Pseudogap temperature $T^{*}$ of cuprate superconductors from the Nernst effect

We use the Nernst effect to delineate the boundary of the pseudogap phase in the temperature-doping phase diagram of hole-doped cuprate superconductors. New data for the Nernst coefficient $\nu \left(T\right)$ of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_y$ (YBCO), ${\mathrm{La}}_{1.8-x}{\mathrm{Eu}}_{0.2}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (Eu-LSCO), and ${\mathrm{La}}_{1.6-x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (Nd-LSCO) are presented and compared with previously published data on YBCO, Eu-LSCO, Nd-LSCO, and ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (LSCO). The temperature $T_{\nu }$ at which $\nu /T$ deviates from its high-temperature linear behavior is found to coincide with the temperature at which the resistivity $\rho \left(T\right)$ deviates from its linear-$T$ dependence, which we take as the definition of the pseudogap temperature $T^{★}$—in agreement with the temperature at which the antinodal spectral gap detected in angle-resolved photoemission spectroscopy (ARPES) opens. We track $T^{★}$ as a function of doping and find that it decreases linearly vs $p$ in all four materials, having the same value in the three LSCO-based cuprates, irrespective of their different crystal structures. At low $p,T^{★}$ is higher than the onset temperature of the various orders observed in underdoped cuprates, suggesting that these orders are secondary instabilities of the pseudogap phase. A linear extrapolation of $T^{★}\left(p\right)$ to $p=0$ yields $T^{★}(p\to 0)\simeq T_{\mathrm{N}}$(0), the Néel temperature for the onset of antiferromagnetic order at $p=0$, suggesting that there is a link between pseudogap and antiferromagnetism. With increasing $p,T^{★}\left(p\right)$ extrapolates linearly to zero at $p\simeq p_{\mathrm{c}2}$, the critical doping below which superconductivity emerges at high doping, suggesting that the conditions which favor pseudogap formation also favor pairing. We also use the Nernst effect to investigate how far superconducting fluctuations extend above the critical temperature $T_{\mathrm{c}}$, as a function of doping, and find that a narrow fluctuation regime tracks $T_{\mathrm{c}}$, and not $T^{★}$. This confirms that the pseudogap phase is not a form of precursor superconductivity, and fluctuations in the phase of the superconducting order parameter are not what causes $T_{\mathrm{c}}$ to fall on the underdoped side of the $T_{\mathrm{c}}$ dome.

Figure 1

Sketch of how the Nernst effect is measured on a sample in the shape of a thin platelet. A longitudinal temperature gradient along $x$ is generated by applying heat to one end of the sample, while the other end is kept cold. A given heat current $\left(\dot{Q}\right)$ produces a temperature difference $\left(\mathrm{\Delta }{T}_x=T^{+}-T^{-}\right)$ that can be measured either with resistance thermometers or thermocouples. When a magnetic field $\left(H\right)$ is applied along $z$, a transverse (Nernst) voltage $\left(\mathrm{\Delta }{V}_y\right)$ is generated. The Nernst signal $N$ is the ratio of $\mathrm{\Delta }{V}_y$ over $\mathrm{\Delta }{T}_x$ [Eq. (3)].

Figure 2

Nernst coefficient $\nu$ of YBCO at a hole doping of $p=0.12$, plotted as $\nu /T$ versus temperature $T$ for different magnetic fields $(H=1$ T to 15 T), as indicated. The thermal gradient is applied in the $b$ direction of the orthorhombic crystal structure. Data are reproduced from Ref. [20]. (a) The vertical line marks the superconducting transition temperature at $H=0,T_{\mathrm{c}}=66.0$ K. (b) Zoom near $T_{\mathrm{c}}$, to show how $T_{\min }$ is defined: It is the temperature at which the Nernst signal at $H=1$ T goes through a minimum, at the foot of the large positive peak due to superconductivity. (c) Zoom at high temperature, where only quasiparticles contribute to the Nernst signal. $T_{\nu }$ (arrow) is defined as the temperature below which $\nu \left(T\right)/T$ starts to deviate downwards from its high-temperature linear behavior.

