Results and Discussion

The nonlocal model outlined in Section 5.2 fitted the $\mu$SR data on LuNi₂B₂C much better than the local London model used in combination with a triangular vortex lattice [51]. This reflects the presence of a highly ordered square flux lattice in LuNi₂B₂C at temperatures T between $T = 2.6\,\mathrm{K}$ and $T = 10\,\mathrm{K}$, achieved through field cooling under an $H = 1.2\,\mathrm{T}$ field applied parallel to the crystal $\hat{c}$ axis. The behaviour of the penetration depth $\lambda$, the nonlocality parameter C and the core radius $\rho$ in LuNi₂B₂C under these field and temperature conditions emerges readily from the fitted parameters.

 

Figure 6.1(a) displays the behaviour of the fitted penetration depth $\lambda$ with temperature T under an external field  $H = 1.2\,\mathrm{T}$. The fitted penetration depth $\lambda$ increases from $\lambda \approx 950\,\textrm{\AA}$ to $\lambda \approx 1100\,\textrm{\AA}$ over the temperature T range from $T = 2.6\,\mathrm{K}$ to $T = 10\,\mathrm{K}$. The solid curve depicts the temperature dependence anticipated from relation (5.4) and plotted in Figure 5.1. Choosing a zero temperature penetration depth  $\lambda(0)$ to optimally fit this BCS temperature variation to the measured penetration depth $\lambda$ indicates a zero temperature penetration depth $\lambda(0) = 949(8)\,\textrm{\AA}$. The observed penetration depth appears fairly consistent with the exponentially-activated form expected for BCS electron-phonon coupling and s-wave order parameter symmetry in LuNi₂B₂C. However, the error bar size also permits the interpretation of the penetration depth temperature dependence  $\lambda (T)$ as a power law, as would be expected [7] if nodes existed in the energy gap. The penetration depth grows much more weakly with temperature in LuNi₂B₂C than in YBa₂Cu₃O_6.95 [52], known to possess an energy gap with line nodes. While the penetration depth $\lambda$ measurements seem as anticipated, the fitted nonlocality parameter C appears a little more surprising.

Figure 6.1(b) depicts the variation of the fitted nonlocality parameter C with temperature T under an $H = 1.2\,\mathrm{T}$ applied field. As described in Section 5.5, reasonable fits to the data require a nonzero C, demonstrating the importance of nonlocal effects in LuNi₂B₂C. The solid curve in Figure 6.1(b) is a fit to the predicted temperature dependence (5.3) in the clean limit. As expected, the best fit values of the nonlocality parameter C are fairly uniform at low temperatures. At higher temperatures the fitted nonlocality parameter C changes little, rather than dropping monotonically as anticipated. Possible explanations for this slight discrepancy include the presence of impurities and the approximate nature of the field distribution model (5.2). Figure 5.1 illustrates the behaviour of parameter C with temperature, as influenced by the nonlocality radius $\rho_{nl}$, when the LuNi₂B₂C sample is completely free of impurities. As the impurity level rises the nonlocality radius $\rho_{nl}$ becomes temperature independent [12], leading to a weaker temperature dependence in parameter C that better agrees with that observed. In fact, the constancy of the fitted nonlocality parameter Cover the studied temperature range agrees with recent small angle neutron scattering (SANS) data [53] that shows the onset field H₂ of the square to hexagonal vortex lattice symmetry transition essentially holds constant below temperature $T = 10\,\mathrm{K}$. Closer agreement between the anticipated and fitted temperature dependences of parameter C might also result from the inclusion of higher order nonlocal terms in expression (5.2) for the spatial field profile  B(r). Refitting the data with the nonlocality parameter C fixed to its expected clean limit temperature variation (5.3) induces negligible change in the remaining fitted parameters and the inferred core radius $\rho$.

 

 

Table: Fitted penetration depth $\lambda$, effective coherence length $\xi$ and nonlocality parameter C for the field distributions n(B) plotted in Figure 6.2.

