High spin polarization of the anomalous Hall current in Co-based Heusler compounds

Spin electronics (or spintronics) [1] has recently become an emergent field because of the exciting promise of such spin-transport devices as magnetic field sensors for reading magnetically stored information based on giant magnetoresistance (GMR) [2, 3], spin-valves-based tunneling magnetoresistance (TMR) [4, 5] and spin-torque switching-based magnetic nanodevices such as magnetic random access memories [6, 7]. The materials, which can provide a highly spin-polarized current, are a key ingredient of spintronics. In this context, half-metallic ferromagnets [8], which are characterized by the coexistence of metallic behavior for one spin channel and insulating behavior for the other, are particularly attractive. Their electronic density of states at the Fermi level is completely spin polarized, and thus they could, in principle, offer a fully spin-polarized current. Therefore, since the first prediction of half-metallicity for the half Heusler compound NiMnSb in 1983 [8], intensive research on half-metallic materials has been carried out [9–15]. Indeed, a large number of materials have been predicted to be half-metallic and the half-metallicity of some of these such as CrO₂ with a Curie temperature T_C = 392 K [16] has also been verified experimentally.

Most Co-based full Heusler compounds in the cubic L2₁ structure are ferromagnetic with a high Curie temperature and a significant magnetic moment [10]. In particular, Co₂MnSi has a Curie temperature as high as 985 K and a large magnetic moment of 4.96 μ_B. Furthermore, many of them were predicted to be half-metallic [11, 14, 15] and hence are of particular interest for spintronics. Therefore, the electronic band structure and magnetic properties of the full Heusler compounds have been intensively investigated both theoretically and experimentally in recent years [10, 11, 14, 15]. For example, the total magnetic moments of these materials were found to follow the Slater–Pauling-type behavior and the mechanism was explained in terms of the calculated electronic structures [11]. The Curie temperatures of Co-based Heulser compounds were also determined ab initio and the trends were related to the electronic structures [14].

In this paper, we study the anomalous Hall effect (AHE) and spin polarization of the Hall current in the Co-based full Heusler compounds Co₂XZ (X = Cr and Mn; Z = Al, Si, Ga, Ge, In and Sn) by ab initio calculations of intrinsic anomalous and spin Hall conductivities. The AHE, discovered in 1881 by Hall [17], is an archetypal spin-related transport phenomenon and hence has received renewed attention in recent years [18]. Indeed, many ab initio studies on the AHE in elemental ferromagnets [19–22] and intermetallic compounds [23, 24] have recently been reported. However, first principles investigations into the AHE in half-metallic ferromagnets, which is interesting on its own account, have been very few [25]. Therefore, the principal purpose of this work is to understand the AHE in Co-based full Heusler compounds, especially those of half-metallic ones. The results may also help experimentally search for the Heusler compounds with a large AHE for applications, e.g., in magnetization sensors [26].

The intrinsic AHE is caused by the opposite anomalous velocities experienced by spin-up and spin-down electrons when they move through the relativistic energy bands in solids under the influence of the electric field [18]. In ferromagnets, where an unbalance of spin-up and spin-down electrons exists, these opposite transverse currents would give rise to the spin-polarized charge current (i.e. anomalous Hall current). In non-magnetic materials where the numbers of the spin-up and spin-down electrons are equal, the same process would result in a pure spin current, and this is known as the intrinsic spin Hall effect [27]. Interestingly, the pure spin current is dissipationless [27] and is thus important for the development of low-energy consumption nanoscale spintronic devices [28]. We note that high spin polarization (P) of the charge current (I_C) from the electrode is essential for large GMR and TMR. However, since the current-induced magnetization switching results from the transfer of spin angular momentum from the current carriers to the magnet [6], a large spin current (I_S) would be needed for the operation of the spin-torque switching-based nanodevices [6, 7], i.e. a large ratio of spin current to charge current [$\eta = |(2e/\hbar ) I_{\mathrm { S}}/I_{\mathrm {C}}|$] would be crucial. For ordinary charge currents, this ratio η varies from 0.0 (spin unpolarized current) to 1.0 (fully spin-polarized current). Interestingly, η can be larger than 1.0 for the charge Hall currents and is ∞ for a pure spin current. Excitingly, spin-torque switching of ferromagnets driven by an intense pure spin current from the large spin Hall effect in tantalum has recently been reported [28]. Therefore, because of this and also the topological nature of the intrinsic AHE [18], it might be advantageous to use the Hall current from ferromagnets for magnetoelectronic devices, rather than the longitudinal current. Another purpose of this work is therefore to investigate the nature and spin polarization of the Hall current in Co-based Heusler compounds, knowledge of which is required for possible spintronic applications of the Hall current.

