8 Experiment: YBa₂Cu₃O_6.60

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 superconductor YBa₂Cu₃O_6.60 are presented.

Figure 8.1 shows the $\mu$SR line shapes for the underdoped compound YBa₂Cu₃O_6.60 (U1) and the optimally doped compound YBa₂Cu₃O_6.95 (O1), at similar temperature and magnetic field. The width of the field distribution in the underdoped compound is considerably smaller due to a larger $\lambda_{ab}$.Furthermore, the high-field cutoff is much more pronounced due to both a larger vortex-core radius and the longer $\lambda_{ab}$. In the analysis that follows, it is found that at $H \! = \! 1.49$ T the fractional volume of the sample occupied by the vortex cores in YBa₂Cu₃O_6.60 is nearly two times greater than in YBa₂Cu₃O_6.95.

Figure 8.2 shows what happens to the internal field distribution in YBa₂Cu₃O_6.60 upon warming the sample in an applied field of 1.49 T. In going from $T \! = \!0.04~T_c$ (see Fig. 8.1) to $T \! = \! 0.42~T_c$ (i.e. 24.8 K), the asymmetry of the line shape suggests that the vortex lattice is comprised of 3D-flux lines arranged in a regular pattern. However, at $T \! = \! 0.59~T_c$the line shape becomes more symmetric, and is completely symmetric at $T \! = \! 0.76~T_c$. The loss of asymmetry in the $\mu$SR line shape is a strong indication that the vortex lattice has melted and/or it has undergone a 3D-to-2D transition. In order for $\lambda_{ab}(H,T)$ and $\xi_{ab}(H,T)$ to be determined in YBa₂Cu₃O_6.60 using $\mu$SR spectroscopy, a rigid 3D-vortex lattice must exist. We will therefore begin by restricting ourselves to the low-temperature region of the H-T phase diagram, and reserve the discussion of the vortex lattice in YBa₂Cu₃O_6.60 at higher temperatures till the end of this section.

 

 

Due to the larger contribution of the vortex cores to the measured internal field distribution, it is unnecessary to relate or fix any of the parameters in the fitting procedure. In this way the data is analyzed exactly as it was for NbSe₂. We note that the theoretical field distribution is generated assuming a triangular vortex lattice. This is reasonable, since we know of no experiments which have imaged the vortex lattice in YBa₂Cu₃O_6.60. As was explained for YBa₂Cu₃O_6.95, in the absence of twin boundary pinning the $\hat{a}$-$\hat{b}$ plane anisotropy will stretch the triangular lattice in a way which does not change the shape of the internal field distribution. Figure 8.3 shows the temperature dependence of the fitted $\lambda_{ab}^{-2}$ in YBa₂Cu₃O_6.60 (U1) using the analytical GL model for the field profile of the vortex lattice. The curves through the data points are merely guides for the eye. At higher temperatures where the vortex lattice is no longer a rigid 3D structure, the fitted value of $\lambda_{ab}^{-2}$ merely characterizes the width of the field distribution and is not the penetration depth as defined at lower temperatures. Note that the transition in the vortex-lattice structure occurs at a lower temperature in the larger magnetic field. Figure 8.3 suggests that measurements of $\lambda_{ab}$ should be restricted to temperatures below 25 K for applied magnetic fields greater than 1.5 T.

 

Figure 8.4 shows the temperature dependence of $\lambda_{ab}^{-2}$in YBa₂Cu₃O_6.60 (U2) at low T, for three of the magnetic fields considered. As in YBa₂Cu₃O_6.95, there is a strong linear decrease in the superfluid fraction with increasing temperature. The solid lines are a fit to Eq. (7.2). The fitted parameters appear in Table 8.6 along with those for an analysis assuming the ML model with a Gaussian cutoff factor. Note that the coefficient $\alpha$ of the term linear in t is weaker than in the optimally doped compound and is at most only weakly dependent on magnetic field. This is consistent with conclusions drawn from earlier $\mu$SR measurements of the muon depolarization rate $\sigma$for various oxygen dopings in YBa₂Cu₃O$_{7-\delta}$ [209]. As shown in the inset of Fig. 8.4, there is also excellent agreement with microwave cavity measurements of $\Delta \lambda_{ab} (T) \! = \! \lambda_{ab} (T) - \lambda_{ab}(1.25$ K) in zero magnetic field [210]. According to the Meissner-phase measurements in Ref. [210], the strength of the term linear in t changes substantially as a function of oxygen doping only in the $\hat{b}$-direction (i.e. the direction of the CuO chains). In particular, the value of the coefficient linear in t for the temperature dependence of $\lambda_b$ is reduced in the underdoped compound. One interpretation is that the oxygen vacancies in the CuO chains act as pair-breaking defects.