Figure 3

Temperature-doping phase diagram of YBCO, showing three characteristic temperatures. The transition temperature $T_{\mathrm{c}}$ (open black circles [34]) marks the onset of superconductivity in zero magnetic field, below which the electrical resistivity is zero. The solid black line is a guide to the eye through the $T_{\mathrm{c}}$ data points. The dotted black line is a smooth extension of this line assuming that the superconducting phase ends at a critical doping $p_{\mathrm{c}2}=0.27$. Blue diamonds mark $T_{\min }$ [defined in Fig. 2], the temperature below which superconducting fluctuations become significant (from $a$-axis data in Ref. [20]). The open diamond shows $T_{\min }$ for a previously measured sample with $p=0.1$ [35]. The solid blue line is a guide to the eye. Red circles mark $T_{\rho }$, the temperature below which the resistivity $\rho \left(T\right)$ deviates from its high-temperature linear dependence (from data in Ref. [12]), a standard definition of the pseudogap temperature $T^{★}$ in YBCO [36] [see Fig. 5]. The open red circle shows $T_{\rho }$ for a sample with $p=0.18$ in which a high level of disorder scattering was introduced by electron irradiation [37]. In this case, $T_{\rho }$ marks the onset of an upturn in $\rho \left(T\right)$ (see text). Red squares mark $T_{\nu }$ [defined in Fig. 2], the temperature below which the quasiparticle Nernst signal departs from its high-temperature behavior (from present work and Ref. [20]). One can see that within error bars, $T_{\nu }\simeq T_{\rho }$, both measures of $T^{★}$. The red dashed line is a linear fit through the $T^{★}$ data points. Beyond $p=0.18$, it is a guide to the eye extending smoothly to reach $p=p^{★}$ at $T=0$ (red diamond). $p^{★}$ is the critical doping where the pseudogap phase ends at $T=0$ in the absence of superconductivity. In YBCO, $p^{★}=0.195\pm 0.005$ [6]. The gray band marks the range of $T^{★}$ values measured in Bi-2212 from spectroscopic probes (ARPES, STS, and SIS) [15], detected up to $p\simeq 0.22$.

Figure 4

High-temperature Nernst coefficient $\nu$ of YBCO at dopings $p=0.078$ (green) and $p=0.085$ (red), plotted as $\nu /T$ versus $T$. The thermal gradient was applied in the $a$ direction. The color-coded arrows mark $T_{\nu }$, the temperature below which $\nu \left(T\right)/T$ starts to deviate downwards from its small, roughly constant value at high temperature: $T_{\nu }=280\pm 20$ K and $260\pm 20$ K for $p=0.078$ and 0.085, respectively. Error bars on $T_{\nu }$ represent the uncertainty in identifying the start of the downturn.

Figure 5

Resistivity $\rho \left(T\right)$ as a function of temperature for four cuprate materials: (a) YBCO at $p=0.13$ [12]; (b) Bi-2201, underdoped with $T_{\mathrm{c}}=27$ K [14]; (c) LSCO at $p=0.143$ (red; [11]); and $p=0.18$ (blue; [10]) (d) Nd-LSCO at $p=0.20$ (red) and $p=0.24$ (blue) [2]. The black line is a linear fit of the high-temperature region and a zoom enables the extraction of $T_{\rho }$(arrow), the temperature below which $\rho \left(T\right)$ deviates from this linear dependence—a standard criterion for the pseudogap temperature $T^{★}$. For LSCO (c) and Nd-LSCO (d), the comparison between two dopings on either side of the pseudogap critical point $p^{★}$ reveals the effect on $\rho \left(T\right)$ of the drop in carrier density (from $n=1+p$ to $n=p)$ caused by the pseudogap present at $p<p^{★}$ [3, 11].