The intrinsic anomalous and spin Hall conductivities of a solid can be evaluated by using the Kubo formalism [19, 21, 22, 29]. The intrinsic Hall effect comes from the static limit (ω = 0) of the off-diagonal elements of the optical conductivity [29]. Here we first calculate the imaginary part of the off-diagonal elements of the optical conductivity. Then we obtain the real part of the off-diagonal elements from the corresponding imaginary part by a Kramers–Kroning transformation. The intrinsic Hall conductivity σ^H_xy is the static limit of the off-diagonal element of the optical conductivity σ⁽¹⁾_xy(ω = 0) (see [22] for more details). We note that the anomalous Hall conductivity (AHC) of bcc Fe [19] and the spin Hall conductivity (SHC) of fcc Pt [30] calculated in this way are in good agreement with that calculated directly by accounting for the Berry phase correction to the group velocity.

Since all the intrinsic Hall effects are caused by the spin–orbit coupling (SOC), first principles calculations must be based on a relativistic band theory. We calculate the relativistic band structure of the Co-based Heusler compounds (Co₂XZ) considered here using the highly accurate full-potential linearized augmented plane wave (FLAPW) method, as implemented in the WIEN2K code [31]. The self-consistent electronic structure calculations are based on the density functional theory (DFT) with the generalized gradient approximation (GGA) for the exchange correlation potential [32]. We consider only the fully ordered cubic Heusler compounds structure (L2₁) and use the experimental lattice constants for all the considered Co₂XZ Heusler compounds except Co₂CrSi, Co₂CrGe and Co₂MnIn, as listed in table 1. Since the lattice constants of Co₂CrSi, Co₂CrGe and Co₂MnIn have not been reported, we have determined their lattice constants theoretically⁴. The SOC is included using the second variation technique [31] with the magnetization along the c-axis in all the present calculations. The wave function, charge density and potential were expanded in terms of the spherical harmonics inside the muffin-tin spheres and the cutoff angular moment ($L_{\max }$) used is 10, 6 and 6, respectively. The wave function outside the muffin-tin spheres was expanded in terms of the augmented plane waves (APW) and a large number of augmented plane waves (about 70 APWs per atom, i.e. the maximum size of the crystal momentum $K_{\max }=8/R_{mt}$) were included in the present calculations. The improved tetrahedron method is used for the Brillouin-zone integration [37]. To obtain accurate ground-state charge density as well as spin and orbital magnetic moments, a fine 36 × 36 × 36 grid with 1240 k-points in the irreducible wedge in the first Brillouin zone was used.

Let us first examine the calculated magnetic properties and band structures near the Fermi level of the considered Co-based Heusler compounds. Since the electronic structure and magnetism in the full Heusler compounds have been extensively studied (see, e.g., [11, 14, 15] and references therein), here we focus on only the salient features which may be related to the anomalous and spin Hall effects as well as spin polarization of the Hall current to be presented in the next two sections. The calculated total spin magnetic moment, local spin and orbital magnetic moments as well as spin-decomposed densities of states (DOSs) at the Fermi level (E_F) of all the considered Co-based Heusler compounds are listed in table 1, together with the available experimental total spin magnetic moments for comparison. The site-decomposed DOSs for the three representative Heusler compounds Co₂CrAl, Co₂MnAl and Co₂MnSi are displayed in figure 1.

