A quantitative vortex-fluid description of Nernst effect in Bi-based cuprate high-temperature superconductors

Since the number of qp inside vortex core (about 10 holes in OP Bi-2212) and the phonon density at low temperature are both small, the scattering between phonon and qp inside vortex core is negligible in a vortex fluid below T_c. In this case, damping inside vortex core is dominated by impurity scattering, which is approximately temperature-independent. Thus, ${\eta }_{0}$ should be a constant for a given material, which can be expressed as ${\eta }_{0}={H}_{{\rm{c}}2}{\phi }_{0}/{\rho }_{a}{c}^{2}$ where ${\rho }_{a}$ is a characteristic resistivity. The steep rises above melting field in experiments [5] indicates that vortices move very fast in a dilute vortex fluid, thus ${\rho }_{a}$ should be much bigger than the normal state resistivity considered in Bardeen–Stephen model [42].

The crucial difference between HTSC and conventional SC is that for HTSC, core–core collisions are more important due to much higher vortex density under small coherence length. While the independent vortex assumption may be valid in low fields, it is certainly inappropriate in high fields in cuprates. When the sample is tested in high fields (say B = 10 T), the distance between vortex cores ${l}_{B}=\sqrt{{{\rm{\Phi }}}_{0}/B}$ becomes of the order of 100 Å, which is much smaller than the penetration depth $\lambda \sim 1000\,\mathring{\rm A}$. Thus, vortices strongly interact with each other, leading to a remarkable increase of damping. In our recent work [26], we propose a new mechanism of core–core collision in dense vortex fluids: two vortex cores merge together to a bigger core, leading to an increase of qp DOS. This process induces a momentum transfer from circular supercurrents to qp, and then a damping. Since the random motion of vortex core originates from quantum fluctuations of holes [26], the mean-free-path of the core–core collision is proportional to reciprocal of core density. Thus, a dimensional argument yields a linear n_v dependence of ${\eta }_{{\rm{c}}}$,

$$\begin{eqnarray}&&{\eta }_{{\rm{c}}}=\displaystyle \frac{{\phi }_{0}^{2}}{{c}^{2}}\displaystyle \frac{{n}_{v}}{{\rho }_{0}},\end{eqnarray} \tag{ 8 }$$

where ${\rho }_{0}$ is a characteristic normal state resistivity. This new mechanism offers an explanation [26] of a longstanding puzzle, i.e. the unconventional vortex dissipation in cuprate superconductors, as well as the nature of the discrepancy of Anderson's damping model.

Express ${\eta }_{0}$ also in terms of ${\rho }_{0}$, we obtain

$$\begin{eqnarray}&&{\eta }_{0}=\displaystyle \frac{{\phi }_{0}}{{c}^{2}}\displaystyle \frac{{B}_{0}}{{\rho }_{0}},\end{eqnarray} \tag{ 9 }$$

where ${B}_{0}={H}_{{\rm{c}}2}{\rho }_{0}/{\rho }_{a}$ is the effective fields to describe the damping strength of impurity scattering relative to core–core collisions.

Pinning effect introduces modification to the transport of a vortex fluid [43]. According to Blatter et al [43], pinning effect can be expressed in an effective damping viscosity. Since pinning and core–core collisions both depend on inter-vortex distance, their joint effect can then be written in terms of a multiplicative factor Γ such that ${\eta }_{{\rm{c}}}+{\eta }_{{\rm{pin}}}={\rm{\Gamma }}{\eta }_{{\rm{c}}}.$ The value of Γ must satisfy two limiting conditions. First, at high temperature and strong fields limit where the pinning effect vanishes, Γ is equal to 1. At low temperature and weak field limit, it is the thermal assisted flux flow (TAFF) described by Arrhenius law $\rho \approx {\rho }_{0}\exp [-{U}_{{\rm{pl}}}/({k}_{B}T)]$, where U_pl is the plastic deformation energy [43, 44]. A simple choice is thus ${\rm{\Gamma }}=\exp ({U}_{{\rm{pl}}}/{k}_{B}T)$, which yields the following expression:

$$\begin{eqnarray}&&{\eta }_{{\rm{pin}}}=\displaystyle \frac{{\phi }_{0}}{{c}^{2}}\displaystyle \frac{B+{B}_{T}}{{\rho }_{0}}[\exp ({U}_{{\rm{pl}}}/{k}_{B}T)-1].\end{eqnarray} \tag{ 10 }$$

The B and T dependence of U_pl may vary with samples [43, 45], which has been parameterized by Geshkenbein et al in terms of an energy involving deformations of the vortex lines on a scale of inter-vortices distance [46]. We introduce a parameter B_p as an effective field strength quantifying pinning. Then, the model of Geshkenbein et al can be expressed as

$$\begin{eqnarray}&&{U}_{{\rm{pl}}}={k}_{B}({T}_{v}-T)\sqrt{{B}_{{\rm{p}}}/(B+{B}_{T})}.\end{eqnarray} \tag{ 11 }$$

Note that this kind of B and T dependence was indeed discovered experimentally in Bi-2212 [47, 48]. Therefore, in the sequel, we use equation (11) to describe the pinning effect in Bi-based cuprates. B_p is an intrinsic material parameter, which can be probed by the melting field H_m as ${B}_{{\rm{p}}}={H}_{{\rm{m}}}/(4{c}_{{\rm{L}}}^{4}{t}^{2})$ according to Lindemann criterion [43, 49], where c_L is the Lindemann number and $t={T}_{v}/T-1$.

Finally, the total damping can be written as

$$\begin{eqnarray}&&\eta =\displaystyle \frac{{\phi }_{0}}{{\rho }_{0}{c}^{2}}\{{B}_{0}+(B+{B}_{T})\exp [t\sqrt{{B}_{p}/(B+{B}_{T})}]\}.\end{eqnarray} \tag{ 12 }$$

In this section, in-plane magnetoresistance is used to validate our damping model. In OP ($p=0.16$) sample of ${\mathrm{Bi}}_{2.1}{\mathrm{Sr}}_{1.9}{\mathrm{CaCu}}_{2}{{\rm{O}}}_{8+\delta }$, ρ is measured in a wide range of temperature (20–120 K) and fields (0–17.5 T) [50]. As is shown in figure 3, ρ first shows a slow increase regime at low temperature, which is a TAFF regime, and then the curves converge at a characteristic temperature. The transition between the two regimes varies with H. Equation (13) is applied to describe the data, see solid lines in figure 3, showing very good agreement at all five magnetic field values. The agreement is specially good at H = 0, where all vortices are thermal. Thus, our model of ρ is clearly validated.

