7 Experiment: YBa₂Cu₃O_6.95

In this chapter, recent $\mu$SR measurements of the $\hat{a}$-$\hat{b}$ plane magnetic penetration depth $\lambda_{ab}$and the vortex core size ($\sim \! \xi_{ab}$) in the high-T_c compound YBa₂Cu₃O_6.95 are presented. The oxygen concentration in this superconductor is that which gives the maximum value of the transition temperature T_c.

Figures 7.1 and 7.2 show the Fourier transforms of the muon precession signal in the optimally doped compound YBa₂Cu₃O_6.95 as a function of temperature and magnetic field, respectively. In Figure 7.2 it was necessary to renormalize the Fourier amplitudes to the same maximum height, because of a reduction in the signal amplitude with increasing magnetic field. Asymmetry loss is due to the finite timing resolution of the counters, a reduction in the radii of the decay positron orbits and a dephasing of the muon beam before it reaches the sample. The last originates from muons with slightly different momenta and/or beam trajectories, which take different times to traverse the magnetic field and therefore precess different amounts prior to arrival at the sample.

The basic features expected for a rigid 3D vortex lattice are observed in these Fourier transforms, although their signal-to-noise ratio is not as good as those for NbSe₂. In addition, the high-field cutoff is not clearly visible at low temperatures, which is partly a result of the much smaller coherence length (and vortex-core radius) in this material. The smaller value of r₀ means that fewer muons stop in the vicinity of the vortex cores, resulting in less signal-to-noise in the high-field tail. Consequently, $\xi_{ab}$ and r₀ are difficult to determine in this material so deep in the superconducting state. Since the signal-to-noise ratio scales with $\sqrt{N}$, where N is the number of counts, it takes an impractical amount of time to make significant improvements in the high-field tail of the measured field distribution. To dramatically improve the signal-to-noise ratio in the high-field tail, it is necessary to go to higher magnetic fields where there are more vortices in the sample. According to the spectra presented in Ref. [50], at low temperatures this means magnetic fields in excess of at least 5 T. Unfortunately, as just mentioned, there are problems associated with the signal amplitude at such large H. Currently, efforts are underway to construct an apparatus which operates effectively in such strong magnetic fields. This ``high-field'' apparatus will include the use of higher timing resolution counters and a reduction in the distance between the decay positron counters and the sample. A high-field cutoff is clearly visible at high temperatures, as shown in Fig. 7.1. This is because r₀ increases with T as was just observed in the case of NbSe₂.

 

 

As noted earlier, oxygen vacancies and twin planes may pin vortices in YBa₂Cu₃O$_{7-\delta}$. The strength of this pinning can be studied by determiming the sensitivity of the $\mu$SR spectrum to small changes in magnetic field. Figure 7.3(a) shows the Fourier transform of the muon spin precession signal in ``detwinned'' YBa₂Cu₃O_6.95 (O3) after field cooling to $T \! = \!5$ K in a magnetic field of $H \! = \! 1.50$ T. When the applied field is decreased by 0.02 T, the residual background signal shifts down to the new applied field $H \! = \! 1.48$ T [see Fig. 7.3(b)]. However, the signal originating from the sample does not shift in response to the small change in applied field. This shows that the vortex lattice is firmly pinned. In addition, the absence of any detectable background peak in the unshifted signal implies that there are no nonsuperconducting inclusions in the sample. As the temperature is increased, the shape of the Fourier transform changes due to the changes in $\lambda_{ab}$ and $\xi_{ab}$. However, the signal remains unshifted indicating that the vortices are still pinned. Eventually, the temperature is large enough that thermal fluctutations depin some of the 3D vortex lines, as shown in Fig. 7.4. Raising the temperature even further results in thermal depinning of the remaining fixed vortex lines. On the other hand, the vortex lattice is not so strongly pinned in NbSe₂. When the applied magnetic field on NbSe₂ was shifted by a small amount at low temperatures, the sample signal always shifted with the background signal.

