DOI: 10.1142/S021797921005380X

March 24, 2010 10:19 WSPC/140-IJMPB S021797921005380X

International Journal of Modern Physics BVol. 24, No. 8 (2010) 967–978c© World Scientific Publishing Company

FIRST PRINCIPLES STUDY OF ELECTRONIC STRUCTURE

AND MAGNETIC PROPERTIES OF HALF-METALLIC

FULL-HEUSLER ALLOYS Co2MnSi and Co2FeSi

JINGSHAN QI∗, HAILIN YU†, XUEFAN JIANG†,‡ and DANING SHI∗

∗Department of Physics,

Nanjing University of Aeronautics and Astronautics,

Nanjing 210016, China†Jiangsu Key Laboratory of Advanced Functional Materials,

Changshu Institute of Technology, Changshu 215500, China‡xfjiang@cslg.edu.cn

Received 2 July 2007

We present a comprehensive investigation of the equilibrium structural, electronic andmagnetic properties of Co2MnSi and Co2FeSi by density-functional theory (DFT) withinthe generalized gradient approximation (GGA) using the projected augmented wave(PAW) method. The on-site Coulomb interaction has also taken into account (GGA+U)approach to unravel the correlation effects on the electronic structure. The change ofthe energy gap, “spin gap”, Fermi energy level and magnetic moments with the latticeparameters is investigated. We found that the on-site correlation interaction in Co2FeSiis stronger than in Co2MnSi. So on-site electronic correlation is necessary for Co2FeSiand the magnetic moments reproduce experimental results well by GGA+U . Further wealso found that a moderate change of the lattice parameters does not change the half-metallic ferromagnet (HMF) behavior for both materials. Appearance of half-metallicityis consistent with the integral magnetic moments, which also agrees with the experimentmeasurements.

Keywords: Half-metallicity; magnetic moments; on-site correlation; Heusler alloys.

PACS numbers: 71.20.Be, 71.20.Lp, 75.50.Cc

1. Introduction

Spin polarization near the Fermi level (EF ) in ferromagnets plays an important

role in spintronics,1 which have attracted great scientific interest in particular for

magneto-electronics.2 Thus, the half-metallic ferromagnets (HMF) are a key mate-

rial for spintronics because they have a band gap at the Fermi level (EF ) for one

spin direction and thus exhibit 100% spin polarization at the EF .3

The Heusler alloys remain attractive for technological applications due to their

relatively high Curie temperatures compared to other half-metallic compounds.4

‡Corresponding author.

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968 J. Qi et al.

Particularly for room temperature devices, one needs to prevent reduction of the

spin polarization and other magnetic properties by thermal effects. From experi-

ments, it should be noted that the Co2 based Heusler compounds exhibit the highest

Curie temperature. The full-Heusler alloy Co2MnSi has attracted particular inter-

est because it is predicted to have a large minority spin band gap of 0.4 eV and very

high Curie temperature of 985 K.5,6 An even higher Curie temperature (1100 K) ac-

companied by a large magnetic moments (6 µB) was found in recent investigations

of Co2FeSi,7–9 which has the highest Curie temperature and the largest magnetic

moments among the known Heusler compounds.

Heusler alloys have already been managed to be applied technologically like

spin-injection devices,10 spin-filters,11 tunnel junctions,12 and GMR devices.13,14

The most successful recent application in spintronics concerns the half-metallic

full Heusler alloys, for example, magnetic tunnel junctions based on Co2MnSi,15–18

spin-polarized current injection from Co2MnGe into a semiconducting structure.19

The report of the tunneling magnetoresistance observation at room temperature

for the Magnetic tunnel junctions using a full-Heusler alloy electrode have been

extensively studied using Co2MnSi20,21 and Co2FeSi films.22

A lot of work has been done by the first-principle calculation by the local spin

density approximation (LSDA) schemes. Recently, Co2FeSi also exhibited to be a

HMF by the first-principles calculation7 and obtained the integer magnetic mo-

ments 6 µB by taking account for on-site correlation (LSDA + U) method, which

was demonstrated experimentally.7,32 Very recently for Co2MnSi and Co2FeSi, the

LSDA + U scheme was also used to represent on-site electron correlation in the

calculations to overcome the shortage of LSDA and generalized gradient approxi-

mation (GGA).9 They also calculated the dependence of the magnetic moments on

the lattice parameter, focusing on the differences between LSDA and GGA treat-

ments. However, they ignored the fact that the Fermi level could change with the

lattice parameter, which could affect the material’s half-metallicity.

