Nernst effect in high-$T_c$ superconductors

The observation of a large Nernst signal $e_N$ in an extended region above the critical temperature $T_c$ in hole-doped cuprates provides evidence that vortex excitations survive above $T_c$. The results support the scenario that superfluidity vanishes because long-range phase coherence is destroyed by thermally created vortices (in zero field) and that the pair condensate extends high into the pseudogap state in the underdoped (UD) regime. We present a series of measurements to high fields $H$ which provide strong evidence for this phase-disordering scenario. Measurements of $e_N$ in fields $H$ up to $45\mathrm{T}$ reveal that the vortex Nernst signal has a characteristic “tilted-hill” profile, which is qualitatively distinct from that of quasiparticles. The hill profile, which is observed above and below $T_c$, underscores the continuity between the vortex-liquid state below $T_c$ and the Nernst region above $T_c$. The upper critical field (depairing field) $H_{c2}$ determined by the hill profile (in slightly UD to overdoped samples) displays an anomalously weak $T$ dependence, which is consistent with the phase-disordering scenario. We contrast the Nernst results and $H_{c2}$ behavior in hole-doped and electron-doped cuprates. Contour plots of $e_N(T,H)$ in the $T$-$H$ plane clearly bring out the continuous extension of the low-$T$ vortex liquid state into the high-$T$ Nernst region in hole-doped cuprates (but not in the electron-doped cuprate). The existence of an enhanced diamagnetic magnetization $M$ that survives to intense $H$ above $T_c$ is obtained from torque magnetometry. The observed $M$ scales accurately like $e_N$ above $T_c$, confirming that the large Nernst signal is associated with local diamagnetic supercurrents that persist above $T_c$. We emphasize implications of the new features in the phase diagram implied by the high-field results and discuss relevant theories.

Figure 1

The vortex-Nernst effect in a type-II superconductor. Concentric circles represent vortices.

Figure 2

Crystal mounting geometry in the Nernst experiment.

Figure 3

Curves of the Nernst signal $e_N$ vs $H$ measured in slightly OV YBCO ($x=6.99$, $T_c=92\mathrm{K}$) at temperatures below $T_c$ [panel (a)] and above $T_c$ [panel (b)]. Below $T_c$, $e_N$ rises nearly vertically at the melting field $H_m$. Above $T_c$ (b), the negative $H$-linear contribution of the qp term $e_N^n$ becomes quite apparent.

Figure 4

Temperature dependence of $e_N$ measured at $14\mathrm{T}$ compared with the resistivity $\rho$ measured on OV YBCO ($x=6.99$, $T_c=92\mathrm{K}$). The onset temperature $T_{onset}$ and $T_c$ are indicated by arrows. The dashed line indicates the negative qp contribution.

Figure 5

(a) The Nernst signal $e_N$ vs $H$ in OP Bi 2212 $(T_c=91\mathrm{K})$ at temperatures $50–130\mathrm{K}$. The oscillations in $e_N$ in weak $H$ are reproducible and may be caused by plastic flow of the vortices. (b) The $T$ dependence of $e_N$ measured at $14\mathrm{T}$ (solid squares) and the Meissner curve (magnetization $M$ measured at $H=10\mathrm{Oe}$) in OP Bi 2212. The dashed line indicates the estimated negative qp contribution.

Figure 6

(a) Curves of $e_N$ vs $H$ in OP Bi 2223 $(T_c=109\mathrm{K})$ measured at selected $T$. (b) The $T$ dependence of $e_N$ at $14\mathrm{T}$ compared with the Meisnner curve ($M$ measured at $10\mathrm{Oe}$) in OP Bi 2223. Dashed line is the negative qp contribution.

Figure 7

(a) Curves of $e_N$ vs $H$ in UD LSCO ($x=0.12$, $T_c=29\mathrm{K}$). The bold curve is nominally at $T_c$. The dashed line at $40\mathrm{K}$ gives the initial slope used to find $\nu$. (b) The $T$ dependence of $e_N$ measured at $14\mathrm{T}$ in UD LSCO (solid squares) compared with the Meissner curve ($M$ measured at $10\mathrm{Oe}$, open circles). Dashed line is the negative qp contribution.

Figure 8

Curves of $e_N$ vs $H$ in heavily UD LSCO ($x=0.07$, $T_c=11\mathrm{K}$). The bold curve is taken at $12\mathrm{K}$, $1\mathrm{K}$ above $T_c$.

