Nernst effect in the electron-doped cuprate superconductor ${\mathrm{Pr}}_{2-x}{\mathrm{Ce}}_x{\mathrm{CuO}}_4$: Superconducting fluctuations, upper critical field $H_{c2}$, and the origin of the $T_c$ dome

The Nernst effect was measured in the electron-doped cuprate superconductor ${\mathrm{Pr}}_{2-x}{\mathrm{Ce}}_x{\mathrm{CuO}}_4$ (PCCO) at four concentrations, from underdoped $(x=0.13)$ to overdoped $(x=0.17)$, for a wide range of temperatures above the critical temperature $T_c$. A magnetic field $H$ up to 15 T was used to reliably access the normal-state quasiparticle contribution to the Nernst signal $N_{\mathrm{qp}}$, which is subtracted from the total signal $N$, to obtain the superconducting contribution $N_{\mathrm{sc}}$. As a function of $H$, $N_{\mathrm{sc}}$ peaks at a field $H^{☆}$ whose temperature dependence obeys $H_{c2}^{☆}ln(T/T_c$), as it does in a conventional superconductor such as ${\mathrm{Nb}}_x{\mathrm{Si}}_{1-x}$. The doping dependence of the characteristic field scale $H_{c2}^{☆}$, shown to be closely related to the upper critical field $H_{c2}$, tracks the domelike dependence of $T_c$, showing that superconductivity is weakened below the quantum critical point where the Fermi surface is reconstructed, presumably by the onset of antiferromagnetic order. Our data at all dopings are quantitatively consistent with the theory of Gaussian superconducting fluctuations, eliminating the need to invoke unusual vortexlike excitations above $T_c$, and ruling out phase fluctuations as the mechanism for the fall of $T_c$ with underdoping. We compare the properties of PCCO with those of hole-doped cuprates and conclude that the domes of $T_c$ and $H_{c2}$ versus doping in the latter materials are also controlled predominantly by phase competition rather than phase fluctuations.

Figure 1

In-plane resistivity of PCCO as a function of temperature for our four thin-film samples. The data at $H=0$ show the superconducting transition and the data at $H=15$ T show the normal-state behavior. For all samples except $x=0.17$, $\rho \left(T\right)$ exhibits a minimum, at a temperature $T_{\min }$ (shown here for $x=0.13$; green arrow). Below $T_{\min }$, $\rho \left(T\right)$ rises as $T\to 0$, a signature of Fermi-surface reconstruction.

Figure 2

Temperature-doping phase diagram of PCCO, showing the superconducting dome delineated by the zero-field critical temperature $T_c$ (black circles). Also shown is $T_{\min }$ (red squares), the temperature at which the resistivity $\rho \left(T\right)$ has a minimum (see Fig. 1). The red line is a linear fit to the $T_{\min }$ data, extrapolated to $T=0$ (open square). The corresponding doping $x_c=0.16$ is the quantum critical point below which the Fermi-surface reconstruction (FSR) onsets, in agreement with the critical doping where the normal-state Hall coefficient $R_H$ at $T\to 0$ exhibits a sharp drop to negative values [22]. Throughout our discussion in Sec. 5, we will use this figure as a map of FSR in PCCO that is derived from resistivity.

Figure 3

Vortex-solid melting lines as a function of temperature, plotted as $H_{\mathrm{vs}}$($T$) vs $T$/$T_c$, for dopings as indicated. $H_{\mathrm{vs}}$($T$) is the field below which the sample resistance is zero (full circles). Solid lines are fits to Eq. (1), used to extrapolate $H_{\mathrm{vs}}$($T$) to $T=0$ and obtain $H_{\mathrm{vs}}$(0), whose value is listed in Table 1. The open circle in the top right panel marks the value of the upper critical field $H_{c2}$ obtained from thermal conductivity data at $x=0.15$ (see Fig. 4).

Figure 4

Thermal conductivity $\kappa$ as a function of magnetic field $H$, measured at $T=0.2$ K on a single crystal of PCCO with $x=0.15)T_c=20$ K) (data from Ref. [35]). The saturation of $\kappa$ vs $H$ marks the end of the vortex state, providing a direct measurement of the upper critical field $H_{c2}$, defined as the field above which vortices disappear [31], giving $H_{c2}$ $=8\pm 1$ T.

Figure 5

Resistivity of our PCCO sample with $x=0.17$. The critical temperature is defined as the temperature where the zero-field data (red) go to zero: $T_c$ $=13.4$ K. The normal-state behavior is given by the data in a field $H=15$ T (blue), which includes a slight rigid upward shift due to positive magnetoresistance. A regime of paraconductivity is detectable below $T\simeq 2$ $T_c$, due to superconducting fluctuations that gradually decrease $\rho \left(T\right)$, before its rapid drop to zero at $T_c$. Inset: zoom on paraconductivity: $\Delta \rho \equiv \rho \left(0\right)-\rho \left(15\mathrm{T}\right)$ vs $T$.

