Linear-response-based DFT+U method for exploring half-metallic Co-based full Heusler alloys

TL;DR

This paper introduces a linear-response-based DFT+U method to accurately investigate and tune the half-metallic electronic structures of Co-based full Heusler alloys, revealing key orbital contributions and stability factors.

Contribution

The study develops and applies a linear-response DFT+U approach to Co-based Heusler alloys, providing insights into their electronic structures and methods to tune their half-metallicity.

Findings

The LR-based DFT+U method yields reasonable correlation parameters for Y sites.

The Co site correlation can lead to unphysical ground states if overestimated.

The half-metallic gap in Co2MnSi originates from specific Co orbitals and hybridizations.

Abstract

The density functional theory (DFT)+U method based on the linear response (LR) theory was applied to investigate the electronic structures of Co-based ternary full Heusler alloy Co$_2Y$Si for exploring half-metallic (HM) ferromagnets with a wide HM gap. The LR-based DFT+U calculations tend to obtain a reasonable correlation parameter for $Y$ site, while the correlation of Co site misleads 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 Co $e_u$ orbital in the conduction state and 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$_2Y$Si 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.

Figures

Tables

Table 1. Table 1: Numerical data of LR calculations for U eff LR ​ ( α ) superscript subscript 𝑈 eff LR 𝛼 U_{\mathrm{eff}}^{\mathrm{LR(}\alpha\mathrm{)}} parameters ( α = 𝛼 absent \alpha= Mn or Co); d 𝑑 d occupation numbers ( n α subscript 𝑛 𝛼 n_{\alpha} ) and changes ( Δ ​ n α Δ subscript 𝑛 𝛼 \Delta n_{\alpha} ) of the on-site α 𝛼 \alpha atom from neutral state ( μ α = 0 subscript 𝜇 𝛼 0 \mu_{\alpha}=0  eV) of KS and SCF terms when the perturbed potential is applied to the on-site α 𝛼 \alpha atom ( μ α ≠ 0 subscript 𝜇 𝛼 0 \mu_{\alpha}\neq 0  eV). The results are obtained from the 3 × 3 × 3 3 3 3 3\times 3\times 3 supercell, in which the well-converged parameters are computed.

	${\mu }_{\alpha }$(eV)	KS		SCF
		$n_{\alpha }$	$\Delta n_{\alpha }$	$n_{\alpha }$	$\Delta n_{\alpha }$
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	$-0.00482$

Table 2. Table 2: Structural parameters of lattice constant a 0 subscript 𝑎 0 a_{0} , bulk modulus B 0 subscript 𝐵 0 B_{0} , and its pressure derivative B 0 ′ superscript subscript 𝐵 0 ′ B_{0}^{\prime} for Co 2 MnSi. The U eff LR superscript subscript 𝑈 eff LR U_{\mathrm{eff}}^{\mathrm{LR}} values determined by the LR approach are employed in the present study: U eff LR ​ ( Mn ) = 3.535 superscript subscript 𝑈 eff LR Mn 3.535 U_{\mathrm{eff}}^{\mathrm{LR(Mn)}}=3.535  eV for GGA+U LR Mn superscript subscript absent Mn LR {}_{\mathrm{Mn}}^{\mathrm{LR}} , U eff LR ​ ( Co ) = 6.570 superscript subscript 𝑈 eff LR Co 6.570 U_{\mathrm{eff}}^{\mathrm{LR(Co)}}=6.570  eV for GGA+U LR Co superscript subscript absent Co LR {}_{\mathrm{Co}}^{\mathrm{LR}} , and both for GGA+U LR Mn , Co superscript subscript absent Mn Co LR {}_{\mathrm{Mn,Co}}^{\mathrm{LR}} .

		$a_0$ (Å)	$B_0$ (GPa)	$B_0^{\prime }$
Present work	GGA	5.639	217.63	4.30
	GGA+U${}_{}{}^{\mathrm{LR}}{}_{\mathrm{Mn}}{}^{}$	5.659	186.29	4.41
	GGA+U${}_{}{}^{\mathrm{LR}}{}_{\mathrm{Co}}{}^{}$	5.910	65.40	1.25
	GGA+U${}_{}{}^{\mathrm{LR}}{}_{\mathrm{Mn},\mathrm{Co}}{}^{}$	6.125	60.70	5.74
Theory	LSDA	5.54a	258.0a
	GGA	5.643b , 5.633c , 5.639d , 5.642e	226b , 212.8c , 214d , 240.89e	4.680c , 4.674d , 4.983e

Table 3. Table 3: Total and atom-resolved magnetic moments (in μ B subscript 𝜇 B \mu_{\mathrm{B}} ) for Co 2 MnSi with the comparison of present and previous theories as well as experiments. The representations in the present paper are the same as those in Table  2 . The first column gives the calculation methods (and types of exchange-correlation functionals in parentheses) for theory and measurement techniques for experiment.

	Total	Co	Mn	Si	Ref.
Present work
GGA	5.01	1.05	2.95	$-0.05$
GGA+U${}_{}{}^{\mathrm{LR}}{}_{\mathrm{Mn}}{}^{}$	5.01	0.72	3.63	$-0.08$
GGA+U${}_{}{}^{\mathrm{LR}}{}_{\mathrm{Co}}{}^{}$	6.95	1.88	3.19	$-0.09$
GGA+U${}_{}{}^{\mathrm{LR}}{}_{\mathrm{Mn},\mathrm{Co}}{}^{}$	8.08	1.94	4.05	$-0.06$
Theorya
FS-KKR (LSDA)	4.94	1.02	2.97	$-0.07$	[prb-66-174429-2002, ]
ASA-ASW (GGA)	5.00	0.93	3.21	$-0.06$	[prb-72-184415-2005, ]
FLAPW (GGA)	5.00	1.06	2.92	$-0.04$	[prb-66-094421-2002, ]
FP-LMTO (GGA+Ub )	5.00	1.08	2.97	$-0.08$	[jmmm-422-13-2017, ]
MLWF-FLAPW (GGA+Uc )	5.00	1.05	3.01	$-0.06$	[prb-88-134402-2013, ]
KKR (LSDA+DMFTd )	4.97				[jphysDapplphys-42-084002-2009, ]
FLAPW-GW (GGA)	5.00				[prb-86-245115-2012, ]
Experimente
Sucksmith	5.07	0.75	3.57		[jpcs-32-1221-1971, ]
	5.01				[jmmm-937-54-1986, ]
SQUID	4.97				[prb-74-104405-2006, ]
SQUID	5.00	0.72	3.34		[prb-80-144405-2009, ]

Table 4. Table 4: Nominal number of valence electrons N val subscript 𝑁 val N_{\mathrm{val}} and calculated spin magnetic moments of total and atom-resolved contributions (in unit of μ B subscript 𝜇 B \mu_{\mathrm{B}} ) for Co Y 2 subscript 𝑌 2 {}_{2}Y Si ( Y = 𝑌 absent Y= Ti, V, Cr, or Fe) and Co 2 ( Y x subscript 𝑌 𝑥 Y_{x} ,Mn 1-x )Si ( x = 𝑥 absent x= 0.25, 0.50, or 0.75). Results are obtained from the LR-based DFT+U method.