Figure 9

(a) Curves of $e_N$ vs $H$ in UD Bi 2212 from 25 to $140\mathrm{K}$. The bold curve is taken at $T_c=50\mathrm{K}$. The curves remain strongly nonlinear up to $70\mathrm{K}$. (b) Comparison of the $T$ dependence of $e_N$ measured at $14\mathrm{T}$ and the Meissner curve ($M$ at $10\mathrm{Oe}$) in UD Bi 2212. The vortex signal deviates from the qp background at $T_{onset}\sim 118\mathrm{K}$.

Figure 10

The Ettingshausen signal ${\mathcal{Q}}_E$ vs $H$ in the type-II superconductor $\mathrm{PbIn}$ (adapted from Ref. 48). The $H$ dependence of ${\mathcal{Q}}_E$ has the characteristic peak profile of the vortex response. The dashed line locates $H_{c2}$ determined by $\rho$ vs $H$. Above $H_{c2}$, the signal displays a weak fluctuation tail.

Figure 11

Curves of $e_N$ vs $H$ in OP ${\mathrm{Bi}}_2{\mathrm{Sr}}_{2-y}{\mathrm{La}}_y{\mathrm{CuO}}_6$ (with La content $y_{La}=0.4$, $T_c=28\mathrm{K}$) measured to intense fields ($45\mathrm{T}$ below $35\mathrm{K}$ and $32\mathrm{T}$ above $35\mathrm{K}$). The depairing field $H_{c2}$ is determined by extrapolation of $e_N$ to zero. Above $60\mathrm{K}$, the weak negative qp contribution becomes apparent.

Figure 12

(a) Curves of $e_N$ vs $H$ in OV ${\mathrm{Bi}}_2{\mathrm{Sr}}_{2-y}{\mathrm{La}}_y{\mathrm{CuO}}_6$ ($y_{La}=0.2$, $T_c=22\mathrm{K}$) measured to intense fields from 4.3 to $40\mathrm{K}$. At all $T$, the curves appear to extrapolate to zero at the same nominal field scale $(42–45\mathrm{T})$. (b) Curves of $e_N$ vs $H$ in UD ${\mathrm{Bi}}_2{\mathrm{Sr}}_{2-y}{\mathrm{La}}_y{\mathrm{CuO}}_6$ ($y_{La}=0.7$, $T_c=12\mathrm{K}$) measured to 45 (32) T for temperatures below (above) $12\mathrm{K}$.

Figure 13

Contour plot of $e_N(T,H)$ in OP Bi 2201 $(y_{La}=0.4)$. The value of $e_N$ is highest in the light-gray region and zero in black regions. The white curve terminating at $15\mathrm{K}$ is $H_m\left(T\right)$. The dashed curve is the ridge-line joining points of maxima of $e_N$ vs $H$. Solid squares are values of $H_{c2}$ estimated from extrapolation of the curves in Fig. 11. The plot emphasizes the smooth continuity of the vortex signal to temperatures high above $T_c$ ($28\mathrm{K})$.

Figure 14

The curve of $e_N$ vs $H$ measured at $T=20\mathrm{K}$ in OP LSCO ($x=0.17$, $T_c=36\mathrm{K}$). The value of $H_{c2}\sim 50\mathrm{T}$ is estimated by the dashed-line extrapolation of the high-field data to zero.

Figure 15

The high-field Nernst curves in optimally doped LSCO $(x=0.17$) from 5 to $40\mathrm{K}$. Below $20\mathrm{K}$, all curves merge to the dashed line at high fields. As $T$ rises above $20\mathrm{K}$, the qp contribution increasingly “pulls” the curves of $e_N$ to negative values in high fields. This effect is much less pronounced at lower $T$.

Figure 16

Extrapolation of the curves of $e_N$ (dashed lines) to determine $H_{c2}$ values in OP Bi 2201 $(y_{La}=0.4)$. The estimated values of $H_{c2}$ are virtually $T$ independent.

Figure 17

Plot of $H_{c2}$ versus the reduced temperature $t=T∕T_c$ in OP and OV Bi 2201 (with $y_{La}=0.4$ and 0.2, respectively). At each $T$, $H_{c2}$ is estimated as in Fig. 16.

Figure 18

Variation of the low-$T$ upper critical field $H_{c2}\left(0\right)$ estimated at $4.2\mathrm{K}$ versus $x$ in LSCO (solid squares). The values are estimated by extrapolation of $e_N^s$ to zero from measurements in $H$ to $45\mathrm{T}$. For comparison, we also plot $T_{onset}$ (open circles) and $T_c$ (solid circles). Lines are guides to the eye.