Figure 6

Resistivity curvature map from Ando et al. [12] showing the second temperature derivative of their resistivity data on YBCO, plotted as a function of temperature (vertical axis) and oxygen doping $x$ (bottom horizontal axis). The green dots mark $T_{\mathrm{c}}$. The top axis shows the approximate hole doping $p$, estimated from the $T_{\mathrm{c}}$ values [34]. White regions correspond to linear, blue ones to sublinear (downward; $d^2\rho /d{T}^2<0)$ and red ones to superlinear (upward; $d^2\rho /d{T}^2>0)$ behavior of resistivity with temperature. The boundary of the pseudogap region $\left(T^{★}\right)$ is the lower limit of the white region in the upper right corner. $T_{\nu },T_{\min }$, and $T_{\mathrm{c}}$ from present work and data of Ref. [20] are added respectively as open squares, triangles, and circles. $T_{\nu }$ points agree reasonably well with the resistivity criterion (as in Fig. 3). The narrow blue region that tracks $T_{\mathrm{c}}$ represents the paraconductivity regime where resistivity drops due to superconducting fluctuations just above $T_{\mathrm{c}}$. $T_{\min }$ (triangles), our criterion for the onset of significant superconducting fluctuations in the Nernst effect, is seen to agree with the onset of paraconductivity, clearly observable at $x<6.8$ (or $p<0.13)$.

Figure 7

Cartoon illustrating the behavior of the Nernst coefficient $\nu$ in cuprate superconductors, plotted as $\nu /T$ vs $T$. The quasiparticle signal (QP, red) goes from small at high $T$ to large at low $T$, with a change of sign. It is independent of magnetic field. The change occurs upon entering the pseudogap phase, by crossing below a temperature $T_{\nu }=T^{★}$ (arrow). In YBCO (and Hg-1201), $\nu$ is positive at high $T$ (left panel), while in LSCO (and Nd/Eu-LSCO), $\nu$ is negative at high $T$. The superconducting signal (SC, blue) develops below a temperature $T_{\mathrm{B}}$ (arrow) slightly above the zero-field $T_{\mathrm{c}}$ (vertical dotted line). It is always positive and it is suppressed by a magnetic field.

Figure 8

Nernst coefficient $\nu$ of Eu-LSCO at dopings $p=0.08$ (a), 0.10 (b) and 0.125 (c) versus $T$, at $H$ = 2 T (open red circles) and 10 T (filled green circles). The data at $p=0.125$ are taken from Ref. [30]. A two-peak structure is seen clearly at $p=0.125$. At the other two dopings, it shows up as a breaking point in the slope of the data, at $T\simeq 35$ K. This two-peak structure reveals the two distinct contributions to the Nernst effect: one from superconducting fluctuations, seen as a narrow positive peak at low temperature (grey shading in bottom panel), and the other from quasiparticles, seen as a broad positive peak at higher temperature. The dashed line is a guide to the eye for delimiting the quasiparticle peak. In panel (c), we also plot LSCO data at $p=0.12$ and $H$ = 1 T (blue; from Ref. [27]), for comparison. In LSCO, we see that the two separate contributions flow smoothly one into the other. The red arrow marks $T_{\mathrm{B}}$, the temperature above which the field dependence of $\nu$ becomes negligible, the signature of a negligible superconducting signal.

Figure 9

Nernst coefficient $\nu$ of Nd-LSCO (red circles) and LSCO (black squares) at $p$ = 0.15, as a function of temperature (data from Ref. [44], at $H=9$ T). Down to 50 K or so, the two data sets are virtually identical (see also Fig. 13). Note the small anomaly in the Nd-LSCO data at $T\simeq 70$ K, due to the LTT transition, a structural transition not present in LSCO. Below 50 K, the superconducting signal in LSCO starts to deviate upwards. The difference between the two curves (cross-hatched region) is attributable to their different $T_{\mathrm{c}}$ values (37 K and 12 K); it is the superconducting contribution to the Nernst signal in LSCO.

