Kenji
Nawa
*^{a} and
Yoshio
Miura
^{abc}
^{a}Research Center for Magnetic and Spintronic Materials (CMSM), National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan. E-mail: nawa.kenji@nims.go.jp
^{b}Center for Materials Research by Information Integration, National Institute for Materials Science (NIMS), 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan
^{c}Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan
First published on 25th September 2019
A density functional theory (DFT)+U method based on linear response (LR) theory was applied to investigate the electronic structures of a Co-based ternary full Heusler alloy Co_{2}YSi to explore half-metallic (HM) ferromagnets with a wide HM gap. The LR-based DFT+U calculations tend to obtain a reasonable correlation parameter for the Y site, while the correlation of the Co site misdirects to the unphysical ground state due to the overestimated parameter value that arises from the delocalized electronic structure of Co. Furthermore, we found that the HM gap of Co_{2}MnSi originates from the Co orbital in the conduction state and the Co–Mn hybridizing t_{2g} orbital in the valence state around the Fermi energy. This means that the HM gap is a tunable property by selecting the Y element and/or mixing several elements into the Y site through t_{2g} atomic-orbital coupling. Our LR-based DFT+U method was extended to other ternary Co_{2}YSi and quaternary Co_{2}(Y,Mn)Si. We found that Co_{2}(Ti_{0.25},Mn_{0.75})Si and Co_{2}(Fe_{0.25},Mn_{0.75})Si show HM nature, with the Fermi energy being at almost the center of the minority band gap, which leads to high thermal stability.
The spin polarization of electrodes in an MTJ device can be evaluated using the Jullièr model^{1} with a simple formula , where P_{1} and P_{2} are the tunneling spin polarizations of the two ferromagnetic electrodes in the MTJ. For a Co_{2}MnSi MTJ with an aluminum oxide (Al–O) barrier, Sakuraba et al. observed the spin polarization to be over 80%.^{19,20} Then, a high value of 95.4%, which may be close to a fully spin-polarized electronic structure, was reported for the MgO barrier MTJ.^{21} However, P_{1(2)} in the Jullièr formula is not the spin polarization in the bulk system but the polarization of the tunneling electrons in the MTJ. The electronic structure of the MTJ electrode differs from that of the bulk material because the band structure is drastically changed due to the interfacial effect arising from the insulating barrier. The tunneling electrons are also influenced by the spin-filtering effect.^{22} These facts imply that there is difficulty in an accurate estimation of purely bulk spin polarization from the TMR of MTJ.
The point contact Andreev reflection (PCAR) technique has also been performed for spin polarization in several Heusler alloys. The conductance of metallic electrons is measured at a cryogenic temperature to evaluate the spin polarization in PCAR; hence, P_{PCAR}, referred to as PCAR-measured spin polarization, is expressed as . Here, N^{σ}(E_{F}) and v^{σ}_{F} are the DOS at the Fermi energy and the Fermi velocity with spin index σ(=↑ or ↓) in a diffusive regime^{23,24} where the current electrons are not assumed to be ballistic because of the mean-free path being shorter than the point contact size in actual experiments. The v^{σ}_{F} is the conductance of the electrons, but the d orbital localized around the Fermi energy is not dominant in the current electrons. This indicates that the spin polarization originating from the d electron is lost in the measured P_{PCAR}. Previous work reported that the current spin polarization deduced by PCAR is only 59% for Co_{2}MnSi^{25} and around 50% for Co_{2}FeSi.^{26–28}P_{PCAR} = 64% is also observed in quaternary Co_{2}(Fe,Mn)Si.^{29}
Another critical subject to overcome is the large temperature dependence of P.^{19–21,26,30–33} Experimental studies have reported that although an extremely high value of TMR ratio is demonstrated at low temperature in the MTJs consisting of Heusler electrodes and an MgO barrier, a significant reduction in TMR at room temperature is observed: for example, in the Co_{2}MnSi/MgO/Co_{2}MnSi MTJ^{21} the TMR at 4.2 K is 2010%, but it decreases to only 335% at 290 K and in Co_{2}(Fe,Mn)Si/MgO/Co_{2}(Fe,Mn)Si^{34} the TMR at 4.2 K is 2610%, but only 429% at 290 K. From the Jullièr model, the spin polarization P = 95% (98%) at low temperature and decreases to P = 79% (82%) at room temperature for MTJs with a Co_{2}MnSi (Co_{2}(Fe,Mn)Si) electrode. A similar situation occurred in a current-perpendicular-to-plane giant MR (CPP-GMR) device composed of Co_{2}(Fe,Mn)Si electrodes and a nonmagnetic Ag spacer.^{30,32} To explain the strong thermal-dependence of TMR and GMR performances, it is known that the spin-flip inelastic tunneling process induced by magnon excitation lowers P in addition to spin-conserving elastic tunneling at increased temperatures.^{21,31} In this sense, the width of the energy band gap in the minority state is also important in the search for HM materials to improve the weak resistivity with respect to temperature.
Ab initio calculations based on density functional theory (DFT)^{35–37} are expected to play a leading role in the understanding of fundamental electronic and magnetic structures in material design using HM Heusler alloys. In the framework of DFT calculation within local spin density approximation (LSDA), Galanakis et al.^{38} presented an energy diagram of the atomic orbital hybridization of the Co_{2}MnGe system to clarify the mechanism of the HM property; the minority energy band gap at the Fermi level originates from the t_{1u} and e_{u} orbitals, which are formed by the d orbital hybridizations between two Co atoms sitting in different sublattices in a unit cell. Numerous other studies have also been performed using DFT calculations.^{39–44}
However, how to deal with correlation effects is a critical issue in the DFT study of a Heusler compound. The standard DFT calculations based on mean-field approximations, such as LSDA and generalized gradient approximation (GGA), often fail to predict the true ground-state electronic structures due to the presence of d orbital localization in the vicinity of transition metal atoms, making the many-body effect problematic. Various approaches introducing the many-body effect into the DFT scheme have been proposed to recover the correlation problem being missed in LSDA and GGA, e.g., dynamical mean field theory (DMFT),^{45,46} hybrid-type PBE0 functional,^{47} GW approximation,^{48,49} and the DFT+U method.^{50,51} However, the obtained electronic structures strongly depend on the employed method. For example, in Co_{2}MnSi, the LDSA+DMFT calculations, where the dynamical correlation effect such as the spin-flip term is considered quantitatively, were performed on the basis of the linear muffin-tin orbital (LMTO) method^{52} and Korringa–Kohn–Rostoker (KKR) method.^{53} The former indicates that the Fermi energy is found at the conduction edge of the minority state, while the latter indicates that it is found at the valence edge. The calculation using the hybrid PBE0 functional also shows that the Fermi level locates at the valence top with a very big HM gap of ∼2 eV.^{54} The GW calculation,^{55} where the electronic self-energy correction is included by many-body perturbation theory, predicts that the Fermi energy lies between the valence and conduction bands of the minority state. For these approaches, the huge computational cost is also a serious problem; applying it to the MTJ model for properties including interfacial magnetocrystalline anisotropy and spin-dependent transport may be difficult. On the other hand, the DFT+U method,^{50,51} in which parametrized on-site coulomb (U) and exchange (J) interactions for d-orbitals are introduced in the manner of the Hubbard model,^{56,57} is a suitable approach on a practical level. Because of the efficient calculation cost, the DFT+U method can be applied to not only simple bulk materials but also large and realistic systems.