 

 

The muon precession signals for YBa₂Cu₃O_6.95 were fit in a manner similar to that for NbSe₂, with the following additional constraints and assumptions:

1.

As explained in Ref. [2], because of the interplay between $\lambda_{ab}$ and $\sigma_{f}$,which arises because of the poor statistics in the high-field tail, it was necessary to fix one of these parameters with respect to the other. In particular, the following linear correlation is assumed, based on the general trend observed in the fits performed without this constraint  

$$\frac{1}{\lambda_{ab}^{2}} = \frac{\sqrt{\sigma_f^2 - \sigma_N^2}}{C} \approx \frac{\sigma_f}{C}$$ (1)

where C is a constant and $\sigma_N$ is the Gaussian muon depolarization rate in the normal state.

2.

An equilateral triangular vortex lattice is assumed for the summation over reciprocal lattice vectors ${\bf K}$ in the theoretical field profile $B ({\bf r})$. As discussed in the previous chapter, this is a reasonable assumption because of the high pinning temperature in YBa₂Cu₃O_6.95. The cutoff in the summation was done in a way which preserves circular symmetry around the vortex cores. The symmetry of the cores themselves is of minor significance in determining $\lambda_{ab}$, since their contribution to the measured field distribution is small in the field range considered here. The theoretical triangular lattice does not include deformations of the vortex-lattice geometry due to mass anisotropy and/or twin planes. The in-plane mass anisotropy ratio $\gamma \! = \! (m_a/m_b)^{1/2}$ has been measured by Basov et al. [172] in YBa₂Cu₃O_6.95 using far infrared reflectance. They find that in single crystals similar to those used in the present study, the ratio of the zero-field penetration depths along the $\hat{a}$ and $\hat{b}$ directions is $\gamma \! = \! \lambda_a/\lambda_b \! = \! 1.47(14)$.A simple scaling argument [196] can be used to show that it is not necessary to incorporate the in-plane anisotropy $\gamma$ into the field profile. The argument is as follows: The orthorhombic crystal structure for YBa₂Cu₃O_6.95 implies that the effective masses along the crystallographic axes are such that $m_c \! \gt \! m_a \! \gt \! m_b$. When a magnetic field is applied along the $\hat{c}$-axis direction, three adjacent vortex lines form a lattice in the $\hat{a}$-$\hat{b}$ plane which is a stretched version of an isotropic triangular lattice [162]. The unit cell is a centered rectangular lattice and the supercurrents flow in an elliptical path around the vortex cores, since $\gamma \! = \! \xi_b / \xi_a \! = \! \lambda_a / \lambda_b$. However, the magnetic field distribution is unaltered from the isotropic case since any change in $\gamma$ is compensated for by rescaling the coordinates in Eq. (4.10) or Eq. (4.13). Of course, this argument does not take into account pinning effects at twin planes. The Fourier transform of the measured muon precession signal in detwinned YBa₂Cu₃O_6.95 (O3) [see Fig. 7.3], does however show the same basic features as that for the twinned crystals. The subtle differences that do exist between the $\mu$SR line shapes of twinned and detwinned crystals, in the field range considered here, can only be determined by fitting the data. Unfortunately, we have yet to carry out a complete study of detwinned YBa₂Cu₃O_6.95. A discussion regarding the effects of twin planes on the outcome of a $\mu$SR experiment in the vortex state will be reserved for the underdoped compound, which is considered later in this report.

3.