Figure 6

Nernst response of PCCO as a function of magnetic field $H$ in our sample with $x=0.17$, at $T=1.08$ $T_c$. (a) Raw Nernst signal $N$ vs $H$ (red). The black dotted line is a polynomial fit to the data above 10 T, of the form $N_{\mathrm{qp}}$ $=a\left(T\right)H+b\left(T\right)H^3$. $N_{\mathrm{qp}}$ is the quasiparticle background of the underlying normal state. (b) Superconducting contribution to the Nernst signal, defined as $N_{\mathrm{sc}}$ $\equiv N-N_{\mathrm{qp}}$ [Eq. (2)]. For any given temperature above $T_c$, $N_{\mathrm{sc}}$ vs $H$ exhibits a peak, at a field labeled $H^{☆}$ (arrow).

Figure 7

Magnetic field-temperature phase diagram of PCCO at $x=0.17$. The peak field $H^{☆}$ (full red dots) is plotted above $T_c$ (vertical black line). The solid red line is a fit of the $H^{☆}$ vs $T$ data, to the formula $H^{☆}$ = $H_{c2}^{☆}ln(T/T_{\mathrm{c}})$ [Eq. (4)]. The vortex-solid melting field $H_{\mathrm{vs}}$($T$) (open circles) is the boundary between the vortex-solid phase (VS), where the electrical resistance is zero, and the vortex-liquid phase (VL), where the resistance is nonzero. The upper critical field $H_{c2}$ (green dotted line) is the boundary between the vortex liquid and the normal state, where there are no vortices. At $T=0$, there is no vortex liquid since $H_{c2}$(0) = $H_{\mathrm{vs}}$(0), as reported for hole-doped cuprates [31], and shown here for PCCO at $x=0.15$ (Figs. 3 and 4). The green dotted line is a schematic representation. Note that the two characteristic field scales for superconductivity, obtained, respectively, at $T\to 0$ and at $T>$ $T_c$, are equal for $x=0.17$, namely, $H_{c2}$(0) = $H_{c2}^{☆}=3.0$ T (Table 1).

Figure 8

Left panels: Raw Nernst data as a function of field for our four samples, at dopings as indicated. For each doping, we present seven isotherms, at $\epsilon =0.0$, 0.05, 0.1, 0.2, 0.4, 0.8, and 1.4, where $\epsilon \equiv (T-T_c$)/$T_c$, with $T_c$ as indicated. Right panels: Superconducting contribution to the Nernst signal $N_{\mathrm{sc}}$, obtained by subtracting the normal-state background $N_{\mathrm{qp}}$, as shown in Figs. 6 and 9, according to Eqs. (2) and (3). Note that with increasing temperature, the magnitude of $N_{\mathrm{sc}}$ decreases and the peak field $H^{☆}$ moves up. $H^{☆}$ is plotted vs $\epsilon$ in Fig. 10.

Figure 9

Nernst response of PCCO vs magnetic field, for $x=0.13$ (green) and $x=0.14$ (magenta), at $T=1.08$ $T_c$. (a) Raw Nernst signal $N$ vs $H$. The black dotted line is a polynomial fit to the data above 10 T, of the form $N_{\mathrm{qp}}$ $=a\left(T\right)H+b\left(T\right)H^3$ [Eq. (3)]. (b) Superconducting contribution to the Nernst signal, defined as $N_{\mathrm{sc}}$ $\equiv N-N_{\mathrm{qp}}$ [Eq. (2)].

Figure 10

Temperature dependence of the peak field $H^{☆}$ in PCCO, at dopings as indicated, plotted as a function of $\epsilon \equiv (T-T_c$)/$T_c$. $H^{☆}$ is the field at which $N_{\mathrm{sc}}$ peaks, in isotherms of $N_{\mathrm{sc}}$ vs $H$ as shown in the right panels of Fig. 8. The lines are a fit of the data to the function $H^{☆}=H_{c2}^{☆}ln(T/T_{\mathrm{c}})$ [Eq. (4)]. The fit allows us to extract a single characteristic field $H_{c2}^{☆}$ from the superconducting fluctuations at each doping, directly from the data. The values of the single fit parameter $H_{c2}^{☆}$ are listed in Table 1 and plotted vs $x$ in Fig. 11. At low $\epsilon$, the data deviate from the fit for reasons given in Sec. 5c.

Figure 11

The two magnetic field scales of superconductivity in PCCO, plotted as a function of doping (Table 1). $H_{\mathrm{vs}}$(0) (blue squares), the zero-temperature value of the upper critical field, is obtained by extrapolating to $T=0$ the vortex-solid melting field $H_{\mathrm{vs}}$($T$) below which the resistance is zero (see Fig. 3). The field scale $H_{c2}^{☆}$ (red circles) is obtained from the superconducting fluctuations above $T_c$ (see Fig. 10). Note that $H_{c2}^{☆}$ = $H_{\mathrm{vs}}$(0) at $x=0.17$.

