Effect of the double-counting functional on the electronic and magnetic properties of half-metallic magnets using the GGA+U method
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n submitted to Physical Review B Effect of the double-counting functional on the electronic and magnetic properties ofhalf-metallic magnets using the GGA+U method
Christos Tsirogiannis and Iosif Galanakis ∗ Department of Materials Science, School of Natural Sciences, University of Paras, GR-26504 Patra, Greece (Dated: April 12, 2018)Methods based on the combination of the usual density functional theory (DFT) codes withthe Hubbard models are widely used to investigate the properties of strongly correlated materi-als. Using first-principle calculations we study the electronic and magnetic properties of 20 half-metallic magnets performing self-consistent GGA+U calculations using both the atomic-limit (AL)and around-mean-field (AMF) functionals for the double counting term, used to subtract the corre-lation part from the DFT total energy, and compare these results to the usual generalized-gradient-approximation (GGA) calculations. Overall the use of AMF produces results similar to the GGAcalculations. On the other hand the effect of AL is diversified depending on the studied material. Ingeneral the AL functional produces a stronger tendency towards magnetism leading in some casesto unphysical electronic and magnetic properties. Thus the choice of the adequate double-countingfunctional is crucial for the results obtained using the GGA+U method.
PACS numbers: 75.50.Cc, 71.20.Lp, 71.15.Mb
I. INTRODUCTION
The rapid expansion of the field of spintronics andmagneto-electronics brought magnetic materials at thenanoscale to the center of attention of modern electron-ics. The spin of the electron offers an additional degree offreedom in electronic devices with respect to conventionalelectronics based on semiconductors. The design of mag-netic nanomaterials with novel properties offers new func-tionalities to future devices, and to this respect ab-initio (also known as first-principles) studies of the electronicstructure within density functional theory (DFT) playa crucial role allowing the modelling of the propertiesof several materials prior to their experimental growth.Among the most studied magnetic materials are theso-called half-metallic (HM) magnets, which presentmetallic behavior for the majority-spin electronic bandstructure and semiconducting for the minority-spin elec-tronic band structure. The ferromagnetic semi-Heuslercompound NiMnSb was the first material for which theHM character was predicted and described, and sincethen several HM compounds have been discovered. The implementation of half-metallic magnets in devicesis an active field of research (see Ref. 8 for a review ofthe literature).DFT-based ab-initio electronic structure calcula-tions using either the local-spin-density approxima-tion (LSDA) or the generalized-gradient-approximation(GGA) for the exchange-correlation functional arequite successful for magnetic materials from weak to in-termediate electronic correlations, but fail for systemswith strong electronic correlations. There are two com-mon ways to include correlations in first-principles elec-tronic structure calculations. The first one is the so-called LDA+ U scheme, in which the local–(spin)-densityapproximation (L(S)DA) of DFT is augmented by anon-site Coulomb repulsion term and an exchange termwith the Hubbard U and Hund exchange J parame- ters, respectively. Such a scheme has been appliedfor example to Co FeSi, showing that correlations re-store the HM character of the compound, and toNiMnSb. When the GGA functional is used insteadof the L(S)DA the method is usually referred to asGGA+U scheme. A more elaborate modern computa-tional scheme, which combines many-body model Hamil-tonian methods with DFT, is the so-called LDA+DMFTmethod, where DMFT stands for Dynamical Mean-FieldTheory.