London model	Temperature T ( K)	$\lambda$ (Å)	$\xi$ (Å)	C
Nonlocal	10	1100(80)	57(1)	0.25(9)
Nonlocal	2.6	940(30)	28(2)	0.17(4)
Local	2.6	1494(9)	41(1)	0

Figure 6.2 compares the internal magnetic field distributions n(B) calculated for the fitted spatial field profile (5.2) at temperatures  $T = 2.6\,\mathrm{K}$and  $T = 10\,\mathrm{K}$ under an  $H = 1.2\,\mathrm{T}$ external field. The best fit values of the penetration depth $\lambda$, the effective coherence length $\xi$ and the nonlocality parameter C for the distributions n(B) shown in this figure appear in Table 6.1. The shape of the nonlocal field distributions n(B) differs qualitatively from that associated with the traditional London model, as illustrated at temperature  $T = 2.6\,\mathrm{K}$ in the inset to this figure. The distinct difference between these shapes explains the vast improvement in fit quality the nonlocal London model (5.2) achieves over the conventional one, as evident in Figure 5.7. The basic London model emerges from the nonlocal model when the nonlocality parameter C = 0, and, in the case of a square vortex lattice, generates a characteristic low field shoulder in the field distribution n(B). The nonlocal corrections greatly reduce the spectral weight of this shoulder, to the point where at temperature  $T = 2.6\,\mathrm{K}$ the shoulder almost vanishes. They also give rise to the small peak appearing at the lowest field B in the distribution n(B). Such a peak is absent in the local London model, and its presence reflects a flatter spatial field profile at the centre of the square unit cell with a vortex in each corner. The clear rise in the maximum field B of the distribution n(B) as the temperature T falls from  $T = 10\,\mathrm{K}$ to  $T = 2.6\,\mathrm{K}$ reflects the shortening of the vortex core radius $\rho$.

 

Figure 6.3 compares the temperature dependence of the core radius $\rho (T)$ measured in LuNi₂B₂C at external field $H = 1.2\,\mathrm{T}$ to that reported for NbSe₂ [6] ( $T_c = 7.0\,\mathrm{K}$), YBaCu₃O_6.95 [7] ( $T_c = 93.2\,\mathrm{K}$) and YBaCu₃O_6.60 ( $T_c = 59\,\mathrm{K}$) at $H=0.5\,\mathrm{T}$. As the reduced temperature T / T_c rises from T/T_c = 0 to  $T/T_c \approx 0.6$, the core radius $\rho$ in YBaCu₃O_6.95and YBaCu₃O_6.60 remains almost constant, consistent with the attainment of the quantum limit as discussed in Section 2.4. In NbSe₂ the quantum limit temperature T₀ occurs around  $1\,\mathrm{K}$, above which the core radius $\rho$ expands linearly with temperature T, at a rate much slower than anticipated from the proposed Kramer-Pesch effect. The Kramer-Pesch effect in LuNi₂B₂C appears equally weak, with no low temperature saturation in its core size evident over the temperature interval studied. Overlapping the LuNi₂B₂C data with that for NbSe₂determines the zero temperature core radius $\rho(0)$ for LuNi₂B₂C to be $\rho(0) = 64(1)\,\textrm{\AA}$, greatly exceeding the predicted value $\rho(0) \sim 1/k_F = 4\,\textrm{\AA}$. The relation $\rho = \rho(0) [1 + \alpha (T-T_0)/T_c]$, displayed as a solid line in Figure 6.3 with the zero temperature core radius $\rho(0)$ set to  $64\,\textrm{\AA}$, best fits the LuNi₂B₂C data for slope  $\alpha = 0.84(5)$ and quantum limit temperature  $T_0 = 1.0(4)\,\mathrm{K}$. This agrees with the expected quantum limit temperature  $T_0 = 0.7(1)\,\mathrm{K}$ calculated in Section 3.2, and is similar to that recorded for NbSe₂. The almost identical reduced temperature T / T_c dependence of the core radius  $\rho / \rho(0)$ in LuNi₂B₂C and NbSe₂ suggests that quasiparticles in these materials behave in the same way as a function of reduced temperature T / T_c, despite the different dimensionalities of these superconductors. It also implies that longitudinal disorder of vortices has negligible effect on $\mu$SR determinations of the core radius $\rho$. As with $\mu$SR studies of the Kramer-Pesch effect in NbSe₂ and YBaCu₃O $_{7-\delta}$, the weak core shrinkage upon cooling and order-of-magnitude larger than expected zero temperature core radius $\rho(0)$ observed in LuNi₂B₂C highlight the need to include interactions between vortices and their zero point motion in theories concerning the Kramer-Pesch effect.