 

Magnetic Beamline/Year

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

 [T]   [Å]   [10^-1]   [Å]   [10^-1] 

0.742 m15/1994 2212(18) 4.1(2) 1698(15) 4.9(2)

1.013 m15/1994 2248(17) 4.0(3) 1739(23) 4.7(2)

1.490 m15/1994 2393(40) 4.5(6) 1810(32) 5.7(8)

0.500 m15/1996 2286(11) 4.1(2) 1754(13) 4.6(2)

0.850 m15/1996 2396(18) 4.1(2) 1815(16) 5.7(3)

1.250 m15/1996 2450(19) 3.9(4) 1846(17) 4.8(4)

 

 

The magnetic-field dependence of $\lambda_{ab}$ extrapolated to $T \! = \! 0$ is shown in Fig. 8.5. There is a clear difference in the magnitude of $\lambda_{ab} (H)$ determined from measurements in the twinned (U1) and detwinned samples (U2). This difference is likely due to vortex-lattice distortions in U1 caused by twin boundary pinning of the vortex lines. Such distortions introduce a systematic uncertainty in the determination of $\lambda_{ab}$. As noted earlier, fitting to a theoretical field distribution which assumes the wrong vortex-lattice geometry seriously affects the magnitude of the extracted $\lambda_{ab}$, but has little effect on the T or H-dependence.

The lines in Fig. 8.5 are fits to Eq. (7.3) assuming $p \! = \! 1$. The fitted parameters are: $\lambda_{ab} (0,0) \! = \! 1586$ Å  and $\beta \! = \! 149$ Å/T in the twinned sample (U1) and $\lambda_{ab} (0,0) \! = \! 1699$ Å  and $\beta \! = \! 121$ Å/T in the detwinned sample (U2). To roughly estimate H_c2(0) in YBa₂Cu₃O_6.60, we can use the ratio of the coherence length between this compound and YBa₂Cu₃O_6.95. It was noted earlier, that at $H \! = \! 1.49$ T the fractional volume of the sample occupied by the vortex cores is nearly two times greater in the underdoped compound. This implies that $\xi_{ab}$ in YBa₂Cu₃O_6.60 is $\approx \! \sqrt{2} \!$times the coherence length in YBa₂Cu₃O_6.95. Taking $H_{c2}(0) \! = \! 120$ T for YBa₂Cu₃O_6.95 and assuming $H_{c2} \! \propto \! \xi_{ab}^{-2}$,we may estimate $H_{c2} (0) \! \approx \! 60$ T in YBa₂Cu₃O_6.60. Using H_c2 (0) in place of H_c2 (0.33 T_c) in Eq. (7.4), the coefficient linear in h (i.e $\varepsilon$) at $T \! = \! 0$ is 5.6 and 4.2 in U1 and U2, respectively. On the other hand, for the YBa₂Cu₃O_6.95 compound treated earlier, $\varepsilon \! = \! 9.0$ in both the O1 and O2 samples. The linear dependence of $\lambda_{ab} (H)$ on H when plotted as a function of reduced field is therefore smaller in YBa₂Cu₃O_6.60 than in YBa₂Cu₃O_6.95. It is predicted that the presence of impurities should reduce the field dependence of the penetration depth in the nonlinear supercurrent response of a d_x²-y²-wave superconductor [36]. It appears as though oxygen vacancies in the chains of the underdoped compound have a similar effect.

 

In the analysis of the $\mu$SR spectra for the optimally doped compound it was assumed that $\sigma_f \! \propto \! \lambda_{ab}^{-2}$. As shown in Fig. 8.6, fits to the underdoped compound without this constraint reveal that this is a reasonable approximation. Moreover, the range of values of $\sigma_{f}$ is small, so that the uncertainty introduced by this assumption into the analysis of the YBa₂Cu₃O_6.95 data is also small.

