LuNi₂B₂C

LuNi₂B₂C has the crystal structure illustrated in Figure 3.1 with lattice parameters $a = 3.4639(1)\,\textrm{\AA}$ and $c = 10.6313(4)\,\textrm{\AA}$ at temperature $T = 2.3\,\mathrm{^{\circ}C}$, leading to a calculated density of $8.488\,\mathrm{g}\,\mathrm{cm}^{-3}$ [35]. Table 3.1 lists the interatomic distances.

 

Table: Interatomic distances within the LuNi₂B₂C structure [35].

Atom pair	Separation ( $\textrm{\AA}$)
B - C	1.47
Lu - C	2.449
Lu - B	2.855
B - B	2.94
Ni - B	2.10
Ni - Ni (in Ni plane)	2.449

The slightly shorter in-plane nickel separation, in comparison to that of metallic nickel ( $2.50\,\textrm{\AA}$), implies a strong metallic bond. Covalent [36] B - C bonds link the LuC and Ni₂B₂ layers together. As previously mentioned, the electronic properties of LuNi₂B₂C are of a strongly 3D metallic character. Band structure calculations indicate an average Fermi velocity $v_F = 2.3 \times 10^7\,\mathrm{cm}\,\mathrm{s}^{-1}$ [37] and a sharp local maximum in the electronic density of states N(E) near the Fermi surface E_F. Normal state heat capacity  C_P = C_e + C_ph measurements [38] yield a Debye temperature of $\Theta_D = 360(3)\,\mathrm{K}$, related to the phononic contribution $C_{ph} \propto T^3$, and a Sommerfeld constant $\gamma_N = 19.5(3)\,\mathrm{mJ}\,\mathrm{mol}^{-1}\,\mathrm{K}^{-2}$, the coefficient of the electronic term $C_e = \gamma_N T$. The Sommerfeld constant gives rise to an estimated Fermi surface density of states $N(E_F) = 11.8\,\mathrm{mJ}\,\mathrm{mol}^{-1}\,{k_B}^{-2}\, \mathrm{K}^{-2}$ [39]. This, together with an experimentally determined plasma energy of $\hbar\omega_{pl} = 4.0\,\mathrm{eV}$, leads to a Fermi velocity $v_F = \sqrt{\smash[b]{3}} \hbar \omega_{pl} / [e \sqrt{\smash[b]{4 \pi N(0)}}] = 2.76 \times 10^7\,\mathrm{cm}\,\mathrm{s}^{-1}$, in near agreement with band theory. In addition to these normal state properties, LuNi₂B₂C also exhibits various superconducting characteristics.

The superconducting properties of LuNi₂B₂C are intriguing. There exists much experimental evidence both for and against an s-wave pairing state [40]. Scanning tunneling microscopy [41] discloses a bulk energy gap of $\Delta = 2.2\,\mathrm{meV}$, and thermal conductivity measurements [42] detect a large gap anisotropy $\Delta_{max}/\Delta_{min} \ge 10$. The average out of plane upper critical field anisotropy holds constant with temperature at $0.5(H_{c2}^{\langle 100 \rangle} + H_{c2}^{\langle 110 \rangle})/H_{c2}^{\langle 001 \rangle} = 1.16$, as found from magnetisation studies [37]. The slight basal plane anisotropy falls from $H_{c2}^{\langle 100 \rangle}/H_{c2}^{\langle 110 \rangle} = 1.1$ at temperature $T = 4.5\,\mathrm{K}$ to $H_{c2}^{\langle 100 \rangle}/H_{c2}^{\langle 110 \rangle} = 1.0$ by the critical temperature T_c. The initial slope of the upper critical field gives an estimated coherence length $\xi_{BCS} = 130\,\textrm{\AA}$. On the other hand small angle neutron scattering (SANS) [43] extracts a coherence length $\xi_{BCS} = 82(2)\,\textrm{\AA}$ and a penetration depth $\lambda = 1060(30)\,\textrm{\AA}$ at temperature $T = 2.2\,\mathrm{K}$.

One of the most fascinating aspects of the superconducting behaviour of LuNi₂B₂C is the occurrence of field driven transitions in its vortex lattice geometry. The evolution of the flux line lattice symmetry in LuNi₂B₂C, as a function of external field H, is clearly evident through Bitter decoration and SANS. Under weak fields H applied parallel to the crystal $\hat{c}$ axis, the decoration method images a hexagonal to square vortex lattice transition [44][45]. As the magnetic field H climbs from $H = 0.002\,\mathrm{T}$ to $H = 0.02\,\mathrm{T}$, triangular flux line domains enlarge and one of their nearest neighbour directions becomes parallel with the $\langle 110 \rangle$ or $\langle 100 \rangle$ orientations. Raising the magnetic field H distorts the hexagonal configuration and local regions of square geometry appear above $H = 0.06\,\mathrm{T}$. Further magnetic field increase up to $H = 0.1\,\mathrm{T}$ reveals an expanding square proportion co-existing with a heavily distorted triangular phase. At fields H upwards of $H = 0.2\,\mathrm{T}$, SANS records [43] a square vortex lattice which slowly becomes completely amorphous by $H = 6\,\mathrm{T}$. SANS also shows another vortex lattice symmetry transition occurring at an external field of $\mathbf{H} = 0.3\,\mathrm{T}\,\mathbf{\hat{a}}$ [46]. The hexagonal lattice reorients from having a nearest neighbour direction along the $\mathbf{\hat{b}}$ axis at lower fields H to having one along the $\mathbf{\hat{c}}$ axis at higher fields H. Whereas the geometrical transitions taking place in LuNi₂B₂C for applied fields  $\mathbf{H} \parallel \mathbf{\hat{c}}$ arise from nonlocal interactions, those for fields $\mathbf{H} \parallel \mathbf{\hat{a}}$ stem from energy gap $\Delta$ anisotropy [46].

The LuNi₂B₂C sample examined in this $\mu$SR experiment was a single crystal $1.3\,\mathrm{cm}$ in diameter and $1\,\mathrm{g}$ in mass. The crystal grew from a mixture of Ni₂B flux and arcmelted and annealed polycrystalline LuNi₂B₂C as the solution cooled from $1500\,^{\circ}\mathrm{C}$ to $1200\,^{\circ}\mathrm{C}$ over several days [47][48][33]. The sample formed as a plate, with the crystalline $\hat{c}$ axis perpendicular to the plate plane. Thermal conductivity measurements [42] performed on this sample find an upper critical field $H_{c2}(0) \approx 7\,\mathrm{T}$. Its residual resistivity is $\rho_0 = 1.30\,\mu\Omega\,\mathrm{cm}$ and its electron mean free path is $l \approx 500\,\textrm{\AA}$. Resistivity data [48] from similarly grown crystals indicate that this sample should have a critical temperature $T_c = 16.0\,\mathrm{K}$.

The expected Kramer-Pesch effect for the sample studied in this experiment is that the vortex core radius $\rho$ should contract linearly with temperature T on cooling from $T \ll 16.0\,\mathrm{K}$ (= T_c) down to $T \gg 0.7(1)\,\mathrm{K}$ [= T₀, assuming $\xi_{BCS} = 100(20)\,\textrm{\AA}$]. Below the quantum limit temperature $T_0 \approx 0.7\,\mathrm{K}$, the core radius $\rho$ should stay constant at $\rho \sim 4\,\textrm{\AA}$ (=1/k_F). The experimental setup employed to investigate this effect in LuNi₂B₂C is described in the following chapter.