LDA+DMFT has been applied to severalHM magnetic systems like Co MnSi, NiMnSb,
FeMnSb, Mn VAl, VAs and CrAs. In the case of both LSDA+U (GGA+U) andLDA+DMFT schemes, the addition of the Hubbard Uinteraction introduces the need for a double-countingcorrection term in the energy functional to account forthe fact that the Coulomb energy between the corre-lated states is already included in the LSDA (GGA)functional. Several double-counting schemes have beenproposed in literature, and in all proposed schemesan averaged energy for the occupation of a selected ref-erence state is subtracted. Among the proposed func-tionals for the double-counting term, two are most com-monly used: the so-called around-mean-field (AMF)functional and the atomic-limit (AL) functional; the lat-ter is also referred to in literature as the fully local-ized limit (FLL) functional. The performance of thesetwo functionals has attracted little attention in litera-ture. In 2009 Ylvisaker and collaborators presented anextensive study on the effect of the two functionals whenperforming self-consistent LSDA+U calculations for sev-eral magnetic materials. They have shown that the useof the LSDA+U interaction term usually enhances spinmagnetic moments, but the AMF double-counting termgives magnetic states a significantly larger energy penaltythan does the AL(FLL) functional and thus AL gives astronger tendency to magnetism than AMF. II. MOTIVATION AND COMPUTATIONALMETHOD
As mentioned above, ab-initio electronic structurecalculations based on the mixed LSDA+U/GGA+Uschemes as well as LDA+DMFT are widely used to studythe influence of electronic correlations on the electronicand magnetic properties of half-metallic magnets. Thusthe study of the influence of the double-counting term onthe calculated properties for these materials is extremelyimportant with respect to their potential use in realisticdevices. The aim of the present study is to explore theeffect of both AL and AMF functionals when perform-ing GGA+U calculations with respect to usual electronicband structure calculations using the GGA functionalfor a wide range of half-metallic magnets (the reader isreferred to Ref. 29 for an extended discussion on theexact formulation of the two functionals). To achieveour goal we have employed the full-potential nonorthog-onal local-orbital minimum-basis band structure scheme(FPLO). For the GGA calculations we have used thePerdew-Burke-Ernzerhof parametrization. In the caseof the GGA+U calculations the on-site Coulomb inter-actions for the correlated d or p orbitals are introducedvia the F , F , F and F Slater parameters. For all cal-culations a dense 20 × ×
20 grid in the reciprocal spacehas been used to carry out the integrals and both thecharge density (up to 10 − in arbitrary units) and thetotal energy (up to 10 − Hartree) have been convergedin each case.In order to cover a wide range of half-metallic mag-nets in a coherent way, we have used in our calculationsthe ab-initio determined Coulomb effective interactionparameters (Hubbard U and Hund exchange J betweenlocalized d or p electrons) calculated in Ref. 32 using theconstrained Random Phase Approximation (cRPA) for 20 half-metallic magnets. We should note that (i)the determination of these parameters from experimen-tal data is a difficult task, and (ii) the constrained local-density approximation (cLDA), although is the mostpopular theoretical approach, it is well known to giveunreasonably large Hubbard U values for the late tran-sition metal atoms due to difficulties in compensatingfor the self-screening error of the localized electrons, and thus cRPA which does not suffer from these dif-ficulties, although numerically much more demandingthan cLDA, offers an efficient way to calculate the effec-tive Coulomb interaction parameters in solids. Wepresent results for all 20 half-metallic magnets studiedin Ref. 32 which include representatives of the (i) semi-Heusler compounds like NiMnSb, (ii) ferrimagnetic full-Heusler compounds like Mn VAl, (iii) inverse full-Heuslercompounds like Cr CoGa, (iv) usual L2 -type ferromag-netic full-Heusler compounds, (v) transition-metal pnic-tides like CrAs, and finally (vi) sp -electron (also called d ) ferromagnets like CaN. We have used the lattice pa-rameters presented in Table 1 of Ref. 32. The Slaterparameters entering the FPLO method are connected to the Hubbard parameter U LDA + U and to the Hund ex-change J presented in Table II of Ref. 32 for the corre-lated p -states though the relations F = U LDA + U , F = 5 × J, F = F = 0 , (1)and for the correlated d -states F = U LDA + U , F + F
