 5

The slope of the $c$-axis magnetoresistance evaluated at $B=50$ T (red circles; left axis) and in-plane Hall coefficient $R_H$ (blue squares; right axis) of (a) YBa${}_2$Cu${}_3$O${}_y$ ($p=0.109$), (b) YBa${}_2$Cu${}_3$O${}_y$ ($p=0.120$) as a function of temperature. $R_H$ data measured at (a) $B=54$ T, and (b) $B=45$ T is taken from Ref. 22 and normalized by its value at $T=100$ K. $T_0$ is the temperature at which $R_H\left(T\right)$ changes sign from positive at high temperature to negative at low temperature (Refs. 11 and 22).

Figure 6

Temperature dependence of the $c$-axis resistivity of YBa${}_2$Cu${}_3$O${}_y$ from which the magnetoresistance has been subtracted using a two-band model (see Sec. 5) for the three samples, as labeled. Solid lines show the resistivity measured in zero magnetic field. Dashed lines are a guide to the eye.

Figure 7

Temperature-doping phase diagram of YBa${}_2$Cu${}_3$O${}_y$, with the superconducting phase in zero magnetic field delineated by the transition temperature $T_c$. Black circles mark the temperature $T^{*}$ below which the in-plane resistivity deviates from its linear temperature dependence at high temperature (from data in Ref. 38), a standard definition for the onset of the pseudogap phase. The coherence temperature $T_{\mathrm{coh}}$ (green circles) is the temperature where ${\rho }_c\left(T\right)$ peaks (see Fig. 6).The coherence crossover splits the phase diagram into two regions: an incoherent two-dimensional (2D) regime above and a coherent 3D regime below. $T_0$ is the temperature at which the normal-state in-plane Hall coefficient $R_H\left(T\right)$ of YBa${}_2$Cu${}_3$O${}_y$ changes sign from positive at high temperature to negative at low temperature (blue diamonds; from Ref. 22).