	$N_{\mathrm{val}}$	Spin magnetic moment
		Total	Co	$Y$	Mn
Co2TiSi	26	1.89	0.97	$-0.02$
Co2(Ti0.75Mn0.25)Si	26.75	2.76	0.94	$-0.09$	3.85
Co2(Ti0.50Mn0.50)Si	27.5	3.54	0.90	$-0.21$	3.75
Co2(Ti0.25Mn0.75)Si	28.25	4.28	0.83	$-0.37$	3.69
Co2VSi	27	3.00	1.26	$0.59$
Co2(V0.75Mn0.25)Si	27.5	3.51	0.84	$1.19$	3.75
Co2(V0.50Mn0.50)Si	28	4.03	0.38	$1.06$	3.70
Co2(V0.25Mn0.75)Si	28.5	4.56	1.10	$0.15$	3.13
Co2CrSi	28	4.03	0.52	$2.90$
Co2(Cr0.75Mn0.25)Si	28.25	4.28	0.58	$2.90$	3.63
Co2(Cr0.50Mn0.50)Si	28.5	4.52	0.62	$2.88$	3.61
Co2(Cr0.25Mn0.75)Si	28.75	4.77	0.67	$2.94$	3.63
Co2(Fe0.25Mn0.75)Si	29.25	5.27	0.94	$2.91$	3.65
Co2(Fe0.50Mn0.50)Si	29.5	5.55	1.15	$2.94$	3.69
Co2(Fe0.75Mn0.25)Si	29.75	5.58	1.25	$2.95$	3.75
Co2FeSi	30	5.42	1.29	$2.92$

Table 5. Table 5: Structural parameters of lattice constant a 0 subscript 𝑎 0 a_{0} , bulk modulus B 0 subscript 𝐵 0 B_{0} , and its pressure derivative B 0 ′ superscript subscript 𝐵 0 ′ B_{0}^{\prime} for Co Y 2 subscript 𝑌 2 {}_{2}Y Si compared with the present study, previous calculations, and experiments. Results of the present study are obtained from the LR-based DFT+U method with the parameters of U eff LR ​ ( Y ) = 2.942 , superscript subscript 𝑈 eff LR 𝑌 2.942 U_{\mathrm{eff}}^{\mathrm{LR(}Y\mathrm{)}}=2.942, 3.979, 3.169, and 3.922 eV for Y = 𝑌 absent Y= Ti, V, Cr, and Fe, respectively. Previous calculation results are from GGA, except for the bottom row for Co 2 FeSi that are from LSDA.

	Present work			Theory			Experiment
	$a_0$ (A)	$B_0$ (GPa)	$B_0^{\prime }$	$a_0$ (A)	$B_0$ (GPa)	$B_0^{\prime }$	$a_{\mathrm{Expt}}$ (A)	$B_0$ (GPa)
Co2TiSi	5.774	189.494	4.191	5.764a	204–244.8304b,a,e,c	4.5151e	5.743g
Co2VSi	5.667	192.408	7.485	5.7609b , 5.679c	216b , 221.5c		5.647h
Co2CrSi	5.694	174.169	5.106	5.6295b , 5.638c	227b , 225.3c
Co2FeSi	5.685	183.263	4.679	5.6431b	203.5–207.1b,f,c	4.62f	5.644i, 5.650j	240j
				5.52d	241.9d

Equations

$$U_{\mathrm{eff}}^{\mathrm{LR}(\alpha )}=\frac{{\partial }^2{E}^{\mathrm{SCF}}[\{q_{\alpha }\}]}{\partial q_{\alpha }^2}-\frac{{\partial }^2{E}^{\mathrm{KS}}[\{q_{\alpha }\}]}{\partial q_{\alpha }^2}.$$

$$E^i[\{q_{\alpha }\}]=\underset{n(\mathbf{r}),{\mu }_{\alpha }}{\min }\left\{E_{\mathrm{GGA}}^i[n(\mathbf{r})]+\sum _{\alpha }{\mu }_{\alpha }(n_{\alpha }-q_{\alpha })\right\},$$

$$\frac{\partial }{\partial q_{\alpha }}{E}^i[\{q_{\alpha }\}]=-{\mu }_{\alpha },\frac{{\partial }^2}{\partial q_{\alpha }^2}{E}^i[\{q_{\alpha }\}]=-\frac{\partial {\mu }_{\alpha }}{\partial q_{\alpha }}.$$

$$({\chi }_{\mathrm{SCF}}{)}_{\beta \alpha }=\frac{\partial n_{\beta }}{\partial {\mu }_{\alpha }},({\chi }_{\mathrm{KS}}{)}_{\beta \alpha }=\frac{\partial n_{\beta }^{\mathrm{KS}}}{\partial {\mu }_{\alpha }},$$

$$U_{\mathrm{eff}}^{\mathrm{LR}(\alpha )}={\left({\chi }_{\mathrm{KS}}^{-1}-{\chi }_{\mathrm{SCF}}^{-1}\right)}_{\alpha \alpha }.$$

$$E(V)=E_0+\frac{B_{0}V}{B_0^{\prime }}\left[\frac{1}{B_0^{\prime }-1}{\left(\frac{V_0}{V}\right)}^{B_0^{\prime }}+1\right]-\frac{B_0}{B_0^{\prime }-1}{V}_0,$$

Full text

††preprint: APS/123-QED

I introduction

A key property in emerging field of spintronics is the so-called half-metallicity (HM); the majority and minority states are completely spin-polarized at the Fermi level, where a finite density of states (DOS) exists for majority spin and an energy band gap is opened for minority spin. For example, the use of HM materials as ferromagnetic electrodes in magnetic tunnel junctions (MTJs) is a straightforward way to enhance tunneling magnetoresistance (TMR) ratio pla-54-225-1975 , leading to high-performance spintronics applications such as non-volatile magnetic random access memories and read-head of ultrahigh-density hard-disk drives. The family of Co-based full Heusler alloys has received considerable attention, as some of these have a potential to possess a high spin polarization ($P$) or ultimately HM ($P=100$ %) in addition to a high Curie temperature, e.g., 985 K for Co2MnSi jmmm-38-1-1983 and 1100 K for Co2FeSi prb-72-184434-2005 ; apl-88-032503-2006 .

The spin polarization of electrodes in an MTJ device can be evaluated using the Jullièr model pla-54-225-1975 with a simple formula $\mathrm{TMR}=\frac{2P_{1}P_{2}}{1-P_{1}P_{2}}\times 100~{}(\%)$, where $P_{1}$ and $P_{2}$ are the tunneling spin polarizations of two ferromagnetic electrodes in the MTJ. For Co2MnSi MTJ with an aluminum oxide (Al-O) barrier, Sakuraba et al. observed the spin polarization to be over $80~{}\%$. jjap-44-L1100-2005 ; apl-88-192508-2006 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 prb-94-094428-2016 . However, $P_{1(2)}$ in the Julliér formula is not the spin polarization in the bulk system but the polarization of tunneling electrons in the MTJ. The electronic structure of 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 spin-filtering effect. prb-63-054416-2001 These facts imply that there is difficulty in an accurate estimation of purely bulk spin polarization from the TMR of MTJ.

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 cryogenic temperature to evaluate the spin polarization in PCAR; hence, $P_{\mathrm{PCAR}}$, referred to as PCAR-measured spin polarization, is expressed as $P_{\mathrm{PCAR}}=\frac{\langle N^{\uparrow}(E_{\mathrm{F}})v_{\mathrm{F}}^{\uparrow}\rangle-\langle N^{\downarrow}(E_{\mathrm{F}})v_{\mathrm{F}}^{\downarrow}\rangle}{\langle N^{\uparrow}(E_{\mathrm{F}})v_{\mathrm{F}}^{\uparrow}\rangle+\langle N^{\downarrow}(E_{\mathrm{F}})v_{\mathrm{F}}^{\downarrow}\rangle}\times 100~{}(\%)$. Here, $N^{\sigma}(E_{\mathrm{F}})$ and $v^{\sigma}_{\mathrm{F}}$ are the DOS at Fermi energy and Fermi velocity with spin index $\sigma$ ($=\uparrow$ or $\downarrow$) in a diffusive regime sci-282-85-1998 ; prb-75-054404-2007 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 $v^{\sigma}_{\mathrm{F}}$ is conductance of electrons, but the $d$ orbital localized around Fermi energy is not dominant in the current electron. This indicates that the spin polarization originating from the $d$ electron is lost in the measured $P_{\mathrm{PCAR}}$. Previous works reported that the current spin polarization deduced by PCAR is only 59 % for Co2MnSi jap-105-063916-2009 and around 50 % for Co2FeSi apl-89-082512-2006 ; jap-102-043903-2007 ; prb-87-220402R-2013 . $P_{\mathrm{PCAR}}=64~{}\%$ is also observed in quaternary Co2(Fe,Mn)Si. prb-91-104408-2015