Figure 10

Temperature-doping phase diagram of LSCO (black), Nd-LSCO (red), and Eu-LSCO (green), showing the pseudogap temperature $T^{★}$ (blue line) and the superconducting transition temperature $T_{\mathrm{c}}$ (of LSCO, gray line). $T_{\nu }$ (filled squares, from this work and Refs. [27, 30, 43, 44, 45]) is the temperature below which the quasiparticle Nernst signal starts to increase toward large positive values (Fig. 15). $T_{\rho }$ (filled circles, from Refs. [2, 3, 46]) is the temperature below which the resistivity $\rho \left(T\right)$ deviates from linearity (Fig. 5). The open triangles show $T^{★}$ detected by ARPES as the temperature below which the antinodal pseudogap opens, in LSCO (black) [47] and Nd-LSCO (red) [5]. We see that $T_{\nu }\simeq T_{\rho }\simeq T^{★}$, within error bars. Note how the pseudogap phase comes abruptly to an end, at a critical doping $p^{★}=0.18\pm 0.01$ for LSCO (black diamond) [10, 11], and at a much higher doping, $p^{★}=0.23\pm 0.01$, for Nd-LSCO (red diamond) [2, 3]. The dashed blue line is a linear extension of the solid blue line.

Figure 11

Nernst coefficient $\nu$ of Nd-LSCO at $p=0.20$ (left axis, red dots, $H=16$ T; this work) and $p=0.24$ (right axis, blue dots, $H=10$ T [30]), plotted as $\nu /T$ vs $T$. The red and blue vertical dashed lines mark $T_{\mathrm{c}}(H=0$ T) at $p=0.20$ (20 K) and 0.24 (17 K), respectively. The black vertical dashed line marks the transition to the low-temperature tetragonal (LTT) structure, at $T_{\mathrm{LTT}}=82$ K for $p=0.20$; the transition only causes a small kink in the (red) data (see also Fig. 9). The solid color-coded lines are linear fits to the data above 82 K, extended down to $T=0$. This comparison shows the effect of the pseudogap on the Nernst coefficient: a large upturn below $T_{\nu }=T^{★}$ (red arrow) at $p=0.20<p^{★}$, in contrast with the continuous linear decrease at $p=0.24>p^{★}$.

Figure 12

Nernst coefficient $\nu$ of Nd-LSCO (red circles; left axis; $H=16$ T) and Eu-LSCO (green squares; right axis; $H=10$ T), both at $p=0.21$, plotted as $\nu /T$ versus $T$. Above 40 K, $\nu$ is independent of magnetic field. Vertical dotted lines mark the structural transitions to the LTT structure at low $T$. The black dashed line is a linear fit to the Nd-LSCO data above 85 K, extended down to $T=0$. Eu-LSCO data also show linearity in the same temperature range. Data deviate upwards from the linear fit below a temperature $T_{\nu }=75\pm 10$ K for Nd-LSCO (blue arrow) and $T_{\nu }=75\pm 10$ K for Eu-LSCO. The very different LTT temperatures of the two materials implies that the upturn in $\nu /T$ observed at roughly the same temperature in both is not caused by this structural transition, but instead by the pseudogap opening.

Figure 13

Nernst coefficient $\nu$ of Nd-LSCO (red circles) and LSCO (black squares) at $p=0.15$, plotted as $\nu /T$ versus $T$ (data from Ref. [44]). Vertical dashed lines indicate structural transition temperatures: from middle-temperature orthorhombic (MTO) to low-temperature tetragonal (LTT) in Nd-LSCO (70 K [53], Fig. 9), and from high-temperature tetragonal (HTT) to low-temperature orthorhombic (LTO) in LSCO (185 K [54]). One can see that the simultaneous rise in $\nu /T$ below $T_{\nu }=120\pm 10$ K (blue arrow) in the two materials cannot be caused by their structural transitions, which take place well below and above, respectively. The gray band marks the location of the pseudogap temperature measured by ARPES in LSCO at $p=0.15$ [47], at $T^{★}=130\pm 20$ K.