14 =
J, F F = 0 . , F = 0 . (2)We should note here that U LDA + U is an effective param-eter depending both on the on-site intra-orbital Coulombrepulsion between electrons occupying the same orbitaland on-site inter-orbital Coulomb repulsion between elec-trons occupying orbitals of the same ℓ character but dif-ferent m ℓ value. Thus, our study covers a wide rangeof half-metallic magnets allowing for a deeper under-standing of the behavior of the AL and AMF double-counting functionals in the GGA+U calculated electronicand magnetic properties of different HM magnetic sys-tems. III. RESULTS AND DISCUSSIONA. Binary Compounds
We will start the presentation of our results fromthe binary compounds. There are two families of half-metallic binary compounds. The first includes theso-called sp -electron ferromagnets (also known as d -ferromagnets). These compounds adopt the rocksaltcubic structure and have no transition-metal atoms intheir chemical formula. We consider the nitrides and thecarbides (CaN, SrN, SrC, and BaC) since they have thelargest calculated Curie temperatures among the stud-ied sp -electron ferromagnets. Their total spin mag-netic moment in units of µ B equals 8 − Z t , where Z t isthe total number of valence electrons in the unit cell;this behavior is known as Slater-Pauling (SP). A de-tailed discussion on the origin of this rule and its con-nection to the half-metallicity can be found in Ref. 42.The usual GGA calculations produced for all four stud-ied compounds a half-metallic state with total spin mag-netic moments of 1 µ B for the nitrides and 2 µ B for thecarbides. The results are gathered in Table I. The spinmoment is carried mainly by the N and C atoms. Our cal-culated GGA results are similar to the GGA ones derivedwith the full-potential linearized augmented plane-wave(FLAPW) method as implemented in the FLEUR codein Ref. 32. The use of the AMF within the GGA+Uscheme leaves intact both the calculated spin magneticmoments and density of states (DOS) with respect toGGA calculations (we do not present the DOS since theyare similar to the ones presented in literature). On thecontrary the use of the AL functional has a tremendouseffect on the calculated results. It produces an unreason-able and unphysical charge transfer from the Ca(Sr,Ba) TABLE I: Atom-resolved and total spin magnetic momentper formula unit for the XY binary compounds. Results havebeen obtained within the FPLO method using the GGAfunctional for the exchange interaction potential and theGGA+U scheme employing both the atomic-limit (AL - alsoknown as fully-localized-limit FLL) and the around-mean-field (AMF) functionals for the double counting term. Val-ues for the on-site Coulomb and exchange parameters are theab-initio determined ones within the constrained Random-Phase-Approximation (cRPA) in Ref. 32. Lattice constantsare the ones presented in Table 1 in the later reference. Notethat for the compounds which do not contain transition metalatoms (known as d -ferromagnets) GGA+U within the ALfunctional gives unrealistic results.Comp. Functional m X m Y m total CaN GGA -0.065 1.065 1.000GGA+U (AL) 11.836 -4.836 7.000GGA+U (AMF) -0.065 1.065 1.000SrN GGA -0.072 1.072 0.999GGA+U (AL) 16.363 -9.362 7.000GGA+U (AMF) -0.072 1.072 0.999SrC GGA -0.004 2.004 1.999GGA+U (AL) 15.578 -9.578 6.001GGA+U (AMF) -0.004 2.004 1.999BaC GGA 0.057 1.943 2.000GGA+U (AL) 3.261 0.378 3.999GGA+U (AMF) 0.057 1.943 2.000VAs GGA 2.427 -0.427 2.000GGA+U (AL) 2.415 -0.415 2.000GGA+U (AMF) 2.151 -0.151 2.000CrAs GGA 3.614 -0.614 3.000GGA+U (AL) 3.880 -0.880 3.000GGA+U (AMF) 3.541 -0.541 3.000MnAs GGA 4.173 -0.311 3.862GGA+U (AL) 4.476 -0.476 3.999GGA+U (AMF) 3.979 -0.326 3.652 atoms to the N(C) atoms resulting to huge values of theatom-resolved spin moments. This state is obviously anartifact of the method. We cannot explain the origin ofthis behavior but starting form various configurations allcalculations involving the AL functional converged to thesame results and thus the breakdown of the AL shouldbe attributed to its characteristics.The second family of binary compounds under studyare the binary VAs, CrAs, and MnAs transition metalpnictides. The first observation of such a compoundsbeing half-metal was made in 2000 when Akinagaand his collaborators managed to grow multilayers ofCrAs/GaAs. CrAs was found to adopt the zincblendestructure of GaAs and was predicted to be a half-metalwith a total spin magnetic moment of 3 µ B in agreementwith experiments. Several studies followed this initialdiscovery, and electronic structure calculations have con-firmed that also similar binary XY compounds, where Xis an early transition-metal atom and Y an sp element,should be half-metals and the total spin magnetic mo-ment follows a SP rule similar to d -ferromagnets being -4,5 -3 -1,5 0 1,5 3 Energy-E F (eV) -10-50510 D O S ( s t a t e s / e V / s p i n ) GGAGGA+U (AMF) -10-50510