Due to the absence of a sharp high-field cutoff in the field distribution at low temperatures, in previous attempts to model the measured field distribution in YBa₂Cu₃O_6.95 (see Refs. [2,3]) the GL parameter $\kappa \! = \! \lambda_{ab}/\xi_{ab}$ was fixed to a constant value. To obtain a reasonable value for $\kappa$ some of the high temperature spectra in which the cutoff was clearly visible were fit and the results averaged to give a value of $\kappa \! = \! 68$. The low temperature data were fit by assuming this value and assuming that $\kappa$ was independent of both temperature and magnetic field. Given the observed field-dependence of $\kappa$ in NbSe₂, the latter assumption is likely invalid. Nevertheless, the fits in Refs. [2,3] were found to be not very sensitive to the value of $\kappa$ anyway. For instance, as noted in Ref. [2], increasing $\kappa$ to 73 changes $\lambda_{ab}(0)$ by less than 3 Å. The data for the underdoped compound, which will be presented later in this report, confirm that the assumption of a T-independent $\kappa$ is reasonable, but that an H-independent $\kappa$ is not. The reason is likely related to the shrinking of the vortex cores with increasing magnetic field. In this thesis I have re-analyzed the $\mu$SR data from Ref. [3] in terms of the analytical GL model and in doing so, I allowed $\kappa$ to vary ``freely'' in the fitting procedure.

 

Figure 7.5 shows the temperature dependence of $\lambda_{ab}^{-2}$in YBa₂Cu₃O_6.95 (O1) at $H \! = \! 0.5$ T, from fits assuming the analytical GL model [see Eq. (4.13)]. The linear temperature dependence at low T supports an unconventional pairing state in which there are nodes in the superconducting energy gap. The solid curve in Fig. 7.5 represents the zero-field microwave measurements of $\Delta \lambda_{ab} (T) \! = \! \lambda_{ab}(T) - \lambda_{ab}(1.35$ K) performed by Hardy et al. [14] on similar high quality YBa₂Cu₃O_6.95 crystals. To plot $\lambda_{ab}^{-2} (T)$ for the microwave data, our extrapolated value of $\lambda_{ab}(1.35$ K) was used. The excellent agreement between the measurements in the vortex state and those in the Meissner state indicate that the variation of the superfluid fraction as a function of temperature is identical in both phases. Furthermore, this confirms that the assumption of a triangular vortex lattice in the fitting procedure introduces at most only a small systematic error in the absolute value of $\lambda_{ab}$. This is reasonable since it has been shown theoretically that including additional terms in the free energy of the vortex state produces only minor changes in the internal field distribution [178]. This can be confirmed by fitting the data to a theoretical field profile which assumes an inappropriate vortex-lattice configuration. For instance, if a square vortex lattice is assumed in the fitting procedure, the quality of the fits is found to be much worse and the absolute value of $\lambda_{ab}(T)$ does change dramatically. However, the temperature dependence of $\lambda_{ab}(T) - \lambda_{ab}(0)/\lambda_{ab}(0)$ from these fits is nearly identical to that obtained assuming a triangular vortex lattice.

Our $\mu$SR measurements of $\lambda_{ab}(T)$ presented in Ref. [2] suggest that the strength of the term linear in T depends on magnetic field. However, as noted in Ref. [3], this effect is artificially created by prematurely cutting off the summation over reciprocal lattice vectors. The problem is easily rectified by increasing the sum until any further increase does not affect the deduced value of $\lambda_{ab}(T)$. Although the term linear in T is found to be H-independent here, the absolute value of $\lambda_{ab}$ does depend strongly on magnetic field. In Ref. [3], $\lambda_{ab}$ was measured in the vortex state of YBa₂Cu₃O_6.95 (O1, O2) as a function of magnetic field. The internal field distribution was fit assuming the ML model with a Gaussian cutoff factor. The results of this study are listed in Table 7.4. Yaouanc et al. [121] suggest that the observed field dependence is probably explained if a more appropriate cutoff function is used. Using the conventional GL equations, they have shown that the variance of the field distribution $\Delta_v^2 \! = \! (\langle B_z^2 \rangle - \langle B_z \rangle^2)$depends on magnetic field. Through rough calculations of the variance from our measurements in Ref. [2], they find good agreement with the field dependence predicted by the conventional GL theory. Unfortunately, as noted above, the results in Ref. [2] are flawed. The field dependence for $\lambda_{ab}$ has since been shown to be much stronger [3]. It is unlikely that this field dependence is related to an improper treatment of the vortex cores, since the small cores in YBa₂Cu₃O_6.95 contribute very little to the variance at low fields. Nevertheless, to properly account for the finite size of the vortex cores, the data has been re-analyzed here using the analytical GL model as suggested by the authors of Ref. [121].