Figure 14

Nernst coefficient $\nu$ of Eu-LSCO (green circles; from Ref. [55]) and LSCO (black squares; from Ref. [44]) at $p=0.125$, plotted as $\nu /T$ versus $T$. Vertical dashed lines indicate structural transition temperatures: from middle-temperature orthorhombic (MTO) to low-temperature tetragonal (LTT) in Eu-LSCO (131 K [56]), and from high-temperature tetragonal (HTT) to low-temperature orthorhombic (LTO) in LSCO (250 K [54]). As in Figs. 12 and 13, the rise in $\nu /T$ below $T_{\nu }=165\pm 20$ K for Eu-LSCO and $T_{\nu }=135\pm 20$ K for LSCO is unrelated to their structural transitions.

Figure 15

Nernst coefficient $\nu$ of Nd-LSCO (top), LSCO (middle), and Eu-LSCO (bottom), at various dopings as indicated, plotted as $\nu /T$ versus temperature. Lines are linear fits of the data at high temperature. Arrows mark the temperature $T_{\nu }$ below which the data start to deviate upward from linearity (see Figs. 11, 12, 13, 14 for a zoomed view of the data from which we can more easily identify $T_{\nu })$. The values of $T_{\nu }$ are (from low to high $p)$: $T_{\nu }=120\pm 10,75\pm 10$, and 0 K in Nd-LSCO, $T_{\nu }=200\pm 25,200\pm 25,135\pm 10,120\pm 10$, and $90\pm 10$ K in LSCO, and $T_{\nu }=190\pm 10,175\pm 10,165\pm 20,115\pm 10$, and $75\pm 10$ K in Eu-LSCO. All values of $T_{\nu }$ are plotted on the phase diagram of Fig. 10. Nd-LSCO with $p=0.15$, LSCO with $p=0.15$, and $p=0.125$ were measured at 9 T (from Ref. [44]); Nd-LSCO with $p=0.20$ and Eu-LSCO with $p=0.21$ at 16 T (present work); Nd-LSCO with $p=0.24$, Eu-LSCO with $p=0.16$ (from Ref. [30]), Eu-LSCO with $p=0.08$ and 0.10 (present work), and Eu-LSCO with $p=0.125$ (from Ref. [55]) at 10 T; LSCO with $p=0.17$ at 8 T (from Ref. [43]) and LSCO with $p=0.05,0.07$ (from Ref. [45]) at $H\to 0$.

Figure 16

Resistivity curvature map from Ando et al. [12] showing the second temperature derivative of their resistivity data on LSCO, with $T_{\mathrm{c}}$ as solid green circles. As in Fig. 6, regions in white correspond to linear, in blue to sublinear (downward; $d^2\rho /d{T}^2<0)$ and in red to superlinear (upward; $d^2\rho /d{T}^2>0)$ behavior of resistivity with temperature. The red ridge inside the white region is due to the HTT-LTO structural transition in LSCO. The boundary of the pseudogap phase $\left(T^{★}\right)$ is the lower border of the white region (the dashed line is a guide to the eye). Our data points for $T_{\nu }$ from Fig. 10 are added, for Nd-LSCO (gray squares), Eu-LSCO (black squares), and LSCO (open squares). The $T_{\nu }$ data points agree reasonably well with the start of the upturn in the resistivity (as in Fig. 10). The narrow blue region that tracks $T_{\mathrm{c}}$ is due to paraconductivity. The values of $T_{\mathrm{B}}$ for LSCO are added as open diamonds (from Fig. 22). They agree well with the onset of paraconductivity. Together they delineate the regime of significant superconducting fluctuations in LSCO, limited to 30 K above $T_{\mathrm{c}}$.