Another critical subject to overcome is large temperature dependence of $P$. jjap-44-L1100-2005 ; apl-88-192508-2006 ; jmsj-38-45-2014 ; apl-93-112506-2008 ; prb-94-094428-2016 ; apl-89-082512-2006 ; apl-101-252408-2012 ; prb-93-134403-2016 Experimental studies have reported that although an extremely high value of TMR ratio is demonstrated at low temperature in the MTJs consisting of the Heusler electrodes and MgO barrier, a significant reduction in TMR at room temperature is observed, for example, the TMR of 2010 % at 4.2 K, but it decreases to only $335~{}\%$ at 290 K in Co2MnSi/MgO/Co2MnSi MTJ prb-94-094428-2016 and 2610 % at 4.2 K, but only 429 % at 290 K in Co2(Fe,Mn)Si/MgO/Co2(Fe,Mn)SijphysDapplphys-48-164001-2015 . From the Jullièr model, the spin polarization $P=95~{}\%$ (98 %) at low temperature decreases to $P=79~{}\%$ (82 %) at room temperature for MTJs with Co2MnSi (Co2(Fe,Mn)Si) electrode. Similar situation occurred in a current-perpendicular-to-plane giant MR (CPP-GMR) device composed of Co2(Fe,Mn)Si electrodes and nonmagnetic Ag spacer. jmsj-38-45-2014 ; apl-101-252408-2012 For explaining the strong thermal-dependence of TMR and GMR performances, it is known that spin-flip inelastic tunneling process induced by magnon excitation lowers $P$ in addition to spin-conserving elastic tunneling at increased temperature. apl-93-112506-2008 ; prb-94-094428-2016 In this sense, a 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.

The ab-initio calculations based on DFT pr-136-b864-1964 ; pr-140-a1133-1965 ; rmp-71-1253-1999 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. prb-66-174429-2002 presented an energy diagram of atomic orbital hybridization of Co2MnGe system to clarify the mechanism of HM property; the minority energy band gap at Fermi level originates from the $t_{1u}$ and $e_{u}$ orbitals, which are formed by the $d$ orbital hybridizations between two Co atoms sitting at different sublattices in a unit cell. Numerous other studies have also been performed by DFT calculations. prl-50-2024-1983 ; jphysFmetphys-16-L211-1986 ; prb-53-1146-1996 ; jpsj-64-2152-1995 ; prb-66-094421-2002 ; jap-93-6844-2003 ; prb-72-184415-2005

However, a 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) jphysCondMat-9-7359-1997 ; prb-57-6884-1998 , GW approximation prb-78-155112-2008 ; prl-102-126403-2009 , and DFT+U method prb-44-943-1991 ; jphysCondMat-9-767-1997 . However, obtained electronic structures strongly depend on the employed method. For example, in Co2MnSi, 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 rmp-80-315-2008 and Korringa-Kohn-Rostoker (KKR) method jphysDapplphys-42-084002-2009 . The former indicates that the Fermi energy is found at the conduction edge of the minority state, while the latter is found at the valence edge. The GW calculation prb-86-245115-2012 , where the electronic self-energy correction is included by the 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 prb-44-943-1991 ; jphysCondMat-9-767-1997 , in which parametrized on-site Coulomb ($U$) and exchange ($J$) interactions for $d$-orbital are introduced in the manner of the Hubbard model ProcRSocA-285-542-1965 ; ProcRSocA-296-82-1967 , 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 prb-71-035105-2005 ; prb-97-035117-2018 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 using the response function of charge density obtained from the standard LSDA or GGA potential with low computational costs. This method has been applied to various correlated systems and succeeded in describing the ground state accurately. prb-97-035117-2018 ; prb-71-035105-2005 ; prb-70-235121-2004 ; prb-84-115108-2011 ; prb-89-085103-2014 ; prb-92-075124-2015 ; jpcs-120-4919-2016 ; prb-94-035136-2016 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. prb-97-035117-2018 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 transition jphysCondMat-24-185501-2012 .

In the present work, the 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 Co2MnSi, we argue the importance of correlation correction in the alloy and the fundamental electronic structure for clarifying the origin of minority HM gap. The LR calculations obtain a reasonable correlation parameter for the $Y$ site ($Y=$ Mn in Co2MnSi) and this correction plays an important role for correlated electrons. On the other hand, the correlation for Co site is unexpectedly overestimated, and thus, the obtained results are misled to a rather unphysical ground state. The failure of an unreliable parameter of Co arises from the fact that the $3d$ electrons of Co site behave itinerary in the alloy, which means that the mean-field approximations such as LSDA and GGA are enough to describe the electronic structure of 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 $Y$ 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${}_{2}Y$Si and quaternary Co2($Y_{x}$,Mn1-x)Si, where a part of Mn is 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 Co2(Ti,Mn)Si, Co2(V,Mn)Si, and Co2(Fe,Mn)Si alloys have a potential to be HM material if the composition of $Y$ is appropriately selected.

This paper is organized as follow. In Sec. II, the model and computational details are described, and the LR calculation procedures for the correlation parameters are overviewed. Sec. III revisits Co2MnSi. The effective on-site Coulomb interaction parameters for Co and Mn are first computed from the LR theory (Sec. III.1). The structural parameters, including equilibrium lattice constant and bulk modulus, are evaluated by standard GGA and GGA+U schemes with LR-determined parameters in Sec. III.2. Using the obtained lattice constant, the electronic structures are investigated to clarify the HM origin within the GGA framework (Sec. III.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 Sec. III.4. The LR-based DFT+U calculations for electronic and magnetic structures are presented and compared with previous theories and experiments in Sec. III.5. Finally, in Sec. IV, systematical results for the other ternary and quaternary compounds are discussed and promising materials for HM ferromagnets are proposed.

II model and method

The full Heusler Co2MnSi compound in $L2_{1}$ structure belongs to $Fm\bar{3}m$ ($O_{h}$) symmetry (space group No. 225). For the modeling, a fcc-primitive cell that contains two Co atoms sitting at Wyckoff position ($1/4,1/4,1/4$) (multiplicity with Wyckoff letter is $8c$), one Mn atom at ($1/2,1/2,1/2$) ($4b$), and one $sp$-element Si at ($0,0,0$) ($4a$) was prepared (a conventional unit cell is shown in Fig. 1 (a)). The detailed crystal structures for the ternary system Co${}_{2}Y$Si, where Mn is replaced with $Y$ of Ti, V, Cr, or Fe, and quaternary Co2($Y$,Mn)Si, where a part of Mn is substituted with $Y$, are described in Sec. IV.

The self-consistent DFT calculations were performed via the ab-initio package of Quantum-ESPRESSOqe-package by implementing the ultra-soft pseudopotentials parametrized by the scheme of Rappe, Rabe, Kaxiras, and Joannopoulos prb-41-1227R-1990 ; note_pp . 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 Monkhorst-Pack special $\mathbf{k}$-point mesh prb-7-5212-1973 of $16\times 16\times 16$ in the first Brillouin zone by Methfessel-Paxton prb-40-3616-1989 smearing method with a broadening parameter of 0.02 Ry.