 

The variation of the additional broadening parameter $\sigma_{f}$ with temperature in YBa₂Cu₃O_6.60 (U2) is shown in Fig. 8.7(a) for $H \! = \! 0.50$ T (open circles) and $H \! = \! 1.25$ T (solid circles). The magnitude of $\sigma_{f}$ is nearly the same at the two magnetic fields shown. As in the optimally doped compound, there is a reduction in the value of $\sigma_{f}$ with increasing temperature and the quality of the fits are independent of temperature and magnetic field [see Fig. 8.7(b)]. Figure 8.8(a) shows the temperature dependence of $\langle s^2 \rangle^{1/2}$ calculated from $\sigma_{f}$ at these fields. The field dependence of $\langle s^2 \rangle^{1/2}$ is weaker in YBa₂Cu₃O_6.60 than in the optimally doped compound. When the RMS displacement of the vortex lines from their ideal positions is expressed as a percentage of the intervortex spacing L [see Fig 8.8(b)], the percent disorder at $H \! = \! 0.50$ T agrees well with that for YBa₂Cu₃O_6.95 [see Fig 7.11(b)]. However, the percent disorder is actually slightly larger at $H \! = \! 1.25$ T in the underdoped compound. This may reflect the quasi-2D nature of the vortex lines in YBa₂Cu₃O_6.60. In particular, there is likely some degree of pinning-induced misalignment between the 2D vortex lattices in adjacent layers at the stronger field, so that the vortex lines are no longer straight. Similar values of $\sigma_{f}$ and $\langle s^2 \rangle^{1/2}$were found in the twinned sample U1.

 

 

 

Figure 8.9 shows the temperature dependence of the GL parameter $\kappa$in YBa₂Cu₃O_6.60 (U2) at three magnetic fields. There is far less scatter in the data at the higher fields compared to the optimally doped samples (note that the vertical scale is larger than in Fig. 7.12). This is due to the increased size of the vortex cores in the underdoped compound over the full field range considered. The horizontal lines in Fig. 8.9 are fits to a T-independent $\kappa$. The field dependence of $\kappa$ obtained from these fits and those for the twinned sample (U1) is shown in Fig. 8.10. The values of $\kappa$ in the twinned and detwinned samples agree well enough that a single fit to Eq. (7.5) was performed. The fit gives $\kappa(0) \! = \! 17(2)$ which is larger than in YBa₂Cu₃O_6.95, and $\eta \! = \! 53(9)$ [assuming $H_{c2} (0) \! = \! 60$ T], which is smaller than in YBa₂Cu₃O_6.95 but still considerably greater than in NbSe₂. The larger value of $\kappa$ results from the substantial increase in the penetration depth in the CuO chain direction, whereas the reduced value of the coefficient linear in h stems from the weaker field dependence of $\lambda_{ab}$.

 

Figure 8.11(a) shows $\xi_{ab}$ as a function of temperature in YBa₂Cu₃O_6.60 (U2), whereas Fig. 8.11(b) shows the temperature dependence of r₀ obtained from the J_s (r) profiles. The net change in $\xi_{ab}$ over the temperature range $T \! = \! 0$ to $T \! = \! 30$ K is greater than in YBa₂Cu₃O_6.95, although when $\xi_{ab}$ is plotted as a function of reduced temperature, the term linear in t is weaker. For instance, fitting to the linear relation

$$\xi_{ab} (T) = \xi_{ab} (0) [1 + \epsilon t] \, ,$$ (1)

at $H \! \approx \! 0.5$ T gives $\xi_{ab} (0) \! = \! 68.7$ Å  and $\epsilon \! = \! 0.27$ in YBa₂Cu₃O_6.60, whereas $\xi_{ab} (0) \! = \! 56.7$ Å  and $\epsilon \! = \! 0.37$ in YBa₂Cu₃O_6.95. The larger value of $\xi_{ab}$ in the underdoped material implies that the energy scale of the quasiparticle bound states in the core is reduced. However, in this conventional picture of a vortex core, the smaller value of $\epsilon$ in YBa₂Cu₃O_6.60 contradicts what is expected from the Kramer-Pesch effect alone. If the quantum limit is realized in both of these materials below $T \! = \! 30$ K, the major source of the T-dependence for $\xi_{ab}$may not be the Kramer-Pesch effect at all. In both compounds, $\epsilon$is essentially independent of magnetic field.