GGAGGA+U (AL)
CrAs
FIG. 1: (color online) Total density of states (DOS) as afunction of the energy for the CrAs compound within theGGA+U method using both the atomic-limit (AL) and thearound-mean-field (AMF) functionals for the double-countingterm. GGA+U results are compared to the GGA calculatedDOS. The zero in the energy axis has been set to the Fermilevel. Positive(negative) values of the DOS correspond to themajority(minority)-spin electrons. now equal to Z t − In Table I we gathered all the calculated spin mag-netic moments. GGA gives a half-metallic state for VAsand CrAs, while for MnAs the Fermi level is slightlyabove the minority-spin energy gap and the total spinmagnetic moment slightly smaller than the ideal valueof 4 µ B for half-metallicity to occur. These results havebeen largely discussed in literature. For both VAsand CrAs, GGA+U self-consistent calculations yield ahalf-metallic state within both AL and AMF function-als with the same total spin magnetic moment but withsubstantial variations of the atom-resolved spin magneticmoments. For MnAs the use of AL functional leads to ahalf-metallic state contrary to AMF for which the Fermilevel is above the minority-spin gap. Overall AL leads tolarger absolute values of the atomic spin moments withrespect to GGA while AMF leads to smaller values. Thisbehavior of the atomic spin magnetic moments confirmsthe conclusion in Ref. 29 that AMF gives the magneticstate a large energy penalty with respect to AL.Since DOS present similar trends between the threetransition metal binary compounds, we present in Fig. 1the calculated DOS per formula unit for CrAs. In theupper panel we compare the GGA+U calculated DOSwithin the AL functional to the usual GGA calculatedDOS, and in the lower panel we present a similar graphfor the AMF case. In the presented energy Cr DOS dom-inates. GGA produces a large minority-spin gap with alarge exchange splitting between the occupied majority-spin bands and the unoccupied minority-spin bands andthus strong tendency to magnetism manifested also bythe large ( ∼ µ B ) Cr spin moment. The use of the ALdouble-counting functional in the GGA+U calculationslead to an almost rigid shift of the minority spin DOStowards higher energies, while in the majority-spin DOSonly the double-degenerate e g states at about -1.5 eVmove lower in energy (see Ref. 54 for a discussion of thecharacter of the bands). In the case of AMF the majority-spin band structure shows a similar behavior with respectto the GGA results as the AL case. But in the minority-spin band structure the tendency is the opposite now.Since AMF does not favor magnetism as strongly as AL,the minority-spin band structure now presents an almostrigid shift towards lower energy values. These findingalso explain the behavior of the MnAs compound. Inthe case of AL the minority-spin band structure movestowards higher energy values and the Fermi level nowmoves within the gap and half-metallicity appears. B. Heusler compounds
Heusler compounds are a huge family of intermetalliccompounds presenting various types of electronic andmagnetic behaviors.
Several among them are half-metallic ferromagnets/ferrimagnets/antiferromagnetsand are of particular interest due to their very highCurie temperatures, which usually exceed 1000 K, mak-ing them ideal for applications. There are four mainfamilies of Heusler compounds: (i) the semi-Heuslersalso known as half-Heuslers like NiMnSb which have thechemical type XYZ with X and Y being transition metalatoms, (ii) the usual full Heuslers like Co MnSi with thechemical type X YZ where the valence of X is larger thanthe valence of Y and the two X atoms are equivalent,(iii) the quaternary Heuslers like (CoFe)MnSi whichpresent similar properties with the full-Heuslers, andfinally (iv) the so-called inverse-Heuslers, like Cr CoGawhich have also the chemical type X YZ but now thevalence of X is smaller than the valence of Y and due tothe change of the sequence of atoms in the unit cell thetwo X atoms are no more equivalent.
We presentresults for all families of compounds with the exceptionof quaternary-Heuslers which present similar behavior tothe full-Heuslers and for which no Hubbard parametershave been derived in Ref. 32.