The GGA functional formulated by Perdew, Burke, and Ernzerhof prl-77-3865-1996 was used for the exchange-correlation term. For the DFT+U method, a 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) prb-52-R5467-1995 ; prb-48-16929-1993 ; prb-49-14211-1994 ; prb-50-16861-1994 , which is also referred as the atomic limit (AL), and around mean-field (AMF) prb-44-943-1991 ; prb-49-14211-1994 approaches are mostly used. The former functional favors the 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 for an intermediate of strongly correlated and itinerant materials. It is still under debate that which of two functionals is a proper approach for Heusler compounds. jmmm-393-297-2015 In this study, the double-counting functional incorporated in the simplified rotationally invariant form prb-57-1505-1998 ; JPhysCondMater-9-767-1997 ; prb-71-035105-2005 , which is equivalent to the FLL approach but $J=0$ (or approximately $U_{\mathrm{eff}}=U-J$, where $U_{\mathrm{eff}}$ stands for 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 Hubbard model; thus, the scaling of $U_{\mathrm{eff}}$ can be simply understood as the strength of electron correlation. The $U_{\mathrm{eff}}$ is computed within the LR theory prb-71-035105-2005 ; prb-97-035117-2018 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 prb-71-035105-2005 , the on-site parameter for an atom $\alpha$, $U_{\mathrm{eff}}^{\mathrm{LR(\alpha)}}$, is evaluated from the second derivatives of the total energy functionals as

[TABLE]

The total energies $E^{\mathrm{SCF}}$ and $E^{\mathrm{KS}}$ correspond to interacting (fully screened) and non-interacting systems. The second term in Eq. (1) is necessary to subtract unphysical contributions in the total energy prl-49-1691-1982 ; prl-51-1884-1983 ; prl-51-1888-1983 , which are caused by the conventional exchange-correlation functionals (LSDA and GGA), where the total energy has a curvature for non-integer occupation $q_{\alpha}$ and often misleads to incorrect energy minima. The total energy derivatives are calculated using the constrained DFT approach:

[TABLE]

where

[TABLE]

The Lagrange multiplier $\mu_{\alpha}$ is a local perturbation potential that constrains the occupations $n_{\alpha}$ ($i=\mathrm{SCF,KS}$). In practice, Eq. (2) is transformed into a tractable representation where the constraint fields are treated as independent variables by Legendre transformation and the variations of $n_{\alpha}$ with respect to $\mu_{\alpha}$ are evaluated. prb-71-035105-2005 Using nonlocal linear response matrices

[TABLE]

Eq. (1) is rewritten to obtain $U_{\mathrm{eff}}^{\mathrm{LR(\alpha)}}$ as

[TABLE]

The matrix elements of the response matrices are numerically computed; $\chi_{\mathrm{SCF}}$ is obtained from the self-consistent (interacting) calculations under the applied local potential $\mu_{\alpha}$ and $\chi_{\mathrm{KS}}$ 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 $\mu_{\alpha}$. The LR approach, in principle, requires a response of electron occupations with regard to the perturbed potentials at a single site in an infinite crystal environment for an accurate $U_{\mathrm{eff}}$ evaluation, where all artifacts due to the periodic boundary conditions are excluded. prb-71-035105-2005 ; prb-97-035117-2018

III Revisit of Co2MnSi

III.1 LR-calculation for effective on-site Coulomb interaction parameter

The LR calculations are performed to determine the correlation $U_{\mathrm{eff}}^{\mathrm{LR(\alpha)}}$ parameters. The response functions of Eq. (4) are evaluated numerically from the gradient of $3d$ electron occupation numbers with regard to the perturbed potential $\mu_{\alpha}$, which constrains the electrons of Co or Mn in Co2MnSi alloy. For the LR calculations, the experimental lattice constant $a_{\mathrm{Expt.}}=5.645$ Åapl-88-032503-2006 is employed. As presented in Figs. 2 (a) and (b), the occupations’ variation of the KS term in $1\times 1\times 1$ primitive cell is slightly off the others ($n\times n\times n$ cell where $n=2$, 3, 4), although the SCF term does not change much (the plotted data are overlapping and the variations for different supercell sizes may not be visible from these figures). The calculated $U_{\mathrm{eff}}^{\mathrm{LR}}$ value is plotted as a function of number of atoms per cell in Fig. 2 (c). We find a $3\times 3\times 3$ fcc supercell including 108 atoms is practically large enough to obtain the well-converged parameters, meaning that the environment of the infinite crystal structure is well-reproduced. The obtained values result in $U_{\mathrm{eff}}^{\mathrm{LR(Mn)}}=3.535$ and $U_{\mathrm{eff}}^{\mathrm{LR(Co)}}=6.570$ eV for Mn and Co, respectively.

The correlation parameter for Co is unexpectedly higher than the typically used empirical values, for instance, the $U_{\mathrm{eff}}(=U-J)$ of 2.5 eV in full-potential (FP)-LMTO jmmm-422-13-2017 and that of 2.1 eV in KKR jphysDapplphys-42-084002-2009 calculations. The constrained random phase approximation (cRPA) approach prb-88-134402-2013 determines a parameter value similar to that determined in our study for Mn (3.07 eV), but almost half value of our LR result for Co (3.28 eV). Table 1 summarizes the numerical data of the $d$ occupations ($n_{\alpha}$) and the changes ($\Delta n_{\alpha}$) induced by $\mu_{\alpha}$ in the LR calculations for $\alpha$ atom ($\alpha=$ Mn and Co). As defined in Eqs. (4) and (5), the $U_{\mathrm{eff}}^{\mathrm{LR}}$ is difference of the inversions of electron occupations’ responses with respect to the applied potential shift $\mu_{\alpha}$ between KS and SCF terms. For both $\alpha=$ Mn and Co cases, the absolute values of $\Delta n_{\alpha}$ in SCF are smaller than those in KS by one order of magnitude, so the inverted response function ($\chi^{-1}\propto\frac{1}{\Delta n_{\alpha}}$) of SCF becomes a main factor in the computed correlation parameters. We also find that, for the SCF term, $\Delta n_{\mathrm{Co}}$ is small compared to $\Delta n_{\mathrm{Mn}}$. Therefore, the unreasonably overestimated parameter for Co originates from the difference in $\Delta n_{\mathrm{Co}}$ of the SCF term. Using the diagonal matrix elements of $\chi^{-1}_{\mathrm{KS}}$ and $\chi^{-1}_{\mathrm{SCF}}$ in Eq. (5), the parameter for Co is calculated as $U_{\mathrm{eff}}^{\mathrm{LR(Co)}}=-1.10114-(-7.67152)=6.570$ eV. For Mn, $U_{\mathrm{eff}}^{\mathrm{LR(Mn)}}=-0.98733-(-4.52196)=3.535$ eV is obtained, where the inverted KS response function’s contribution (the first term) is almost the same as the $U_{\mathrm{eff}}^{\mathrm{LR(Co)}}$ case, while the SCF one (the second term) is significantly different.

From above discussion, we conclude that the overestimation of $U_{\mathrm{eff}}^{\mathrm{LR}}$ 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. note_ulrt_alpha-size This can be attributed to the delocalized electronic structures of Co compared to Mn, which originates from 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 atom, as discussed in Sec. III.3. The localized characters of Co electronic states compared to those of Mn are consistent with the fact that the spin magnetic moment of Co ($1.05~{}\mu_{\mathrm{B}}$) is much smaller than that of Mn ($2.95~{}\mu_{\mathrm{B}}$). Recently, an extended LR theory pccp-19-8008-2017 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 ($\mu_{\alpha}\neq 0$), which might be canceled incompletely in the current LR calculation. In the present study, the DFT+U method incorporated by FLL formalism is used, but another approach for solving the failure in estimating $U_{\mathrm{eff}}^{\mathrm{LR}}$ for a weakly correlated Co site might be the use of AMF approximationprb-44-943-1991 ; prb-49-14211-1994 or hybrid approach prb-67-153106-2003 , where the AMF and FLL approximations are linearly interpolated.

