---
abstract: 'The magnetization of three high-quality single crystals of YBa$_{2}$Cu$_{3}$O$_{6+x}$, from slightly overdoped to heavily underdoped, has been measured using torque magnetometry. Striking effects in the angular dependence of the torque for the two underdoped crystals, a few degrees above the superconducting transition temperature ($T_c$) are described well by the theory of Gaussian superconducting fluctuations using a single adjustable parameter. The data at higher temperatures ($T$) are consistent with a strong cut-off in the fluctuations for $T\gtrsim1.1T_c$. Numerical estimates suggest that inelastic scattering could be responsible for this cut-off.'
author:
- 'I. Kokanović$^{1,2}$, D. J. Hills$^{1}$, M. L. Sutherland$^{1}$, R. Liang$^{3}$ and J. R. Cooper$^1$'
title: 'Diamagnetism of YBa$_{2}$Cu$_{3}$O$_{6+x}$ crystals above $T_c$ : evidence for Gaussian fluctuations '
---

Cuprate superconductors show much stronger thermodynamic fluctuations than classical ones because of their higher transition temperatures ($T_c$), shorter Ginzburg-Landau (GL) coherence lengths and quasi-two dimensional layered structures with weakly interacting CuO$_2$ planes [@Bulaevskii; @Larkin]. Observations of diamagnetism [@LuLi] and large Nernst coefficients over a broad temperature ($T$) range well above $T_c$ for several types of cuprate [@Xu; @Wang06] are intriguing [@Kivelson]. They are often cited as evidence for pre-formed Cooper pairs without the long-range phase coherence needed for superconductivity. In contrast, in Ref.  it is argued that phase and amplitude fluctuations set in simultaneously. However the fluctuations are still considered to be strong in that the mean-field transition temperature $T_c^{MF}$, obtained by applying entropy and free energy balance considerations to heat capacity data, is substantially larger than $T_c$ especially for underdoped cuprates. In standard GL theory the coefficient of the $|\psi|^2$ term in the free energy, where $\psi$ is the order parameter, changes sign at $T_c^{MF_1}$, as explained in footnote . If $|\psi|^4$ and higher order terms are neglected, $T_c^{MF_1}$ can be obtained from a Gaussian fluctuation (GF) analysis of the magnetic susceptibility and other physical properties [@Bulaevskii].

One difficulty in this area is separating the fluctuation (FL) contribution to a given property from the normal state (N) background. Recently this has been dealt with for the in-plane electrical conductivity $\sigma_{ab}(T)$ of YBa$_{2}$Cu$_{3}$O$_{6+x}$ crystals by applying very high magnetic fields ($B$) [@Alloul]. When analyzed using GF theory, $\sigma_{ab}^{FL}(T)$ was found to cut off even more rapidly above $T\gtrsim1.1T_c$ than previously thought [@Genova; @Vidal]. It was also strongly reduced at high $B$ and the fields needed to suppress $\sigma_{ab}^{FL}(T)$ extrapolated to zero between 120 and 140 K depending on $x$, which tends to support a vortex or Kosterlitz-Thouless scenario. Therefore questions such as the applicability of GF theory $vs.$ a phase fluctuation or mobile vortex scenario and the extent to which $T_c$ is suppressed below $T_c^{MF_1}$ by strong critical fluctuations, are still being discussed. They are of general interest because superconducting fluctuations could limit the maximum $T_c$ that can be obtained in a given class of material [@Tallon2011], and moreover [@Alloul] the fluctuation cut-off could be linked in some way to the pairing mechanism.

Here we report torque magnetometry data measured [@exptldetails] from $T_c$ to 300 K for tiny YBa$_{2}$Cu$_{3}$O$_{6+x}$ (YBCO) single crystals from overdoped (OD) to heavily underdoped (UD). These were grown in non-reactive BaZrO$_3$ crucibles from high-purity (5N) starting materials. Evidence for the quality of the UD crystals includes extremely sharp x-ray peaks [@Liang2000], and substantial mean free paths from quantum oscillation measurements [@Audouard2009]. The OD89 crystal is from another preparation batch which had narrow superconducting transitions and a maximum $T_c$ of 93.8 K [@Kirby2005]. We analyze the results using GF theory which, unlike some other approaches, predicts the *magnitude* of the observed effects as well as their $T$-dependence. We show that it gives excellent single-parameter fits to the striking angular dependence of the torque, which has previously been attributed to the presence of a very large magnetic field scale [@LuLi]. We also show that inelastic scattering is a plausible mechanism for cutting off the fluctuations at higher $T$ and a possible alternative to strong fluctuations for limiting $T_c$.