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Since the formula involves seven parameters, a systematic procedure must be developed to determine them. A two-step procedure is developed, with a rough estimate (RE) first and then a fine tuning (see appendix A.1 for a detailed presentation). Note that this determination is not an arbitrary fitting, but involves a careful asymptotic analysis, which ensures the uniqueness of the outcome. The main idea can be summarized as follows. First, a RE is conducted using asymptotical analysis at large and small limits of T and B; at each limit, only one or two parameters appear, so that the parameter values can be easily determined. Applying all limiting data yields a complete set of RE values, which are then subjected to a fine tuning (FT) process, involving a search for optimal parameters around the RE values, minimizing the errors between the set of predictions and data. At the end of the FT step, precise parameter values are obtained, and its uncertainties are assessed, see table 1.

Table 1.  Material parameters in equation (13) determined in two steps, rough estimate (RE) and fine tuning (FT), described in appendix, for ${\mathrm{Bi}}_{2.1}{\mathrm{Sr}}_{1.9}{\mathrm{CaCu}}_{2}{{\rm{O}}}_{8+\delta }$. [50] The uncertainty is calculated by first setting all parameters to be their FT-determined values, and then varying only one given parameter to observe the root mean square fitting error equal to the experimental uncertainty, i.e., $5\,\mu {\rm{\Omega }}$ cm.

Step	Tc (K)	${\rho }_{0}$ (mΩ cm)	Tv (K)	B0 (T)	Bp (T)	Bl (T)	b	RMSE ($\mu {\rm{\Omega }}$ cm)
RE	87	0.116	112	4	30.8	68.1	0.42	10
FT	87	0.099 ± 0.11	104 ± 3	0.4 ± 0.4	89 ± 34	68.1	0.56 ± 0.11	5

Let us discuss the reliability of each parameter determination. First, the upper critical temperature T_v is the onset temperature of short-range coherence and thermal vortices, when decreasing the temperature. In the vortex-fluid model, it indicates the temperature, beyond which the vortex magnetoresistance collapses and vortex Nernst signal vanishes. Using the data of Usui et al, T_v = 112 K is obtained in RE and corrected to 104 K in FT, and the two values are very close, indicating a consistency. Besides, using the magnetoresistance data and Nernst signal measured by Ri et al [51], we find that T_v is restricted in the range of $[100,110]$ K. In the next section, we will show that T_v derived from the Nernst signal of Wang et al is also within the range (e.g. $107\pm 1.6$ K). All these results indicate that T_v is a well-defined, intrinsic parameter in HTSC samples.

B_p is the parameter to determine pinning strength and melting field, and thus can be determined with data in TAFF regime. The significant difference between RE and FT is attributed to measurement error due to low signal to noise ratio of the exponentially small signal, which amplifies the quantitative imperfection of Geshkenbein et al's pinning model, i.e., equation (11). Note that more precise data in TAFF regime can improve the quality of B_p, as we show below with the fitting of Nernst signal.

B₀ is the contribution from the qp scattering (inside vortex core) by defects. Comparing to pinning strength B_p and ${H}_{{\rm{c}}2}$, B₀ is weak, which indicates a fast-vortices behavior of cuprates superconductors [52, 53], i.e., the steep rise of ρ at low fields and near T_c with weak damping characterized by B₀ [5]. For this sample, ${B}_{0}\ll {B}_{{\rm{p}}}$, thus the determination of B₀ is strongly influenced by fluctuations of B_p, which results in a substantial difference between the RE value (4 T) and FT value (0.4 ± 0.4 T).

b is the unique parameter to determine the T dependence of thermal vortex activation. And, the value $b=0.42$ is obtained in the RE, and 0.56 ± 0.11 in the FT. Note that the classic procedure [54] yields an estimate of $b=0.55;$ thus, our determination is quite accurate.

In summary, the magnetoresistance measured by Usui [50] is well described by the vortex-fluid model, equation (13), with T_v and ${\rho }_{0}$ well determined, as well as the Kosterlitz coefficient b. But, the quantitative deviation is still present for the description of pinning effect in the TAFF regime below T_c, which results in an ill-determination of B_p. In addition, B₀ is also not very reliable, because it is too small in this sample.

Wang et al have made systematic measurement of Nernst signal in cuprates superconductor, see figure 4 [5, 55, 58]. These observations reveal a characteristic tilted-hill profile of vortex-Nernst signal. At low T and H, the exponentially small signal indicate a typical TAFF behavior. On the other hand, a linear decay of Nernst signal in strong field regime is found, due to vortex–vortex interaction which dominates in η. Between these two regimes, a peak of maximum Nernst signal appears, and the peak vanishes at a characteristic temperature, called the upper critical temperature T_v in the present work.

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The Nernst signal encompasses a rich set of properties which enable us to determine a set of physical parameters. Comparing to the magnetoresistance ρ in equation (13), the Nernst signal comprises extra complexity from S_ϕ which introduces two T-dependent parameters e_T and ${H}_{{\rm{c}}2}$ in equation (14). However, our two-step procedure still applies, see appendix A.2 for a discussion of the sample OP Bi-2201.

The accurate agreements between the predictions and data are shown in figure 4, which yields a determination of the parameters shown in table 2. What is important is that the precise agreement extends to the whole range of H, T and doping in both monolayer (Bi-2201) and bilayer (Bi-2212) samples. The minimum root mean square error (RMSE) between predictions and data is less than 0.1 μV K⁻¹ in most figures, excepts in figure 4(a) where RMSE = 0.152 μV K⁻¹. This is considered to be remarkable, supporting the validity of this vortex-fluid model and the reliability of the parameter values. In the following, we discuss T_v, B_p, B₀ and b, while n_s and ${H}_{{\rm{c}}2}$ are discussed in section 3.2.