Figure 17

Temperature-doping phase diagrams of YBCO (a) and Nd/Eu-LSCO (b) showing the pseudogap temperature $T^{★}(T_{\nu }$, squares), the Néel temperature $T_{\mathrm{N}}$ (brown line), and the superconducting transition temperature $T_{\mathrm{c}}$ (gray line). The blue line is a linear guide to the eye showing that $T^{★}$ extrapolates to $T_{\mathrm{N}}$ at half filling on the underdoped side $\left(p=0\right)$ while it merges with $T_{\mathrm{c}}$ on the overdoped side where superconductivity disappears. Note that the $T^{★}$ line of YBCO is proportional to but higher than that of LSCO: $T_{\mathrm{YBCO}}^{★}\simeq 1.5T_{\mathrm{LSCO}}^{★}$. Roughly the same scaling applies to $T_{\mathrm{N}}$ at $p=0$: $T_{\mathrm{N}}^{\mathrm{YBCO}}\left(0\right)\simeq 450$ K [57] and $T_{\mathrm{N}}^{\mathrm{LSCO}}\left(0\right)\simeq 280$ K [54]. Diamonds mark the pseudogap critical points for YBCO (red) at $p^{★}=0.195\pm 0.005$ [6], LSCO (black) at $p^{★}=0.18\pm 0.01$ [10, 11], and Nd-LSCO (red) at $p^{★}=0.23\pm 0.01$ [3]. $T_{\nu }$ are taken from Fig. 3 for YBCO and from Fig. 10 for LSCO; $T_{\mathrm{N}}$ is taken from Ref. [57] for YBCO and from Ref. [58] for LSCO.

Figure 18

Temperature-doping phase diagram of YBCO showing the Néel temperature $T_{\mathrm{N}}$ (brown line), the superconducting transition temperature $T_{\mathrm{c}}$ (black line), and the pseudogap temperature $T^{★}$ (red line) and critical point $p^{★}$ (red diamond), all from Figs. 3 and 17. In addition, we show the charge-density-wave phase (CDW; green), delineated by the temperature $T_{\mathrm{CDW}}$ below which short-range CDW correlations are detected by x-ray diffraction (up triangles [69]; down triangles [70]). The two green diamonds mark the critical dopings at which the CDW phase begins $(p_1^{\mathrm{CDW}}$ = 0.08 [71]) and ends $(p_2^{\mathrm{CDW}}$ = 0.16 [6]) at $T=0$ in the absence of superconductivity, as detected by high-field Hall effect measurements. $T_{\mathrm{SDW}}$ (purple squares) marks the temperature below which incommensurate short-range spin-density-wave (SDW) correlations are detected by neutron diffraction (in zero field) [72]. Gray symbols mark $T_{\mathrm{nem}}$, the onset temperature of nematicity, an electronic in-plane anisotropy detected in the resistivity (circles [73, 74]), the Nernst coefficient (squares [20, 74]), and the spin fluctuation spectrum measured by inelastic neutron scattering (triangles [72]). $T_{\mathrm{mag}}$ (blue circles) is the onset temperature of intra-unit-cell magnetic order detected by polarized neutron diffraction [75, 76, 77]. The blue line highlights the drop in $T_{\mathrm{mag}}$ below $p=0.09$.

Figure 19

Temperature-doping phase diagram of LSCO showing the Néel temperature $T_{\mathrm{N}}$ (brown line), the superconducting transition temperature $T_{\mathrm{c}}$ (black line), and the pseudogap temperature $T^{★}$ (red line) and critical point $p^{★}$ (red diamond), all from Figs. 10 and 17. In addition, we show the charge-density-wave phase (CDW; green), delineated by the temperature $T_{\mathrm{CDW}}$ below which short-range CDW correlations are detected by x-ray diffraction (up triangles [78]; down triangles [79]). The two green diamonds mark the critical dopings at which the CDW phase begins $(p_1^{\mathrm{CDW}}$ = 0.085) and ends $(p_2^{\mathrm{CDW}}$ = 0.15) at $T=0$ in the absence of superconductivity, as detected by high-field thermopower measurements [80]. $T_{\mathrm{SDW}}$ (purple squares) marks the temperature below which incommensurate short-range SDW order is detected by neutron diffraction [65, 81, 82, 83, 84]. The blue circle at $p=0.085$ marks $T_{\mathrm{mag}}$, the onset temperature of intra-unit-cell magnetic order, detected by polarized neutron diffraction [85].