Although measurements of the London penetration depth [@Kamal] below $T_c$ and thermal expansion [@Meingast] above and below $T_c$ for optimally doped (OP) YBCO crystals, give evidence for critical fluctuations described by the 3D-XY model, up to $\pm$ 10 K from $T_c$, we argue later that these do not alter our overall picture.

A crystal with magnetization $M$ in an applied magnetic field $B$ attached to a piezoresistive cantilever causes a change in electrical resistance proportional to the torque density $\tau\equiv\underline{M}\times\underline{B}$. If $B$ is parallel to the $c$-axis of a cuprate crystal, then in the low field limit the contribution to $M$ in the $c$-axis direction from Gaussian fluctuations ($M_c^{FL}$) is given by [@Larkin]: $${M_c^{FL}(T)=-\frac{\pi k_BTB}{3 \Phi_0^2}\frac{
\xi_{ab}^{2}(T)}{s\sqrt{1+[2\xi_{ab}(T)/\gamma s)]^2}}} \label{1}$$ Here $\gamma = \xi_{ab}(T)/\xi_{c}(T)$ is the anisotropy, defined as the ratio of the $T$-dependent coherence lengths $\parallel$ and $\bot$ to the layers, i.e. $\xi_{ab,c}(T) = \xi_{ab,c}(0)/\epsilon^{1/2}$ with $\epsilon=\ln(T/T_c^{MF_1})$ [@Larkin; @Alloul]. The distance between the CuO$_2$ bi-layers is taken as $s$ = 1.17 nm, and $ \Phi_0$ and $k_B$ are the pair flux quantum and Boltzmann’s constant respectively. For $B\perp c$ the fluctuation magnetization is negligibly small.

As the angle $\theta$ between the applied field and CuO$_2$ planes is altered, $\tau(\theta)$ will vary as $\tau(\theta)= \frac{1}{2}\chi_D(T)B^2\sin2\theta$, as long as $M\propto B$. Thus, fits to $\tau(\theta) \propto B^2\sin2\theta$ give $\chi_D(T)\equiv\chi_c(T)-\chi_{ab}(T)$, which is the susceptibility anisotropy. Fig. 1 shows torque data for UD57 up to 15 K above the low-field $T_c$ of 57 K. Much of our data, including the two curves for UD57 in Fig. 1 at higher $T$ follow a $\sin2\theta$ dependence very closely, however there are striking deviations at lower $T$ arising from non-linearity in $M(B)$ that we discuss later.

![ Color online. Angular dependence of the torque density for the UD57 YBa$_{2}$Cu$_{3}$O$_{6.5}$ crystal in 10 T at $T$= 58.1, 60.3, 61.5, 66.9 and 72.2 K after correcting for a fixed instrumental offset of 10$^\circ$ and subtracting the gravitational term [@exptldetails]. The solid lines show single parameter fits to the formula for 2D GF derived from Eq. 2 plus $\chi_D^N(T)$ shown in Fig. 2a. Note the $\sin2\theta$ behavior at higher $T$. []{data-label="rawdata1"}](Fig1mod.eps){width="7.0cm"}

Fig. 2a shows $\chi_D(T)$ obtained from $\sin2\theta$ fits for three doping levels at high enough $T$ so that $M$ remains $\propto B$. The solid lines for OD89 and UD57 are fits up to 300 K that include $\chi_c^{FL}(T)$ from Eq. 1, with the strong cut-off described below, plus the normal state background anisotropy $\chi_D^N(T)$ which arises from the $g$-factor anisotropy of the Pauli paramagnetism [@KokanovicEPL]. For UD crystals the $T$-dependence of $\chi_D^N(T)$ is caused by the pseudogap, see footnote , plus a smaller contribution from the electron pocket [@KokanovicEPL] observed in high field quantum oscillation studies [@TailleferReview]. We used the same pseudogap energies ($k_BT^*$) and other parameters defining $\chi_D^N(T)$ as in our recent work on larger single crystals [@KokanovicEPL], e.g. $T^*$ = 435 K for UD57. OD89 has no pseudogap and presumably no pockets, so we represent the weak variation of $\chi_D^{N}(T)$ with $T$ by the second order polynomial shown in Fig. 2a.

