Anisotropic superconducting gaps in ${\mathrm{YNi}}_2{\mathrm{B}}_2\mathrm{C}$: A first-principles investigation

We calculate superconducting gaps and quasiparticle density of states of ${\mathrm{YNi}}_2{\mathrm{B}}_2\mathrm{C}$ in the framework of the density functional theory for superconductors to investigate the origin of highly anisotropic superconducting gaps in this material. Calculated phonon frequencies, the quasiparticle density of states, and the transition temperature show good agreement with experimental results. From our calculation of superconducting gaps and orbital character analysis, we establish that the orbital character variation of the Fermi surface is the key factor of the anisotropic gap. Since the electronic states that consist of mainly Ni $3d$ orbitals couple weakly with phonons, the superconducting gap function is suppressed for the corresponding states, which results in the anisotropy observed in the experiments. These results are hints to increase the transition temperature of materials in the borocarbide family.

Figure 1

Schematic illustration of the energy dependence of $Z$. This function has a strong energy dependence when $|{\xi }_{nk}|$ are equal to or lower than the phonon frequency.

Figure 2

Schematic illustration of the auxiliary energy grid. We use a smooth uniform $k$ grid and dense nonuniform energy grid to solve this gap equation; the latter has much more points in the vicinity of $\xi =0$.

Figure 3

A schematic illustration of the energy grid. It is represented in Eqs. (13) and (14); we can easily control the accuracy by tuning parameters in those equations.

Figure 4

(a) Crystalline structure of ${\mathrm{YNi}}_2{\mathrm{B}}_2\mathrm{C}$ (using vesta [62]). (b) Brillouin zone and $k$ path.

Figure 5

(a) Band structure of ${\mathrm{YNi}}_2{\mathrm{B}}_2\mathrm{C}$. Sizes of red squares, green circles, blue upward triangles, and magenta downward triangles indicate the amount of components of atomic orbitals of Y $4d$, Ni $3d$, B $2s2p$, and C $2s2p$, respectively. (b) Partial and total density of states. The black solid line, the red dashed line, the green dotted line, the blue dashed-dotted line, and the magenta dashed-two dotted line indicate the total Y $4d$, Ni $3d$, B $2s2p$, and C $2s2p$ DOS, respectively.

Figure 6

Fermi velocity of the electronic states on Fermi surfaces. The red, green, and blue region have a high, middle, and low Fermi velocity, respectively. Fermi surfaces in this paper are drawn by using the FermiSurfer [65] program which is developed by us.

Figure 7

Projection of atomic orbitals Y $4d$, Ni $3d$, B $2s2p$, and C $2s2s$ on Fermi surfaces ($|〈{\phi }_{\mathrm{Atom}}|{\phi }_{nk}〉{|}^2$).

Figure 8

Phonon dispersion. The radii of circles indicate magnitude of ${\lambda }_{q\nu }$. Green filled circles indicate results of the neutron diffraction [32].

Figure 9

Phonon dispersion. Sizes of red squares, green circles, blue upward triangles, and magenta downward triangles indicate magnitude of components of Y, Ni, B, and C, respectively, of the displacement pattern.

Figure 10

Electron-phonon renormalization $Z_{nk}$ on Fermi surfaces.

Figure 11

Phonon dispersions and Eliashberg functions computed in three different conditions. The red dashed-dotted line, the blue solid line, and the yellow dashed line indicate those computed by using the GGA functional with the GGA geometry (a geometry optimized with the GGA functional), the LDA functional with the LDA geometry, and the GGA functional with the LDA geometry, respectively. Green and gray filled circles indicate results of the neutron diffraction experiment in Refs. [32, 70], respectively.

Figure 12

Calculated and experimental superconducting transition temperature. Red squares, green triangles, and blue circles indicate the maximum, the averaged, and the minimum superconducting gaps on Fermi surfaces; solid lines are a fit of these gaps with a function $\mathrm{\Delta }\left(T\right)={\mathrm{\Delta }}_0{\{1-{(T/T_{\mathrm{c}})}^p\}}^{1/q}$ via ${\mathrm{\Delta }}_0,T_{\mathrm{c}},p$, and $q$.

Figure 13

Superconducting gap functions ${\mathrm{\Delta }}_{nk}$ on Fermi surfaces at 0.1 K.

Figure 14

The $nk$ dependent Coulomb potential defined in Eq. (28).

Figure 15

Calculated superconducting quasiparticle density of states at 0.1 K and experimental tunnel conductance spectrum at 0.5 K [11].

Figure 16

The electron-phonon renormalization $Z$ calculated by using the bare electron-phonon vertex on Fermi surfaces.