The suitable values of U and J for the DFT+U method are unknown; they depend on the atomic species and surroundings of the atom. A linear response (LR) approach^{58,59} is an advanced way to determine the correlation terms theoretically and to exclude the ad hoc selection of the parameter values. The +U values at respective localized atom sites can be evaluated with low computational costs using the response function of charge density obtained from the standard LSDA or GGA potential. This method has been applied to various correlated systems and succeeded in describing the ground state accurately.^{58–65} A recent study has also reported that the parameters are not transferable among different calculation methods due to non-negligible dependence on computational setups even in theoretically determined values.^{58} This implies that the optimal correlation parameters for the system of interest must be estimated by the method used for the calculation; however, the application of this LR-based DFT+U method to Heusler compounds has been limited to structural phase transitions.^{66}
In the present work, electronic band calculations based on the DFT+U method are carried out for an L2_{1} Co-based full Heusler alloy to address these issues. Focusing on the typical model Co_{2}MnSi, we argue the importance of correlation correction in the alloy and the fundamental electronic structure for clarifying the origin of the minority HM gap. The LR calculations obtain a reasonable correlation parameter for the Y site (Y = Mn in Co_{2}MnSi) and this correction plays an important role for correlated electrons. On the other hand, the correlation for the Co site is unexpectedly overestimated, and thus, the obtained results are misdirected to a rather unphysical ground state. The failure for this unreliable parameter of Co arises from the fact that the 3d electrons of the Co site behave in an itinerant fashion in the alloy, which means that the mean-field approximations such as LSDA and GGA are enough to describe the electronic structure of the Co site with high accuracy. We also revealed an important d-orbital hybridization between Co and Mn that mainly dominates the minority HM gap. The energy diagram proposed in this study suggests that the HM gap is tunable by a selected Y atom and/or mixing several elements into the Y site. The results obtained from the LR-based DFT+U method, where the determined correlation parameter is incorporated into only the strongly correlated Y site, are consistent with the experimental observations. Moreover, this methodology is superior to the standard GGA calculation, especially in terms of electronic and magnetic properties. This study is further extended to the other ternary Co_{2}YSi and quaternary Co_{2}(Y_{x},Mn_{1−x})Si, where some Mn atoms are substituted with 3d transition metal Y (Y = Ti, V, Cr, and Fe) to explore the potential for the HM ferromagnet with a wide band gap. The systematical calculations indicate that the ternary alloys are found to be ordinary ferromagnets, whose minority bands do not have a finite gap at the Fermi energy, but quaternary Co_{2}(Ti,Mn)Si, Co_{2}(V,Mn)Si, and Co_{2}(Fe,Mn)Si alloys have the potential to be an HM material if the composition of Y is appropriately selected.
This paper is organized as follows. In Section 2, the model and computational details are described, and the LR calculation procedures for the correlation parameters are overviewed. Section 3 revisits Co_{2}MnSi. The effective on-site coulomb interaction parameters for Co and Mn are first computed from the LR theory (Section 3.1). The structural parameters, including the equilibrium lattice constant and bulk modulus, are evaluated by standard GGA and GGA+U schemes with LR-determined parameters in Section 3.2. Using the obtained lattice constant, the electronic structures are investigated to clarify the HM origin within the GGA framework (Section 3.3). The understanding of a fundamental band structure in GGA is essential for discussing the effects of correlation correction on Mn and Co, which is given in Section 3.4. The LR-based DFT+U calculations for electronic and magnetic structures are presented and compared with previous theories and experiments in Section 3.5. Finally, in Section 4, systematical results for the other ternary and quaternary compounds are discussed and promising materials for HM ferromagnets are proposed.
The self-consistent DFT calculations were performed via the ab initio package of Quantum-ESPRESSO^{67} by implementing the ultra-soft pseudopotentials parametrized by the scheme of Rappe, Rabe, Kaxiras, and Joannopoulos^{68,69} that were taken from the PS Library. The plane wave basis sets for the wave function and charge density had cutoff energies of 40 and 400 Ry, respectively. The self-consistent procedures were achieved until the iterative total energy difference became less than the convergence criterion of 10^{−8} Ry, by using the Monkhorst–Pack special k-point mesh^{70} of 16 × 16 × 16 in the first Brillouin zone by the Methfessel–Paxton^{71} smearing method with a broadening parameter of 0.02 Ry. Previous works discussed the role of spin–orbit coupling (SOC) in the Heusler alloys and found that the orbital magnetic moment induced by the SOC is completely quenched.^{42,72} It is also reported that although a non-vanishing DOS appears in the minority gap by the effect of SOC, the reduction on P is very small.^{72–74} In present work, therefore, the SOC is not incorporated into all our calculations. Importantly, it is expected that the wide-gap half-metallic Heusler alloy is robust against the non-vanishing DOS arising from the SOC.
The GGA functional formulated by Perdew, Burke, and Ernzerhof^{75} was used for the exchange-correlation term. For the DFT+U method, the choice of “double-counting” correction term is also crucial to subtract the electron coulomb energy that is already included in the LSDA or GGA functional. This correction is conceptually desired to be the same energy contribution as that defined in LSDA or GGA. So far, however, an appropriate prescription for the double-counting term has not been established, but the so-called fully localized limit (FLL),^{76–79} which is also referred as the atomic limit (AL), and around mean-field (AMF)^{50,78} approaches are mostly used. The former functional favors integer electron occupation numbers at a localized site, and thus, might be useful for strongly correlated materials such as insulating oxide systems. The latter might be useful for an intermediate between strongly correlated and itinerant materials. It is still under debate which of the two functionals is a proper approach for Heusler compounds.^{80} In this study, the double-counting functional incorporated in the simplified rotationally invariant form,^{59,81,82} which is equivalent to the FLL approach but J = 0 (or approximately U_{eff} = U − J, where U_{eff} stands for the effective on-site coulomb interaction), was employed. We expect that this approach can easily address the underlying physics of correlated electronic structures, compared to AMF, because the electron-localization limit in FLL corresponds to the concept of the Hubbard model; thus, the scaling of U_{eff} can be simply understood as the strength of electron correlation. The U_{eff} is computed within the LR theory^{58,59} for all transition metal atoms, where we assume that the coulomb interaction is more dominant than the exchange at localized electron sites.