Since T_v is the temperature where vortex Nernst signal vanish, it can be determined from the Nernst signal from the vanishing point of the peak (i.e. maximum Nernst signal). For instance, in OP Bi-2212, we have determined ${T}_{v}=107\pm 1.6$ K from e_N, which is consistent with both the result determined from ρ (see section 2.1) and the data from Ri et al [51]. This convergence of the multiple estimations strongly support the validity of the concept of T_v. Our final determined T_v for Bi-2212 are shown in figure 5, which increases slowly in the UD regime, but decreases quickly in the OV regime. Wang et al defined T_v as the onset temperature where the signal deviates from a background linear T dependence at a given magnetic field [5], see black open hexagons in figure 5. Note that Wang et al's definition may involve contributions of qp and contributions of amplitude fluctuations; in contrast, our definition is more precise for vortex since it involves only the onset effect of vortex signal.

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In OP Bi-2201, B_p is found to be more precise than what is determined from ρ, varying only $16 \%$ from RE and FT. The resulted uncertainty of B_p is around ±10% for all samples, as shown in table 2. This reliability of B_p is due to the clear existence of the TAFF regime in OV Bi-2212, OP and OV Bi-2201, as shown in figure 4. On the other hand, the data at low temperature in UD Bi-2201, UD and OP Bi-2212 are lacking, B_p in these samples requires further investigation.

B₀ in Bi-2212 is found to be bigger than in Bi-2201, which is an interesting feature to be explored. It implies that the damping force of impurity scattering is stronger in Bi-2212, for which a qualitative explanation is proposed as follows. As explained in section 2.3.1, damping contributed by impurity scattering can be expressed by resistivity of impurity scattering of the holes inside vortex core, i.e., ${\rho }_{a}$. Based on Drude model of ${\rho }_{a}$, one finds: ${B}_{0}\propto {H}_{{\rm{c}}2}{n}_{h}\tau$, where τ is the relaxation time. Since n_h and ${H}_{c2}$ are significantly bigger in Bi-2212 than in Bi-2201, the higher B₀ or damping in Bi-2212 is expected, although a quantitative understanding requires a more careful estimate of τ which is also sample-dependent.

The determination of b in various samples under different doping is made for the first time. In Bi-2201 and Bi-2212, b decreases monotonically from 0.672 to 0.310, and 0.490 to 0.181 respectively, with increasing doping. The uncertainties of b in UD Bi-2212, UD and OP Bi-2201 are around ±11%, for which the determination of b is reliable. On the other hand, for OV Bi-2201, OP and OV Bi-2212, the uncertainties of b are bigger (around $30 \%$), due to the fact that the signals above T_c are too small. However, since the result b = 0.186 ± 0.066 in OP Bi-2212 is very close to the results determined from magnetoresistance: 0.183 in Bi-2212 (${T}_{{\rm{c}}}=84.7$ K) [54], we consider that the estimates are reliable.

The identification of the dominant component of Nernst effect slightly above T_c is important to clarify the physics in pseudogap phase. In this section, we carry out a comparison between Gaussian theory and the present vortex-fluid model, and conclude that the Gaussian fluctuations can not be, but the vortex fluid is the dominant contribution to Nernst signal in Bi-based cuprates.

First, let us carry out an estimate of the thermoelectric coefficient ${\alpha }_{{xy}}^{{\rm{VF}}}={e}_{N}{c}_{0}/\rho$ in 2D system using our vortex Nernst model equation (14). Taking the same η as in magnetoresistance equation (13), we obtain

$$\begin{eqnarray}&&{\alpha }_{{xy}}^{{\rm{VF}}}=\displaystyle \frac{{C}_{1}{\rm{c}}}{{\phi }_{0}}\displaystyle \frac{{n}_{{\rm{s}}}}{T}\displaystyle \frac{B}{B+{B}_{T}}\mathrm{ln}\displaystyle \frac{{H}_{{\rm{c}}2}}{B}.\end{eqnarray} \tag{ 16 }$$

In addition, we make an assumption that n_s decays linearly from ${n}_{{\rm{s}}0}$ (at zero temperature) to zero (at T_v), i.e., ${n}_{{\rm{s}}}={n}_{{\rm{s}}0}(1-T/{T}_{v})$.

The comparison is made with the data (red open circles) at low field (B = 1 T) and above ${T}_{{\rm{c}}}=85$ K of (nearly) OP Bi-2212 measured by Ri et al [51]; figure 6 shows a precise agreement between the prediction (red solid line) of the present vortex-fluid model and data at $\mathrm{ln}(T/{T}_{{\rm{c}}})\leqslant 0.2$. Note that every parameter is determined independently: ${T}_{v}=106$ K is determined from the onset temperature of B dependence of magnetoresistance, and ${H}_{{\rm{c}}2}=68.1$ T is estimated from ${\phi }_{0}/2\pi {\xi }^{2}$ with the coherence length determined as in [5], ${n}_{{\rm{s}}0}={n}_{h}/2$ is determined with data of ${\alpha }_{{xy}}$ at T_c, and $b=0.5$ is determined from the T dependence above T_c. Note that an extra confirmation of the vortex-fluid model in Bi-2212 was obtained in Ginzburg–Landau–Lawrence–Doniach study of diamagnetism, where a mean field of order parameters was introduced to revise the discrepancy between Gaussian theory and data [59].

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On the other hand, in the limit of $B/{\tilde{H}}_{{\rm{c}}2}(0)\ll \mathrm{ln}(T/{T}_{{\rm{c}}})\ll 1$, for 2D superconductors, Gaussian-fluctuation theory predicts ${\alpha }_{{xy}}$ as [16]

$$\begin{eqnarray}&&{\alpha }_{{xy}}^{{\rm{GF}}}=\displaystyle \frac{2e}{3h}\displaystyle \frac{B}{{\tilde{H}}_{{\rm{c}}2}(0)}\displaystyle \frac{1}{{\rm{ln}}(T/{T}_{{\rm{c}}})}.\end{eqnarray} \tag{ 17 }$$

Previously, equation (17) was claimed to agree with data in NbSi (conventional superconductor) [16, 17], OV LSCO (hole doped) [8], UD Eu-LSCO (hole doped) [9] and PCCO (electron doped) [18]. We have made a quantitative comparison between the fitting parameter ${\tilde{H}}_{{\rm{c}}2}(0)$ from Gaussian-fluctuation theory and low temperature ${H}_{{\rm{c}}2}$ measured in those samples. It turns out that the ratio ${\tilde{H}}_{{\rm{c}}2}(0)/{H}_{{\rm{c}}2}$ is 1.8 for nearly OP (p = 0.15) PCCO, 1.9 for OV (p = 0.17) LSCO, 2.8 for NbSi, 0.25 for UD (p = 0.12) LSCO and 14 for UD (p = 0.11) Eu-LSCO. While the first three measured upper critical fields are slightly below the predicted one, the last two (UD LSCO and UD Eu-LSCO) shows a considerable departure.