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Firstly, table 1 shows that Co₂CrZ (Z = Al, Si and Ge) and Co₂MnZ (Z = Si and Ge) are half-metallic, since the spin-down DOS at the Fermi level [N^↓(E_F)] for these compounds is zero. Therefore, their calculated total spin magnetic moments are almost an integer, as all the half-metallic ferromagnets should be. The tiny deviation of the total spin magnetic moment from an integer for these compounds is due to the inclusion of the SOC in the present calculations. Note that, in principle, including the SOC could induce a non-vanishing N^↓(E_F) in the spin-down band gap of half-metals and hence could destroy the half-metallicity. Nevertheless, the N^↓(E_F) was found to be very small in the Heusler compound NiMnSb although it could be large in heavy element compounds such as MnBi [38]. Indeed, the N^↓(E_F) in the above-mentioned Co₂XZ compounds obtained with the SOC is negligible (see table 1). Co₂CrGa, Co₂MnAl, Co₂MnGa, Co₂MnSn and Co₂MnIn are not half-metallic (see table 1 and figure 1). Nevertheless, in Co₂CrGa, Co₂MnAl and Co₂MnSn, because the E_F is located only slightly below the top of the spin-down valence band (figure 1), the total spin magnetic moment deviates only slightly from an integer (see table 1 and figure 1).

Secondly, we find that the local spin magnetic moments on the 3d transition metal sites, namely Co and X (X = Cr and Mn), are large and coupled ferromagnetically (see table 1). The Co atoms have a spin magnetic moment ranging from 0.7 to 1.0 μ_B, and the spin magnetic moment of the Mn (Cr) atoms is around 3.0  (2.0) μ_B. In contrast, the local spin magnetic moments on the non-transition metal atoms Z (Z = Al, Si, Ga, Ge, In and Sn) are small and aligned antiparallel to that of the Co and X atoms. Not surprisingly, the orbital magnetic moments on the transition metal atoms (Co, Mn and Cr) are rather small, being about two orders of magnitude smaller than the spin magnetic moments, because of the weakness of the SOC in these 3d transition metal compounds. All the non-transition metal (Z) atoms have a practically zero orbital magnetic moment (i.e. ⩽0.0001 μ_B atom⁻¹) and thus are not listed in table 1. The calculated total spin magnetic moments are in good agreement with previous theoretical calculations [11, 14] and also the experimental results [10].

Thirdly, we note that the calculated total spin magnetic moments for all the Heusler compounds more or less follow the so-called n_v-24 rule (n_v is the number of valence electrons), as has already been reported in [11]. For 3d transition metals and their binary compounds, the total spin magnetic moment (m^tot_s) shows the well-known Slater–Pauling behavior [39]. This m^tot_s = n_v-24 rule is essentially a generalized Slater–Pauling rule for the full-Heusler compounds. The reason is that in these compounds, the number of occupied spin-down states (n^↓) is found to remain fixed at 12, at least when they are in the half-metallic state [11]. Therefore, m^tot_s = n_v − 2n^↓ = n_v − 24. For example, the n_v for Co₂CrAl and Co₂CrGa is 27, and the calculated m^tot_s are 3 μ_B. Similarly, the total spin magnetic moment is 4 μ_B for Co₂MnAl, and 5 μ_B for Co₂MnSi, Co₂MnGe, Co₂MnSn, also following this n_v-24 rule. The obvious exceptions are Co₂MnGa and Co₂MnIn because they deviate strongly from the half-metallicity (table 1).