1. Semi-Heuslers
The first family of Heusler compounds for which wewill present results are the semi-Heuslers. The first com-pound that was predicted to be a half-metal was actu-ally a semi-Heusler, NiMnSb. Their total spin magneticmoment follows also a SP rule being Z t -18 (for an ex- -6 -3 0 3 Energy-E F (eV) -4-2024 D O S ( s t a t e s / e V / s p i n ) -4-2024 GGAGGA+U (AL) -6 -3 0 3
GGAGGA+U (AMF)
NiMnSb NiNiMn Mn
FIG. 2: (color online) Ni and Mn atom-resolved DOS inNiMnSb. Details as in Fig. 1. tended discussion see Ref. 58). In Table II we have gath-ered our calculated spin magnetic for all studied casesand for three compounds FeMnSb, CoMnSb and NiMnSb(note that for FeMnSb we were not able to converge theGGA+U calculations using the AMF functional). As wemove from one compound to the other, the total num-ber of valence electrons increases by one and so does theGGA calculated total spin magnetic moment. Mn atomsin all case posses a large value of spin magnetic momentwhich starts from ∼ µ B in FeMnSb and exceeds 4 µ B in NiMnSb. As we increase the total number of valenceelectrons the spin magnetic of the X atoms also increasesbeing ∼ -1.3 µ B for Fe, -0.34 µ B for Co and 0.14 µ B forNi in the corresponding compounds. The GGA calcu-lated DOS, presented in Fig. 2 for NiMnSb, has beenstudied in detail in literature and it is mainly character-ized by the large exchange splitting between the occupiedmajority-spin and the unoccupied minority-spin d -statesat the Mn site which together with the very small weightof the occupied minority-spin states are responsible forthe large Mn spin magnetic moments. This feature iscommon for all three studied compounds and has beenalready observed in literature. The self-consistent GGA+U calculations using theAMF functional for the CoMnSb and NiMnSb com-pounds produced a similar picture to the GGA calcu-lations. The total spin magnetic moment, as shown inTable II remains identical to the GGA case and the atom-resolved spin magnetic moments only scarcely changed.This is also reflected on the Ni and Mn resolved DOS forNiMnSb in Fig. 2 where the GGA+U within AMF cal-culated DOS is almost identical to the GGA calculatedDOS. The effect of the use of the AL functional is more
TABLE II: Similar to Table I for the semi-Heusler compoundscrystallizing in the C1 b lattice structure having the XYZchemical formula.Comp. Functional m X m Y m Z m total FeMnSb GGA -1.279 3.374 -0.095 2.000GGA+U(AL) -2.274 4.559 0.042 2.327CoMnSb GGA -0.340 3.568 -0.227 3.000GGA+U (AL) -1.264 4.520 -0.186 3.068GGA+U (AMF) -0.358 3.535 -0.177 3.000NiMnSb GGA 0.143 4.031 -0.174 3.999GGA+U (AL) -0.087 4.553 -0.301 4.164GGA+U (AMF) 0.144 4.026 -0.171 3.999 drastic. As shown in Fig. 2 GGA+U within the ALfunctional compared to usual GGA leads to large mod-ifications of the DOS of the transition metal atoms. Inthe case of Mn, the exchange splitting between the oc-cupied majority-spin and the unoccupied minority-spinstates increase considerably. In the majority-spin bandstructure of Mn, the occupied states move lower in en-ergy and as a result they are no more in the same en-ergy with the Ni majority-spin states. This leads to aweaker hybridization between the d states of Ni and Mnatoms and in the Ni DOS the width of the majority bandsbecomes smaller as a result of the weaker hybridizationeffects. Almost all the weight of the occupied minority-spin band structure is located at the Ni atoms while al-most all unoccupied minority-spin states are located atthe Mn atom. Thus there is almost no hybridization be-tween the minority-spin d -states of Ni and Mn and theformer are not affected by the shift of the later to largerenergy values being identical to the GGA case. Thesechanges in DOS are also reflected on the spin magneticmoments in Table II. The larger tendency to magnetismwithin AL compared to AMF leads to slightly larger to-tal spin magnetic moments which deviate from the idealinteger values of the SP rule and the Fermi level is lo-cated slightly below the minority-spin energy gap. In thecase of FeMnSb and CoMnSb, both the absolute valuesof the Fe(Co) and Mn spin magnetic moments increaseby about 1 µ B almost cancelling each other. In the caseof NiMnSb the variations in the atomic spin magneticmoments are considerably smaller since almost all Ni d -states are occupied in all studied cases. But even forNiMnSb the Mn spin moment increases by ∼ µ B andthe Ni spin moment decrease by about 0.3 µ B . The Sbatoms in all cases also present changes in their atomicspin magnetic moments between the various calculationsalthough these variations are considerably smaller thanfor the transition metal atoms.