Figure 20

Schematic phase diagram of cuprates, showing the antiferromagnetic phase (AF, below the blue line), the pseudogap phase (PG, below the red line), and the superconducting dome between $p_{\mathrm{c}1}$ and $p_{\mathrm{c}2}$ (SC, below the black dotted line). As in Fig. 17, a linear extension of the $T^{★}$ line (red) extrapolates to zero at $p_{\mathrm{c}2}$. In the region between the solid red line and the dashed green line, the normal-state resistivity $\rho \left(T\right)$ of cuprates is predominantly linear in temperature. This linearity appears below $p_{\mathrm{c}2}$, along with the superconductivity (see text). At $T\to 0$, it persists down to $p^{★}$. Above $p_{\mathrm{c}2},\rho \propto T^2$ at low $T$, the signature of a Fermi-liquid-like metallic state (FL). The green dashed line is drawn to capture the behavior of the linearity of resistivity as observed in LSCO in Fig. 3 of Ref. [10] (the so-called “red foot” contour plot).

Figure 21

Nernst coefficient of YBCO at doping $p=0.12$ with thermal gradient applied along the $a$ axis [panel (a)] and $b$ axis [panel (b)], plotted as $\nu /T\left(H\right)-\nu /T$ (15 T), versus magnetic field, at various temperatures as indicated. In (a), the isotherms above $T_{\mathrm{c}}\left(0\right)=66\mathrm{K}$ show a field-induced suppression, for $T<T_{\mathrm{B}}=95\pm 5$ K. Above $T_{\mathrm{B}}$, the field dependence of $\nu$ is negligible. We use $T_{\mathrm{B}}$ as a second criterion to define the onset of superconducting fluctuations, in addition to $T_{\min }$. In (b), isotherms immediately above $T_{\mathrm{c}}\left(0\right)$ (70 K and 75 K) also show a field-induced suppression of the superconducting signal. Isotherms far above $T_{\mathrm{c}}$ (90 K and 106 K) show a field-induced growth of $\nu \left(H\right)$, proportional to $H^2$ (dashed lines), due to a “magnetoresistance” in the quasiparticle contribution to the Nernst signal (see text). At low $H$, a superconducting signal is seen above the $H^2$ background (dashed line) at $T=80$ K, for example. The temperature above which $\nu \left(H\right)$ is purely quadratic is $T_{\mathrm{B}}=90\pm 5$ K.

Figure 22

Field dependence of the Nernst coefficient in LSCO at doping $x=p=0.07$ [(a); $T_{\mathrm{c}}=11$ K], 0.10 [(b); $T_{\mathrm{c}}=28$ K], and 0.12 [(c); $T_{\mathrm{c}}=29$ K] at various temperatures above $T_{\mathrm{c}}$ (color-coded legend). These curves show that above a certain temperature, defined as $T_{\mathrm{B}}$, the Nernst coefficient $\left(\nu \equiv N/H\right)$ is essentially field independent. The strong field dependence associated with superconducting fluctuations gone, the quasiparticle field-independent contribution dominates the signal. We find $T_{\mathrm{B}}=40\pm 10,50\pm 10$, and $50\pm 10$ K for $x=0.07$, 0.10, and 0.12, respectively. These $T_{\mathrm{B}}$ values are plotted on the curvature map of Fig. 16 and are seen to fall on the boundary of the paraconductivity region. Data at $p=0.07$ and 0.12 are taken from Ref. [27], at $p=0.10$ from Ref. [43].