Displayed in figure 1 are the total and site decomposed DOSs of three selected Heusler compounds Co₂CrAl (a), Co₂MnAl (b) and Co₂MnSi (c). Figure 1 shows clearly that Co₂CrAl and Co₂MnSi are half-metallic with the E_F falling in the spin-down insulating gap, while Co₂MnAl is a normal ferromagnetic metal with the E_F being located just below the top of the spin-down valence band. The DOS spectra for Co₂MnAl and Co₂MnSi are, in general, similar except for the location of the Fermi level. As one goes from Co₂MnSi to Co₂MnAl, the n_v decreases by 1 and hence the E_F is lowered from the middle of the spin-down insulating gap to that just below the top of the spin-down valence band. Similar situations occur for Co₂MnGa and Co₂MnGe, and thus their DOS spectra are not shown here. However, pronounced differences in the band structure between Co₂CrAl and Co₂MnAl exist. These differences are mainly caused by the different exchange splittings of the Cr and Mn 3d bands. The Mn atoms have a larger spin moment of ∼3.0 μ_B and hence a larger 3d band exchange splitting, whilst the Cr atoms have a smaller spin moment of ∼2.0 μ_B and hence a smaller exchange splitting (see table 1 and figure 1). Consequently, the Fermi level is located at the center of the spin-up 3d band in Co₂CrAl (figure 1(a)), while the spin-up 3d band in Co₂MnAl is mostly below the Fermi level (figure 1(b)). In short, the Co-based Heusler compounds considered here can be divided into two families, namely Co₂CrZ and Co₂MnZ. Within each family, the electronic structure and other physical properties for one member can be approximately obtained from another member by appropriately shifting the Fermi level. One exception is the pair of Co₂MnSn and Co₂MnIn.

A dense k point mesh would be needed for obtaining accurate anomalous and spin Hall conductivities [19, 21, 29]. Therefore, we use several fine k-point meshes with the finest k-point mesh being 58 × 58 × 58. To obtain the theoretical anomalous and spin Hall conductivities, we first calculate the AHC and SHC as a function of the number (N_k) of k-points in the first Brillouin zone. The calculated anomalous (σ^A_xy) and spin (σ^S_xy) Hall conductivities versus the inverse of the N_k are then plotted and fitted to a first-order polynomial to obtain the converged theoretical σ^A_xy and σ^S_xy (i.e. the extrapolated value at N_k = ∞) (see [21, 22]). The theoretical σ^A_xy and σ^S_xy determined this way are listed in table 2.

Table 2 shows that the calculated σ^A_xy and σ^S_xy are large for Co₂MnAl, Co₂MnGa and Co₂MnIn, but are five to ten times smaller for the other compounds. This suggests that Co₂MnAl, Co₂MnGa and Co₂MnIn may find applications in, e.g., magnetic sensors [26]. Interestingly, the calculated σ^S_xy seems to be about half of the σ^A_xy for every compound considered here except Co₂CrGa. This may be attributed to the half-metallic behavior of these Heusler compounds. In the half-metallic materials, the charge current would flow only in one spin channel, and no charge current in the other spin channel, resulting in σ^A_xy being twice as large as σ^S_xy. We will further discuss this point in the next section.