In the framework of LR theory,^{59} the on-site parameter for an atom α, U^{LR(α)}_{eff}, is evaluated from the second derivatives of the total energy functionals as
(1) |
The total energies E^{SCF} and E^{KS} correspond to interacting (fully screened) and non-interacting systems. The second term in eqn (1) is necessary to subtract unphysical contributions in the total energy,^{83–85} which are caused by the conventional exchange-correlation functionals (LSDA and GGA), where the total energy has a curvature for non-integer occupation q_{α} and often misdirects to incorrect energy minima. The total energy derivatives are calculated using the constrained DFT approach:
(2) |
(3) |
The Lagrange multiplier μ_{α} is a local perturbation potential that constrains the occupations n_{α} (i = SCF, KS). In practice, eqn (2) is transformed into a tractable representation where the constraint fields are treated as independent variables by Legendre transformation and the variations of n_{α} with respect to μ_{α} are evaluated.^{59} Using nonlocal linear response matrices
(4) |
eqn (1) is rewritten to obtain U^{LR(α)}_{eff} as
U^{LR(α)}_{eff} = (χ_{KS}^{−1} − χ_{SCF}^{−1})_{αα}. | (5) |
The matrix elements of the response matrices are numerically computed; χ_{SCF} is obtained from the self-consistent (interacting) calculations under the applied local potential μ_{α} and χ_{KS} is obtained from the first iteration in a self-consistent cycle after the end of GGA ground-state calculations – the latter is occupation changes that arise from noninteracting hybridization due to μ_{α}. The LR approach, in principle, requires a response of the electron occupations to the perturbed potentials at a single site in an infinite crystal environment for an accurate U_{eff} evaluation, where all artifacts due to the periodic boundary conditions are excluded.^{58,59}
The correlation parameter for Co is unexpectedly higher than the typically used empirical values, for instance, the U_{eff}(=U − J) of 2.5 eV in full-potential (FP)-LMTO^{86} and that of 2.1 eV in KKR^{53} calculations. The constrained random phase approximation (cRPA) approach^{87} determines a parameter value similar to that determined in our study for Mn (3.07 eV), but determines a value of almost half our LR result for Co (3.28 eV). Table 1 summarizes the numerical data of the d occupations (n_{α}) and the changes (Δn_{α}) induced by μ_{α} in the LR calculations for α atoms (α = Mn and Co). As defined in eqn (4) and (5), the U^{LR}_{eff} is difference of the inversions of electron occupations’ responses with respect to the applied potential shift μ_{α} between the KS and SCF terms. For both the α = Mn and Co cases, the absolute values of Δn_{α} in SCF are smaller than those in KS by one order of magnitude, so the inverted response function of SCF becomes a main factor in the computed correlation parameters. We also find that, for the SCF term, Δn_{Co} is small compared to Δn_{Mn}. Therefore, the unreasonably overestimated parameter for Co originates from the difference in Δn_{Co} of the SCF term. Using the diagonal matrix elements of χ_{KS}^{−1} and χ_{SCF}^{−1} in eqn (5), the parameter for Co is calculated as U^{LR(Co)}_{eff} = −1.10114 − (−7.67152) = 6.570 eV. For Mn, U^{LR(Mn)}_{eff} = −0.98733 − (−4.52196) = 3.535 eV is obtained, where the inverted KS response function’s contribution (the first term) is almost the same as in the U^{LR(Co)}_{eff} case, while the SCF one (the second term) is significantly different.
μ _{α} (eV) | KS | SCF | |||
---|---|---|---|---|---|
n _{α} | Δn_{α} | n _{α} | Δn_{α} | ||
Mn | −0.04 | 6.35202 | 0.02908 | 6.33086 | 0.00792 |
0.00 | 6.32294 | 0.00000 | 6.32294 | 0.00000 | |
0.04 | 6.29357 | −0.02937 | 6.31500 | −0.00794 | |
Co | −0.04 | 8.36249 | 0.02682 | 8.34047 | 0.00480 |
0.00 | 8.33567 | 0.00000 | 8.33567 | 0.00000 | |
0.04 | 8.30830 | −0.02737 | 8.33085 | −0.00482 |
From the above discussion, we conclude that the overestimation of U^{LR}_{eff} for the Co site arises from the fact that the charge density response of Co is insensitive compared to Mn or is still insufficient to evaluate the parameters through the SCF iteration cycles under the applied potential shift.^{88} This can be attributed to the delocalized electronic structures of Co compared to Mn, which originate from the fact that the Co d orbital distribution is spatially-spread due to the d orbital hybridization with first (Mn) and second (Co) neighboring atoms, whereas the Mn d orbital distribution is spatially-narrow due to the d hybridization with only first (Co) neighboring atoms, as discussed in Section 3.3. The localized characters of the Co electronic states compared to those of Mn are consistent with the fact that the spin magnetic moment of Co (1.05 μ_{B}) is much smaller than that of Mn (2.95 μ_{B}). Recently, an extended LR theory^{89} has been proposed to overcome the insufficiency of response of charge density; the second response of charge density is additionally included, which is required for complete cancelation of the electron–electron coulomb interaction (Hartree energy) term changed by the external potential (μ_{α} ≠ 0), which might be canceled incompletely in the current LR calculation. In the present study, the DFT+U method incorporated by the FLL formalism is used, but another approach for solving the failure in estimating U^{LR}_{eff} for a weakly correlated Co site might be the use of the AMF approximation^{50,78} or a hybrid approach,^{90} where the AMF and FLL approximations are linearly interpolated.
