Role of chemical disorder in tuning the Weyl points in vanadium doped ${\mathrm{Co}}_2\mathrm{TiSn}$

The lack of time-reversal symmetry and Weyl fermions give exotic transport properties to Co-based Heusler alloys. In the present study, we have investigated the role of chemical disorder on the variation of Weyl points in ${\mathrm{Co}}_2{\mathrm{Ti}}_{1-\mathrm{x}}{\mathrm{V}}_{\mathrm{x}}\mathrm{Sn}$ magnetic Weyl semimetal candidate. We employ the first principle approach to track the evolution of the nodal lines responsible for the appearance of Weyl node in ${\mathrm{Co}}_2\mathrm{TiSn}$ as a function of V substitution in place of Ti. By increasing the V concentration in place of Ti, the nodal line moves toward Fermi level and remains at Fermi level around the middle composition. Further increase of the V content, leads shifting of nodal line away from Fermi level. Density of state calculation shows half-metallic behavior for the entire range of composition. The magnetic moment on each Co atom as a function of V concentration increases linearly up to $x=0.4$, and after that, it starts decreasing. We also investigated the evolution of the Weyl nodes and Fermi arcs with chemical doping. The first-principles calculations reveal that via replacing almost half of the Ti with V, the intrinsic anomalous Hall conductivity increased twice as compared to the undoped composition. Our results indicate that the composition close to the 50% V doped ${\mathrm{Co}}_2\mathrm{TiSn}$ will be an ideal composition for the experimental investigation of Weyl physics.

Figure 1

Total energy vs volume of the unit cell for ${\mathrm{Co}}_2{\mathrm{Ti}}_{1-x}{\mathrm{V}}_x\mathrm{Sn}$ ($x=0.0$, 0.2, 0.4, 0.6, 0.8, 1.0). The Birch-Murnaghan equation of state is used to determine the equilibrium values [66].

Figure 2

(a) Crystal structure of ${\mathrm{Co}}_2\mathrm{TiSn}$. Co atoms are represented by green spheres, Ti atoms by blue spheres, and Sn atoms by yellow spheres. (b) Brillouin zone showing the high-symmetry points and the $k$ path followed in the band structures for the primitive unit cell. (c), (e) Calculated band structures of ${\mathrm{Co}}_2\mathrm{TiSn}$ and ${\mathrm{Co}}_2\mathrm{VSn}$ with spin-orbit coupling in the [110] quantization direction. (d), (f) Calculated band structures of ${\mathrm{Co}}_2\mathrm{TiSn}$ and ${\mathrm{Co}}_2\mathrm{VSn}$ with spin-orbit coupling in the [001] quantization direction. The red circles indicate the location of the crossings in the Co-Y (Y = Ti, V) hybridized $3d$ bands.

Figure 3

Bloch spectral function plots of ${\mathrm{Co}}_2{\mathrm{Ti}}_{1-\mathrm{x}}{\mathrm{V}}_{\mathrm{x}}\mathrm{Sn}$ for the primitive unit cell. The majority and minority spin states are represented by the blue and red lines, respectively. The nodal line can be seen shifting downwards with respect to the Fermi level as the concentration of V increases, tuning with the Fermi level at $x=0.2$ and $x=0.4$.

Figure 4

Fermi surface plots of ${\mathrm{Co}}_2{\mathrm{Ti}}_{1-\mathrm{x}}{\mathrm{V}}_{\mathrm{x}}\mathrm{Sn}$ with chemical potential set at the Fermi energy for the spin polarized calculations without SOC. Majority spin states are presented in blue. Since all the compositions are half-metallic, there are no minority spin states at the Fermi level. On the right is a diagram of the Brillouin zone (BZ) in the conventional unit cell setting showing the high-symmetry lines and points, along with the Fermi surface cross-section (cut through the BZ at $k_z=0$). The dotted shaded square region shows the first BZ.

Figure 5

Density of states of ${\mathrm{Co}}_2{\mathrm{Ti}}_{1-\mathrm{x}}{\mathrm{V}}_{\mathrm{x}}\mathrm{Sn}$ ($x=0.0$, 0.2, 0.4, 0.6, 0.8, and 1.0). For each compound, the positive $y$ axis represents the density of the majority spin states, and the negative $y$ axis represents the minority spin states.