Figure 12

(a) Raw Nernst coefficient $\nu \equiv N/H$ of PCCO at $x=0.15$, for selected isotherms above $T_c$, as indicated. In the limit $H\to 0$, $\nu \left(H\right)$ becomes flat, reaching a constant value ${\nu }_0$ given by the dashed line. (b) Temperature dependence of ${\nu }_0$ (dots), at dopings as indicated, plotted as a function of $\epsilon$. The solid lines are the normal-state contribution to ${\nu }_0$, i.e., ${\nu }_{\mathrm{qp}}\equiv$ $N_{\mathrm{qp}}/H$ in the limit $H\to 0$. Note that ${\nu }_0$ saturates below $\epsilon \simeq 0.1$, as expected from Gaussian theory in the limit $T\to$ $T_c$ (see Sec. 5c).

Figure 13

Sketch of the doping evolution of the Fermi surface in PCCO, based on ARPES measurements performed on NCCO, a closely related material [42, 43]. At high $x$, in the overdoped regime, the Fermi surface is a single large holelike nearly circular Fermi cylinder (drawn in red). Below a critical doping $x_c$ $\simeq 0.16$, the Fermi surface undergoes a reconstruction, into small hole (red) and electron (blue) pockets. At low $x$, the small hole pockets eventually disappear, leaving only the electron pockets centered at $(\pm \pi ,0)$ and $(0,\pm \pi )$.

Figure 14

$H_{\mathrm{vs}}$(0) divided by $T_c{}^2$ vs doping, for the electron-doped cuprate PCCO (red circles, top horizontal axis) and the hole-doped cuprates (bottom horizontal axis; data from Ref. [31]) YBCO (full blue squares) and Tl-2201 (open blue squares).

Figure 15

Comparison of measured [full red circles; Eq. (8)] and calculated [full blue squares; Eq. (7)] values of ${\alpha }_{xy}^{\mathrm{sc}}/H$ vs $x$. Open blue squares are the values calculated using $H_{\mathrm{vs}}$(0) instead of $H_{c2}^{☆}$ in Eq. (7). The agreement between experiment and theory, with no adjustable parameter, is remarkable.

Figure 16

Superconducting Nernst signal of PCCO in the absence of FSR, plotted as a function of magnetic field for the overdoped $(x=0.17$; red) and optimally doped $(x=0.15$; blue) samples, measured at $T=1.08$ $T_c$ in both cases. The data are compared to the theoretical calculation of ${\alpha }_{xy}^{\mathrm{sc}}$ (green curve, from Ref. [20]). For both experimental data and calculated curve, the $x$ axis is normalized by the peak field $H^{☆}$ and the $y$ axis is normalized by the magnitude of the superconducting Nernst signal at $H^{☆}$. The agreement between theory and experiment is excellent.

Figure 17

Upper critical field $H_{c2}$ in the hole-doped cuprate YBCO, obtained in two different ways. First, directly from resistive measurements of $H_{\mathrm{vs}}$($T$) at low temperature in high magnetic fields, extrapolated to $T=0$, giving $H_{c2}$ = $H_{\mathrm{vs}}$(0) (blue symbols) [31]. The open circles are for ${\mathrm{YBa}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8)p=0.14)$ and Tl-2201 $(p>0.21)$ [31]. Second, we plot $0.59{\tilde{H}}_{\mathrm{c}2}\left(0\right)$ (red symbols) with ${\tilde{H}}_{\mathrm{c}2}\left(0\right)={\Phi }_0/2\pi {\xi }^2\left(0\right)$, where the coherence length $\xi \left(0\right)$ is obtained from Gaussian Aslamazov-Larkin theory applied above $T_c$ to either the conductivity (full squares from Ref. [60], open squares from Ref. [61]) or the magnetization (open diamonds, from Ref. [62]). The agreement between the two ways of determining $H_{c2}$ is remarkable.

Figure 18

Superconducting Nernst signal of electron-doped PCCO (blue) and hole-doped Eu-LSCO (red), measured at $T=1.08$ $T_c$ in both cases, for dopings as indicated. The $x$ axis is normalized by the peak field $H^{☆}$ and the $y$ axis is normalized by the magnitude of the superconducting Nernst signal at $H^{☆}$. The agreement between the two is excellent, showing that the nature of superconducting fluctuations is basically the same whether cuprates are electron doped or hole doped, and both are well described by Gaussian theory (see Fig. 16).

Figure 19

Comparison of $H^{☆}$ vs $\epsilon$ for electron-doped PCCO (green circles) and hole-doped Eu-LSCO (red circles), at dopings as indicated. The PCCO data are multiplied by a factor 3. The black dotted line is a fit of the data to the function $H^{☆}$ = $H_{c2}^{☆}ln(T/T_{\mathrm{c}})$ [Eq. (4)].