2. Full-Heuslers
The second family of Heusler compounds which maypresent half-metallicity are the so-called usual full-Heuslers crystallizing in the cubic L2 structures. Half- TABLE III: Similar to Table I for the full-Heusler compoundscrystallizing in the L2 lattice structure having the X YZchemical formula.Comp. Functional m X m Y m Z m total Mn VAl GGA -1.670 1.227 0.113 -1.999GGA+U (AL) -4.102 3.169 0.483 -4.551GGA+U (AMF) -1.798 1.527 0.079 -1.990Mn VSi GGA -0.801 0.557 0.060 -0.985GGA+U (AL) -2.248 2.759 0.376 –1.362Co CrAl GGA 0.737 1.684 -0.160 2.999GGA+U (AMF) 0.965 1.222 -0.153 3.000Co CrSi GGA 0.934 2.242 -0.111 4.000GGA+U (AL) 0.871 2.525 -0.267 4.000GGA+U (AMF) 0.890 2.187 0.031 4.000Co MnAl GGA 0.673 2.910 -0.231 4.025GGA+U (AL) 1.381 4.176 -0.501 6.438GGA+U (AMF) 1.048 1.998 -0.096 3.999Co MnSi GGA 0.972 3.195 -0.140 4.999GGA+U (AL) 0.732 3.954 -0.338 5.080GGA+U (AMF) 0.987 3.020 -0.004 5.000Co FeAl GGA 1.163 2.870 -0.203 4.999GGA+U (AL) 1.188 3.326 -0.520 5.177GGA+U (AMF) 1.216 2.673 -0.105 4.999Co FeSi GGA 1.327 2.926 -0.042 5.539GGA+U (AL) 1.375 3.450 -0.201 5.999 metallicity can be combined either with the appearanceof ferrimagnetism, when the X atoms in the X YZ is theMn one, or with ferromagnetism when X is Co. In allcases the total spin magnetic moment in µ B follows aSP rule being Z t -24. In Table III we have gathered thecalculated spin magnetic moments for all studied com-pounds with both GGA and GGA+U methods usingboth the AL and AMF double-counting functional in thelater case. When one case is missing in the table, this isdue to the fact that we were not able to get convergenceirrespectively of the starting input which we have used.First, we will discuss our results on the half-metallicferrimagnetic Mn VAl and Mn VSI compounds wherethe total spin magnetic moments is negative since thetotal number of valence electrons is less than 24. More-over the Mn spin magnetic moments are antiferromag-netically coupled to the V spin moments due to theirsmall distance. In the case of Mn VAl, GGA+U cal-culations within AMF produced similar spin momentsand DOS to the GGA case; we were not able to con-verge GGA+U within AMF for the Mn VSi compound.For both compounds the use of AL double-counting func-tional produced unphysical results similar to the case of d -ferromagnets in Sec. III B. The use of AL tripled,with respect to the GGA case, the absolute values of thespin magnetic moments of the transition metal atoms inMn VA; in the case of V in Mn VSi the increase is almost600% . Thus the use of AL for the half-metallic ferrimag-netic Heusler compounds obviously is inadequate.In the case of ferromagnetic full-Heuslers containingCo the effect of using both AMF and AL on the calcu-lated electronic and magnetic properties is more complex -6 -3 0 3
Energy-E F (eV) -4-2024 D O S ( s t a t e s / e V / s p i n ) -4-2024 GGAGGA+U (AL) -6 -3 0 3
GGAGGA+U (AMF) Co MnSi CoCoMn Mn
FIG. 3: (color online) Co and Mn atom-resolved DOS inCo MnSi. Details as in Fig. 1. than in all the previously studied cases. When Y is Cr(Co CrAl and Co CrSi) both AL and AMF yielded a per-fect half-metallic state with the total spin magnetic mo-ment being equal to the ideal values predicted by the SPrule as shown in Table III. When Y is Mn (Co MnAl andCo MnSi) AMF produced a half-metallic states and bothatom-resolved and total spin magnetic moments whereclose to the GGA case, but AL led to a considerableincrease of the Mn spin moment similarly to the semi-Heuslers. The increase of the Mn spin moment withinAL led to an increase also of the total spin magnetic mo-ment which is only 0.80 µ B for Co MnSi but reachesthe ∼ µ B for Co MnAl. When Y is Fe (Co FeAland Co FeSi) the behavior of the spin moments with re-spect to the GGA results is similar within both AL andAMF to the case where Y is Mn. Moreover in the caseof Co FeSi which is not half-metallic within GGA, theuse of GGA+U combined with AL leads to a total spinmagnetic moment of 6 µ B and to a half-metallic stateas shown also in Ref. 13. Although the GW scheme produced similar results to the GGA+ calculations, cor-relations in this materials are still an open issue sincerecent results by Meinert and collaborators show that aself-consistent calculation fixing the total spin magneticmoment to 6 µ B reproduces more accurately the posi-tion of the band with respect to available experimentaldata. To understand the behavior of the spin moments wehave to examine in detail the behavior of the DOS. Sincethe trends when either AMF or AL is employed are simi-lar for all six ferromagnetic Co-based full-Heuslers understudy, we will use Co MnSi as an example and in Fig.3 we present the Co and Mn resolved DOS. When the -6 -3 0 3