This result can be interpreted as following. Since coherence length in conventional superconductors (e.g., NbSi) and electron-doped cuprate (e.g., PCCO) are large, signal above T_c in these samples is dominated by Gaussian fluctuations. However, signals due to Gaussian fluctuations must be smaller in hole-doped cuprates due to their smaller coherence lengths, and thus the contribution of phase fluctuations may not be neglected. Indeed, in OV LSCO, experimental signals are smaller, as consistent with Gaussian-fluctuation theory [8]. On the other hand, in UD samples, strong phase fluctuations result in a larger ${\alpha }_{{xy}}/B$. So, Gaussian-fluctuation theory generally underestimates the signal in UD samples (e.g., UD LSCO) [8]. Surprisingly, Gaussian-fluctuation theory overestimates the signal in Eu-LSCO, indicating that the theory predicting only the dependence on coherence length is too simplified, with some physics left over in strong stripe order [15]. In summary, in hole-doped cuprates, Gaussian-fluctuation theory is incapable to describe the UD samples, despite its successful application in OV samples.

Furthermore, we consider the application of equation (17) in Bi-cuprates, e.g. taking Bi-2212 specifically. In figure 6, the black dash line is predicted by Gaussian model with ${\tilde{H}}_{{\rm{c}}2}(0)=10.5$ T, which only describe the data in $0.03\ll \mathrm{ln}(T/{T}_{{\rm{c}}})\ll 0.085$, but significantly higher outside that range. Unfortunately, the fitting value (${\tilde{H}}_{{\rm{c}}2}(0)=10.5$ T) is far from being reasonable, since it is much smaller than the estimate from ${H}_{{\rm{c}}2}\approx {\phi }_{0}/2\pi {\xi }^{2}\,=68.1$ T with ξ measured with STM [60]. If the physical parameter ${\tilde{H}}_{{\rm{c}}2}(0)=68.1$ T is used, the Gaussian theory's prediction (see, the blue dash dot line in figure 6) is much lower than the data. Note that previously, the Gaussian-fluctuation theory of diamagnetism for (nearly) OP Bi-2212 also yields unreasonable fitting parameter (e.g., ${H}_{{\rm{c}}2}=330$ T) and irrational B dependence [61, 62]. We thus conclude that Gaussian fluctuation alone is unable to describe the physics in pseudogap state, at lest for Bi-based cuprates, for which the phase fluctuations are dominant.

Anderson pointed out that, in a vortex fluid, 'n_s is not the conventional macroscopic quantity giving the penetration depth, which is its average over phase fluctuations, but is defined microscopically...' [25]. Indeed, in the BKT scenario of 2D superconductor, thermal vortices emerge above T_c and results in a dramatic increase of the penetration depth [63] and the macroscopic quantity of ${n}_{{\rm{s}}}={m}_{e}{c}^{2}/4\pi {\lambda }^{2}{e}^{2}$ (see, open hexagon in figure 7) quickly decreases to zero. However, the local superfluid density which flows within interstitial puddles between vortex cores remains finite, which can induce a vortex Nernst effect. An advantage of the present vortex-fluid model is to allow one to determine the local superfluid density from Nernst signal, and then discuss its unconventional temperature dependence in pseudogap state.

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Equation (15) yields an expression for the local superfluid density ${n}_{{\rm{s}}}$:

$$\begin{eqnarray}&&{n}_{{\rm{s}}}=\displaystyle \frac{{\phi }_{0}{c}_{0}}{{C}_{1}c{\rho }_{0}}{{Te}}_{T},\end{eqnarray} \tag{ 18 }$$

where ${c}_{0}$ = 24.6 Å for Bi-2201, 30.9 Å for Bi-2212. In order to extract ${n}_{{\rm{s}}}$, ${\rho }_{0}$ should be estimated. Since ρ for the samples corresponding to the present Nernst data is not available in literature, ${\rho }_{0}$ needs to be estimated from ρ of other (similar) samples with the same T_cs. According to equation (13), ${\rho }_{0}$ is also the resistivity of holes in normal fluid. Based on the Drude model of normal metal, the conductivity is proportional to hole concentration p, thus

$$\begin{eqnarray}&&{\rho }_{0}\propto 1/p.\end{eqnarray} \tag{ 19 }$$

Hence, ${\rho }_{0}$ can be determined for each sample, once a value of ${\rho }_{0}$ at some doping is obtained. The results are shown in table 3, using the data ${\rho }_{0}\approx 0.11$ mΩ cm in OP (p = 0.16) ${\mathrm{Bi}}_{2.1}{\mathrm{Sr}}_{1.9}{\mathrm{CaCu}}_{2}{{\rm{O}}}_{8+\delta }$ [50] and ${\rho }_{0}\approx 0.13$ mΩ cm in the OV (p = 0.24) Bi-2201 [64].

Table 3.  Characteristic resistivity ${\rho }_{0}$ estimated with equation (19).

Sample	UD Bi-2201	OP Bi-2201	OV Bi-2201	UD Bi-2212	OP Bi-2212	OV Bi-2212
${\rho }_{0}$ (mΩ cm)	0.41	0.20	0.15	0.19	0.11	0.076

In this work, we find that local superfluid density indeed exhibits interesting behaviors. As shown in figure 7, ${n}_{{\rm{s}}}$ of OV Bi-2201 and OV Bi-2212 determined from equation (18) nearly collapse on a straight line ${n}_{{\rm{s}}}={n}_{{\rm{s}}0}(1-T/{T}_{v})$, where ${n}_{{\rm{s}}0}$ is a fitting parameter. Two distinct features different from the temperature dependence of macroscopic superfluid density (see, open hexagon in figure 7) are noteworthy: a finite n_s above T_c and a universal linear temperature dependence below T_v. The finite n_s above T_c and vanishing of n_s at T_v but not T_c is very indicative of the nature of pseudogap state: full of thermal vortices. This indicates that the thermal effects above T_c and below T_v are restricted within the vortex core, perhaps in the form of qp holes, while a local superfluid around the (thermal) vortex core still survives. Note that a linear T dependence of n_s at low temperature has been widely observed in HTSC [65] and can be described by the standard BCS theory of d-wave superconductor; but the presently observed linear behavior of local superfluid density above T_c is exotic, and requires further theoretical explanation. For more discussion, see section 3.2.4.