Figure 19

Comparison of the field profiles of the flux-flow resistivity $\rho$ and the Nernst signal $e_N$ measured on the same sample, an overdoped crystal of LSCO $(x=0.20)$ at $T=22$ and $12\mathrm{K}$. Above $H_m$, $\rho$ quickly approaches saturation to the resistivity value extrapolated from above $T_c$ (this occurs near $H_{ridge}$ defined by the peak in $e_N$). However, $e_N$ decreases to zero at the depairing field $H_{c2}$ which lies much higher $(H_{c2}\sim 45\mathrm{T})$.

Figure 20

The phase diagram of LSCO showing the Nernst region between $T_c$ and $T_{onset}$ (numbers on the contour curves indicate the value of the Nernst coefficient $\nu$ in nV/KT). The curve of $T_{onset}$ vs $x$ has end points at $x=0.03$ and $x=0.26$ and peaks conspicuously near 0.10. The dashed line is $T^{*}$ estimated from heat-capacity measurements.

Figure 21

The phase diagram of Bi 2212 showing the Nernst region between $T_{onset}$ and $T_c$ (based on Nernst measurements on five crystals). As in LSCO (Fig. 20), the Nernst region does not extend to the pseudogap temperature $T^{*}$ on the OP and OV sides. In the UD regime, $T_{onset}$ shows a decreasing trend as $x$ decreases below 0.15.

Figure 22

Comparison of the Nernst curves in severely UD LSCO [$x=0.05$, panel (a)] and heavily OV LSCO [$x=0.26$, panel (b)]. In both samples, $T_c<2\mathrm{K}$ and the observed Nernst signal is weak. However, in the UD sample, $e_N$ retains the “tilted-hill” profile characteristic of vortices, whereas $e_N$ in the OV sample shows only the negative, $H$-linear qp contribution. The resistivity profile $\rho$ vs $T$ of the UD sample is shown in panel (c).

Figure 23

Comparison of curves of $e_N$ vs $H$ in LSCO samples with (from left panel to right) $x=0.05$, 0.07, 0.12, 0.17, 0.20, and 0.26. The same $x$ and $y$ scales are used in all panels. In each panel, the envelope curve provides a measure of the overall magnitude of the vortex signal.

Figure 24

The variation with $x$ of the maximum vortex-Nernst signal $e_N^{max}$ (solid symbols), $T_c$ (open circles), and $T_{onset}$ (dashed line) in LSCO. Solid lines are guides to the eye.

Figure 25

Magnetization curves $M$ vs $H$ in UD Bi 2212 $(T_c=50\mathrm{K})$ at temperatures from 40 to $200\mathrm{K}$. The magnetization is extracted as a strongly $T$-dependent diamagnetic contribution to the total torque signal in a Si cantilever (Ref. 81). The bold curve is taken at $T_c=50\mathrm{K}$. Even at $70\mathrm{K}$, $M\left(H\right)$ displays nonlinearity in $H$.

Figure 26

The $T$ dependence of $e_N^s$ (open circles) and the diamagnetic signal $-M$ (solid squares) in UD Bi 2212. Both quantities were measured at $14\mathrm{T}$ in the same crystal ($e_N^s$ is the vortex-Nernst signal obtained after subtracting the small negative qp contribution $e_N^n$ from the total Nernst signal). The dashed line is the weak-field Meissner effect of this sample measured at $H=10\mathrm{Oe}$ measured by SQUID.

Figure 27

(a) Curves of the observed Nernst signal $e_N$ vs $H$ in OP NCCO ($x=0.15$ and $T_c=24.5\mathrm{K}$) at temperatures $5\mathrm{K}$ to $30\mathrm{K}$. The dashed lines are fits of the high-field segments to a qp term of the form $e_N^n(T,H)=c_{1}H+c_3{H}^3$. (b) The vortex-Nernst signal $e_N^s$ extracted from panel (a) by subtracting $e_N^n(T,H)$ from the observed signal. The contour plot of this vortex signal is displayed in Fig. 28.

Figure 28

The contour plot of the vortex-Nernst signal $e_N^s(T,H)$ in NCCO ($x=0.15$, $T_c=24.5\mathrm{K}$). The magnitude of $e_N^s$ is highest in the light-gray region and zero in the black region below the melting-field curve $H_m\left(T\right)$ (white curve). The dashed curve is the ridge joining the maxima of the curves of $e_N^s$ vs $H$. The upper critical field values $H_{c2}$ estimated from where $e_N^s\to 0$ are shown as white symbols. The absence of a pseudogap correlates with the vanishing of the vortex-Nernst signal at $T_c$ and the termination of the $H_{c2}$ curve at $T_c$ (contrast with Fig. 13 for Bi 2201).