(6) |
We present bulk modulus B_{0} and its pressure derivative in Table 2, through comparisons of theoretical literature.^{42,92–95} Among the previous reports, the B_{0} of LSDA is greater than that of GGA. Our GGA result is almost similar to the reported values in B_{0} and , while the GGA+U^{LR}_{Mn} result is slightly smaller in B_{0}. On the other hand, the B_{0} calculated by the GGA+U^{LR}_{Co} and GGA+U^{LR}_{Mn,Co} methods is one order of magnitude smaller than the other calculations. Because the experimental data of the bulk modulus and its derivative are not available for Co_{2}MnSi at this moment, we cannot conclude the validity of our method. However, at least focusing on the lattice constant, these results indicate that the introduction of U^{LR}_{eff} to the Mn atom tends to obtain a reasonable result, as well as GGA, from the comparison with experiments, while the inclusion of U^{LR}_{eff} to Co fails to evaluate the a_{0} of Co_{2}MnSi.
a _{0} (Å) | B _{0} (GPa) | |||
---|---|---|---|---|
a Ref. 92. b Ref. 42. c Ref. 93. d Ref. 94. e Ref. 95. f Ref. 18. | ||||
Present work | GGA | 5.639 | 217.63 | 4.30 |
GGA+U^{LR}_{Mn} | 5.659 | 186.29 | 4.41 | |
GGA+U^{LR}_{Co} | 5.910 | 65.40 | 1.25 | |
GGA+U^{LR}_{Mn,Co} | 6.125 | 60.70 | 5.74 | |
Theory | LSDA | 5.54^{a} | 258.0^{a} | |
GGA | 5.643^{b}, 5.633^{c}, 5.639^{d}, 5.642^{e} | 226^{b}, 212.8^{c}, 214^{d}, 240.89^{e} | 4.680^{c}, 4.674^{d}, 4.983^{e} | |
Experiment | 5.645^{f} |
We again start with the results of the standard GGA calculations to discuss the underlying electronic structure of Co_{2}MnSi. Fig. 4(a) shows the band structures for the minority spin states projected into the Co e_{g(u)} (d_{z2}, d_{x2−y2}) and t_{2g(1u)} (d_{xz}, d_{yz}, d_{xy}), Mn e_{g} (d_{z2}, d_{x2−y2}) and t_{2g} (d_{xz}, d_{yz}, d_{xy}), and Si t_{1u} (p_{x}, p_{y}, p_{z}) orbitals. The lattice constant is set to the theoretically obtained value of 5.639 Å. The d states of Co and Mn are visible around the Fermi energy, while the Si t_{1u} state can be seen only very far from the Fermi energy. To discuss the orbital hybridization mechanism, the eigenstates at the Γ point in the Brillouin zone are focused on. At 0.4 eV above the Fermi energy, the Co state (this state is an anti-bonding state as discussed in the next paragraph) appears but the other orbital components are not included in these eigenstates, which means that Co does not hybridize with the other atomic orbitals. We can find t_{2g} hybridization between Co and Mn that forms a bonding Mn t_{2g} state at −1.4 eV and an anti-bonding Co at 0.3 eV. As a result, a minority band gap originates from the anti-bonding and non-bonding states of the Co atom. Another essential orbital hybridization is found in the t_{1u} symmetry character between Co and Si. The eigenstate components of Co and Si exist at energy levels of 3.8 and 3.9 eV, respectively, so that the Co and Si atoms contribute to the anti-bonding state and to the bonding state (t_{1u}), respectively. We do not mention the e_{g} hybridization between Co and Mn as it has already been discussed previously.^{38}
Fig. 4(b) presents an energy diagram of Co atoms with O_{h} site symmetry. Due to the crystal field, the e_{g} and t_{2g} orbitals are formed and hybridize with the same character orbitals of Co at the other site. These hybridizations arise from the bonding states of e_{g} and t_{2g} orbitals and anti-bonding states of and orbitals. The t_{2g} orbital hybridization, including d_{xz} − d_{xz}, d_{yz} − d_{yz}, and d_{xy} − d_{xy}, is expected to form a π-like bonding in the O_{h} atomic positions, and the e_{g} hybridization, including d_{z2} − d_{z2} and d_{x2–y2} − d_{x2–y2}, is expected to form a σ-like bonding, whose orbital coupling is stronger than that of π-like bonding. Accordingly, the energy gap between bonding e_{g} and anti-bonding states arising from e_{g} hybridization becomes wide compared to that from the bonding t_{2g} and anti-bonding states from t_{2g} hybridization.
Next, the orbital interactions between the first-neighboring atoms are discussed by focusing on T_{d} site symmetry. Before that, we mention here the correlation between the O_{h} and T_{d} site symmetries and the possibility of atomic orbitals hybridizing. The T_{d} site symmetry, which is a subgroup of the O_{h} site symmetry, has the same irreducible representations as the O_{h} site symmetry except for an absence (presence) of inversion symmetry in T_{d} (O_{h}) site symmetry. The Co–Co d orbital’s character in the O_{h} site symmetry can be transformed into the T_{d} notation; the doublet e_{g} and orbitals in O_{h} are represented as the e character in T_{d}, and the triplet t_{2g} and orbitals as t_{2} character. The T_{d} site symmetry also gives the e (d_{z2}, d_{x2–y2}) and t_{2} (d_{xz}, d_{yz}, d_{xy}) characters for Mn and t_{2} (p_{x}, p_{y}, p_{z}) for Si. These augments allow Co–Mn and Co–Si to interact in the atomic orbitals in T_{d} site symmetry, i.e., t_{2g} orbital hybridization of Co–Mn and t_{1u} orbital hybridization of Co–Si in O_{h} site symmetry.
Fig. 4(c) illustrates the possible energy diagram between Co–Co and Mn or Si. The a_{1g} orbital corresponds to the Si s orbital, which does not appear in the band structure of Fig. 4(a) because the energy level is very low. The anti-bonding Co state dominates the highest orbital state in the valence band, which hybridizes with the bonding Mn t_{2g}. The Co is pushed up to quite a higher energy through hybridization with the Si t_{1u} (p) orbital and the non-bonding Co is left at above the Fermi level. This energy diagram, thus, suggests that the main contributions to constructing the minority band gap arise from the t_{2g} coupling of Co and Mn atoms and the t_{1u} orbital of Co no longer contributes to the gap; this conclusion is different from that of a previous study,^{38} where the band gap in the minority state is mostly dominated by Co e_{u} and t_{1u} orbitals (these orbital characters are used in previous work). Instead, more importantly, our diagram proposes that the HM property and electronic structure near the Fermi level can be tuned by selection of the Y atom and/or a mix of several atoms in the Y site through t_{2g} coupling in L2_{1} Heusler alloys.