![Color online: (a) Main: $\chi_D(T)$ for the three crystals, solid lines show fits to $\chi_c^{FL}(T)+\chi_D^N(T)$ for OD89 and UD57, dashed lines show $\chi_D^N(T)$. Insert: Symbols show $M$ calculated for various values of $\epsilon$, using Eq. 2, when the anisotropy parameter $r\equiv(2\xi_c(0)/s)^2$ = 0. For $r$= 0.13 symbols show $M$ given by the 2D-3D form of Eq. 2, which contains $r$ and an extra integral [@Larkin]. The lines show formulae used [@insertnote] to represent these values of $M$ when fitting $\tau(\theta)$ data.\
(b) to (d) - plots of $1/|\chi_c^{FL}(T)|$ vs. $T$ for the three crystals. GF fits based on Eq. 1, are shown by short dashed lines, without a cut-off and by solid lines, with a strong cut-off [@cutoffnote]. Red triangles for UD57 show $\xi_{ab}(0)^2/\epsilon$ obtained by fitting $\tau(\theta)$ to the full 2D GF formula when $M(B)$ is non-linear, and converted to $1/|\chi_c^{FL}(T)|$ using Eq. 1. For UD22 the full GF formula was used for all the points shown in Fig. 2b.[]{data-label="chidata"}](Fig2amod2.eps "fig:"){width="7.5cm"} ![Color online: (a) Main: $\chi_D(T)$ for the three crystals, solid lines show fits to $\chi_c^{FL}(T)+\chi_D^N(T)$ for OD89 and UD57, dashed lines show $\chi_D^N(T)$. Insert: Symbols show $M$ calculated for various values of $\epsilon$, using Eq. 2, when the anisotropy parameter $r\equiv(2\xi_c(0)/s)^2$ = 0. For $r$= 0.13 symbols show $M$ given by the 2D-3D form of Eq. 2, which contains $r$ and an extra integral [@Larkin]. The lines show formulae used [@insertnote] to represent these values of $M$ when fitting $\tau(\theta)$ data.\
(b) to (d) - plots of $1/|\chi_c^{FL}(T)|$ vs. $T$ for the three crystals. GF fits based on Eq. 1, are shown by short dashed lines, without a cut-off and by solid lines, with a strong cut-off [@cutoffnote]. Red triangles for UD57 show $\xi_{ab}(0)^2/\epsilon$ obtained by fitting $\tau(\theta)$ to the full 2D GF formula when $M(B)$ is non-linear, and converted to $1/|\chi_c^{FL}(T)|$ using Eq. 1. For UD22 the full GF formula was used for all the points shown in Fig. 2b.[]{data-label="chidata"}](Fig2bdJuly.eps "fig:"){width="7.0cm"}

Figs. 2b to 2d show plots of $1/|\chi_c^{FL}(T)|$ vs. $T$ where $\chi_c^{FL}(T) \equiv \chi_D(T)- \chi_D^{N}(T)$. The short-dashed lines for UD22 and UD57 in Figs. 2b and 2c show the contribution from Eq. 1 in the 2D limit ($\gamma
\rightarrow\infty$) with the two adjustable parameters $T_c^{MF_1}$ and $\xi_{ab}(0)$ given in Table 1. The solid lines show the effect of the same type of cut-off used in previous studies of the the conductivity $\sigma_{ab}^{FL}(T,B)$, as summarized in footnote . For OD89 we use the full 2D-3D form of Eq. 1 with $\xi_{ab}(0)$ = 1.06 nm and $\gamma$ = 5, [@Babic] shown by the short-dashed line, with the solid line again including the cut-off [@cutoffnote]. The high quality of these fits could be somewhat fortuitous in view of our neglect of any charge density wave (CDW) [@CDWnote], but other subtraction procedures give similar values of $1/|\chi_c^{FL}(T)|$. Heat capacity studies give a very similar value $\xi_{ab}(0)$ = 1.12 nm for OD88 YBCO [@LoramPhilMag] while our values for UD57 and UD22 agree with previous work [@Alloul; @AndoHc2] for the same $T_c$ values. For UD57, setting $\gamma = 45$ [@Pereg], rather than the 2D limit of Eq. 1 ($
\gamma\rightarrow\infty$) has no significant effect.