III.2 Structural property

Here, we consider three schemes of LR-based DFT+U calculations, in addition to the standard GGA: the determined $U_{\mathrm{eff}}^{\mathrm{LR}}$ values are applied to only Mn (referred as GGA+U${}_{\mathrm{Mn}}^{\mathrm{LR}}$) or Co (GGA+U${}_{\mathrm{Co}}^{\mathrm{LR}}$) and to both of them (GGA+U${}_{\mathrm{Mn,Co}}^{\mathrm{LR}}$). First, the GGA calculations are performed for evaluating the equilibrium lattice constant. The total energies at different volume sizes of a primitive cell are obtained as shown in Fig. 3 (a). The energy minimum is searched by energy fitting to the Murnaghan equation of states pnasusa-30-244-1944 as a function of volume $V$,

[TABLE]

where $E_{0}$ is the ground-state total energy at equilibrium volume $V_{0}$, $B_{0}$ bulk modulus, and $B_{0}^{\prime}$ pressure derivative of the bulk modulus. The obtained lattice constant is 5.639 Å, which agrees with the experimental value. apl-88-032503-2006 The error value between the calculated lattice constant $a_{0}$ and experiment, defined as $(a_{0}-a_{\mathrm{Expt.}})/a_{\mathrm{Expt.}}\times 100~{}(\%)$, is only $-0.104~{}\%$. In the GGA+U${}_{\mathrm{eff}}^{\mathrm{LR}}$ case, the obtained lattice constant of 5.695 Å is similar to the GGA result and the error from the experiment is less than 1 % (0.866 %), as shown in Fig. 3 (b). On the other hand, $a_{0}$ is significantly overestimated by the errors of 4.701 and 8.506 % in the GGA+U${}_{\mathrm{Co}}^{\mathrm{LR}}$ and GGA+U${}_{\mathrm{Mn,Co}}^{\mathrm{LR}}$ cases. Figure 3 (c) shows a local energy minimum around the experimental value but the global minimum is found at 51.62 Å3, corresponding to $a_{0}=5.910~{}\mathrm{\AA}$. Note also that in the GGA+U${}_{\mathrm{Mn,Co}}^{\mathrm{LR}}$ scheme (Fig. 3 (d)), a jump of total energy change around the volume of $\sim 60~{}\mathrm{\AA}^{3}$ occurs due to a magnetic phase transition, but we confirm that an energy minimum, corresponding to $a_{0}=6.125$ Å, exists at less than the volume where this magnetic transition is induced.

We present bulk modulus $B_{0}$ and its pressure derivative $B_{0}^{\prime}$ in Table 2, through comparisons of theoretical literature. prb-66-094421-2002 ; IEEETransMagn-50-1301104-2014 ; JAlloyCompd-560-25-2013 ; EuroPhysJB-76-321-2010 ; intermetal-44-26-2014 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 $B_{0}^{\prime}$, while the GGA+U${}_{\mathrm{Mn}}^{\mathrm{LR}}$ result is slightly smaller in $B_{0}$. On the other hand, the $B_{0}$ calculated by GGA+U${}_{\mathrm{Co}}^{\mathrm{LR}}$ and GGA+U${}_{\mathrm{Mn,Co}}^{\mathrm{LR}}$ 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 Co2MnSi 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 $U_{\mathrm{eff}}^{\mathrm{LR}}$ to the Mn atom tends to obtain a reasonable result, as well as GGA, from the comparison of experiments, while the inclusion of $U_{\mathrm{eff}}^{\mathrm{LR}}$ to Co fails to evaluate the $a_{0}$ of Co2MnSi.

III.3 Origin of half-metallicity

As mentioned in Sec. II, the full Heusler alloy of $L2_{1}$ structure belongs to the octahedral ($O_{h}$) space group symmetry. In this whole-crystal symmetry, we first focus on the Co lattice by ignoring the first-neighboring Mn and Si atoms. The lattice is assumed to be a simple cubic composed by the second-neighboring Co at different sublattices in the primitive cell, which leads to the Co sitting at $O_{h}$ site symmetry. Second, our focus is turned on the tetrahedral ($T_{d}$) site symmetry. Neglecting the chemical atomic species, every atom forms a bcc lattice structure and is surrounded by a tetrahedral environment. The hybridization diagram of atomic orbital energy is discussed by following these two steps. Note that, for avoiding the confusion regarding the notations, the symmetric characters of the atomic orbital are unified using only representations for the $O_{h}$ site symmetry, which corresponds to the space group of the $L2_{1}$ full Heusler compound, as done also in the previous research prb-66-174429-2002 .

We again start with the results of the standard GGA calculations for discussing the underlying electronic structure of Co2MnSi. Figure 4 (a) shows the band structures for the minority spin states projected into the Co $e_{g(u)}$ ($d_{z^{2}},d_{x^{2}-y^{2}}$) and $t_{2g(1u)}$ ($d_{xz},d_{yz},d_{xy}$), Mn $e_{g}$ ($d_{z^{2}},d_{x^{2}-y^{2}}$) 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 Fermi energy, while the Si $t_{1u}$ state can be seen only at very far from Fermi energy. To discuss the orbital hybridization mechanism, the eigenstates at $\Gamma$ point in Brillouin zone are focused. At 0.4 eV above the Fermi energy, the Co $e_{u}$ state appears but the other orbital components are not included in these eigenstates, which means that Co $e_{u}$ 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 $t_{2g}^{*}$ at $-0.3$ eV. As a result, a minority band gap is originated by the anti-bonding $t_{2g}^{*}$ and non-bonding $e_{u}$ states of 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 ($t_{1u}^{*}$) 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. prb-66-174429-2002

Figure 4 (b) presents an energy diagram of Co atoms under $O_{h}$ 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 $e_{u}^{*}$ and $t_{1u}^{*}$ 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 $\pi$-like bonding in the $O_{h}$ atomic positions, and $e_{g}$ hybridization, including $d_{z^{2}}$–$d_{z^{2}}$ and $d_{x^{2}-y^{2}}$–$d_{x^{2}-y^{2}}$, is expected to form a $\sigma$-like bonding, whose orbital coupling is stronger than that of $\pi$-like bonding. Accordingly, the energy gap between bonding $e_{g}$ and anti-bonding $e_{u}^{*}$ states arising from $e_{g}$ hybridization becomes wide compared to the bonding $t_{2g}$ and anti-bonding $t_{1u}^{*}$ states from $t_{2g}$ hybridization.

Next, the orbital interactions between the first-neighboring atom 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 to hybridize. 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 $e_{u}^{*}$ orbitals in $O_{h}$ are represented as the $e$ state in $T_{d}$, and the triplet $t_{2g}$ and $t_{1u}^{*}$ orbitals as $t_{2}$, where the asterisk symbol indicating the anti-bonding state is omitted for simplicity. The $T_{d}$ site symmetry also gives the $e$ ($d_{z^{2}},d_{x^{2}-y^{2}}$) 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.

Figure 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 $t_{2g}^{*}$ state dominates the highest orbital state in the valence band, which hybridizes with bonding Mn $t_{2g}$. The Co $t_{1u}^{*}$ is pushed up to a quite higher energy through the hybridization with Si $t_{1u}$ ($p$) orbital and the non-bonding Co $e_{u}$ 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 prb-66-174429-2002 , where the band gap in the minority state is mostly dominated by Co $e_{u}$ and $t_{1u}$ orbitals. Instead, more importantly, our diagram proposes that the HM property and electronic structure near the Fermi level can be tuned by a selection of $Y$ atom and/or a mix of several atoms into $Y$ site through $t_{2g}$ coupling in $L2_{1}$ Heusler alloy.

Even though our diagram is at variance with the previously reported one prb-66-174429-2002 , the 12 valence electrons for Co2MnSi are confirmed to occupy three Co $t_{2g}^{*}$, 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 prb-66-174429-2002 : The magnetic moment of the system, $m_{\mathrm{spin}}$, is obeyed by $m_{\mathrm{spin}}=N_{\mathrm{val}}-24$, where $N_{\mathrm{val}}$ is the total number of valence electrons. The calculations obtain a total magnetic moment of 5.01 $\mu_{\mathrm{B}}$, which is very close to the integer value expected by the Slater-Pauling rule and in agreement with a previous theory within LSDA prb-66-174429-2002 and GGA prb-72-184415-2005 ; prb-66-094421-2002 , as well as the experiments prb-74-104405-2006 ; jmmm-937-54-1986 ; jpcs-32-1221-1971 ; prb-80-144405-2009 , as summarized in Table 3.

Note that the previous study prb-66-174429-2002 was carried out for Co2MnGe, where the number of valence electrons is equivalent to that of Co2MnSi. Thus, Co2MnGe is confirmed to be similar to Co2MnSi. 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 ($m_{\mathrm{spin}}=5.00~{}\mu_{\mathrm{B}}$).