Energy-E F (eV) -8-4048 D O S ( s t a t e s / e V / s p i n ) -8-4048 GGAGGA+U (AL) -6 -3 0 3
GGAGGA+U (AMF) Co MnSiCo MnAl
FIG. 4: (color online) Total DOS per formula unit for theCo MnSi and Co MnAl compounds. Details as in Fig. 1.
GGA+U combined with AMF is used (left panel) there isa significant change in the DOS unlikely all other familiesof half-metallic compounds discussed above. AMF en-hances the tendency to magnetism with respect to GGA.For the Mn atom the occupied majority spin states shiftlower in energy but the minority-spin energy gap in theMn DOS remains unchanged. Through hybridizationalso the Co majority-spin DOS shifts lower in energyand so do also the occupied Co minority-spin states. Thisleads to an increase of the energy gap in the Co minority-spin band structure. Since as explained in Ref. 61 Coatoms present in usual GGA calculations a much smallergap than the Mn atoms and thus determine the energygap in the total DOS, the opening of the former furtherstabilizes the half-metallic state.The GGA+U method combined with AL even fur-ther enhances the tendency to magnetism with respect toAMF as concluded in Ref. 29. In the case of Mn atomsthe exchange splitting between the occupied majority-spin and unoccupied minority-spin states is greatly en-hanced as for Mn in NiMnSb and thus the energy gapbecomes much larger. As a side-effect some weight inthe minority-spin band structure appears just below theFermi level. Thus although with respect to the GGAcase, AL opens the gap the Fermi level is located close tothe left edge of the gap instead of the middle in the GGAcase. Co DOS follows through hybridization the behaviorof the Mn d -states and the gap is now also much largerbut the occupied minority-spin states move closer to theFermi level which now just crosses the states just belowthe low-energy edge of the gap and the total spin mag-netic moment within AL is slightly larger than the idealvalue of 5 µ B . TABLE IV: Similar to Table I for the full-Heusler compoundscrystallizing in the inverse XA lattice structure having theCr YZ chemical formula, where the two Cr atoms occupy sitesof different symmetry (see text).Comp. Functional m Cr A m Cr B m Y m Z m total Cr FeGe GGA -1.454 1.753 -0.313 0.027 0.012GGA+U(AL) -4.162 4.672 -1.054 0.551 0.006Cr CoGa GGA -2.680 1.973 0.379 -0.014 0.069GGA+U (AL) -4.860 4.137 1.160 -0.082 0.520
In the case of Co MnAl the change in the spin mag-netic moments is larger within both AL and AMF func-tionals. As shown in Fig. 4, although we just changeAl for Si in Co MnSi, the AMF DOS shows a differenttendency with respect to the energy gap. The exchangesplitting between occupied majority and unoccupied mi-nority spin states is smaller, and within AMF the gapis smaller than within GGA showing the contrary ten-dency to Co MnSi where AMF produced a larger gapwith respect to GGA. For Co MnAl within usual GGAthe Fermi level is close to the left edge of the gap whilefor Co MnSi it is located at the middle of the gap. Thusin the case of AL based calculations the shift of the Cooccupied minority spin states towards higher energies forCo MnAl, discussed just above also for Co MnSi, leadsto the loss of the half-metallicity since now the Fermi levelcrosses the occupied minority-spin states. The other Al-based Heuslers (Co CrAl and Co FeAl) exhibit withinGGA a DOS around the minority-spin energy gap simi-lar to Co MnSi and not Co MnAl and thus the increasein their total spin magnetic moment within AL is muchsmaller than for Co MnAl.