As the critical region is approached from above $T_c$ the exponent of $\xi_{ab}(T)$ is expected to change from the MF value of -1/2 to the 3D-XY value of -2/3 [@Bulaevskii]. It is very likely that this will also apply to strongly 2D materials, including UD57, since heat capacity data above and below $T_c$ [@LoramLnt] do show the $\ln|\epsilon|$ terms associated with the 3D-XY model. We have addressed this by repeating our GF fits in Figs. 2b and 2c with $\epsilon\geq0.20$ (UD22) or $0.15$ (UD57) without altering the cut-off [@cutoffnote]. The only significant change is that $\xi_{ab}(0)$ becomes 15$\%$ larger for UD57. For OD89 fits with $T_c^{MF_1}$ = 90 K and $\epsilon\geq0.05$ do not alter $\xi_{ab}(0)$ within the quoted error. This is expected since the width of the critical region for OD89 is much smaller than for OP YBCO [@Kamal; @Meingast] because of the extra 3D coupling from the highly conducting CuO chains [@LoramPhilMag].

![ Color online: Magnetic field dependence of the magnetization obtained from the torque data for UD57 at $T$ = 58.1, 60.3, 61.5, 66.9 and 72.2 K. Solid lines show fits to the 2D GF formula for $M$ plus the same normal state contribution used in Figs. 1, 2a and 2c.[]{data-label="magdata"}](fig3){width="7.0cm"}

Fig. 3 shows plots of $\tau/B\cos\theta$ vs $B\sin\theta$ at fixed $T$ for UD57. We use this representation of the data and MKS units, A/m, for comparison with Ref. . If $\chi_D^{N}(T)$ is subtracted, which has not been done for Fig. 3, then since $M_{ab}^{FL}$ is small this would be the same as plotting $M_c^{FL}$ vs. $B\parallel c$. Near $T_c$ there is clear non-linearity which is remarkably consistent with GF in the 2D limit, for which the free energy density at all $B$ is [@Larkin]: $${F = \frac{k_BT}{2\pi\xi_{ab}^2s}\{b\ln[\Gamma(\frac{1}{2}+
\frac{\epsilon}{2b})/\sqrt{2\pi}]+ \frac{\epsilon}{2}\ln(b)\} } \label{2DFree}$$ using the standard $\Gamma$ function, with $b= B/\tilde{B}_{c2}(0)$, where $\tilde{B}_{c2}(0)=\Phi_0/2\pi\xi_{ab}(0)^2$, and as before $\epsilon=\ln(T/T_c^{MF_1}(B=0))$. The magnetization $M=-\partial F/\partial B$ obtained by numerical differentiation of Eq. 2 for three typical values of $\epsilon$ is shown in the insert to Fig. 2a. $M$ scales with $b/\epsilon$ to within a few $\%$ and for $0.01<\epsilon<1$ can be adequately represented by the simple formula $-bk_BT/[\Phi_0s(3b+6\epsilon)]$, that has a single unknown parameter $\xi_{ab}(0)^2/\epsilon$. We note that GF formulae will be approximately valid in the crossover region to 3D-XY behavior [@Bulaevskii], because to first order the main effect is the change in the exponent of $\xi_{ab}(T)$.