III.4 Correlation effects on Mn and Co

As mentioned in the introduction, the behaviors of electron localizations are supposed to be different at Co and Mn sites. This fact motivates us to investigate the effects of the correlations for each site. To discuss the influence of +U on the atomic energy diagram, modifications of magnetic moment and band structure are studied by performing DFT+U calculations with varying $U_{\mathrm{eff}}$ parameters for Co and Mn atoms independently. Here, we refer to the case where the varying $U_{\mathrm{eff}}$ is applied to only Mn (Co) site as GGA+UMn (GGA+UCo) representation, where the lattice constant is set to the theoretical value of 5.695 Å (5.910 Å) obtained in Sec. III.2.

We first mention the GGA+UMn case. The total $m_{\mathrm{spin}}$ is constant but Mn (Co) $m_{\mathrm{spin}}$ monotonically increases (decreases) when the correlation parameter for Mn, $U_{\mathrm{eff}}^{\mathrm{Mn}}$, increases (see Fig. 5 (a)). Note that two of the Co atoms exist in the primitive cell, so the variation of Co $m_{\mathrm{spin}}$ is estimated to be twice. The increased $m_{\mathrm{spin}}$ 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 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 $U_{\mathrm{eff}}^{\mathrm{Mn}}$ value, we trace modifications in the band structures of these orbitals. Figure 6 (a) presents the minority spin band structures around $\Gamma$ point calculated by standard GGA and the GGA+UMn with small ($U_{\mathrm{eff}}^{\mathrm{Mn}}=3$ eV) and large (6 eV) parameter values. The GGA results indicate that the anti-bonding Co $t_{2g}^{*}$ is dominant just below the Fermi energy and bonding Mn $t_{2g}$ is visible at $-1.5$ eV in a minority state. Interestingly, increasing the $U_{\mathrm{eff}}^{\mathrm{Mn}}$ 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 $U_{\mathrm{eff}}^{\mathrm{Mn}}$, but the anti-bonding $t_{2g}^{*}$ becomes dominant by Mn compared to Co and $t_{2g}^{*}$ shifts above the Fermi energy at a large $U_{\mathrm{eff}}^{\mathrm{Mn}}$. Schematic diagrams are illustrated in Figs. 6 (b) and (c). In the majority spin, the valence Mn $t_{2g}$ atomic orbital is shifted to a lower energy by the $U_{\mathrm{eff}}^{\mathrm{Mn}}$ effect and the anti-bonding state Co $t_{2g}$ is drawn to 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 $U_{\mathrm{eff}}^{\mathrm{Mn}}$ increases, and the anti-bonding Co $t_{2g}^{*}$ orbital gradually touches the Fermi energy. When the Mn $t_{2g}$ state becomes energetically higher than Co at a large $U_{\mathrm{eff}}^{\mathrm{Mn}}$, the component of the anti-bonding $t_{2g}^{*}$ is switched from Co to Mn in minority spin.

Surprisingly, $U_{\mathrm{eff}}^{\mathrm{Mn}}$ 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) vary to increase by the applied +U effect, but simultaneously, the occupations in the opposite spin (minority state) are also changed to reduce sensitively for the total occupations constant at each atom. This argument is based on the energy diagrams in Figs. 6 (b) and (c), and is consistent with the behaviors of spin magnetic moment and electron occupations at each atomic site [Figs. 5 (a) and (b)]; thus, this scenario can be concluded to be behind the effects of +U on Mn $d$ orbitals.

Second, the GGA+UCo result is considered. In the range of $U_{\mathrm{eff}}^{\mathrm{Co}}$ less than around 4 eV, an increase in the spin magnetic moment at the Co site is not significant, but suddenly increases afterward [Fig. 5 (c)]. In Fig. 7, the $U_{\mathrm{eff}}^{\mathrm{Co}}$-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 $t_{2g}^{*}$ is switched from Co to Mn [Fig. 7 (a)]; in contrast, the anti-bonding $e_{g}^{*}$ state is switched from Mn to Co [Fig. 7 (b)] with increasing $U_{\mathrm{eff}}^{\mathrm{Co}}$. The $e_{u}$ state in Fig. 7 (b) moves to higher energy by U${}_{\mathrm{eff}}^{\mathrm{Co}}$, but does not hybridize with Mn.

To understand the behavior of changing $m_{\mathrm{spin}}$ and electron numbers in GGA+UCo in Figs. 5 (c) and (d), the possible energy diagrams for the majority and minority states are illustrated in Figs. 7 (c) and (d). The majority Co $t_{2g}$ simply goes to lower energy by the introduction of $U_{\mathrm{eff}}^{\mathrm{Co}}$, so that the Co $d$ spin-up occupation increases and is saturated at larger $U_{\mathrm{eff}}^{\mathrm{Co}}$ values ($\sim 7$ eV). For minority state, the $d$ bands’ behaviors of Co and Mn are intricate, but it can be understood by going back to the principle view that first focus is paid to the hybridization between Co atoms at different sublattices and that between Mn and Co–Co states afterward, as discussed in Sec. III.3 and a previous report prb-66-174429-2002 . The Co $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ ($d_{xz},d_{yz}$, and $d_{xy}$) orbitals are pushed up (down) due to $U_{\mathrm{eff}}^{\mathrm{Co}}$, and hybridize with Mn $e_{g}$ ($t_{2g}$) state [Fig. 7 (d)]. Increasing $U_{\mathrm{eff}}^{\mathrm{Co}}$ affects the energy gap, and most notably, the Co $e_{g}$ orbital becomes an un-occupied anti-bonding state at a large $U_{\mathrm{eff}}^{\mathrm{Co}}$ value, while it is an occupied bonding state at a small $U_{\mathrm{eff}}^{\mathrm{Co}}$ (see blue band of 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 $m_{\mathrm{spin}}$ in the range over $\sim 4$ eV of $U_{\mathrm{eff}}^{\mathrm{Co}}$, 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 GGA+UMn and GGA+UCo cases. Thus, the consistency of our energy diagram proposed in Fig. 4 (c) is demonstrated successfully.

III.5 Electronic and magnetic properties

We now discuss the electronic and magnetic properties obtained from the band calculations that incorporate the LR-determined correlation parameters (3.535 eV for Mn and 6.570 eV for Co). First, the $m_{\mathrm{spin}}$ obtained from the GGA+U${}_{\mathrm{Mn}}^{\mathrm{LR}}$ method is compared with the GGA in Table 3. The value of total $m_{\mathrm{spin}}$ is same as that of GGA and agrees with previous reports. jmmm-422-13-2017 ; prb-88-134402-2013 ; jphysDapplphys-42-084002-2009 ; prb-86-245115-2012 ; prb-74-104405-2006 ; prb-80-144405-2009 ; jpcs-32-1221-1971 ; jmmm-937-54-1986 On the other hand, regarding the atom-resolved contributions, the results of $m_{\mathrm{spin}}$ of Co (0.72 $\mu_{\mathrm{B}}$) and Mn (3.63 $\mu_{\mathrm{B}}$) are not in agreement with GGA and previous calculations, but in good agreement with the experiments jpcs-32-1221-1971 ; prb-80-144405-2009 . Thus, the GGA+U${}_{\mathrm{Mn}}^{\mathrm{LR}}$ calculation results are superior to the standard GGA results.

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 Fermi energy to $-5$ eV for majority state and from $-0.5$ eV to $-4.5$ eV for 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 majority (minority) state, respectively. As expected from Figs. 4 (b) and (c), we also confirm the $e_{u}$ and $t_{2g}^{*}$ orbital characters of Co $d$ states above and below the Fermi energy, as shown by arrows in Fig. 8 (a). The value of spin polarization referred as $P_{\mathrm{DOS}}$ is estimated by $P_{\mathrm{DOS}}=\frac{D^{\uparrow}(E_{\mathrm{F}})-D^{\downarrow}(E_{\mathrm{F}})}{D^{\uparrow}(E_{\mathrm{F}})+D^{\downarrow}(E_{\mathrm{F}})}\times 100~{}(\%)$, where $D^{\sigma}(E_{\mathrm{F}})$ is the DOS of the majority ($\sigma=\uparrow$) or minority ($\sigma=\downarrow$) spin state at the Fermi energy. A 100 % $P_{\mathrm{DOS}}$ value is obtained, and the energy band gap in the minority spin state $E_{\mathrm{gap}}^{\downarrow}$ is around 0.8 eV. The GGA+U${}_{\mathrm{Mn}}^{\mathrm{LR}}$ calculation modifies the DOS from GGA. The energy level of Mn occupied (unoccupied) states is shifted to lower (higher) level due to the exchange splitting induced by the $U_{\mathrm{eff}}^{\mathrm{LR(Mn)}}$ 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 $P_{\mathrm{DOS}}=90.5~{}\%$ is obtained.