Strictly speaking, the concentration of oxygen deficiency in the CuO₂ plane are different in each sample so there exists a certain deviation of this estimated ${\rho }_{0}$, and hence of n_s [64]. However, the relative values of n_s with respect to temperature are considered to be reliable.

Upper critical field ${H}_{{\rm{c}}2}$ represents a crossover from a vortex fluid to the normal state due to overlap of magnetic vortex cores, and it is also a probe of the coherence length ξ because ${H}_{{\rm{c}}2}={\phi }_{0}/(2\pi {\xi }^{2})$. Since core–core collisions are enhanced with increasing magnetic field (see, equation (13)) in cuprate HTSC, magnetoresistance presents a 'knee' behavior, instead of following a Bardeen–Stephen law [5, 42, 53]. Therefore, one can no longer make an exact definition for ${H}_{{\rm{c}}2}$ from magnetoresistance as in conventional superconductor. Indeed, empirically determined '${H}_{{\rm{c}}2}$' from the 'knee' feature (see crosses in figure 8(a)) displays a sharply different behavior from that described by the conventional Werthamer–Helfand–Hohenberg (WHH) theory for BCS superconductors [66]. Recently, distinct progress has been achieved using Nernst signal [5, 55] and diamagnetism [67, 68], in which ${H}_{{\rm{c}}2}$ is defined as the field where e_N and M versus B extrapolates to zero. Our model enables one to make this determination in a wider range of T and B, since it may be considered as a more sophisticated version of the simple method by Wang et al, especially when the measured profile is imperfect.

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The T dependence of ${H}_{{\rm{c}}2}$ (see, section A.2) as shown in figure 8 displays a slow variation near T_c, consistent with the results of linear extrapolation of Nernst and diamagnetism versus B [5, 68]. For instance, at T = 5–40 K in OP Bi-2201 (or T = 70–100 K in OP Bi-2212), ${H}_{{\rm{c}}2}$ is presently in [43.3, 54.2] T (or [40.3, 81.8] T), as compared to [42, 52] T (or [43, 90] T) according to extrapolations of Nernst or diamagnetism [5, 68]. This behavior of finite ${H}_{{\rm{c}}2}$ above T_c is different from the canonical BCS behavior described by WHH theory, in which ${H}_{{\rm{c}}2}=0$ at T_c due to gap-closing. If ${H}_{{\rm{c}}2}=0$ above T_c in Bi-based cuprates, the strong and positive e_N above T_c in OP Bi-2212 must be interpreted by Gaussian fluctuations which is shown above (in section 3.1.3) to be unsuccessful, and the linear decaying of e_N in high-field in OP Bi-2201 cannot be explained by the logarithmic decay of Gaussian fluctuations. Furthermore, the finite ${H}_{{\rm{c}}2}$ $={\phi }_{0}/(2\pi {\xi }^{2})$ also implies finite core size or coherence length ξ, which has a weak T dependence near T_c. This is highly consistent with the Pippard coherence length ${\xi }_{{\rm{P}}}\propto 1/{{\rm{\Delta }}}_{0}$ determined from gap amplitude ${{\rm{\Delta }}}_{0}$ which is also observed to be finite and varies slowly above T_c in ARPES measurement [69]. In conclusion, the finite ${H}_{{\rm{c}}2}$ above T_c is physically sound in Bi-based cuprates, indicating that vortices with finite core size survive above T_c.

On the other hand, in figure 8(b), anomalous increase of ${H}_{{\rm{c}}2}$ with increasing temperature occurs above T_c in OV Bi-2201, which might be the evidence of the contribution of amplitude fluctuations. The signal above T_c is one order of magnitude lower than the maximum signal at low temperature, thus can be influenced by weak amplitude fluctuations tail [5], leading to an overestimate of ${H}_{{\rm{c}}2}$. In addition, for extremely UD samples, anomalous increase of ${H}_{{\rm{c}}2}$ is also present at T near and above T_c. A possible interpretation is exotic qp excitation due to density-wave order in this regime [12]. While the regular behaviors of ${H}_{{\rm{c}}2}$ in most regimes ensure the validity of the vortex-fluid model, the irregular behaviors of ${H}_{{\rm{c}}2}$ deduced with vortex fluid model provide a sensitive probe for additional contributions of qp or amplitude fluctuations. The present study thus serves a basis for developing a more comprehensive theory in the future.

Equation (3) is able to predict the temperature dependence of thermal vortex density ${n}_{T}$, as shown in figure 9(a). Note that ${n}_{T}$ in UD sample is much larger than in OP and OV samples, in both Bi-2201 and Bi-2212. We argue that the energy cost to excite a thermal vortex-antivortex pair is proportional to n_s. Thus, thermal vortex is easier to be excited in UD sample. Furthermore, it is interesting to observe a similar temperature dependence for each sample. In figure 9(b), normalized densities ${n}_{T}/{n}_{T}{({T}_{v})=\exp [-2b({T}_{v}/{T}_{{\rm{c}}}-1)}^{-1/2}({t}^{{\prime} -1/2}-1)]$ collapse into three groups, where ${n}_{T}({T}_{v})$ is the maximum density at T_v and ${t}^{{\prime} }=(T-{T}_{{\rm{c}}})/({T}_{v}-{T}_{{\rm{c}}})$. The difference is controlled by one material parameters ${S}_{{\rm{c}}}=b{({T}_{v}/{T}_{{\rm{c}}}-1)}^{-1/2}$ which may be treated as a normalized coherence strength to resist thermal vortex activation. For Bi-2212, ${n}_{T}/{n}_{T}({T}_{v})$ is nearly doping independent with S_c ranging from 0.42 to 0.47, indicating a universal normalized resistance of thermal vortex activation for different doping of Bi-2212. For Bi-2201, however, two behaviors of ${n}_{T}/{n}_{T}({T}_{v})$ appear, one in UD and OV sample with ${S}_{{\rm{c}}}\approx 0.31$, indicating that the normalized coherence strength in bilayer system is stronger than in monolayer bilayer system. But, an exception is found for OP Bi-2201 that S_c reaches a maximum (i.e., 0.55) with a strong decline of ${n}_{T}/{n}_{T}({T}_{v});$ this last observation requires further theoretical explanation.