Figs. 1 and 3 show that this formula fits our data for UD57 very well and importantly, as shown by the red triangles in Fig. 2c, the corresponding values of $1/\chi_c^{FL}(T)$ obtained via Eq. 1 agree well with points from $\sin2\theta$ fits at lower $B$ or higher $T$. For OD89 strong deviations from $\sin2\theta$ behavior only occur within $\sim$ 1 K of $T_c$ and these [@OD89note] are not properly described by GF theory. For UD22 there were small jumps in $\tau(\theta)$ at $\theta =0$ between 35 and 26 K of size $M_c =
0.01-0.03k_BT/(3\Phi_0s)$ that were fitted by including an extra contribution from Eq. 2 in the $\epsilon\ll b$ limit. This is ascribed to small regions, 1 to 3$\%$ of the total volume, with higher $T_c$ [@Lascialfari] that are not detected in low-field measurements of $T_c$ because they are much smaller than the London penetration depth. Fig. 2b shows that the values of $\xi_{ab}(0)^2/\epsilon$ \[or equivalently $1/\chi_c^{FL}(T)$\] obtained from full GF fits to $\tau(\theta)$ data at 2, 5 and 10 T agree well, which supports this conclusion.

The good description of our data by this GF analysis suggests that the high critical fields proposed in Refs.  for $0.01<\epsilon\lesssim 0.2$ are associated with vortex-like excitations. In the present picture 2D GF give $M_c^{FL} \simeq -0.33 k_BT/\Phi_0s = -0.112$ emu/cm$^3$ or -112 A/m at 60 K for $B\gtrsim \phi_0/[2\pi\xi_{ab}(T)^2]$. We expect this to be suppressed for $B\gtrsim B_{c2}(0)$ where the magnetic length becomes smaller than $\xi_{ab}(0)$ and the slow spatial variation approximation of GL theory breaks down. However it may also fall when $\epsilon\gtrsim0.1$ because of the GF cut-off discussed below. So in the first approximation the high fields are $\simeq
 B_{c2}(0)$. Precise analysis of these effects at very high fields might need to allow for small changes in $\chi_D^N(T)$ with $B$ that depend on the ratio of the Zeeman energy to the pseudogap. We note that the present results are consistent with a recent study of $B_{c2}$ for YBCO [@Taillefer2] and that recent torque magnetometry data [@Barisic2012] for HgBa$_2$CuO$_{4+x}$ and other single layer cuprates, show similar exponential attenuation factors to those for YBCO [@Alloul; @cutoffnote].

An intriguing question about the present results and those of Ref.  is the origin of the strong cut-off in the GF above $\sim1.1T_c$. If the weakly $T$-dependent $\chi^N_D(T)$ behavior for OD89 shown in Fig. 2a is correct then our $\chi^{FL}_D(T)$ data and $\sigma_{ab}^{FL}(T)$ [@Alloul] both decay as $\exp[-(T-1.08T_c)/T_0]$ above $T\sim1.08T_c$ with $T_0\sim9$ K. If instead $\chi^N_D(T)$ were constant below 200 K then our $\chi^{FL}_D(T)$ data would give $T_0\sim$25 K, a slower decay than Ref. . In either case the presence of this cut-off for OD YBCO rules out explanations connected with the mean distance between carriers. This is much less than $\xi_{ab}(0)$ for hole concentrations of $\simeq1.2$ per CuO$_2$ unit, the value found directly from quantum oscillation studies of OD Tl$_2$Ba$_2$CuO$_{6+x}$ crystals [@Rourke].

Assuming there are no unsuspected effects caused by $d$-wave pairing, one hypothesis is that the GF and possibly $T_c$ itself are suppressed by inelastic scattering processes. In a quasi-2D Fermi liquid the inelastic mean free path, $l_{in}$, can be found from the $T$-dependence of the electrical resistivity and the circumference of the Fermi surface. For OD YBCO the measured $a$-axis resistivity [@AndoHc2] gives $l_{in}$ = $2.5(100/T)$ nm, but values for UD samples are less certain because of the pseudogap. The BCS relation $\xi_{ab}(0)= \hbar v_F/\pi\Delta(0)$, where $\Delta(0)$ is the superconducting energy gap at $T=0$, implies that irrespective of the value of the Fermi velocity $v_F$, the usual pair-breaking condition for significant inelastic scattering, $\hbar/\tau_{in} \gtrsim \Delta(0)$ is equivalent to $l_{in}\lesssim\pi\xi_{ab}(0)$. Taking $\xi_{ab}(0)$ from Table I and the above value of $l_{in}$ shows that this is satisfied at 100 K for OD YBCO. So some suppression of GF and indeed $T_c$ by inelastic scattering is entirely plausible. If $T_c$ is suppressed then $\Delta(T)$ will fall more quickly than BCS theory as $T_c$ is approached from below, which would affect the analysis of Ref. .