By contrast, GGA+U${}_{\mathrm{Co}}^{\mathrm{LR}}$ and GGA+U${}_{\mathrm{Mn,Co}}^{\mathrm{LR}}$ seem to fail to obtain the total magnetic moment reasonably consistent with the experimental observations jpcs-32-1221-1971 ; prb-80-144405-2009 because of the overestimated $U_{\mathrm{eff}}^{\mathrm{LR(Co)}}$ parameters (see Table 3). Figure 8 (c) indicates the fact that the exchange splitting arising from the large $U_{\mathrm{eff}}^{\mathrm{LR(Co)}}$ induces a fully-occupied Co $d$ state in majority spin states, which leads to a Co $m_{\mathrm{spin}}$ of 1.88 $\mu_{\mathrm{B}}$ and total $m_{\mathrm{spin}}$ of 6.95 $\mu_{\mathrm{B}}$. The energy gap does not appear in the minority channel and the top of valence states around $-2$ eV from Fermi energy is composed of the Mn $d$ orbital of majority states. Similarly, in the GGA+U${}_{\mathrm{Mn,Co}}^{\mathrm{LR}}$ case, the overestimated value of the total $m_{\mathrm{spin}}$ of $8.08~{}\mu_{\mathrm{B}}$ arises from that the majority electrons of Co and Mn are fully occupied at low energy ($-4$ eV and below) through both $U_{\mathrm{eff}}^{\mathrm{LR(Mn)}}$ and $U_{\mathrm{eff}}^{\mathrm{LR(Co)}}$, as shown in Fig. 8 (d). In this scheme, the half-metallic electronic structure is broken by a few DOS that is widely broad around the Fermi energy. The spin polarizations are found to be negative and small absolute values, $P_{\mathrm{DOS}}=-26.97$ and $-33.82$ % for the respective GGA+U${}_{\mathrm{Co}}^{\mathrm{LR}}$ and GGA+U${}_{\mathrm{Mn,Co}}^{\mathrm{LR}}$ methods.

From the experimental viewpoint, the hard X-ray photoelectron spectroscopy measurements reported that the valence band structure in the binding energy region from Fermi energy to $\sim 1.2$ eV (corresponding to $-1.2$ eV in calculated DOS) is mostly contributed by Co $3d$ electrons prb-79-100405R-2009 and the Mn $d$ state does exist in this binding-energy region, while the number of electrons is very few compared to Co. prv1 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 Sec. III.2), we can conclude that the static many-body correlation +U at $Y$ site ($Y=\rm{Mn}$ for Co2MnSi) plays an important role in 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 Co site, which is not reliable for accurate band calculations. In other words, for 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 $U_{\mathrm{eff}}^{\mathrm{LR}}$ only for $Y$ site; i.e., correlation correction is excluded for Co. We here explicitly mention that the energy diagram obtained from the GGA+U${}_{\mathrm{Mn}}^{\mathrm{LR}}$ calculation corresponds to Figs. 4 (b) and (c), which are obtained from the GGA results.

IV Search for HM materials of other ternary and quaternary alloys

For considering the ternary Co${}_{2}Y$Si alloys, where $Y$ is replaced from Mn to Ti, V, Cr, or Fe atom, the LR calculations for $U_{\mathrm{eff}}^{\mathrm{LR(}Y\mathrm{)}}$ are first carried out using the lattice constants assumed in the experiments jmmm-38-1-1983 ; apl-88-032503-2006 ; spmmc1-262-1962 , as in the case of Co2MnSi. To the best of our knowledge, there are no experimental data for Co2CrSi, so the lattice constant obtained from Murnaghan fitting pnasusa-30-244-1944 by the GGA potential is employed for $U_{\mathrm{eff}}^{\mathrm{LR(Cr)}}$ calculation. In this study initial magnetization for the SCF calculation is assumed to the ferromagnetic state in all ternary models. The determined parameters are around $3\sim 4$ eV depending on the materials; the $U_{\mathrm{eff}}^{\mathrm{LR(}Y\mathrm{)}}=2.942,$ 3.979, 3.169, and 3.922 eV for Co2TiSi, Co2VSi, Co2CrSi, and Co2FeSi, respectively. note_ulrt_Co2YSi

Calculated total DOSs are shown in Figs. 9 (a)$\sim$(d), and the results of spin magnetic moments are summarized in Table 4. The Co2TiSi is not the HM ($P_{\mathrm{DOS}}=25.8~{}\%$), where the Fermi energy is located at the minority conduction edge state. For the Co2VSi and Co2CrSi, 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 $P_{\mathrm{DOS}}=98.2$ and 89.3 %, respectively. On the other hand, negative spin polarization, $P_{\mathrm{DOS}}=-62.3~{}\%$, is obtained in the Co2FeSi, where the minority DOS is much compared to the majority state at Fermi energy. Note that, as the $Y$ atom is changed from a large atomic number ($Z_{\mathrm{Fe}}=26$) to small ($Z_{\mathrm{V}}=23$), the Fermi energy position seems to move away from the conduction state of minority spin, but this is not the case for $Y=\mathrm{Ti}$. This exception is attributable to the fact that the Ti spins in Co2TiSi 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 $\mu_{\mathrm{B}}$ for Co2TiSi, Co2VSi, Co2CrSi, and Co2FeSi, respectively.

The structural properties are also investigated as summarized in Table 5. The estimated lattice constants are in good agreement with the experiments jmmm-38-1-1983 ; apl-88-032503-2006 ; spmmc1-262-1962 ; jap-100-083523-2011 and their error values from the experiments are less than 1 % for Co2TiSi, Co2VSi, and Co2FeSi. In Co2CrSi, the lattice constant of 5.694 Å is close to the previous calculation. jap-100-113901-2006 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 Co2MnSi case, and might come from that the previous studies were conducted by standard LSDAphysicaB-407-3339-2012 ; jap-100-113901-2006 ; jap-125-082523-2019 ; UnvJMechanEng-7-16-2019 ; jpcs-75-391-2014 and GGA IEEETransMagn-50-1301104-2014 . The experimentally measured $B_{0}$ is available only for $Y=$ Fe ($B_{0}=240$ GPa). jap-100-083523-2011 From our calculations, the $B_{0}$ and $B_{0}^{\prime}$ in Co2FeSi are found to be 183.263 GPa and 4.679, respectively. The LSDA calculation IEEETransMagn-50-1301104-2014 shows a reasonably consistent value of $B_{0}=241.9$ GPa with the experiment, although the GGA calculation jap-100-113901-2006 ; jap-125-082523-2019 ; jpcs-75-391-2014 underestimates $B_{0}$ ($B_{0}=203.5\sim 207.1$ GPa). Zhu et al. jpcs-75-391-2014 also performed the 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 are employed, and obtained $B_{0}=209.3$ GPa and $B_{0}^{\prime}=4.67$ (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 Co2FeSi IEEETransMagn-50-1301104-2014 . However, the LR-based DFT+U method provides reasonable results at least for $a_{0}$ values.