 

Figure 7.6 shows the low temperature behaviour of $\lambda_{ab}^{-2}$in YBa₂Cu₃O_6.95 (O2) obtained from this new analysis for three of the magnetic fields considered. As in Ref. [3], excellent fits are obtained to a linear relation  

$$\lambda_{ab}^{-2}(T) \! = \! \lambda_{ab}^{-2} (0)[1 - \alpha t] \, .$$ (2)

where $t \! = \! T/T_c$ and T_c is the transition temperature at zero magnetic field. The results of these fits appear in Table 7.4. The term linear in T is essentially independent of magnetic field, so that $\alpha$ agrees well with the microwave cavity measurements of Ref. [14] at all magnetic fields considered.

 

Magnetic Beamline/Year

$$\lambda_{ab}(T \! = \! 0) \alpha \lambda_{ab}(T \! = \! 0) \alpha$$ Field C

$$^2/\mu s$$  [T]   [Å]   [10^-1]   [Å]   [10^-1] 

0.191 m20/1993 1.955 1188(8) 6.0(3) 1115(7) 4.8(4)

0.192 m15/1993 1.943 1181(7) 6.5(6) 1114(6) 6.2(2)

0.498 m15/1992 1.835 1208(13) 6.6(3) 1129(12) 6.3(2)

0.731 m15/1994 1.827 1222(20) 6.3(5) 1165(18) 6.3(4)

1.003 m15/1993 1.625 1228(16) 6.3(4) 1168(12) 6.0(4)

1.488 m15/1992 1.784 1272(7) 5.8(4) 1195(6) 5.9(4)

1.952 m15/1993 2.275 1351(37) 7.3(7) 1261(34) 5.5(8)

0.103 m20/1995 1.195 1149(6) 6.2(2) 1069(6) 6.4(4)

0.497 m20/1995 1.485 1171(9) 7.5(4) 1099(9) 7.5(3)

1.500 m20/1995 1.833 1277(14) 6.6(6) 1192(12) 6.1(7)

 

Figure 7.7 shows the magnetic field dependence of $\lambda_{ab}$ extrapolated to $T \! = \! 0$ for sample O1 (open circles) and sample O2 (solid circles). The solid and dashed curves are fits to the power-law relation  

$$\lambda_{ab}(H,T\!=\!0) \! = \! \lambda_{ab} (0,0) + \beta H^p \, .$$ (3)

Table 7.5 shows the parameters from these fits together with those from the analysis in Ref. [3]. Over this narrow field range the data is reasonably described by a relation which depends linearly on H. Moreover, the strength of this linear term is essentially the same in both types of analysis, as found for NbSe₂. This is reasonable here, since the fits are not very sensitive to the field distribution near the vortex cores. The scatter in the data is remarkably small given the variety of conditions under which they were recorded (see Table 7.4). This demonstrates the reliability of the fitting procedure for extracting a consistent value of $\lambda_{ab}$, despite variations in the experimental arrangement which cause changes in the size of the background signal. Although the fits to Eq. (7.3) suggest that $\lambda_{ab} (T \! = \! 0) \! \propto \! H$ in the vortex state, the measurements are only for very small values of reduced field, $0.0009\!<\!h\!<\!0.016$ (assuming that $H_{c2}(0) \! = \! 120$ T), so that it is difficult to draw any firm conclusion about the precise way in which $\lambda_{ab}$ varies with H.