 

 

The values of $\xi_{ab} (H)$ and r₀(H) extrapolated to $T \! = \! 0$ are shown in Fig. 8.12 for both the twinned (U1) and detwinned (U2) samples. Note that there is good agreement between both samples, which implies that the twin planes have little effect on the vortex-core size. The magnetic field dependence of $\xi_{ab} (T \! = \! 0)$ in YBa₂Cu₃O_6.60 is shown compared to that in YBa₂Cu₃O_6.95 in Fig. 8.13. The coherence length in the vortex state is larger in the underdoped material at all fields considered. The solid curve through the data points of the optimally doped sample was described in the previous section. The solid curve through the data points for YBa₂Cu₃O_6.60 was obtained by averaging the linear best-fit lines for $\lambda_{ab} (H)$ and $\kappa (H)$ of samples U1 and U2, so that $\xi_{ab} (H) \! = \! 107[1+0.082H]/[1+1.095H]$.As was explained in the discussion for NbSe₂, the curves in Fig. 8.13 should flatten out at low fields, when $\lambda_{ab} \! < \! L$ (i.e. there should be no change in the vortex core radius when the vortices are isolated from each other). From our $\mu$SR results, the coherence length should stop rising below $H \! \approx \! 0.19~T$ in YBa₂Cu₃O_6.95 and below $H \! \approx \! 0.09~T$ in YBa₂Cu₃O_6.60.

 

Let us now return to the discussion of vortex-lattice melting in YBa₂Cu₃O_6.60, which occurs well below the superconducting-to-normal phase transition. As noted earlier, the sudden loss of asymmetry in the $\mu$SR line shape as the temperature is raised may be interpreted as a melting transition for the vortex lattice (see Fig. 8.2). In the melted phase the vortices are mobile, giving rise to a vortex structure with no long range spatial order. As indicated in Fig. 8.3, the melting temperature T_m depends strongly on magnetic field. By monitoring the changes in shape of the measured field distribution with increasing temperature, at several different magnetic fields, we find that the melting transition can be described by  

$$H_m(T_m) \! \approx \! \nu(T_c - T_m)^n \, ,$$ (2)

where $\nu \! = \! 0.059(4)$ T/K, $n \! \approx \! 1$ and T_c is the zero-field critical temperature. Although this gives a power-law exponent of 1, the temperature increments for these measurements were too large to determine T_m any better than $\pm 5$ K. Furthermore, n is usually greater than 1.0 as discussed in section 4.2.

The dimensionality of the ``melted'' vortex lattice is of equal interest here. The question is whether the ``melted'' lattice consists of straight 3D-vortex lines, highly flexible 3D-vortex lines or pancake vortices in which the phase coherence is destroyed across neighboring CuO₂ planes. The strong magnetic field dependence of T_m favours the latter scenario. Within the LD model, the pancake vortices are aligned and weakly coupled between neigboring CuO₂ layers at low temperatures in the absence of disorder. As the temperature is increased, thermal fluctutations of the strings of pancake vortices becomes significant. The displacement of the 2D-pancake vortices within a layer by thermal fluctuations is opposed by the strength of the Josephson-coupling between pancakes in neighboring CuO₂ layers. At low magnetic fields, the vortex-vortex interactions within a layer are weak, so that the displacement of an individual vortex does not necessarily affect the other vortices in the layer. Thus, for small thermal fluctuations a 3D structure is maintained in the majority of the sample. It has been suggested that at low fields, the vortex lattice first melts into a state consisting of mobile 3D vortex lines, which are destroyed upon further increase in the temperature as the pancakes which makeup the vortex lines breakup [110].

On the other hand, at high magnetic fields, the interaction between pancake vortices within a CuO₂ layer is stronger than the interaction between pancake vortices between neighboring layers. In this case, the vortex lattice behaves in a quasi-2D manner, consisting of weakly interacting 2D-vortex lattices in different CuO₂ layers. Thermal fluctuations cause the 2D lattices to move back and forth over top of each other. For larger fluctuations the pancakes within the layers become mobile. The effect of temperature and magnetic field on the $\mu$SR line shape in YBa₂Cu₃O_6.60 is similar to that previously observed in $\mu$SR experiments on Bi₂Sr₂CaCu₂O$_{8+\delta}$ [107,111] and arises here due to a reduction in the $\hat{c}$-axis coherence length relative to that in the optimally doped compound YBa₂Cu₃O_6.95.