We note that the difference in the AHC between Co₂MnAl (Co₂MnGa) and Co₂MnSi (Co₂MnGe) is rather dramatic, while the number of valence electrons in Co₂MnAl (Co₂MnGa) differs from that in Co₂MnSi (Co₂MnGe) only by 1 (see table 1). In particular, the AHC in Co₂MnAl (Co₂MnGa) is about six times larger than that in Co₂MnSi (Co₂MnGe). This is somewhat surprising because one would expect a half-metallic metal with a larger magnetization to have a larger AHC than a normal ferromagnet with a smaller magnetization. To better understand this, we display the relativistic band structure as well as the AHC and n_v as a function of the E_F for three selected compounds Co₂CrAl, Co₂MnAl and Co₂MnSi in figure 2. Figures 2(d) and (g) suggest that the band structures of Co₂MnAl and Co₂MnSi are rather similar. The key difference is the location of the E_F due to the difference in n_v in these compounds. In other words, the band structure and other physical properties of Co₂MnSi may approximately be obtained from that of Co₂MnAl by raising the E_F by about 0.5 eV due to one extra p valence electron. Interestingly, there is a pronounced peak sitting at the E_F in the σ^A_xy spectrum of Co₂MnAl. This feature may be attributed to the prominent contributions to the AHC from the spin-down band pocket at the Γ point and also the narrow spin-up bands along the Brillouin zone edges X–W and W–K (see figure 2, and also figure 5 in [25]). These band features also appear in the isoelectronic compound Co₂MnGa (not shown here), which has a large AHC too (table 2). However, when the Fermi level is raised to the highly dispersive Co and Mn d-orbital hybridized spin-up band region to accommodate one more valence electron as one goes from Co₂MnAl (Co₂MnGa) to Co₂MnSi (Co₂MnGe), these band features are now significantly below the E_F (figure 2) with much diminished contributions to the AHC, thus resulting in the much reduced σ^A_xy and σ^S_xy in Co₂MnSi (Co₂MnGe) (table 2). Interestingly, the calculated σ^A_xy is larger in Co₂MnGa than in Co₂MnAl. This difference in the σ^A_xy could be attributed to the larger SOC in the Ga atoms than in the Al atoms. This finding prompted us to calculate the σ^A_xy and σ^S_xy for Co₂MnIn. Unfortunately, the calculated σ^A_xy in Co₂MnIn is even smaller than that of Co₂MnAl.

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However, as described in the preceding section, significant differences in the band structure between Co₂MnAl and Co₂CrAl exist and they are not simply due to the different locations of the E_F in these compounds (figure 2). In particular, the top of the spin-down valence band which sticks out of the Fermi level in Co₂MnAl becomes completely submerged below the Fermi level in Co₂CrAl (figures 2(a) and (d)). These result in rather pronounced differences in the σ^A_xy at the E_F and below. For example, the large broad peak in the σ^A_xy spectrum between −0.8 and 0.0 eV in Co₂MnAl (figure 2(e)) becomes a much reduced narrow peak in Co₂CrAl (figure 2(b)), and Co₂CrAl has a much smaller σ^A_xy value of 241 S cm⁻¹ (table 2). Nevertheless, the calculated σ^A_xy spectra above the E_F in the two compounds are rather similar, both becoming negative at ∼1.0 eV.

The AHC for Co₂CrAl was experimentally found to follow closely the magnetization with the ratio (σ¹_H) of the two quantities being constant over a large temperature range [40], suggesting the intrinsic origin. Nonetheless, the measured AHC conductivity for Co₂CrAl is ∼125 S cm⁻¹ [40], being only about half of our theoretical value (table 2). We note that the measured magnetization is only 1.65 μ_B f.u.⁻¹ [40], being about two times smaller than our theoretical value of 3.0 μ_B f.u.⁻¹ (table 1). Previous experimental studies and ab initio calculations showed that the intrinsic AHC is proportional to the magnetization [23, 40]. Therefore, the smaller experimental AHC may be attributed to the smaller observed magnetization in Co₂CrAl due to the presence of imperfections such as structural disorder [41]. If we use the measured σ¹_H and extrapolate the linear relation σ^A_xy = σ¹_H × M to the theoretical magnetization (M) value, we obtain an 'experimental' σ^A_xy value of 227 S cm⁻¹, being quite close to the theoretical σ^A_xy value of 241 S cm⁻¹. The AHE in Co₂MnAl was recently measured by Vidal et al [26], and the Hall resistivity ρ_xy was found to be about 20 and 15 μΩ cm in clean and dirty conditions, respectively. Using these experimental values together with the longitudinal resistivity of about 100 μΩ cm, one can estimate the AHC to be in the range of 1500–2000 S cm⁻¹. We note that our theoretical σ^A_xy of 1264 S cm⁻¹ for Co₂MnAl is smaller than the estimated experimental values [26] by about 15–40%. The discrepancy could be due to the fact that the Co₂MnAl samples used in the experiments contain mostly the B2 phase [26], instead of the L2₁ structure considered here. Note that the measured AHC values would contain both the intrinsic σ^A_xy and the contributions from extrinsic skew scattering and side jump mechanisms [18]. Therefore, another source of the discrepancy could come from the extrinsic AHC which is not addressed here. In fact, recent ab initio calculations based on the short-range disorder in the weak scattering limit [42] indicated that the scattering-independent side jump contribution is of the order of ∼100 S cm⁻¹ in 3d transition metal ferromagnets and L1₀ FePd and FePt. Similar values of the extrinsic AHC in L1₀ FePd_1−xPt_x alloys were derived recently by comparing the measured AHC with the calculated intrinsic AHC [24, 43].