 

$$\lambda_{ab}(0,0) \beta \lambda_{ab}(0,0) \beta$$ p p

 [Å]   [Å/T]   [Å]   [Å/T] 

1181(4) 52(4) 1.5(1) 1084(3) 81(1) 1.0(1)

1147(5) 71(8) 1.5(2) 1063(2) 80(3) 1.16(12)

 

 

The strong field dependence for $\lambda_{ab}$ in YBa₂Cu₃O_6.95 is obtained from both types of analysis, and is considerably stronger than that found in NbSe₂. Figure 7.8 shows the field dependence of $\lambda_{ab}$ at $T \! = \! 0.33~T_c$ for both of these materials. The solid lines are a fit to the equation  

$$\frac{\lambda_{ab} (H)}{\lambda_{ab} (0)} = 1+ \varepsilon \left[ \frac{H}{H_{c2} (0.33~T_c)} \right] \, ,$$ (4)

where $\varepsilon \! = \! 1.6$ with $H_{c2} (0.33~T_c) \! = \! 2.9$ T for NbSe₂ and $\varepsilon \! = \! 5.4$for YBa₂Cu₃O_6.95, assuming $H_{c2} (0.33~T_c) \! = \! 95$ T. Undoubtedly some of the field dependence is due to the effects of a nonlinear supercurrent response, similar to that observed in the Meissner state. However, as noted earlier, nonlocal effects associated with nodes at the Fermi surface should be more important in YBa₂Cu₃O_6.95.

 

Figure 7.9 shows the temperature dependence of the linear coefficient $\beta_2 (T)$ determined from Eq. (2.50) for YBa₂Cu₃O_6.95 (O1). The large error bars are due to the scatter and to temperature variations between the different data sets. The scatter in the data at each temperature was too large to deduce the precise field dependence, so a linear dependence on H was assumed. The finite value of $\beta_2 (T)$ at $T \! = \! 0$is consistent with the field-induced pair breaking effects expected in a superconductor with nodes in the energy gap. We note that $\beta_2 (T)$ is approximately 30 times smaller at low T and about 10 times smaller at $T \! \approx 0.5~T_c$ than the values reported by Maeda et al. [38] for microwave cavity perturbation measurements in the Meissner state of YBa₂Cu₃O$_{7-\delta}$. However, very recently Bidinosti et al. [208] have determined the field dependence of $\Delta \lambda_{ab}$ in YBa₂Cu₃O_6.95 in the Meissner state from AC susceptibility measurements. They find that the coefficient of the term linear in H is approximately an order of magnitude smaller than that reported by Maeda et al. Nevertheless, the different definition of the penetration depth in the $\mu$SR experiment (which was discussed earlier) makes a comparison to these Meissner state experiments very difficult.

It is possible that some of the measured field dependence for $\lambda_{ab}$ is due to changes in the vortex-lattice geometry with increasing magnetic field--which is predicted in a number of theoretical studies [178,179,180]. It is currently unknown if such geometry changes actually occur. However, if they do, the question is whether these changes are subtle over the narrow field range considered here. We now show that the fits to the data suggest that there are no significant changes in the vortex-lattice geometry. This does not necessarily imply that the theories are wrong, since the strong pinning of the vortex lines in the YBa₂Cu₃O_6.95 samples studied here likely prevents such geometrical changes from occurring.

Figure 7.10(a) shows the temperature dependence of the additional broadening parameter $\sigma_{f}$ in YBa₂Cu₃O_6.95 (O1) at $H \! = \! 0.19$ T (open circles) and $H \! = \! 1.48$ T (solid circles). Due to the imposed constraint of Eq. (7.1), $\sigma_f(T)$ exhibits the same linear dependence on T as $\lambda_{ab}^{-2} (T)$. Despite this constraint, we find that $\sigma_f(0.19$ T$) \! \gt \! \sigma_f(1.48$ T), while $\lambda_{ab}^{-2}(0.19$ T$) \! \gt \! \lambda_{ab}^{-2}(1.48$ T)--which implies that the line width of the measured internal field distribution is definitely larger at smaller fields.