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In the BKT scenario, long range phase coherence is destroyed by the proliferation of thermal vortices. Here, the quantification of the Nernst signal allows us to examine the dynamical properties of thermal vortices, and to propose a mechanism for the vanishing of short-range phase coherence, namely the lifetime of thermal vortices versus the cycle period of Cooper pairs around the core, as we explain now.

For a vortex, short-range phase coherence can be quantified by the cycle period of Cooper pairs around the core, i.e., ${\tau }_{{\rm{c}}}\sim 4\pi {m}_{e}{\xi }^{2}/{\rm{\hslash }}$. This coherence may be destroyed by the activation and annihilation of thermal vortices, whose characteristic time defines the average lifetime of thermal vortices (or coherence time) during which the local wavefunction becomes totally dephased. This lifetime can be estimated by ${\tau }_{v}={s}_{v}/{v}_{R}$, where s_v is the moving path length of the thermal vortex during its lifetime, determined by ${s}_{v}=\pi {\xi }_{+}$, the perimeter of a circle with diameter ${\xi }_{+}$, and v_R is the random velocity of the vortex. The latter can be obtained by an energy estimate [26] due to quantum fluctuations of holes inside vortex core, ${v}_{R}={\rm{\hslash }}/({m}_{e}\xi )$. It follows that ${\tau }_{v}=\pi {m}_{e}\xi {\xi }_{+}/{\rm{\hslash }}$, and the ratio of the two time scales is

$$\begin{eqnarray}&&\displaystyle \frac{{\tau }_{v}}{{\tau }_{{\rm{c}}}}=\sqrt{\displaystyle \frac{\pi {H}_{{\rm{c}}2}}{8{B}_{T}}}.\end{eqnarray} \tag{ 20 }$$

Now, we argue that vanishing of short-range phase coherence requires that ${\tau }_{v}/{\tau }_{{\rm{c}}}$ equal to 1 at the upper critical temperature T_v. Using the parameters near T_v, we can obtain ${\tau }_{v}/{\tau }_{{\rm{c}}}$ = 1.13 ± 0.07, 0.91 ± 0.10 and 0.98 ± 0.30 for three samples, OP Bi-2201, OP Bi-2212 and OV Bi-2212, respectively. For other samples, ${H}_{{\rm{c}}2}$ at T_v is either unavailable or is unreliable due to too big errors. Thus, we consider that available data confirm our proposal. Physically, the near unity is originated from the key fact that motions of thermal vortices and Cooper pairs near the core both depend on the same core size ξ.

Furthermore, we speculate that this understanding may also help in interpreting the temperature dependence of n_s. This is because that the lifetime for a vortex (magnetic or thermal) can be defined as ${\tau }_{v}$. When temperature increases, both ${\tau }_{v}$ and n_s decays. If we assumed that the decay of n_s is induced by the dephasing effect of vortex activation and annihilation, a simple relation, i.e., ${n}_{{\rm{s}}}/{n}_{{\rm{s}}}(0)=1-{\tau }_{{\rm{c}}}/{\tau }_{v}$ can be obtained, where ${n}_{{\rm{s}}}(0)$ is the superfluid density at zero temperature. At zero temperature, ${\tau }_{v}$ is infinite thus ${n}_{{\rm{s}}}={n}_{{\rm{s}}}(0)$. While at T_v, n_s vanishes. This interesting idea will be explored elsewhere in the future.

First, let us discuss the RE. In section 2.1, we assume that the high T limit of thermal vortex density is as dense as magnetic vortices at ${H}_{{\rm{c}}2}^{{\prime} }$, thus ${B}_{l}\approx {H}_{{\rm{c}}2}^{{\prime} }$. ${H}_{{\rm{c}}2}^{{\prime} }$ can be estimated as ${\phi }_{0}/(2\pi {\xi }^{2})$. Thus, ${B}_{l}\approx {\phi }_{0}/(2\pi {\xi }^{2})=68.1$ T as $\xi =22$ Å is measured by STM [60].

Near T_v, core–core collisions are dominant in η since the density of thermal vortices is dense, thus B₀ can be neglected. Then, ρ can be expressed as

$$\begin{eqnarray}&&\rho \approx {\rho }_{0}\exp \left[-(\displaystyle \frac{{T}_{v}}{T}-1)\sqrt{\displaystyle \frac{{B}_{{\rm{p}}}}{B+{B}_{T}}}\right],\end{eqnarray} \tag{ A1 }$$

which predicts that ρ increases with B for $T\lt {T}_{v}$ and converge to ${\rho }_{0}$ for T approaching T_v. We use the convergent regime of the experimental curves (see figure 3) to carry out a RE of T_v and ${\rho }_{0}$. The convergent regime corresponds to the range where the curves at the highest (17.5 T) and lowest (0) become close (or overlapped). In practice, we consider the overlap starts where the difference of the two curves equals the scatters of each curve, which is about 5 $\mu {\rm{\Omega }}$ cm. Applying this criteria, we obtain ${T}_{v}\in [104,120]$ K and ${\rho }_{0}\in [0.098,0.133]$ mΩ cm. The middle values, e.g., ${T}_{v}=112$ K and ${\rho }_{0}=0.116$ mΩ cm, are taken to be the roughly estimated values.

B_p is determined from the low T limit (i.e., the TAFF regime) of the resistivity profiles. At $T\ll {T}_{{\rm{c}}}$, pinning effect dominates and B₀ is neglected. We define $y=\mathrm{ln}({\rho }_{0}/\rho )$, $x={T}_{v}/T-1$, then a linear formula is obtained,

$$\begin{eqnarray}&&y={a}_{0}x+{b}_{0},\end{eqnarray} \tag{ A2 }$$

where ${a}_{0}=\sqrt{{B}_{{\rm{p}}}/B}$, and b₀ contains the contributions from B₀ and data error. A least square fitting of the line yields ${a}_{0}\approx 1.85$ so that ${B}_{{\rm{p}}}\approx 30.8$ T, as shown in figure 10.

