Effect of electron-phonon coupling on the superconducting transition temperature in dodecaboride superconductors: A comparison of ${\text{LuB}}_{12}$ with ${\text{ZrB}}_{12}$

We report a detailed study of specific heat, electrical resistivity, and optical spectroscopy in the superconducting boride ${\text{LuB}}_{12}$ $\left(T_c=0.4\text{K}\right)$, and compare it to the higher $T_c$ compound ${\text{ZrB}}_{12}$ $\left(T_c=6\text{K}\right)$. Both compounds have the same structure based on enclosed metallic Lu or Zr ions in oversized boron cages. The infrared reflectivity and ellipsometry in the visible range allow us to extract the optical conductivity from 6 meV to 4 eV in the normal state from 20 to 280 K. By extracting the superconducting properties, phonon density of states, and electron-phonon coupling function from these measurements, we discuss the important factors governing $T_c$ and explain the difference between the two compounds. The phonon density of states seems to be insignificantly modified by substitution of Zr with Lu. However, the soft vibrations of the metal ions in boron cages, responsible for the relatively high $T_c$ in ${\text{ZrB}}_{12}$, have almost no contribution to the electron-phonon coupling in ${\text{LuB}}_{12}$.

Figure 1

(Color online) (a) Temperature dependence of the reflectivity. The dashed line corresponds to the reflectivity measured by Okamura et al. (Ref. 25). (b) Temperature dependence of the dielectric function measured using ellipsometry. Open symbols represent frequencies where the full temperature dependence is measured.

Figure 2

(Color online) Temperature dependence of the real part of the optical conductivity. dc values from resistivity measurements are shown in the left panel.

Figure 3

(Color online) Total specific heat $C/T$ of ${\text{LuB}}_{12}$ (closed symbols) in the superconducting states (zero-field data) and the normal states (superconductivity has been suppressed by a magnetic field of 1 T) in comparison to ${\text{ZrB}}_{12}$ (open symbols) (Ref. 7), showing the superconducting transitions at 0.42 and 6 K, respectively. The dotted line models a Schottky contribution that we attribute to nuclear moments. 1 gat represents here 1/13 mol.

Figure 4

(Color online) Total specific heat divided by temperature for ${\text{ZrB}}_{12}$ and ${\text{LuB}}_{12}$.

Figure 5

(Color online) Lattice specific heat divided by $T^3$ of ${\text{ZrB}}_{12}$ and ${\text{LuB}}_{12}$ showing its decomposition into Einstein terms. Our model uses a set of Einstein functions where characteristic temperatures are equally spaced on a logarithmic scale.

Figure 6

(Color online) Total resistivity divided by temperature in ${\text{ZrB}}_{12}$ (Ref. 7) (upper panel) and ${\text{LuB}}_{12}$ (lower panel), showing the decomposition into Einstein terms and the residual term. The largest Einstein component in ${\text{ZrB}}_{12}$ is centered on $\omega =170\mathrm{K}$. The corresponding mode in ${\text{LuB}}_{12}$ at $\omega =162\mathrm{K}$ only shows a small amplitude.

Figure 7

(Color online) Electron-phonon transport coupling function ${\alpha }_{\text{tr}}^{2}F\left(\omega \right)$ (closed circles) of (a) ${\text{ZrB}}_{12}$ and (b) ${\text{LuB}}_{12}$, deconvolved from the resistivity, in comparison to the phonon density of states $F\left(\omega \right)$, deconvolved from the specific heat (histogram of rectangles). Fits are performed with $\delta$ functions ${\left({\alpha }_{\text{tr}}^{2}F\right)}_k\delta \left(\omega -{\omega }_k\right)$ on a basis of Einstein frequencies ${\omega }_{k+1}=1.75{\omega }_k$.

Figure 8

(Color online) Low frequency part of the reflectivity data at 25 K for ${\text{ZrB}}_{12}$ and 20 K for ${\text{LuB}}_{12}$ (symbols). Dashed curves are fits to experimental data with a simple Drude model. Solid lines are fits taking into account the electron-phonon interaction given by the transport electron-phonon coupling function presented in Fig. 7.