 

The RMS displacement $\langle s^2 \rangle^{1/2}$ of the vortex lines from their ideal positions in a perfect triangular lattice [determined from Eq. (6.3)], is plotted as a function of temperature in Fig. 7.11(a) . The value of $\langle s^2 \rangle^{1/2}$is much larger at $H \! = \! 0.19$ T than at $H \! = \! 1.48$ T. This is most likely due to an enhancement in the random pinning of vortex lines by point defects at the smaller magnetic field. Figure 7.11(b) shows $\langle s^2 \rangle^{1/2}$ as a percentage of L in YBa₂Cu₃O_6.95 (O1). The close agreement at the two different magnetic fields suggests that at low temperatures, where thermal fluctuations are small, the disorder in the vortex lattice scales with the nearest-neighbor distance between vortex lines, as was found in NbSe₂. This result is inconsistent with a dramatic change in the vortex-lattice geometry in going from $H \! = \! 0.19$ T to $H \! = \! 1.48$ T at low T. Further evidence that there are no significant distortions in the vortex lattice over this narrow field range is given by the consistency in the quality of the fits assuming a triangular vortex lattice. Figure 7.10(b) shows that $\chi^2$ normalized to the number of degrees of freedom is essentially independent of magnetic field and temperature in the region of the phase diagram considered in this experiment. Certainly this would not be the case if there were a sharp transition e.g. from a triangular to a fourfold-symmetric vortex lattice.

 

 

Figure 7.12 shows the temperature dependence of $\kappa$ in YBa₂Cu₃O_6.95 (O1) at the different magnetic fields considered. The scatter in the data reflects the uncertainty which arises in fitting a field distribution which has a small signal-to-noise ratio in the high-field tail. Surprisingly, there is less scatter in the data at the lower fields where there are fewer vortices in the sample. This suggests that the vortex-core radius must be significantly larger at smaller H, as was the case in NbSe₂. The data in Fig. 7.12 suggests that $\kappa$ is either independent of temperature or increases very weakly with increasing T. However, $\kappa$ depends strongly on magnetic field in YBa₂Cu₃O_6.95. Figure 7.13 shows the best fits to the data sets at the different magnetic fields in Fig. 7.12, assuming a T-independent value of $\kappa$.The solid line in Fig. 7.13 is a fit to the linear relation  

$$\kappa (H) = \kappa (0) [1 + \eta h ] \, ,$$ (5)

where $h \! = \! H/H_{c2}(0)$ and $H_{c2}(0) \! = \! 120$ T. The best fit is obtained for $\kappa (0)\! = \! 10.6(3)$ and $\eta \! = \! 212(10)$.

 

Figure 7.14 shows the first 1.5 $\mu$s of a typical muon precession signal in YBa₂Cu₃O_6.95 displayed in a reference frame rotating at about 3.3 MHz below the Larmor precession frequency of a free muon. The curves through the data points are examples of fits (actually performed over the first 6 $\mu$s) to the theoretical polarization function for fixed values of $\xi_{ab}$.The only additional constraint in these fits was that $\sigma_f \! \propto \! \lambda_{ab}^{-2}$. Differences in the quality of the fits for the various values of $\xi_{ab}$ are most noticeable at early times. This is seen more clearly in Fig. 7.15 which shows the difference between the data points and the fitted curve for fits similar to those in Fig. 7.14. The ratio of $\chi^2$ to the number of degrees of freedom (NDF) is shown in Fig. 7.16(a) as a function of $\xi_{ab}$ for two of the magnetic fields considered. Figure 7.16(b) shows the values of the free parameter $\kappa$ obtained from the same fits as in Fig. 7.16(a). Note that the distribution of data points around the minimum value of $\chi^2$/NDF is asymmetric. Since $\lambda_{ab}$ is essentially unchanged in the fits for different values of $\xi_{ab}$, this asymmetry reflects the lack of statistics from the vortex cores in the measured internal field distribution. In particular, the fits can tolerate a smaller value of $\xi_{ab}$ and a longer high-field tail. At the lower field in Fig. 7.16, the minimum is much sharper because of the increased size of the vortex cores.

