In this chapter, recent SR measurements of the - plane magnetic penetration depth and the vortex core size () in the high-T_{c} compound YBa_{2}Cu_{3}O_{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 YBa_{2}Cu_{3}O_{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 NbSe_{2}. 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 r_{0} means that fewer muons stop in the vicinity of the vortex cores, resulting in less signal-to-noise in the high-field tail. Consequently, and r_{0} are difficult to determine in this material so deep in the superconducting state. Since the signal-to-noise ratio scales with , 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 r_{0} increases with T as was just observed in the case of NbSe_{2}.
As noted earlier, oxygen vacancies and twin planes may pin vortices in YBa_{2}Cu_{3}O. The strength of this pinning can be studied by determiming the sensitivity of the SR spectrum to small changes in magnetic field. Figure 7.3(a) shows the Fourier transform of the muon spin precession signal in ``detwinned'' YBa_{2}Cu_{3}O_{6.95} (O3) after field cooling to K in a magnetic field of T. When the applied field is decreased by 0.02 T, the residual background signal shifts down to the new applied field 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 and . 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 NbSe_{2}. When the applied magnetic field on NbSe_{2} was shifted by a small amount at low temperatures, the sample signal always shifted with the background signal.
The muon precession signals for YBa_{2}Cu_{3}O_{6.95} were fit in a manner similar to that for NbSe_{2}, with the following additional constraints and assumptions:
(1) |
Figure 7.5 shows the temperature dependence of in YBa_{2}Cu_{3}O_{6.95} (O1) at 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 K) performed by Hardy et al. [14] on similar high quality YBa_{2}Cu_{3}O_{6.95} crystals. To plot for the microwave data, our extrapolated value of 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 . 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 does change dramatically. However, the temperature dependence of from these fits is nearly identical to that obtained assuming a triangular vortex lattice.
Our SR measurements of 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 . Although the term linear in T is found to be H-independent here,^{} the absolute value of does depend strongly on magnetic field. In Ref. [3], was measured in the vortex state of YBa_{2}Cu_{3}O_{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 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 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 YBa_{2}Cu_{3}O_{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 in YBa_{2}Cu_{3}O_{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
(2) |
2c| | 2c| | |||||
Magnetic | Beamline/Year | |||||
Field | C | |||||
[T] | [10^{6} Å] | [Å] | [10^{-1}] | [Å] | [10^{-1}] | |
1|l|(i) | ||||||
---|---|---|---|---|---|---|
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) |
1|l|(ii) | ||||||
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 extrapolated to for sample O1 (open circles) and sample O2 (solid circles). The solid and dashed curves are fits to the power-law relation
(3) |
3|c| | 3c| | ||||
p | p | ||||
[Å] | [Å/T] | [Å] | [Å/T] | ||
1|l|(i) | |||||
---|---|---|---|---|---|
1181(4) | 52(4) | 1.5(1) | 1084(3) | 81(1) | 1.0(1) |
1|l|(ii) | |||||
1147(5) | 71(8) | 1.5(2) | 1063(2) | 80(3) | 1.16(12) |
The strong field dependence for in YBa_{2}Cu_{3}O_{6.95} is obtained from both types of analysis, and is considerably stronger than that found in NbSe_{2}. Figure 7.8 shows the field dependence of at for both of these materials. The solid lines are a fit to the equation
(4) |
Figure 7.9 shows the temperature dependence of the linear coefficient determined from Eq. (2.50) for YBa_{2}Cu_{3}O_{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 at is consistent with the field-induced pair breaking effects expected in a superconductor with nodes in the energy gap. We note that is approximately 30 times smaller at low T and about 10 times smaller at than the values reported by Maeda et al. [38] for microwave cavity perturbation measurements in the Meissner state of YBa_{2}Cu_{3}O. However, very recently Bidinosti et al. [208] have determined the field dependence of in YBa_{2}Cu_{3}O_{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 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 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 YBa_{2}Cu_{3}O_{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 in YBa_{2}Cu_{3}O_{6.95} (O1) at T (open circles) and T (solid circles). Due to the imposed constraint of Eq. (7.1), exhibits the same linear dependence on T as . Despite this constraint, we find that T T), while T T)--which implies that the line width of the measured internal field distribution is definitely larger at smaller fields.
The RMS displacement 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 is much larger at T than at 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 as a percentage of L in YBa_{2}Cu_{3}O_{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 NbSe_{2}. This result is inconsistent with a dramatic change in the vortex-lattice geometry in going from T to 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 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 in YBa_{2}Cu_{3}O_{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 NbSe_{2}. The data in Fig. 7.12 suggests that is either independent of temperature or increases very weakly with increasing T. However, depends strongly on magnetic field in YBa_{2}Cu_{3}O_{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 .The solid line in Fig. 7.13 is a fit to the linear relation
(5) |
Figure 7.14 shows the first 1.5 s of a typical muon precession signal in YBa_{2}Cu_{3}O_{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 s) to the theoretical polarization function for fixed values of .The only additional constraint in these fits was that . Differences in the quality of the fits for the various values of 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 to the number of degrees of freedom (NDF) is shown in Fig. 7.16(a) as a function of for two of the magnetic fields considered. Figure 7.16(b) shows the values of the free parameter obtained from the same fits as in Fig. 7.16(a). Note that the distribution of data points around the minimum value of /NDF is asymmetric. Since is essentially unchanged in the fits for different values of , 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 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 , a smooth plot for the temperature dependence of can be generated from Eq. (7.2) for and the fitted constant values of . Such plots are shown in Fig. 7.17 at different magnetic fields where is given by the following relation
(6) |
The magnetic field dependence of extrapolated to is shown in Fig 7.18. The solid curve represents the combination of the fitted relations for and , namely, Eq. (7.3) and Eq. (7.5). Recall that in the data analysis assuming the analytical GL model it was found that in Eq. (7.3), so that the relation for at is
(7) |
Our findings are most easily interpreted in terms of vortex cores which contain discrete quasiparticle bound states. At T, which is the field at which the STM experiment [197] on YBa_{2}Cu_{3}O was performed, Eq. (7.7) gives Å and with the help of Eq. (7.6) gives K Å. Using the formula [124] and taking to be the BCS coherence length ,the lowest bound energy level is estimated to be meV. This estimate agrees well with the STM result of meV and the value of 9.5 meV obtained from an infrared absorption experiment [198] on YBa_{2}Cu_{3}O thin films. The STM measurement implies that 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 is extremely small. The results herein imply that the spacing between energy levels becomes larger with increasing magnetic field because of the reduction in --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 YBa_{2}Cu_{3}O_{6.95} at low fields (i.e. T).