3. Inverse-Heuslers
The last family of potential half-metallic Heusler com-pounds ar the so-called inverse Heusler compounds. Among these half-metals the most interesting are theso-called fully-compensated ferrimagnets (also knownas half-metallic antiferromagnets) like Cr FeGe andCr CoGa. These materials are of special inter-est since they combine half-metallicity to a zero to-tal net magnetization and thus are ideal for spin-tronic/magnetoelectronic devices due to the vanishingexternal stray fields created by them. We should notethat films of Cr CoGa have been grown experimentally and this compounds has been predicted to exhibit ex-tremely large Curie temperature. As shown in Table IVGGA yields for both Cr FeGe and Cr CoGa compoundsa total spin magnetic moment close to zero (for an ex-tended discussion on the half-metallic inverse Heuslers see Ref. 64). Note that we have two inequivalent Cratoms in these compounds denoted by the superscripts A and B in Table IV. We were not able to convergethe GGA+U self-consistent calculations using the AMFdouble-counting functional. For the AL functional al-though the total spin magnetic moment stays close tozero, the absolute values of the Cr spin magnetic mo-ments are about doubled leading to an unphysical sit-uations. Thus for these materials the use of GGA+Ucombined with AL is not able to produce a reasonabledescription of the electronic structure as was also the casefor the semi-Heuslers and the ferrimagnetic full-Heuslers. IV. CONCLUSIONS
We have studied the electronic and magnetic proper-ties of 20 half-metallic magnets performing self-consistentGGA+U calculations using both the atomic-limit (AL)and around-mean-field (AMF) functionals for the dou-ble counting term and compared them to the usual GGAcalculations. Overall the use of AMF produced resultssimilar to the usual GGA calculations. The effect of ALwas diversified depending on the studied material. Inthe case of d -ferromagnets, semi-Heuslers, ferrimagneticfull-Heuslers and inverse Heuslers the use of AL leads tounrealistic electronic and magnetic properties of the stud-ied compounds and thus its use is not justified. On theother hand in the case of transition-metal binary com-pounds and usual ferromagnetic full-Heusler compoundsthe use of AL enhanced the tendency towards magnetismwith respect to both GGA and GGA+U combined withAMF. Depending on the position of the Fermi level, therewere cases like MnAs and Co FeSi for which AL produceda half-metallic state contrary to GGA and GGA+U com-bined with AMF, cases like VAs, CrAs and Co CrSiwhere all three methods produced a half-metallic state,and cases like Co MnAl, Co MnSi and Co FeAl wherethe use of AL led to the loss of half-metallicity.Methods based on the combination of the usual densityfunctional theory (DFT)-based codes and of the Hubbard U - Hund’s exchange J are widely used to investigate theproperties of strongly correlated materials. Our resultssuggest that especially in the case of half-metallic mag-nets the choice for the double counting functional usedto subtract the part from the DFT total energy, whichis associated to the Coulomb repulsion between the cor-related orbitals, plays a decisive role on the obtained re-sults. 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