We finally investigate the quaternary Heusler compounds of a chemical formula Co2($Y_{x}$,Mn1-x)Si ($Y=\mathrm{Ti}$, 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 Figs. 1 (b)$\sim$(d), cubic primitive cells consisting of 16 atoms are considered. The lattice constant is given by Vegard’s law ZPhysik-5-17-1920 ; pra-43-3161-1991 using the obtained equilibrium lattice constants for Co${}_{2}Y$Si ($a^{\mathrm{C}Y\mathrm{S}}$) and Co2MnSi ($a^{\mathrm{CMS}}$) as $a(x)=xa^{\mathrm{C}Y\mathrm{S}}+(1-x)a^{\mathrm{CMS}}$. The correlation parameters of Mn and $Y$ atoms for quaternary systems at all compositions are assumed to be the values of $U_{\mathrm{eff}}^{\mathrm{LR(Mn)}}$ and $U_{\mathrm{eff}}^{\mathrm{LR(}Y\mathrm{)}}$, which are determined by the LR theory in ternary Co2MnSi and Co${}_{2}Y$Si. For the quaternary compounds, in which the atomic position of the different elements is not symmetric as the ternary system, the structures are geometrically relaxed under the equilibrium lattice constants by force calculations using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm SIAP-6-76-291970 ; ComputJ-J13-317-1970 ; MathComputModell-24-23-1970 ; MathComputModell-24-647-1970 until the forces acting on each atom are minimized below the criterion of $10^{-3}$ Ry/bohr.

The calculated $m_{\mathrm{spin}}$ values for quaternary alloys are also available in Table 4. In case of only $Y=\mathrm{Ti}$, the Ti spins are ferrimagnetically coupled with Co and Mn similar to that in the ternary model. The Mn $m_{\mathrm{spin}}$ is a large value over 3 $\mu_{\mathrm{B}}$ in all systems. Figure 10 plots the total $m_{\mathrm{spin}}$ for ternary and quaternary Co-based full Heusler compounds under study as a function of $N_{\mathrm{val}}$ in the system. The Slater-Pauling relation is satisfied in the range of less than 29.5 in $N_{\mathrm{val}}$ while being slightly underestimated for the range over $N_{\mathrm{val}}=29.5$, which corresponds to Co2(Fe0.75,Mn0.25)Si, and Co2FeSi.

The results on DOS for quaternary alloys are shown in Figs. 9 (e)$\sim$(h). A perfectly HM electronic structure ($P_{\mathrm{DOS}}$ is equal to 100 %) is found in Co2(Ti0.25,Mn0.75)Si, Co2(V0.25,Mn0.75)Si, Co2(V0.50,Mn0.50)Si, and Co2(Fe0.25,Mn0.75)Si. Among them, the Co2(V0.75,Mn0.25)Si has the largest minority band gap $E_{\mathrm{gap}}^{\downarrow}=0.5$ eV, and thus, this material can be a good candidate for a wide-gap HM ferromagnet. The Co2(Ti0.25,Mn0.75)Si and Co2(Fe0.25,Mn0.75)Si are also HM candidates because of the advantage in Fermi energy position that 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 ($P_{\mathrm{DOS}}$ is almost 100 %) is found in Co2(Ti0.50,Mn0.50)Si ($P_{\mathrm{DOS}}=99.9$ %) and Co2(Fe0.50,Mn0.50)Si (99.4 %). Figure 11 presents the composition dependence of $P_{\mathrm{DOS}}$. Although the $Y=$ Cr system does not show the HM property at each composition, an interesting trend we observed is that a high $P_{\mathrm{DOS}}$ is independent of the composition, where $Y=$ V is also the same, whereas a large reduction of $P_{\mathrm{DOS}}$ occurs with an increase in $x$, especially in the Co(Fex,Mn1-x)Si.

Finally, we states the results of systems including Fe by comparing the previous studies. The Co2FeSi compound is still under debate to judge whether its electronic structure shows HM for past few decades from theories with and without the correlation effects. prb-72-184434-2005 ; prb-73-094422-2006 Our LR-based DFT+U calculations indicate that it is not a HM ferromagnet. However, we emphasize that tuning the composition in quaternary Co2(Fe,Mn)Si demonstrates that the electronic structure can be HM. This conclusion is supported by a consistency in anisotropic magnetoresistance (AMR) measurement. apl-104-172407-2014 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 Fermi level, is a signature of HM. jpsj-81-024705-2012 ; amr-750-978-2013 ; pssc11-1026-2014 Based on this model analysis, a positive behavior of AMR is found in the Co2FeSi apl-104-172407-2014 that indicates a ferromagnetic without the minority band gap but the negative sign is confirmed in the Co2(Fe,Mn)Si prb-86-020409R-2012 , leading to HM. Note that the composition range of Fe and Mn for Co2(Fe,Mn)Si, showing HM, is different between our study and the AMR experiment, which may be because the present quaternary models [Figs. 1 (b)$\sim$(d)] are assumed to be a periodic structure missing a perfectly disordered property of Fe and Mn, and/or the ordering parameter of $L2_{1}$ structure in the experiment apl-104-172407-2014 is rather low at all compositions. Nonetheless, we suggest that the quaternary Co2(Fe0.25,Mn0.75)Si is one of the most promising candidates as an HM Heusler ferromagnet because of the sizable $E_{\mathrm{gap}}^{\downarrow}(=0.4~{}\mathrm{eV})$ and Fermi energy position being at almost the center of the gap. We believe our present results encourage the 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.

V Summary

In summary, we revisited the fundamental electronic structure and effects of correlation parameters for $3d$ electrons in a Co-based full Heusler Co${}_{2}Y$Si alloy via the LR-based DFT+U method, where the correlation correction $U_{\mathrm{eff}}$ parameters were determined from the LR approach and the +U formalism was incorporated as the FLL form. Focusing on Co2MnSi ($Y=$ Mn), we considered the origin of the minority HM gap from the projected band structures calculated by the standard GGA, and found that the $t_{2g}$ hybridization between Co and Mn is important for the gap. The energy diagram of atomic-orbital hybridizations revealed that the HM gap originates from Co $e_{u}$ of the conduction state and Co–Mn hybridizing $t_{2g}$ orbitals of the valence state at the Fermi energy. Thus, the gap is tunable by selecting a $Y$ element and/or mixing different elements into $Y$ site through $t_{2g}$ atomic orbital coupling. The LR calculations tend to obtain a reasonable value as a correlation parameter for $Y$ site ($Y=$ Mn in Co2MnSi) but an unexpectedly large value for Co site, which misleads to an unphysical ground state. The failure in determining $U_{\mathrm{eff}}^{\mathrm{LR}}$ for Co site arises from the fact that the $d$ electrons of Co site behave rather itinerant in the alloy. This means that the mean-field approximations such as LSDA and GGA are enough to describe the ground-state properties with high accuracy; thus, we propose the LR-based DFT+U method, where the determined $U_{\mathrm{eff}}^{\mathrm{LR}}$ parameters are incorporated into only strongly-correlated $Y$ site, as a suitable methodology on a practical level for $L2_{1}$ Co-based full Heusler alloys. For Co2MnSi, our results are consistent with the experimental observations and superior to the standard GGA calculation, particularly in terms of electronic and magnetic properties. It is also indicated that Co2MnSi is not HM but a highly spin-polarized ferromagnet. Further investigations were carried out for the other ternary and quaternary Co2(Y,Mn)Si to explore the potential for HM ferromagnets. The results showed that the Co2(Ti,Mn)Si, Co2(V,Mn)Si, and Co2(Fe,Mn)Si compounds are expected to be HM materials when the composition of $Y$ element is appropriately selected. The Co2(Cr,Mn)Si does not show HM property at every composition, but a notable tendency is that the high spin polarization is independent of the composition. However, for using spintronics applications, Co2(Fe0.25,Mn0.75)Si, in which the HM nature is consistent with the experimental AMR study, is one of the most promising candidates because of the sizable HM gap in the minority state and the Fermi energy position being at almost the center of the gap.

Acknowledgements.

The authors are grateful to K. Masuda, H. Sukegawa, and S. Mitani for critical comments and suggestions. K. N. also thanks K. Nakamura, T. Oguchi, and M. Weinert for fruitful discussions. This work was supported in part by Grants-in-Aid for Scientific Research (S) (Grant Numbers JP16H06332 and JP17H06152) from the Japan Society for the Promotion of Science, the ImPACT Program of Council for Science, Technology and Innovation, ’Materials research by Information Integration Initiative ($Mi^{2}i$)’ project of the Support Program for Starting Up Innovation Hub from Japan Science and Technology Agency (JST), and Center for Spintronics Research Network (CSRN), Osaka University. The computations in this study were performed on a Numerical Materials Simulator at NIMS.