Model for the Internal Field Distribution

The $\mu$SR data is fitted to a London model [11] that incorporates the effects of nonlocality though a kernel  Q(k), an extension described in Section 2.1. As mentioned in that section, the London model is inapplicable within a vortex core. Despite this the theory can yield the vortex lattice structure belonging to high $\kappa$superconductors, such as LuNi₂B₂C, under weak external fields $H \ll H_{c2}$, because in these situations the intervortex spacing greatly surpasses the diameter of a core [10]. LuNi₂B₂C exhibits both square and triangular vortex structures according to the field  H applied, as discussed in Chapter 3. This nonlocal theory is chosen to fit the $\mu$SR data since it accounts for an observed vortex lattice transformation between these two geometries in LuNi₂B₂C, and because the data are collected under low field ( $H \ll H_{c2}$) conditions.

Expanding [11] the BCS kernel Q(k), for weak currents J, to first order in the small term $k^2\xi_{BCS}^2$ leads to the internal spatial field profile B(r) of a nonlocal London model:

$$\mathbf{B}(\mathbf{r}) = \mathbf{\hat{z}}\overline{B} \sum_{\ . . . . . . 2/2)} {1+\lambda^2k^2+\lambda^4C(0.0705k^4 + 0.675k_x^2k_y^2)}$$ (7.1)

where the co-ordinates (x, y, z) coincide with the crystal frame (a, b, c) and the external field  H is in the $\mathbf{\hat{z}}$direction. The average magnetic field inside the superconducting sample is  $\overline{B}$. The constant co-efficients 0.0705 and 0.675 arise from the evaluation of Fermi surface averages of products of Fermi velocities according to LuNi₂B₂C band structure. The Gaussian cutoff factor $\exp( - k^2\xi^2/2)$ in the sum over the reciprocal lattice vectors k of the vortex lattice compensates for the failure of the London approach within the core region. This model neglects vortex-vortex interactions. The parameter C reflects the strength of the nonlocal effects, and contains several poorly known factors.

The nonlocality parameter C varies theoretically with temperature T as [11]

$$C(T) \varpropto \left(\frac{\rho_{nl}(T)}{\lambda(T)}\right)^2$$ (7.2)

where $\rho_{nl}(T)$ is the nonlocality radius, on the order of the BCS coherence length $\xi_{BCS}$ [12]. Figure 5.1 shows the numerically calculated behaviour of the penetration depth $\lambda (T)$, the nonlocality radius $\rho_{nl}(T)$ and the nonlocality parameter C(T)as functions of temperature.
 
The relation [12]

$$\frac{1}{\lambda^2(T)} \varpropto T\Delta^2(T) \sum_{\hbar\om . . . . . . T) > 0} \left\{\Delta^2(T) + [\hbar\omega(T)]^2\right\}^{-3/2}$$ (7.3)

where $\hbar\omega(T) = \pi k_B T(2n+1)$ are the Matsubara frequencies and n is a non-negative integer, gives the expected temperature dependence of the penetration depth $\lambda$ in the clean limit. Solving [13] the equation

$$\frac{1}{N(0)V} = \int_0^{k_B \Theta_D} \frac {\tanh\left[\fr . . . . . . xi^2 + \Delta^2(T)}\right]} {\sqrt{\xi^2 + \Delta^2(T)}}\,d\xi$$ (7.4)

determines the BCS energy gap $\Delta(T)$, where $k_B\Theta_D$ is the Debye energy, N(0) is the density of states at the Fermi level for electrons of one spin orientation, and V describes the strength of the interaction potential for scattering a Cooper pair. Evaluating the above integral at temperature T = T_c reveals the constant N(0)V to be

(7.5)

where $\gamma = 0.5772\dots$ is Euler's constant. The calculations plotted in Figure 5.1 assume a critical temperature $T_c = 16\,\mathrm{K}$ and a Debye temperature $\Theta_D = 360\,\mathrm{K}$, values appropriate for LuNi₂B₂C. The expression [12]

$$\rho_{nl}^2(T) \varpropto \frac {\displaystyle\sum_{\hbar\ome . . . . . . > 0}\left\{\Delta^2(T) + [\hbar \omega(T)]^2\right\}^{-3/2}}$$ (7.6)

generates the clean limit behaviour of the nonlocality radius $\rho_{nl}(T)$ shown in this figure. The above sums over $\hbar\omega(T)$ converge by $\hbar \omega(T) \approx 2k_B\Theta_D$. Combining the temperature dependence of the penetration depth  $\lambda (T)$ and the nonlocality radius $\rho_{nl}(T)$ according to (5.3) produces the temperature variation of the internal field model parameter C, displayed in Figure 5.1. The next section describes the software utilised to fit this internal field model to the experimental data.