 

 

 

Despite the scatter in the data for $\kappa (T)$, a smooth plot for the temperature dependence of $\xi_{ab}$ can be generated from Eq. (7.2) for $\lambda_{ab}(T)$ and the fitted constant values of $\kappa (T)$. Such plots are shown in Fig. 7.17 at different magnetic fields where $\xi_{ab} (T)$ is given by the following relation  

$$\xi_{ab} (T) = \frac{\lambda_{ab}(T)}{\kappa (T)} \! = \! \frac{\lambda_{ab} (0)}{\kappa (0) \sqrt{1 - \alpha t}} \, ,$$ (6)

The values of $\alpha$ are given in Table 7.4. The strength of the T-dependence of $\xi_{ab}$ is considerably weaker than in NbSe₂. It should be noted that thermal fluctuations of the vortex lines will lead to an increase in the measured size of the vortex cores, as explained in section 4.2. As in NbSe₂, there is a clear reduction in the magnitude of $\xi_{ab}$ with increasing magnetic field, which is consistent with a shrinking of the vortex cores due to the increased strength of the vortex-vortex interactions.

The magnetic field dependence of $\xi_{ab}$ extrapolated to $T \! = \! 0$is shown in Fig 7.18. The solid curve represents the combination of the fitted relations for $\lambda_{ab} (H)$ and $\kappa (H)$, namely, Eq. (7.3) and Eq. (7.5). Recall that in the data analysis assuming the analytical GL model it was found that $p \! \approx \! 1$ in Eq. (7.3), so that the relation for $\xi_{ab} (H)$ at $T \! = \! 0$ is  

$$\xi_{ab} (H,0) = \frac{\lambda_{ab} (H,0)}{\kappa (H,0)} = \xi_{ab} (0,0) \frac{[1 + \beta^\prime h]}{[1 + \eta h ]} \, ,$$ (7)

where $\xi_{ab} (0,0) \! = \! \lambda_{ab} (0,0)/\kappa (0,0) \! = \! 102$ Å, $\beta^\prime \! = \! \beta H_{c2}(0)/\lambda_{ab}(0) \! = \! 8.97$and $\eta \! = \! 212$ using the values in Table 7.5.

 

Our findings are most easily interpreted in terms of vortex cores which contain discrete quasiparticle bound states. At $H \! = \! 6$ T, which is the field at which the STM experiment [197] on YBa₂Cu₃O$_{7-\delta}$ was performed, Eq. (7.7) gives $\xi_{ab} (T \!= \! 0) \! = \! 12.8$ Å  and with the help of Eq. (7.6) gives $\xi_{ab} (T \! = \! 4.2$ K$) \! = \! 13.0$ Å. Using the formula $E_{\mu} \! = \! 2 \mu \Delta_0^2/E_F$ [124] and taking $\xi_{ab} (T \! = \! 0)$ to be the BCS coherence length $\xi_0 \! = \! \hbar v_f/ \pi \Delta_0$,the lowest bound energy level is estimated to be $E_{1/2} \! = \! 2 \hbar^2/m_e \pi^2 \xi_{ab}^2 \! \approx \! 9.1$ meV. This estimate agrees well with the STM result of $E_{1/2} \! = \! 5.5$ meV and the value of 9.5 meV obtained from an infrared absorption experiment [198] on YBa₂Cu₃O$_{7-\delta}$ thin films. The STM measurement implies that $\xi_{ab} \! \approx \! 17$ Å  at 6 T. The moderate agreement found here strongly supports our assertion that the coherence length rises appreciably with decreasing magnetic field. This is one of the most important findings of this study, because theoretical predictions and the interpretation of experiments on the high-T_c materials are often based on the assumption that $\xi_{ab}$ is extremely small. The results herein imply that the spacing between energy levels becomes larger with increasing magnetic field because of the reduction in $\xi_{ab}$--which is analogous to a reduction in the radius of a cylindrical potential well. In this picture numerous bound states should exist in the vortex cores of YBa₂Cu₃O_6.95 at low fields (i.e. $< \! 1$ T).