 

Pinning will of course also lead to vortex displacements. As discussed earlier, the effects of pinning on the vortex structure in the low and high-field regimes will be considerably different in the pancake-vortex model. Figure 8.14 shows the $\mu$SR line shape in ``twinned'' YBa<sub>2</sub>Cu<sub>3</sub>O<sub>6.60</sub> (U1), obtained by cooling the sample through the transition to low temperature in different magnetic fields. The field distributions at $H \! = \! 0.74$ T and $H \! = \! 1.49$ T exhibit the characteristic features of a 3D-ordered vortex lattice. However, the field distribution which arises from field cooling in a magnetic field of 2.91 T does not exhibit the expected high-field tail for a 3D-ordered vortex lattice, but rather shows a low-field tail. From the melting-line expression obtained above, the lattice should melt at $T_m \! \approx \! 9.5$ K at $H \! = \! 2.91$ T, which is higher than $T \! = \! 2.5$ K in Fig. 8.14. The most natural interpretation is that the 2D-vortex lattices become misaligned with respect to each other due to pinning brought about by pointlike defects within the layers. The origin of the low-field tail in Bi<sub>2</sub>Sr<sub>2</sub>CaCu<sub>2</sub>O$_{8+\delta}$ has been previously attributed to the geometry of the sample [112,211,212]. In a flat rectangular sample, the average magnetic field penetrating near the crystal edges is smaller than that in the center of the sample, due to non-uniform demagnetization. As a result, the density of vortices is smaller near the sample edges. The sample-geometry effect of course also exists when the vortex lattice is 3D and ordered. However, the variations of the internal field distribution due to the vortex lattice ``swamp'' out the effect. Also, in the case of YBa₂Cu₃O_6.95 and at low fields in YBa₂Cu₃O_6.60, the high pinning temperature likely freezes in a nearly uniform density of vortices in the sample, which remains uniform upon cooling. A tiny low-field tail is visible in all of the line shapes shown earlier.

 

The effect of the sample geometry on the measured internal field distribution can occur without vortex pinning--as evidenced by the existence of a large low-field tail in the $\mu$SR line shape for Bi₂Sr₂CaCu₂O$_{8+\delta}$ in the melted phase [107]. The top panel of Figure 8.15 shows the $\mu$SR line shape measured in the twinned sample of YBa₂Cu₃O_6.60 (U1) obtained by field cooling in a magnetic field $H \! \approx \! 2.89$ T to $T \! = \! 2.5$ K, followed by an increase of approximately 0.01 T in the applied field. Note that the small background signal shifts to the new applied field at $H \! \approx \! 2.90$ T, but the signal from the sample does not. This verifies that the vortex lattice is not melted and strongly supports the picture of a dimensional transition which is induced at high magnetic fields in the presence of pinning. When the temperature is raised, the 2D-vortex lattices depin within the layers and the lattice melts (see middle and bottom panels in Fig. 8.15). Although not shown, at $T \! \approx \! 15$ K the lattice is still pinned. Since this temperature is greater than the estimated T_m at this field (i.e. 9.5 K), it is likely that $n \! \gt \! 1$ in the expression for the melting line transition.

Figure 8.16 shows the measured field distributions in ``detwinned'' YBa₂Cu₃O_6.60 (U2) as a function of magnetic field at low temperatures. The asymmetries in the line shapes in the top two panels are similar to those for the twinned sample, showing the features characteristic of a 3D vortex lattice. However, at $H \! = \! 2.99$ T where the line shape should exhibit the features associated with the misaligned 2D-vortex lattices, the line shape still shows a high-field tail. Note that the temperature in the bottom panel is $T \! = \! 10$ K, compared to $T \! = \! 2.5$ K in the bottom panel of Figure 8.14--which makes it even more surprising that the line shape is still asymmetric. Also, when the detwinned sample is field cooled in a magnetic field of 3.4 T to $T \! = \! 10$ K, the measured field distribution still shows a small high-field tail and the background signal can be field shifted. The dimensional crossover observed in the twinned sample should not arise from pinning by the twin boundaries, since they extend the full depth of the sample and displace the 2D-vortex lattices equivalently in all layers. Since the two samples were not from the same batch, the absence of the dimensional crossover in the detwinned sample (U2) at fields comparable to those in the twinned sample (U1), is likely related to differences in the pointlike defects in the samples. In particular, these results suggest that defects in the CuO₂ layers are greater in the twinned sample.