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B₀ is supposed to dominate in low fields where vortices flow individually in materials with weak pinning. However, for ${\mathrm{Bi}}_{2.1}{\mathrm{Sr}}_{1.9}{\mathrm{CaCu}}_{2}{{\rm{O}}}_{8+\delta }$ [50], the melting field ${H}_{{\rm{m}}}\geqslant 17.5$ T indicates a significant pinning effect, thus B₀ is nearly invisible at low temperature. The contribution of core–core collisions and pinning effect is presented by f,

$$\begin{eqnarray}&&f=B\exp [({T}_{v}/T-1)\sqrt{{B}_{{\rm{p}}}/B}].\end{eqnarray} \tag{ A3 }$$

To suppress this contribution and avoid the effect of thermal vortices, we propose to determine B₀ at T_c and near H', where H' is the minimum of f at T_c. At T_c, the equation (13) is simplified to

$$\begin{eqnarray}&&\rho ={\rho }_{0}\displaystyle \frac{B}{{B}_{0}+f}.\end{eqnarray} \tag{ A4 }$$

Since T_v and B_p are determined, the minimum of f is found to be ${f}_{{\rm{\min }}}\,=\,{({e}^{2}/4)({T}_{v}/{T}_{{\rm{c}}}-1)}^{2}{B}_{{\rm{p}}}$ = 4.7 T at $H^{\prime} =0.64$ T. According to calculation, f is approximately a constant near $H^{\prime}$, thus the data at $H=0.5{\rm{T}}$ ($\rho =10.9$ $\mu {\rm{\Omega }}$ cm) and 1 T ($\rho =16.6$ $\mu {\rm{\Omega }}$ cm) are used in the linear fitting. Finally, ${B}_{0}\approx 4$ T is obtained.

b is determined near and above T_c with Kosterlitz model [37]. At $T\gt {T}_{{\rm{c}}}$, we have the following relation:

$$\begin{eqnarray}&&{B}_{0}=({B}_{i}+{B}_{T})\left\{\displaystyle \frac{{\rho }_{0}}{{\rho }_{i}}-\exp \left[t\sqrt{\displaystyle \frac{{B}_{{\rm{p}}}}{{B}_{i}+{B}_{T}}}\right]\right\},\end{eqnarray} \tag{ A5 }$$

where i = a, b. Equaling the case a and b, we obtain B_T. B_T calculated from ${B}_{a}=0.5$ T and ${B}_{b}=9$ T in the range $T\in [90,100]$ K is shown in figure 11. According to equation (3), the Kosterlitz coefficient $b=0.42$ is fitted from the data at $T\in [90,94]$ K.

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In this step, we vary the five parameters (${\rho }_{0}$, T_v, B₀, B_p and b) around their roughly estimated values to get a precise fitting of the experimental curves, while keeping ${B}_{l}=68.1$ T unchanged, following the argument in the beginning of this section. This fine tuning process needs to set up a ranges for each parameter. First, the result of RE already yields ${T}_{v}\in [104,120]$ K and ${\rho }_{0}\in [0.098,0.133]$ mΩ cm. Secondly, B₀, B_p and b are set to be positive, as it is evident from their physical meaning. Then, we treat the empirical data of five curves at H = 0, 0.5, 3, 9 and 17.5 T for $T\leqslant 104$ K as an ensemble, and evaluate the RMSE of the predictions from the data, when the five parameters vary in the range specified. Finally, the smallest RMSE ≈ 3.5 $\mu {\rm{\Omega }}$ cm is reached and the fitting is shown in figure 3.

According to the procedure introduced in the beginning of this section, the error bar can be determined from the uncertainty of the original data [50], which approaches ±2.5 $\mu {\rm{\Omega }}$ cm. We thus use 5 $\mu {\rm{\Omega }}$ cm as the upper boundary of the RMSE function. The error bar for any one parameter is calculated when the others are set to their FT-determined values, and the results are shown in table 1.

In RE, the asymptotic analysis at high and low limits of T and B is also the key strategy. T_v can be determined from the zero point of the linear extrapolation of the peak at very high temperature. Using similar linear extrapolation in high fields, ${H}_{{\rm{c}}2}$ is obtained. ${e}_{T}$ and B_p can be determined from field dependence of the signal in the TAFF regime. The nearly linear field dependence of e_N near the minimum of f below and near T_c can be used to determine B₀. Meanwhile, b can be determined from the peaks of the profiles above T_c.

On the profile of ${e}_{N}$ versus B, when $T\to {T}_{v}$,

$$\begin{eqnarray}&&{e}_{N}\to {e}_{T}\displaystyle \frac{B}{{B}_{0}+B+{B}_{T}}\mathrm{ln}\left(\displaystyle \frac{{H}_{{\rm{c}}2}}{B}\right).\end{eqnarray} \tag{ A6 }$$

Since n_s approaches 0 at T_v, the peak ${e}_{N}^{m}$ of the profile approaches zero. Since ${e}_{N}^{m}$ versus T is a nonlinear curve at large T range, only the data (50 and 65 K) nearest to zero are used in the linear extrapolation, shown in figure 12(a). Finally, ${T}_{v}=71.5$ K is obtained. The uncertainty of the data is estimated to be ±100 nV K⁻¹ (discussed in the end of this section), and the error bar of T_v is found to be ±6.2, thus ${T}_{v}=71.5\pm 6.2$ K.

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When $B\to {H}_{{\rm{c}}2}$, a linear dependence of B can be obtained from equation (14),

$$\begin{eqnarray}&&{e}_{N}\approx {e}_{T}\displaystyle \frac{{H}_{{c}{2}}-{B}}{{B}_{0}+({H}_{{\rm{c}}2}+{B}_{T})\exp [t\sqrt{{B}_{{\rm{p}}}/({H}_{{\rm{c}}2}+{B}_{T})}]}.\end{eqnarray} \tag{ A7 }$$

${H}_{{\rm{c}}2}$ represents the zero point of the signal. In figure 12(b), a linear extrapolation of the high fields data, applied to the curve at 20 K, yields ${H}_{{\rm{c}}2}=45\pm 0.7$ T. Results of ${H}_{{\rm{c}}2}$ at other temperatures are shown in table 4. Since ${B}_{l}\approx {H}_{{\rm{c}}2}^{{\prime} }$ (T_c = 28 K), ${B}_{l}$ can be estimated as ${H}_{{\rm{c}}2}$ (30 K) = 48 T.

Table 4.  ${H}_{{\rm{c}}2}$ determined from linear extrapolation of e_N.