Even though our diagram differs from the previously reported one,^{38} the 12 valence electrons for Co_{2}MnSi are confirmed to occupy three Co , three Mn t_{2g}, two Co e_{g}, three Si t_{1u}, and one Si a_{1g} orbitals in the down-spin state. This means our diagram satisfies the well-known Slater–Pauling relation:^{38} the magnetic moment of the system, m_{spin}, obeys m_{spin} = N_{val} − 24, where N_{val} is the total number of valence electrons. The calculations obtain a total magnetic moment of 5.01 μ_{B}, which is very close to the integer value expected by the Slater–Pauling rule and in agreement with a previous theory within LSDA^{38} and GGA,^{42,44} as well as experiments,^{96–99} as summarized in Table 3.
Total | Co | Mn | Si | |
---|---|---|---|---|
a FS-KKR: full-potential screened Korringa–Kohn–Rostoker Green’s function method; ASA: atomic sphere approximation; ASW: augmented spherical waves method; FLAPW: full-potential linearized augmented plane wave method; FP-LMTO: full-potential liner muffin-tin orbital method; MLWF: maximally localized Wannier functions; GW: GW approximation. b The U and J values of 3.5 (5.0) and 1.0 (0.9) eV for Co (Mn), respectively, are chosen to reproduce the total spin magnetic moment observed experimentally. c The respective U_{eff} values of 3.28 and 3.07 eV for Co and Mn are determined by cRPA. d The U and J values of 3.0 and 0.9 eV, which have been reported as average values of the determined parameters by theory for pure bulk 3d transition metals, are used. e Sucksmith: Sucksmith ring-balance measurement by Faraday method; SQUID: superconducting quantum interface device magnetometry. | ||||
Present work | ||||
GGA | 5.01 | 1.05 | 2.95 | −0.05 |
GGA+U^{LR}_{Mn} | 5.01 | 0.72 | 3.63 | −0.08 |
GGA+U^{LR}_{Co} | 6.95 | 1.88 | 3.19 | −0.09 |
GGA+U^{LR}_{Mn,Co} | 8.08 | 1.94 | 4.05 | −0.06 |
Theory ^{a} | ||||
FS-KKR^{38} (LSDA) | 4.94 | 1.02 | 2.97 | −0.07 |
ASA-ASW^{44} (GGA) | 5.00 | 0.93 | 3.21 | −0.06 |
FLAPW^{42} (GGA) | 5.00 | 1.06 | 2.92 | −0.04 |
FP-LMTO^{86} (GGA+U^{b}) | 5.00 | 1.08 | 2.97 | −0.08 |
MLWF-FLAPW^{87} (GGA+U^{c}) | 5.00 | 1.05 | 3.01 | −0.06 |
KKR^{53} (LSDA+DMFT^{d}) | 4.97 | |||
FLAPW-GW^{55} (GGA) | 5.00 | |||
Experiment ^{e} | ||||
Sucksmith^{96} | 5.07 | 0.75 | 3.57 | |
Ref. 97 | 5.01 | |||
SQUID^{98} | 4.97 | |||
SQUID^{99} | 5.00 | 0.72 | 3.34 |
Note that the previous study^{38} was carried out for Co_{2}MnGe, where the number of valence electrons is equivalent to that of Co_{2}MnSi. Thus, Co_{2}MnGe is confirmed to be similar to Co_{2}MnSi. The energy diagram obtained from the band structure calculations corresponds to Fig. 4(c) and the integer value of the total spin magnetic moment is calculated (m_{spin} = 5.00 μ_{B}).
We first mention the GGA+U_{Mn} case. The total m_{spin} is constant but the Mn (Co) m_{spin} monotonically increases (decreases) when the correlation parameter for Mn, U^{Mn}_{eff}, increases (see Fig. 5(a)). Note that two of the Co atoms exist in the primitive cell, so the variation of the Co m_{spin} is estimated to be twice as great. The increased m_{spin} of Mn arises from a significant reduction in minority spin electron occupations, as shown in Fig. 5(b). This reflects the following behavior: a large +U value intensifies the coulomb interaction contributions and allows electrons to occupy not the same but different orbitals with parallel spins from Pauli exclusion principles and Hund’s rule, leading to a gain in kinetic energy.
As the t_{2g} orbitals of Co and Mn change the most noticeably depending on the U^{Mn}_{eff} value, we trace modifications in the band structures of these orbitals. Fig. 6(a) presents the minority spin band structures around the Γ point calculated by standard GGA and the GGA+U_{Mn} with small (U^{Mn}_{eff} = 3 eV) and large (6 eV) parameter values. The GGA results indicate that the anti-bonding Co is dominant just below the Fermi energy and the bonding Mn t_{2g} is visible at 1.5 eV in a minority state. Interestingly, increasing the U^{Mn}_{eff} value modifies the spectral weights of the minority components; the Mn and Co orbital weights in bonding and anti-bonding states are almost identical at small U^{Mn}_{eff}, but the anti-bonding becomes dominated by Mn compared to Co and shifts above the Fermi energy at a large U^{Mn}_{eff}. Schematic diagrams are illustrated in Fig. 6(b) and (c). In the majority spin, the valence Mn t_{2g} atomic orbital is shifted to a lower energy by the U^{Mn}_{eff} effect and the anti-bonding state Co is drawn to a lower energy by the hybridization with Mn t_{2g}. On the other hand, in the minority state, the energy level of the Mn valence state becomes higher as U^{Mn}_{eff} increases, and the anti-bonding Co orbital gradually touches the Fermi energy. When the Mn t_{2g} state becomes energetically higher than that of Co at a large U^{Mn}_{eff}, the major component of the anti-bonding is switched from Co to Mn in minority spin.
Surprisingly, U^{Mn}_{eff} shifts the minority occupied state of Mn upward energetically; this shifting is an opposite tendency to the well-known fact of DFT+U study. In general, the +U term opens the band gap with the valence (conduction) state being lower (higher) energy in the insulating and semi-conducting materials regardless of the spin channels. However, ferromagnetic materials, including the Heusler alloy, are different from insulators and semi-conductors because a finite DOS lies at the Fermi energy in ferromagnets. In principle, the total number of valence electrons at each atom site must be preserved even though the +U effect is introduced. Accordingly, the upward shifting in the valence state of the minority Mn d orbital can be understood as follows: the occupations in a spin channel (majority state) increase due to the applied +U effect, but simultaneously, the occupations in the opposite spin (minority state) are reduced to keep the total occupations constant at each atom. This argument is based on the energy diagrams in Fig. 6(b) and (c), and is consistent with the behaviors of the spin magnetic moment and electron occupations at each atomic site [Fig. 5(a) and (b)]; thus, this scenario can be concluded to be behind the effects of +U on the Mn d orbitals.
