###### Abstract  
Based on density functional theory, we studied the electronic, magnetic, and mechanical properties of Co\({}_{3-x}\)Cr\({}_{x}\)Al (x = 0, 1, 2, 3) Heusler compounds with the generalized gradient approximation (GGA) for the exchange-correlation potential. In this study, we report two principal spin-related phenomena, namely, the anomalous Hall effect and current spin polarization of the Co\({}_{3-x}\)Cr\({}_{x}\)Al Heusler compounds in the \(L2_{1}\) crystal structure. Heusler compounds, both ideally and inversely ordered, were considered. We found that the calculated magnetic moment of Co\({}_{3-x}\)Cr\({}_{x}\)Al decreased with an increase in the Cr concentration for both ideally and inversely ordered structures, except for Cr\({}_{3}\)Al. We also found that the spin polarization for all Co\({}_{3-x}\)Cr\({}_{x}\)Al was larger than 50%, except for Cr\({}_{2}\)CoAl in the inverse structure. All the considered Heusler compounds were mechanically stable except for the regular Cr\({}_{2}\)CoAl. The Hall current spin polarization was also calculated. We found that Co\({}_{2}\)CrAl in the \(XA\) structure had the largest spin Hall conductivity of 370 (\(l\)S/e cm), and the spin polarization of the induced Hall current was high. first-principles calculation; anomalous Hall conductivity; spin Hall conductivity +
Footnote †: journal: Applied Physics Letters  
0000-0002-22001-8800]S.S.  
crystal structure for half-Heusler compounds are XYZ and C1b, respectively. The chemical formula and crystal structure for full-Heusler compounds are X2 YZ and \(L2_{1}\), respectively. The X and Y atoms are transition metals, while the Z atoms are the main group elements. The original discovery of Heusler compounds occurred with the unveiling of ferromagnetic Cu2 MnAl by [PERSON] in 1903. Since then, Heusler compounds have been investigated for various purposes. The NiMnSb half-Heusler compound with half-metallicity was presented in 1983 [1], and afterward, intensive research on the half-metallic materials was conducted. To date, a large number of materials with half-metallicity have theoretically been predicted [5] or verified experimentally, such as CrO2 [6; 7]. For example, previous studies on Fe2 YSi (Y = Cr, Mn, Fe, Co and Ni), Ti2 YAl, and Ti2 YGa (Y = Co, Fe; Y\({}^{\prime}\)= Cr, Fe) reported theoretical or experimental half-metals [8; 9; 10; 11]. Our previous theoretical studies on full-Heusler alloys showed that most Co-based full-Heusler compounds [12] in the ideal cubic \(L2_{1}\) structure are ferromagnetic and their magnetic moments are significantly large. For example, the spin magnetic moments for Co2 CrAl, Co2 CrSi, Co2 CrGa and Co2 CrGe were 2.998, 3.997, 3.051 and 3.997 \(\mu_{\beta}\), respectively. Furthermore, Co2 CrAl, Co2 CrSi and Co2 CrGe were theoretically predicted to be half-metallic [12]. Accordingly, the electronic band structure and magnetic properties of full-Heusler compounds were intensively investigated both theoretically and experimentally [13] in previous decades. It was found that the total magnetic moments of these Heusler compounds follow Slater-Pauling-type behavior, where the total spin magnetic moments for these Heusler compounds follow a simple rule that relates to the total valence electrons of the Heusler compounds, i.e., \(M_{t}=N_{e}-24\). Let us describe the Heusler structure together with the regular and inverse structures. The ideal X2 YZ full-Heusler alloy is a cubic \(L2_{1}\) structure with point group 225 (Fm3 m symmetric; see Figure 1). It consists of four interpenetrating _fcc_ lattices with two X atoms placed on A(0, 0, 0) and B(1/2, 1/2, 1/2), a Y atom on C(1/4, 1/4, 1/4) and a Z atom on D(3/4, 3/4, 3/4). The two X atoms are environmentally equivalent in the conventional cell, as shown in Figure 1a. In regular full-Heusler compounds, the atomic arrangement is X-Y-X-Z along the diagonal direction of the cubic cell. In this manuscript, we denote this X-Y-X-Z atomic order as the regular structure.