Another possibility [@Alloul] which might account for the observations, is that the pairing strength itself falls sharply outside the GL region, for example when the in-plane coherence length becomes comparable to, or less than, the correlation length of spin fluctuations. From Figs. 2b to 2d we can read off the values of $T$ where the solid and dashed lines differ by (say) a factor of two. At these points $\xi_{ab}(T)\equiv\xi_{ab}(0)/\ln(T/T_c^{MF_1})$ = 15.6, 9.5 and 7.9 nm for UD22, UD57 and OD89 respectively. Neutron scattering studies [@Hayden; @Stock] typically give a full width half maximum of 0.17$\frac{2\pi}{a}$ for the scattering intensity from spin fluctuations. Although this does vary with composition and scattering energy it corresponds to a correlation length [@Kittel] of just over 6 lattice constants, $a$, or 2.5 nm, similar to $\xi_{ab}(0)$ but much smaller than the $\xi_{ab}(T)$ values for which $\chi_c^{FL}$ is reduced by a factor two. It remains to be seen whether theory could account for this.

In these two pictures the effective $T_c$ describing the strength of the GF would fall for $T>1.1T_c$ either because of inelastic scattering or because of a weakening of the pairing interaction. If it could be shown theoretically that $\tilde{B}_{c2}(0)$ falls in a similar way, this would account naturally for the fact [@Alloul] that the magnetic fields needed to destroy the GF fall to zero in the temperature range 120-140 K, where the fluctuations become very small. In summary, Gaussian superconducting fluctuations, plus a strong cut-off that seems to be linked to a reduction in the effective value of $T_c$, provide a good description of the diamagnetism of our superconducting cuprate crystals above $T_c$.

  ---------- ------------ -------------- --------------- --------------------------------- --------------------
  $Sample$    $^\S T_c $   $T_c^{MF_1}$   $\xi_{ab}(0)$   $0.59\tilde{B}_{c2}(0)^{\ddag}$   $\Delta(0)^{\dag}$
  $ $           $ (K)$       $  (K)$         $(nm)$                   $ (T)$                      $ (K)$
  OD89          $89.4$        $89.7$      $1.06\pm0.1$                 $173$                      $448$
  UD57          $56.5$         $59$       $2.02\pm0.1$                 $48$                       $234$
  UD22          $21.6$        $24 $        $4.5\pm0.5$                 $10$                       $105$
  ---------- ------------ -------------- --------------- --------------------------------- --------------------

  : Summary of results. $^\S T_c$ defined by sharp onsets of SQUID signal at 10G and torque data at $\pm$50G. $^{\ddag}$2D clean limit formula [@Larkin] for $B_{c2}(0)$. $^{\dag}$From the BCS relation $\xi_{ab}(0)=\frac{\hbar
  v_F}{\pi\Delta(0)}$, which may not hold exactly for $d$-wave pairing, with $v_F$=2x10$^{7}$ cm/sec. []{data-label="summary"}

We are grateful to D. A. Bonn, A. Carrington, W. N. Hardy, G. G. Lonzarich, J. W. Loram and L. Taillefer for several helpful comments. This work was supported by EPSRC (UK), grant number EP/C511778/1 and the Croatian Research Council, MZOS project No.119-1191458-1008.