Our theoretical σ^A_xy value for Co₂MnSi agrees rather well with the theoretical value obtained recently by computing Berry curvatures [25] (table 2). However, for Co₂CrAl, Co₂MnAl and Co₂MnSn, our theoretical σ^A_xy value differs significantly from that reported in [25] by about 30–100% (see table 2). We note that in [25], only 2000 k-points in the Brillouin zone were used and the convergence of the σ^A_xy was reported to be about 20%. Also, here we used the accurate FLAPW method while, in contrast, the atomic spherical wave method with the atomic sphere approximation (ASA) was used in [25]. Since the calculated AHC is sensitive to both the number of k points in the Brillouin zone used and the details of the energy bands in the vicinity of the Fermi level [19, 21, 22, 29], the discrepancies in the calculated σ^A_xy between the two theoretical studies may perhaps be due to the fewer k points in the Brillouin zone and the ASA used in [25].

The AHE has recently received intensive renewed interest mainly because of its close connection with spin-transport phenomena [18]. Indeed, it could be advantageous to use the Hall current from a ferromagnet as a spin-polarized current source, instead of the longitudinal current, as mentioned before. Therefore, it would be interesting to know the spin polarization of the Hall current. The spin polarization P^H of the Hall current may be written as

$$\begin{equation} P^{\mathrm{H}} =\frac{ \sigma^{H\uparrow}_{xy}-\sigma^{H\downarrow}_{xy}}{\sigma^{H\uparrow}_{xy}+\sigma^{H\downarrow}_{xy}}, \end{equation} \tag{ 1 }$$

where σ^H↑_xy and σ^H↓_xy are the spin-up and spin-down Hall conductivities, respectively. The σ^H↑_xy and σ^H↓_xy can be obtained from the calculated AHC and SHC via the relations

$$\begin{equation} \sigma^{\mathrm{A}}_{xy} = \sigma^{H\uparrow}_{xy}+\sigma^{H\downarrow}_{xy}, \end{equation} \tag{ 2a }$$

$$\begin{equation} 2\frac{e}{\hbar}\sigma^{\mathrm{S}}_{xy} = \sigma^{H\uparrow}_{xy}-\sigma^{H\downarrow}_{xy}. \end{equation} \tag{ 2b }$$

The calculated σ^H↑_xy, σ^H↓_xy, P^H and η are listed in table 2. Interestingly, table 2 indicates that unlike the longitudinal charge current, the spin-up and spin-down Hall currents would flow in opposite directions in all the Co-based Heusler compounds considered except Co₂MnIn. Nonetheless, the spin-up Hall conductivity is more than ten times larger than the spin-down Hall conductivity in all these compounds except Co₂CrGa. In other words, the Hall current carriers are mostly of spin-up particles. Remarkably, this gives rise to the nearly 100% spin polarization of the Hall current in all the Heusler compounds except Co₂CrGa (table 2). Furthermore, the spin current to charge current ratio η in these compounds is also very high, being just over 1.0. Therefore, both the P^H and η of the Hall current in all the Co-based Heusler compounds except Co₂CrGa are close to or even better than the corresponding values of the longitudinal current from an ideal half-metallic ferromagnet, even though some of them are not half-metallic. This suggests that the charge Hall current from these compounds is promising for spintronic applications. Surprisingly, the spin-down Hall conductivity is non-negligible even in the half-metallic metals, corroborating that the occupied states well below the E_F would also contribute to the AHE [19].