Secondly, the GGA+U_{Co} result is considered. In the range of U^{Co}_{eff} less than around 4 eV, the increase in the spin magnetic moment at the Co site is not significant, but it suddenly increases afterward [Fig. 5(c)]. In Fig. 7, the U^{Co}_{eff}-dependent electronic band structures and hybridization behaviors of Co and Mn are summarized. In the valence states, it can be seen that the contribution to the anti-bonding is switched from Co to Mn [Fig. 7(a)]. In contrast, the anti-bonding state is switched from Mn to Co [Fig. 7(b)] with increasing U^{Co}_{eff}. The state in Fig. 7(b) moves to a higher energy on increasing U^{Co}_{eff}, but does not hybridize with Mn.
Fig. 7 Dependence of the minority band structures on the varying +U_{Co} parameter values, i.e., GGA (U_{eff} = 0 eV for the Co site), small (3 eV), and large (6 eV), where the projected spectral weights for the (a) Co and Mn t_{2g} states are shown in the left (red) and right (orange) panels, and for the (b) Co and Mn e_{g} states are shown in the left (blue) and right (sky-blue), respectively. Note that the energy ranges in (a and b) are different. Schematic summary of the changes in atomic orbital hybridizations for (c) majority and (d) minority states. Notation is the same in Fig. 6. |
To understand the behavior of the changing m_{spin} and electron numbers in GGA+U_{Co} in Fig. 5(c) and (d), the possible energy diagrams for the majority and minority states are illustrated in Fig. 7(c) and (d). The majority Co simply goes to a lower energy on the introduction of U^{Co}_{eff}, so that the Co d spin-up occupation increases and is saturated at larger U^{Co}_{eff} values (∼7 eV). For the minority state, the d bands’ behaviors of Co and Mn are intricate, but can be understood by going back to the principle view that first attention is paid to the hybridization between Co atoms in different sublattices and then that between the Mn and Co–Co states afterward, as discussed in Section 3.3 and a previous report.^{38} The Co d_{z2} and d_{x2–y2} (d_{xz}, d_{yz}, and d_{xy}) orbitals are pushed up (down) due to U^{Co}_{eff}, and hybridize with the Mn e_{g} (t_{2g}) state [Fig. 7(d)]. Increasing U^{Co}_{eff} affects the energy gap, and most notably, the Co e_{g} orbital becomes an un-occupied anti-bonding state at a large U^{Co}_{eff} value, while it is an occupied bonding state at a small U^{Co}_{eff} (see the blue band of the energy diagram in Fig. 7(d)). This event induces a significant reduction in the minority Co occupations [red down-pointing triangle in Fig. 5(d)], resulting in an increase in the total m_{spin} in the range of U^{Co}_{eff} over ∼4 eV, as shown by black plots in Fig. 5(c).
From the above discussions, the underlying physics of the correlation effects on the magnetic moment can be addressed from the viewpoint of electronic structure for both the GGA+U_{Mn} and GGA+U_{Co} cases. Thus, the consistency of our energy diagram proposed in Fig. 4(c) is demonstrated successfully.
In the GGA-calculated DOS in Fig. 8(a), we can clearly see that the Co d orbital is broad over a wide energy region (from the Fermi energy to 5 eV for the majority state and from 0.5 eV to 4.5 eV for the minority state). Contrarily, the Mn d orbital is relatively localized compared to the Co one and splits into two peaks located around 3 and 1 eV (1.5 and 1.8 eV) in the majority (minority) state, respectively. As expected from Fig. 4(b) and (c), we also confirm the and orbital characters of the Co d states above and below the Fermi energy, as shown by arrows in Fig. 8(a). The value of spin polarization referred to as P_{DOS} is estimated by , where D^{σ}(E_{F}) is the DOS of the majority (σ = ↑) or minority (σ = ↓) spin state at the Fermi energy. A 100% P_{DOS} value is obtained, and the energy band gap in the minority spin state E^{↓}_{gap} is around 0.8 eV. The GGA+U^{LR}_{Mn} calculation modifies the DOS from GGA. The energy level of the Mn occupied (unoccupied) states is shifted to a lower (higher) level due to the exchange splitting induced by the U^{LR(Mn)}_{eff} effect. As a result, the valence and conduction edges are dominated mainly by Co d components and only a few Mn d states appear around the Fermi energy. Due to the presence of a few Co d DOSs at the Fermi energy, the half-metallicity is broken but high spin polarization P_{DOS} = 90.5% is obtained.
By contrast, GGA+U^{LR}_{Co} and GGA+U^{LR}_{Mn,Co} seem to fail to obtain the total magnetic moment reasonably consistently with the experimental observations^{96,99} because of the overestimated U^{LR(Co)}_{eff} parameters (see Table 3). Fig. 8(c) indicates the fact that the exchange splitting arising from the large U^{LR(Co)}_{eff} induces a fully-occupied Co d state in majority spin states, which leads to a Co m_{spin} of 1.88 μ_{B} and total m_{spin} of 6.95 μ_{B}. The energy gap does not appear in the minority channel and the top of the valence states around 2 eV from the Fermi energy is composed of the Mn d orbital of the majority states. Similarly, in the GGA+U^{LR}_{Mn,Co} case, the overestimated value of the total m_{spin} of 8.08 μ_{B} arises from the fact that the majority electrons of Co and Mn are fully occupied at low energy (4 eV and below) through both U^{LR(Mn)}_{eff} and U^{LR(Co)}_{eff}, as shown in Fig. 8(d). In this scheme, the half-metallic electronic structure is broken by a few DOS that are widely broad around the Fermi energy. The spin polarizations are found to be negative and small absolute values: P_{DOS} = −26.97 and 33.82% for the respective GGA+U^{LR}_{Co} and GGA+U^{LR}_{Mn,Co} methods.
From the experimental point of view, hard X-ray photoelectron spectroscopy measurements reported that the valence band structure in the binding energy region from the Fermi energy to ∼1.2 eV (corresponding to −1.2 eV in calculated DOS) is mostly contributed to by Co 3d electrons^{100} as while the Mn d state does exist in this binding-energy region, the number of electrons is very few compared to Co.^{101} Based on the above comparative discussions between our calculations and experiments on the electronic structure and magnetic moment (as well as the equilibrium lattice constant in Section 3.2), we can conclude that the static many-body correlation +U at the Y site (Y = Mn for Co_{2}MnSi) plays an important role in obtaining ground-state properties that are in good agreement with the experiments. On the other hand, the Co d electrons are rather itinerant in the alloy; thus, the LR approach tends to overestimate the correlation parameter for the Co site, which is not reliable for accurate band calculations. In other words, for the Co site, correlation correction may not be necessary and mean-field approximation (GGA or LSDA) is enough to treat the itinerant Co d electrons. Thus, hereafter, all LR-based DFT+U calculations are performed with U^{LR}_{eff} only for the Y site; i.e., correlation correction is excluded for Co. We here explicitly mention that the energy diagram obtained from the GGA+U^{LR}_{Mn} calculation corresponds to Fig. 4(b) and (c), which are obtained from the GGA results.