T (K)	5	11	20	30	40
${H}_{{\rm{c}}2}$ (T)	50.0 ± 3.3	45.0 ± 1	45.0 ± 0.7	48.0 ± 1.3	50.0 ± 3.6

At low temperature and fields (i.e., the TAFF regime), pinning effect is dominant in damping viscosity. Thus, B₀ can be neglected in e_N,

$$\begin{eqnarray}&&{e}_{N}\approx {e}_{T}\mathrm{ln}\left(\displaystyle \frac{{H}_{{\rm{c}}2}}{B}\right)\exp \left[-\left(\displaystyle \frac{{T}_{v}}{T}-1\right)\sqrt{\displaystyle \frac{{B}_{{\rm{p}}}}{B}}\right].\end{eqnarray} \tag{ A8 }$$

Taking the logarithm of the two sides of e_N in equation (A8), we can obtain

$$\begin{eqnarray}&&y={a}_{1}x+{b}_{1},\end{eqnarray} \tag{ A9 }$$

where $y={\rm{ln}}[{\rm{ln}}({H}_{{\rm{c}}2}/B)]-{\rm{ln}}{e}_{N}$, ${a}_{1}=({T}_{v}/T-1)\sqrt{{B}_{{\rm{p}}}}$, ${b}_{1}=-\mathrm{ln}{e}_{T}$, $x={B}^{-1/2}$. The linear fitting of the data allows us to determine the parameters,

$$\begin{eqnarray}{e}_{T} & = & \exp (-{b}_{1}),\\ {B}_{{\rm{p}}} & = & {a}_{1}^{2}/{({T}_{v}/T-1)}^{2}.\end{eqnarray} \tag{ A10 }$$

The fitting at T = 20 K is shown in figure 13, and ${B}_{{\rm{p}}}=2.5$ T, ${e}_{T}=11.0$ μV K⁻¹ are obtained. Data in very low fields are not used because the signals are very weak thus the relative error are big.

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A nearly linear equation can be obtained from the model equation (14) of the Nernst effect at the minimum of f and below T_c,

$$\begin{eqnarray}&&g=\displaystyle \frac{B}{{B}_{0}+f},\end{eqnarray} \tag{ A11 }$$

where $g={e}_{N}/[{e}_{T}\mathrm{ln}({H}_{{\rm{c}}2}/B)]$. Similar to equation (A4), the data in low fields and high temperature should be used to determine B₀. In figure 14(a), g versus B at T = 20 K shows a linear regime in low fields (from 1 to 8 T), which supports a linear fit ($g=0.032\ast H$). But the minimum of f is so big (31.1 T) that B₀ is found to be very small (0.1 T). In addition, the error of the slope yields that B₀ varies in ±1 T. Thus, B₀ can be reasonably chosen as 0.

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Since ${B}_{l}\approx 48$ T is estimated, b can be determined from the peak value ${e}_{N}^{m}$ above T_c, where

$$\begin{eqnarray}&&{e}_{N}^{m}={e}_{T}\displaystyle \frac{{B}^{m}\mathrm{ln}({H}_{{\rm{c}}2}/{B}^{m})}{{B}_{0}+({B}^{m}+{B}_{T})\exp \phantom{\rule[-0000]{}{3ex}}(t\sqrt{\tfrac{{B}_{{\rm{p}}}}{{B}^{m}+{B}_{T}}}\phantom{\rule[-0000]{}{3ex}})}.\end{eqnarray} \tag{ A12 }$$

B_T can be calculated from this formula directly. According to Korsterlitz model equation (3), a linear model can be obtained as

$$\begin{eqnarray}&&y={a}_{2}x+{b}_{2},\end{eqnarray} \tag{ A13 }$$

where $y=\mathrm{ln}({B}_{l}/{B}_{T})$, ${a}_{2}=2b$, $x={(T/{T}_{{\rm{c}}}-1)}^{-1/2}$ and b₂ contains the errors. b can be determined from the linear fit of equation (A13). In figure 14(b), the data near T_c (30 and 35 K) are used to conduct the linear fitting and $b=0.27$ is obtained. The results of RE are shown in table 2.

Since both e_T and ${H}_{{\rm{c}}2}$ in equation (14) are T-dependent, the FT procedure of the Nernst effect is a little more complicated, which requires to run an iteration program. First, the parameters are divided into two groups. One group is made up of intrinsic parameters B₀, B_p, B_l, b and T_v. The other contains the T-dependent parameters ${e}_{T}$ and ${H}_{{\rm{c}}2}$. The RE helps to define the ranges of the fitting parameters, e.g., ${T}_{v}\in [65.3,77.7]$ K, ${H}_{{\rm{c}}2}\in [44,53.6]$ T, ${B}_{l}\in [46.7,49.3]$ T and ${B}_{0}=0$. Besides, ${B}_{{\rm{p}}}\gt 0$, $b\gt 0$ and ${e}_{T}\gt 0$ are set to be constraints. Second, we treat the empirical data of five curves at T = 5, 11, 20, 30 and 40 K for $H\leqslant 30$ T as an ensemble. Using the iteration program, the FT can be conducted, one group after another, with the former results substituted into the later calculation. This iterative process runs until a local minimum (RMSE ≈ 0.0993 μV K⁻¹) in the parameter space specified as above is reached.

In the Nernst experiment [5], the error of the voltage measured by nanovoltmeter is ±5 nV, thus the uncertainty of the Nernst signal $\delta {e}_{N}=\delta E/{\rm{\nabla }}T$ is estimated to be ±25 nV K⁻¹. By private communication, Y. Wang indicates that the noise in strong fields should be larger, thus $\delta {e}_{N}$ is larger than ±50 nVK⁻¹ (the maximum value may be ±0.1 μV K⁻¹ at 45 T), consistent with the typical fluctuations of the data. In the present work, only data below 30 T are used. We choose a rational fitting error of ±100 nV K⁻¹ as the upper boundary of the error function for the samples except for UD Bi-2201. For UD Bi-2201, the local minimum error is 0.152 μV K⁻¹, thus we set 0.2 μV K⁻¹ as the upper boundary, to reveal the sensitivity of the model. The procedure to determine the error bar is similar to that above for ρ, except B_l is assumed as equals to ${H}_{{\rm{c}}2}$ at T_c (with error bar). The final results of FT are shown in table 2 and the multi-curves fitting figure is shown in figure 4(b).