The spin polarization of a magnetic material is usually described in terms of the spin-decomposed DOSs at the Fermi level as follows:

$$\begin{equation} P^{\mathrm{D}} =\frac{N_{\uparrow}(E_{\mathrm{F}})-N_{\downarrow}(E_{\mathrm{F}})}{N_{\uparrow}(E_{\mathrm{F}})+N_{\downarrow}(E_{\mathrm{F}})}, \end{equation}\noindent \tag{ 3 }$$

where N_↑(E_F) and N_↓(E_F) are the spin-up and spin-down DOSs at the E_F, respectively. The spin polarization P^D would then vary from −1.0 to 1.0 only. For the half-metallic materials, P^D equals either −1.0 or 1.0. The calculated N_↑(E_F) and N_↓(E_F) for the Heusler compounds are listed in table 1, and the corresponding P^D are listed in table 2. In terms of P^D, only Co₂CrAl, Co₂CrSi, Co₂CrGe, Co₂MnSi and Co₂MnGe are half-metallic. Interestingly, we note that in the non-half-metallic metals Co₂MnAl, Co₂MnGa, Co₂MnIn and Co₂MnSn, the P^D is significantly smaller than the P^H, which is close to 100% (see table 2). Therefore, we may perhaps regard these compounds as anomalous Hall half-metals. As pointed out by researchers previously [45], the spin polarization P^D defined by equation (3) is not necessarily the spin polarization of the transport currents measured in the experiments. This can be clearly illustrated by magnetically anisotropic materials such as hcp Co. In hcp Co, the spin-decomposed DOSs at the E_F and hence P^D are independent of the magnetization direction, while, in contrast, the Hall conductivities and hence P^H change dramatically as the magnetization rotates [22]. From the viewpoint of spintronic applications, only the current spin polarizations such as P^H, instead of the P^D, count.

The anomalous and spin Hall conductivities as well as the electronic and magnetic properties of Co-based full Heusler compounds Co₂XZ (X = Cr and Mn; Z = Al, Si, Ga, Ge, In and Sn) have been calculated within the DFT with the GGA. The highly accurate FLAPW method is used. Interestingly, it is found that the spin-up and spin-down Hall currents would flow in opposite directions in all the Co-based Heusler compounds considered except Co₂MnIn, although the spin-up Hall conductivity is more than ten times larger than the spin-down Hall conductivity in these compounds except Co₂CrGa. As a result, the charge Hall current in all the Heusler compounds considered except Co₂CrGa would be almost fully spin-polarized, even though Co₂MnAl, Co₂MnGa, Co₂MnIn and Co₂MnSn are not half-metallic from the viewpoint of the electronic structure. Moreover, the ratio of the accompanying spin current to the charge Hall current is slightly larger than 1.0. Based on these results, therefore, these Heusler compounds may be called anomalous Hall half-metals. These anomalous Hall half-metals, especially Co₂MnAl, Co₂MnGa and Co₂MnIn which have large anomalous and spin Hall conductivities, could find valuable applications in spintronics such as magnetoresistive and spin-torque switching nanodevices as well as spin valves. The calculated electronic band structures and magnetic moments as well as anomalous and spin Hall conductivities as a function of the Fermi level, are used to analyze these interesting findings. It is hoped that these interesting results would stimulate further experimental investigations into the AHE and also the characteristics of the Hall current in these Co-based Heusler compounds.

The authors acknowledge support from the National Science Council and the NCTS of Taiwan, as well as the Center for Quantum Science and Engineering, National Taiwan University (CQSE-10R1004021). They also acknowledge the National Center for High-performance Computing of Taiwan for providing CPU time.