Calculated total DOSs are shown in Fig. 9(a)–(d), and the results for the spin magnetic moments are summarized in Table 4. Co_{2}TiSi is not HM (P_{DOS} = 25.8%), where the Fermi energy is located at the minority conduction edge state. For Co_{2}VSi and Co_{2}CrSi, a few broad minority DOSs are found around the Fermi energy; thus, the electronic structure is not HM, but the highly spin-polarized values are estimated as P_{DOS} = 98.2 and 89.3%, respectively. On the other hand, negative spin polarization, P_{DOS} = −62.3%, is obtained in Co_{2}FeSi, where the minority DOS is much greater compared to the majority state at Fermi energy. Note that, as the Y atom is changed from a large atomic number (Z_{Fe} = 26) to small (Z_{V} = 23), the Fermi energy position seems to move away from the conduction state of minority spin, but this is not the case for Y = Ti. This exception is attributable to the fact that the Ti spins in Co_{2}TiSi couple with those of Co with anti-parallel direction and the ferrimagnetic structure is obtained in our calculations, while the other systems favor the ferromagnetic structure (see Table 4). Total spin magnetic moments are calculated as 1.89, 3.00, 4.03, and 5.42 μ_{B} for Co_{2}TiSi, Co_{2}VSi, Co_{2}CrSi, and Co_{2}FeSi, respectively.
Fig. 9 (a–d) Total DOS for ternary Co_{2}YSi (Y = Ti, V, Cr, or Fe), and (e–h) total DOS dependence on composition x for quaternary Co_{2}(Y_{x},Mn_{1−x})Si (x = 0.25, 0.5, or 0.75) calculated by the LR-based DFT+U method. In each panel, the upper (bottom) region shows the DOS for up- (down-) spin states, and the Fermi energy is set to zero. Note that the vertical axis range of DOS in (a–d) is different from that in (e–h) as the number of atoms per primitive cell of the ternary system is a quarter of the quaternary one (see Fig. 1). |
N _{val} | Spin magnetic moment | ||||
---|---|---|---|---|---|
Total | Co | Y | Mn | ||
Co_{2}TiSi | 26 | 1.89 | 0.97 | −0.02 | |
Co_{2}(Ti_{0.75}Mn_{0.25})Si | 26.75 | 2.76 | 0.94 | −0.09 | 3.85 |
Co_{2}(Ti_{0.50}Mn_{0.50})Si | 27.5 | 3.54 | 0.90 | −0.21 | 3.75 |
Co_{2}(Ti_{0.25}Mn_{0.75})Si | 28.25 | 4.28 | 0.83 | −0.37 | 3.69 |
Co_{2}VSi | 27 | 3.00 | 1.26 | 0.59 | |
Co_{2}(V_{0.75}Mn_{0.25})Si | 27.5 | 3.51 | 0.84 | 1.19 | 3.75 |
Co_{2}(V_{0.50}Mn_{0.50})Si | 28 | 4.03 | 0.38 | 1.06 | 3.70 |
Co_{2}(V_{0.25}Mn_{0.75})Si | 28.5 | 4.56 | 1.10 | 0.15 | 3.13 |
Co_{2}CrSi | 28 | 4.03 | 0.52 | 2.90 | |
Co_{2}(Cr_{0.75}Mn_{0.25})Si | 28.25 | 4.28 | 0.58 | 2.90 | 3.63 |
Co_{2}(Cr_{0.50}Mn_{0.50})Si | 28.5 | 4.52 | 0.62 | 2.88 | 3.61 |
Co_{2}(Cr_{0.25}Mn_{0.75})Si | 28.75 | 4.77 | 0.67 | 2.94 | 3.63 |
Co_{2}(Fe_{0.25}Mn_{0.75})Si | 29.25 | 5.27 | 0.94 | 2.91 | 3.65 |
Co_{2}(Fe_{0.50}Mn_{0.50})Si | 29.5 | 5.55 | 1.15 | 2.94 | 3.69 |
Co_{2}(Fe_{0.75}Mn_{0.25})Si | 29.75 | 5.58 | 1.25 | 2.95 | 3.75 |
Co_{2}FeSi | 30 | 5.42 | 1.29 | 2.92 |
The structural properties are also investigated as summarized in Table 5. The estimated lattice constants are in good agreement with the experiments^{16,18,102,104} and their error values from the experiments are less than 1% for Co_{2}TiSi, Co_{2}VSi, and Co_{2}FeSi. In Co_{2}CrSi, the lattice constant of 5.694 Å is close to the previous calculation.^{105} The bulk moduli in all models estimated from the LR-based DFT+U method are slightly smaller than those in the previous calculations. This trend is similar to the Co_{2}MnSi case, and might come from the fact that the previous studies were conducted by standard LSDA^{105–109} and GGA.^{92} The experimentally measured B_{0} is available only for Y = Fe (B_{0} = 240 GPa).^{104} From our calculations, the B_{0} and in Co_{2}FeSi are found to be 183.263 GPa and 4.679, respectively. The LSDA calculation^{92} shows a reasonably consistent value of B_{0} = 241.9 GPa with the experiment, although the GGA calculation^{105,107,109} underestimates B_{0} (B_{0} = 203.5–207.1 GPa). Zhu et al.^{109} also performed GGA+U calculations, where the empirical parameters of U = 3.5 and J = 0.9 eV for Co and those of U = 3.4 and J = 0.9 eV for Fe were employed, and obtained B_{0} = 209.3 GPa and (the GGA+U results are not shown in Table 5). Therefore, the LSDA calculations might be suitable for the bulk modulus compared to the GGA+U approaches, while it seems to underestimate the lattice constant from the experiments, for example, a_{0} = 5.52 Å in Co_{2}FeSi.^{92} However, the LR-based DFT+U method provides reasonable results at least for a_{0} values.