Figure 23

(a) Nernst coefficient $\nu$ of Nd-LSCO (red circles) and LSCO (black squares) at $p=0.15$ versus temperature, measured with $H=9$ T (data from Ref. [44]). At high $T$, there is good agreement between the two data sets (see also Fig. 9), until the superconducting fluctuations in LSCO start to grow, below $\sim 50$ K. This difference (cross-hatched region) is attributable to their different $T_{\mathrm{c}}$ (37 K for LSCO and 12 K for Nd-LSCO) and can be seen as the superconducting fluctuations contribution for LSCO. (b) Difference between $\nu$ of LSCO and Nd-LSCO [cross-hatched region of panel (a)], plotted as $\mathrm{\Delta }\nu /T$, normalized at maximum value, versus normalized temperature $T/T_{\mathrm{c}}$ (where $T_{\mathrm{c}}$ is the $T_{\mathrm{c}}$ of LSCO). Subtracting Nd-LSCO from LSCO has the effect of taking away the quasiparticle contribution and revealing the superconducting contribution to the Nernst signal in LSCO. This superconducting contribution is seen to decrease rapidly with increasing temperature, vanishing around $1.35T_{\mathrm{c}}$ (see inset).

Figure 24

Temperature-doping phase diagram of Bi-2201 as a function of La doping $x$, showing $T_{\mathrm{N}}$ (brown line), $T_{\mathrm{c}}$ (black line), and $T^{★}$ detected by NMR (dashed red line) (from Ref. [17]). Blue diamonds mark the onset of a finite Nernst signal $(T^{\mathrm{onset}}$; Ref. [131]), attributed to SC fluctuations. Also shown is the onset temperature for SC fluctuations in Bi-2201 detected by torque magnetometry near optimal doping (blue circle; Refs. [132, 133]).

Figure 25

Temperature-doping phase diagram of YBCO, showing the pseudogap temperature $T^{★}$ (dashed blue line) and the onset of SC fluctuations detected in the Nernst signal $(T_{\min }$, blue diamonds; from Fig. 3) and in the microwave conductivity (blue square; from Ref. [135]). In addition, the onset temperature for SC fluctuations in Bi-2212 (green) and Hg-1201 (red) is also displayed, for samples with a $T_{\mathrm{c}}$ value given by the solid black line, from Nernst (red diamond, $T_{\min }$; Ref. [21]), microwave (green square; Ref. [134]) and magnetization (red circle; Refs. [132, 133]) data. Three different measurements on three different cuprates are seen to yield a very similar regime of SC fluctuations, close to $T_{\mathrm{c}}$ and well below $T^{★}$.

Figure 26

Temperature-doping phase diagram of LSCO from the Princeton group (gray). The early version of their phase diagram [43] plots an onset temperature, labeled $T_1^{\mathrm{onset}}$ (gray squares), defined as the temperature below which $\nu \left(T\right)$ starts to rise upon cooling. In a later version of their phase diagram [27, 45], they plot a revised onset temperature, which we label $T_2^{\mathrm{onset}}$ (gray diamonds). For $p\ge 0.10,T_2^{\mathrm{onset}}\equiv T_1^{\mathrm{onset}}$; for $p<0.10,T_2^{\mathrm{onset}}$ is the temperature below which $\nu$+$\mu S$ starts to rise upon cooling, where $\mu$ is the mobility and $S$ the Seebeck coefficient (see text). For comparison, we also plot the $T_{\nu }$ values reported here for LSCO (black squares, Fig. 10), i.e., the pseudogap temperature $T^{★}$ defined as the deviation from linearity in $\nu /T$ vs $T$ (Fig. 15). In red, we show the various signatures of superconductivity: $T_{\mathrm{c}}$ (solid line); $T_{\mathrm{B}}$ (open diamonds, Fig. 16); $T_{\mathrm{para}}$, the onset of paraconductivity (dashed line, Fig. 16); the onset of superconducting fluctuations probed by terahertz spectroscopy (pink shading [137]) and by torque magnetization (red circle [132, 133]).