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We use the notation $T_c^{MF_1}$ because the standard proof (Ref. ) that the GL equations follow from the microscopic Bardeen, Cooper, Schrieffer (BCS) theory of superconductivity, uses a pairing interaction that is confined to energies within $k_B\Theta_D$ of the Fermi energy, where $\Theta_D$ is the Debye temperature. There is a corresponding spread in coordinate space of $\hbar v_F/(k_B\Theta_D)$, where $v_F$ is the electron velocity. In this case $T_c^{MF_1}$ in GL theory and the GF formulas is the same as $T_c$ from BCS theory (Ref. ). These conditions may not be satisfied in the cuprates and other unconventional superconductors and could cause $T_c^{MF_1}$ to be lower than the mean field $T_c$ obtained from a microscopic theory such as the $t-J$ model \[G. G. Lonzarich, (private communication)\]. Critical superconducting fluctuations will suppress the measured value of $T_c$ below $T_c^{MF_1}$ by an amount related to the Ginzburg parameter, $\tau_G$ (Ref. ). For our UD57 crystal, taking the electronic specific heat coefficient to be 2 mJ/gm.at/K$^2$, $\xi_{ab}(0)$ from Table 1 and using formulas in Refs. ,  and , we find $\tau_G=0.01$ in the 2D limit. Using the 2D formula $\delta T_c/T_c=-2 \tau_G
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The crystal is glued to the end of a commercial piezolever with its CuO$_{2}$ planes parallel to the flat surface of the lever. A dummy lever compensates background magneto-resistance signals, using a 3-lead Wheatstone bridge circuit driven by a floating 77 Hz current source. The chip is mounted on a single-axis rotation stage inside a He$^4$ cryo-magnetic system providing stable temperatures from 1.4 K up to 400 K and fields up to 15 T. The bridge signal arising from the gravitational torque on the crystal when the sample stage is rotated in zero magnetic field gives the $T$-dependent sensitivity of the piezolever. Because the masses of the glue and the lever are much less than that of the crystal, the calibration constant relating the out-of balance bridge signal to the angular dependent torque density $\tau(\theta)$ in J/m$^3$ or $\chi_D(T)$ [@emumole], only depends on the distance between the center of mass of the crystal and the base of the lever at the silicon chip, measured to $\pm5\%$.

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We fitted the normalized $\sigma_{ab}^{FL}(T)$ data in Fig. 25 of Ref.  to an empirical formula $(\exp[(T- \alpha T_c)/\beta] +1)^{-0.1}$ which is $\approx
1$ for $\epsilon\lesssim0.1$ and $\approx\exp[-(T-\alpha T_c)/10\beta]$ at higher $T$. This formula was used to cut off $\chi_c^{FL}(T)$ with $\alpha$ = 1.078, 1.1 and 1.12 and $\beta$ = 0.869, 1.234 and 0.70 K for OD89, UD57 and UD22 respectively and $T_c=T_c^{MF_1}$ shown in Table 1. For OD89, $\alpha$ and $\beta$ values correspond to OD92.5 in Ref. , for UD57 we used UD85 data in Ref.  which are similar to UD57 but have less scatter.

The solid line for $r=0$ shows our empirical 2D formula $b/(3b+6\epsilon)$, where $b=2\pi\xi_{ab}(0)^2B/\Phi_0$. The dashed line shows the 2D limit of Eq. 1 with $\xi_{eff}(b)$ given by $\xi_{eff}(b)^{-4}=\xi_{ab}(T)^{-4}+l_B^{-4}$, where $l_B=(\hbar/eB)^{1/2}$, the formula used to analyze Nernst data for NbSi films [@Pourret]. For $r=0.13$, $b<r$ and $\epsilon<r$, our empirical 3D formula is $-M/\sqrt{\epsilon}=(k_BT/s\Phi_0)0.68b/\sqrt{\epsilon(b+1.94\epsilon)}$.

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Units: 1 J/m$^3$ = 10 ergs/cm$^3$ and using CGS units for $\tau(\theta)= \frac{1}{2}\chi_DB^2\sin2\theta$ with $B$ in gauss gives $\chi_D$ in emu/cm$^3$. Complete flux exclusion corresponds to $\chi$ = -1/4$\pi$ emu/cm$^3$, or $\chi$ = -1 in MKS units. For YBCO $\chi_D$ in emu/cm$^3$, is multiplied by the volume per mole, 666/6.38 cm$^3$ to convert to emu/mole.

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