## 3 Results and Discussion
### Spin and Anomalous Hall Conductivities  
The calculated anomalous (\(\sigma_{xy}^{A}\)) and spin (\(\sigma_{xy}^{S}\)) Hall conductivity, as well as the Hall current conductivity (\(\sigma_{xy}^{H\dagger}\), \(\sigma_{xy}^{H\dagger}\)) and spin polarization (\(P^{D}\), \(P^{H}\)), are listed in Table 4. To obtain the theoretical AHC and SHC, we first calculated the AHC and SHC as a function of the number of \(k\) points (\(N_{k}\)), and the results are displayed in Figure 3. The theoretical AHC and SHC were obtained from \(1/N_{k}=\)0 (\(N_{k}=\infty\)). We can note that Cr\({}_{2}\)CoAl had the largest \(\sigma_{xy}^{A}\); this is because Cr\({}_{2}\)CoAl had the largest spin magnetic moment and the intrinsic anomalous Hall effect was proportional to the spin magnetic moment of a solid. We can also note that the calculation results of AHC and SHC at specific \(k\) points fluctuated in Cr\({}_{2}\)CoAl for the \(XA\) structure. The fluctuation may be owing to the insufficient usage of \(k\) points. There are two main sources of \(k\) points; one for obtaining the self-consistent charge density, and the other for obtaining the AHC and SHC. Table 4 also shows that \(\sigma_{xy}^{S}\) was smaller than \(\sigma_{xy}^{H}\), except for Co\({}_{2}\)CrAl in the inverse structure. In a half metal, the charge current will flow in only one spin channel, with no charge current in the other spin channel, thus, resulting in \(\sigma_{xy}^{A}\) being twice as large as \(\sigma_{xy}^{S}\). To determine how the anomalous Hall conductivities vary with the number of valence electrons, as illustrated in Figure 4, we explored the band structures, anomalous Hall conductivities and the number of valence electrons of \(XA\) Co\({}_{2}\)CrAl and regular Cr\({}_{3}\)Al. Figure 4b,e represent the anomalous Hall conductivity as a function of energy for the Fermi energy. We used so-called frozen band approximation to calculate \(\sigma_{xy}^{A}\) for different Fermi energies, finding that the band structures remained unchanged as the valence electrons increased or decreased. Figure 4b shows that if N\({}_{v}\) is increased, which can be achieved by electron doping of Co\({}_{2}\)CrAl, \(\sigma_{xy}^{A}\) decreased rapidly to around \(-500\) S/cm. There was a large energy range, from 0 to \(-1\) eV, where \(\sigma_{xy}^{A}\) was large, from around 400 to 1400 S/cm. This implies that electron-hole doping of Co\({}_{2}\)CrAl was beneficial to the larger anomalous Hall conductivities. Compared with our previous studies on regular Co\({}_{2}\)CrAl [12], we concluded that this significant change in \(\sigma_{xy}^{A}\) was due to the band pocket at the \(\Gamma\) point. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**X\({}_{2}\)YZ**} & \multirow{2}{*}{**Structure**} & \(\rho\) & \(v_{I}\) & \(v_{t}\) & \(v_{m}\) & \(\Theta_{D}\) \\ \cline{3-6}  & & **g/cm\({}^{3}\)** & & **(m/s)** & & **K** \\ \hline Co\({}_{3}\)Al & regular & 4.397 & 6069 & 3099 & 3472 & 459 \\ Co\({}_{2}\)CrAl & regular & 4.182 & 6816 & 3678 & 4106 & 540 \\  & \(XA\) & 4.135 & 6013 & 3075 & 3445 & 451 \\ Cr\({}_{2}\)CoAl & regular & 3.668 & 8115 & 3502 & 3956 & 504 \\  & \(XA\) & 3.993 & 7164 & 3882 & 4332 & 568 \\ Cr\({}_{3}\)Al & regular & 3.492 & 6866 & 4293 & 4729 & 600 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The calculated density \(\rho\) (g/cm\({}^{3}\)), longitudinal sound speed \(v_{l}\) (m/s), transverse sound speed \(v_{l}\) (m/s), average sound speed \(v_{m}\) (m/s) and Debye temperature \(\Theta_{D}\) (K), d from the calculated elastic moduli. The band pocket (see Figure 2a in [12]) at \(\Gamma\) point was below the Fermi energy in regular Co\({}_{2}\)CrAl, while the Fermi level lies between the spin-up and spin-down band pockets in _XA_ Co\({}_{2}\)CrAl--see Figure 4a--resulting in a large \(\sigma_{xy}^{A}\) away from the Fermi energy. Figure 3: (**a**) Anomalous Hall conductivity and (**b**) spin Hall conductivity for Co\({}_{3-x}\)Cr\({}_{x}\)Al.