Figure 6

Calculated Weyl nodes and Fermi arcs for the are shown for ${\mathrm{Co}}_2\mathrm{TiSn}$ with chemical potential 278 meV above the Fermi energy. Panel (a) shows the Weyl nodes of two opposite Chern numbers ($\mathrm{WP}+$ and $\mathrm{WP}-$) connected by Fermi arcs in the momentum space in the primitive Brillouin zone (shown in inset). (b) Normalized Berry curvatures show source and sink type of flux and the flow chart of the average position of Wannier charge centers (WCC) obtained by the Wilson-loop method applied on a sphere that encloses the two nodes of opposite chirality.

Figure 7

Calculated Weyl nodes and Fermi arcs are shown for ${\mathrm{Co}}_2\mathrm{VSn}$ with chemical potential 227 meV below the Fermi energy. Panel (a) shows the Weyl nodes of two opposite Chern numbers ($\mathrm{WP}+$ and $\mathrm{WP}-$) connected by Fermi arcs in the momentum space in the primitive BZ. (b) Normalized Berry curvatures show source and sink type of flux and the flow chart of the average position of Wannier charge centers (WCC) obtained by the Wilson-loop method applied on a sphere that encloses the two nodes of opposite chirality.

Figure 8

Calculated Weyl nodes and Fermi arcs are shown for the ${\mathrm{Co}}_2{\mathrm{Ti}}_{0.5}{\mathrm{V}}_{0.5}\mathrm{Sn}$ in the twice supercell in primitive setting. Panel (a) shows the two equivalent pairs of Weyl nodes with two opposite chirality [WP1+ (WP2+) and WP1- (WP2-)] of +1 and $-1$, respectively. Insets show the WCC for the two paris separately, running in the opposite direction in the K-space. Panels (b) and (c) show the normalized Berry curvatures for opposite chirality (source and sink type of flux) for WP1 and WP2, respectively.

Figure 9

(a) Comparison of Wannier interpolated band structure (red) with the full electronic band structure (blue) of ${\mathrm{Co}}_2\mathrm{TiSn}$. The Fermi energy is set to 0 eV. (b) The calculated intrinsic anomalous Hall conductivity at different energies. The conductivity is found to be constant in the vicinity of $E_F$ ($E_F\pm$ 0.2 eV).

Figure 10

Spin-polarized band structures of ${\mathrm{Co}}_2\mathrm{TiSn}$ and ${\mathrm{Co}}_2\mathrm{VSn}$ with (a), (c) calculated lattice parameters and (b), (d) experimental lattice parameters. The black lines represent majority spin states, and the red lines represent minority spin states. The blue circles show the majority band crossings that form nodal lines in the momentum space.

Figure 11

(a) Defect formation energy of ${\text{Co}}_2{\text{Ti}}_{0.5}{\text{V}}_{0.5}\text{Sn}$ varies with the chemical potential of $Ti$. The enthalpy of formation for two end-compounds, ${\text{Co}}_2\text{TiSn}$ and ${\text{Co}}_2\text{VSn}$, are also shown in the same plot. (b) Phonon dispersion relations of ${\text{Co}}_2{\text{Ti}}_{0.5}{\text{V}}_{0.5}\text{Sn}$ crystal along the high-symmetry lines of the optimized structure.

Figure 12

Calculated surface states using slab calculations for the stoichiometric compositions (a) ${\mathrm{Co}}_2\mathrm{TiSn}$ and (b) ${\mathrm{Co}}_2\mathrm{VSn}$ for the [100] surface. The surface spectral function crossing one pair of Weyl points of opposite chirality in the BZ for ${\mathrm{Co}}_2\mathrm{TiSn}$ and ${\mathrm{Co}}_2\mathrm{VSn},$ respectively. The dashed line represents the chemical potentials where the WP's are appearing. The spectral function responsible for showing Fermi arcs are marked in the figure. Fermi energy set at zero in the energy axis.