Present work | Theory | Experiment | ||||||
---|---|---|---|---|---|---|---|---|
a _{0} (A) | B _{0} (GPa) | a _{0} (A) | B _{0} (GPa) | a _{Expt} (A) | B _{0} (GPa) | |||
a Ref. 106. b Ref. 105. c Ref. 107. d Ref. 92. e Ref. 108. f Ref. 109. g Ref. 16. h Ref. 18. i Ref. 102. j Ref. 104. | ||||||||
Co_{2}TiSi | 5.774 | 189.494 | 4.191 | 5.764^{a} | 204–244.8304^{a}^{,}^{b}^{,}^{c}^{,}^{e} | 4.5151^{e} | 5.743^{g} | |
Co_{2}VSi | 5.667 | 192.408 | 7.485 | 5.7609^{b}, 5.679^{c} | 216^{b}, 221.5^{c} | 5.647^{h} | ||
Co_{2}CrSi | 5.694 | 174.169 | 5.106 | 5.6295^{b}, 5.638^{c} | 227^{b}, 225.3^{c} | |||
Co_{2}FeSi | 5.685 | 183.263 | 4.679 | 5.6431^{b} | 203.5–207.1^{b}^{,}^{c}^{,}^{f} | 4.62^{f} | 5.644^{i}, 5.650^{j} | 240^{j} |
5.52^{d} | 241.9^{d} |
We finally investigate the quaternary Heusler compounds of chemical formula Co_{2}(Y_{x},Mn_{1−x})Si (Y = Ti, V, Cr, or Fe) with a composition x (x = 0.25, 0.50, or 0.75). To model these systems in L2_{1} structure, as illustrated in Fig. 1(b)–(d), cubic primitive cells consisting of 16 atoms are considered. The lattice constant is given by Vegard’s law^{110,111} using the obtained equilibrium lattice constants for Co_{2}YSi (a^{CYS}) and Co_{2}MnSi (a^{CMS}) as a(x) = xa^{CYS} + (1 − x)a^{CMS}. The correlation parameters of the Mn and Y atoms for quaternary systems at all compositions are assumed to be the values of U^{LR(Mn)}_{eff} and U^{LR(Y)}_{eff}, which are determined by the LR theory for ternary Co_{2}MnSi and Co_{2}YSi. For the quaternary compounds, in which the atomic position of the different elements is not symmetric as in the ternary system, the structures are geometrically relaxed under the equilibrium lattice constants by force calculations using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm^{112–115} until the forces acting on each atom are minimized below the criterion of 10^{−3} Ry per bohr.
The calculated m_{spin} values for the quaternary alloys are also available in Table 4. In the case of only Y = Ti, the Ti spins are ferrimagnetically coupled with Co and Mn similar to that in the ternary model. The Mn m_{spin} is a large value over 3 μ_{B} in all systems. Fig. 10 plots the total m_{spin} for ternary and quaternary Co-based full Heusler compounds under study as a function of N_{val} in the system. The Slater–Pauling relation is satisfied in the range of less than 29.5 in N_{val} while being slightly underestimated for the range over N_{val} = 29.5, which corresponds to Co_{2}(Fe_{0.75},Mn_{0.25})Si, and Co_{2}FeSi.
The results for DOS for quaternary alloys are shown in Fig. 9(e)–(h). A perfectly HM electronic structure (P_{DOS} is equal to 100%) is found in Co_{2}(Ti_{0.25},Mn_{0.75})Si, Co_{2}(V_{0.25},Mn_{0.75})Si, Co_{2}(V_{0.50},Mn_{0.50})Si, and Co_{2}(Fe_{0.25},Mn_{0.75})Si. Among them, Co_{2}(V_{0.75},Mn_{0.25})Si has the largest minority band gap E^{↓}_{gap} = 0.5 eV, and thus, this material can be a good candidate for a wide-gap HM ferromagnet. Co_{2}(Ti_{0.25},Mn_{0.75})Si and Co_{2}(Fe_{0.25},Mn_{0.75})Si are also HM candidates because of the advantage in Fermi energy position, as it locates at almost the center of the valence and conduction states in minority states. These HM characters lead to the robustness of spin polarization due to the broadening of valence and conduction states at finite temperature. Nearly HM (P_{DOS} is almost 100%) is found in Co_{2}(Ti_{0.50},Mn_{0.50})Si (P_{DOS} = 99.9%) and Co_{2}(Fe_{0.50},Mn_{0.50})Si (99.4%). Fig. 11 presents the composition dependence of P_{DOS}. Although the Y = Cr system does not show the HM property at each composition, an interesting trend we observed is that a high P_{DOS} is independent of the composition, and Y = V is also the same, whereas a large reduction of P_{DOS} occurs with an increase in x in the other systems, especially for Co(Fe_{x},Mn_{1−x})Si.
Fig. 11 Spin polarization P_{DOS} dependence on composition x in Co_{2}(Y_{x},Mn_{1−x})Si. Red, blue, green, and pink plots are Y of Ti, V, Cr, and Fe, respectively. |
Finally, we state the results of systems including Fe by comparing with previous studies. Whether the electronic structure of the Co_{2}FeSi compound shows HM is still under debate and has been for the past few decades, considering theories both with and without correlation effects.^{17,116} Our LR-based DFT+U calculations indicate that it is not a HM ferromagnet. However, we emphasize that tuning the composition in quaternary Co_{2}(Fe,Mn)Si demonstrates that the electronic structure can be HM. This conclusion is supported by a consistency in anisotropic magnetoresistance (AMR) measurement.^{117} According to an extended model for AMR formulated by Kokado et al., the negative sign of the AMR effect arising from the empty DOS, either spin-up or -down states, at the Fermi level is a signature of HM.^{118–120} Based on this model analysis, positive AMR behavior is found in Co_{2}FeSi^{117}, which indicates a ferromagnetic without the minority band gap, but the negative sign is confirmed in Co_{2}(Fe,Mn)Si,^{121} leading to HM. Note that the composition range of Fe and Mn for Co_{2}(Fe,Mn)Si, showing HM, is different between our study and the AMR experiment, which may be because the present quaternary models [Fig. 1(b)–(d)] are assumed to be a periodic structure not including the disordered properties of Fe and Mn, and/or the ordering parameter of the L2_{1} structure in the experiment^{117} is rather low at all compositions. Nonetheless, we suggest that the quaternary Co_{2}(Fe_{0.25},Mn_{0.75})Si is one of the most promising candidates as an HM Heusler ferromagnet because of the sizable E^{↓}_{gap}(=0.4 eV) and the Fermi energy position being at almost the center of the gap. We believe our present results encourage experiments to improve the degree of crystallinity of bulk Heusler alloys and/or to fabricate a clean interface without any atomic inter-diffusion in MTJ and CPP-GMR devices for the enhancement of MR performances in the future.
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