In order to further investigate Co2MnSi and Co2FeSi in this work, we present a

comprehensive investigation of the equilibrium structural, electronic and magnetic

properties of Co2MnSi and Co2FeSi. Especially, we investigated the dependence

of half-metallicity and the magnetic moments on the lattice parameter in detail,

focusing on the differences between GGA and GGA+ U treatments.

2. Computational Details

Co2MnSi and Co2FeSi belong to a group of ternary intermetallics with the stoichio-

metric composition X2YZ which crystallizes in the L21-type structure, and consists

of four FCC sublattices, two of which are occupied by the same type of X-atoms. In

general, the X and Y atoms are transition metals and Z is a main group element.

In some cases, Y is replaced by a rare earth element. The X atoms are placed on

8a (1/4, 1/4, 1/4) Wyckoff positions and the Y and Z atoms on 4a (0, 0, 0) and 4b

(1/2, 1/2, 1/2). A super cell containing 16 atoms was used in our computation.

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First Principles Study and Heusler Alloys 969

The calculations were performed within the framework of DFT, using the

projector-augmented wave (PAW) method23 as implemented in the Vienna ab initio

simulation package (VASP).24–27 The spin polarized calculations were performed

within GGA given by Perdew-Wang28 and the VWN (Vosko, Wilk and Nusair)

interpolation formula29 is used. The cutoff energy for the plane wave was set

400 eV with a convergence in the total energy of 10−4 eV and automatic generation

11× 11× 11 k-points (56 irreducible k-points).

The GGA + U method30 was used to account for on-site Coulomb interaction at

the transition metal sites and UCo = 3.0 eV, UFe = UMn = 4.1 eV, JCo = 1.0 eV,

JMn = JFe = 1.1 eV. It will be shown that the GGA+U method gives qualitative

and quantitative improvements compared to GGA approaches for Co2FeSi. The

same U and J for both materials were used for comparing the different effect which

was brought by the same correlation. A variation of those U and J parameters in

the calculations was omitted here because it would not bring more insight into the

nature of the problem, at present.

3. Results and Discussion

3.1. Structure properties

First, a structural optimization was performed for Co2FeSi and Co2MnSi to find

out whether the experimental lattice parameter minimizes the total energy. Table 1

summarizes the optimized lattice constant and the experimental values. It was

found that the optimized lattice constant from the calculation agreed very well

with the experimental values of aexp = 5.64 A for Co2FeSi and aexp = 5.645 A for

Co2MnSi. For Co2FeSi, the energy minimum (ferromagnetic) was found to appear

at a = 5.625 A (corresponding to ∆a/aexp = −0.26%) by GGA and 5.651 A

(corresponding to ∆a/aexp = +0.20%) by GGA+ U . Similar calculations revealed

for Co2MnSi that the energy minimum appears at a = 5.635A by GGA and a =

5.655 A by GGA + U with the error ∆a/aexp = ±0.18%. Those results show that

Table 1. Optimized lattice constant a in A and calculated mag-netic moments M in µB for Co2MnSi and Co2FeSi by GGA andGGA+ U , respectively. The experimental value aexp and Mexp

and the error (a) = ∆a/aexp and the error (M) = ∆M/Mexp

of the calculated value and experimental value is also presentedhere.

Co2FeSi Co2MnSiGGA GGA+ U GGA GGA+ U

a 5.625 5.651 5.635 5.655aexp 5.64 5.64 5.645 5.645error(a) −0.26% +0.20% −0.18% +0.18%M 5.4 6.0 5.0 5.2Mexp 6.0 6.0 5.0 5.0error(M) +10% 0% 0% +4%

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both GGA and GGA+U schemes are essential to accurately produce the equilibrium

structural properties of Co2FeSi and Co2MnSi. At the same time, we found that in

the process of relaxation, the internal parameter and cell shape are not changed,

only volumes changed. So when we investigate the dependence of half-metallicity

and the magnetic moments on the lattice parameter, we can select different lattice

constants to calculate electronic and magnetic properties.

3.2. Electronic properties

We showed spin-polarized total and projected density of states (DOS and PDOS)

at equilibrium lattice constant from GGA and GGA + U for Co2FeSi in Fig. 1

and for Co2MnSi in Fig. 2. The lowest DOS (from −12 eV to −9 eV in both the

majority and minority spin states) is ignored because it is almost entirely due to

Fig. 1. Total spin resolved density of states (DOS) and the partial density of states (PDOS) of

Co2FeSi by GGA and GGA + U . The upper part of the picture is for spin majority (solid line)and lower part for spin minority (dotted line).

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First Principles Study and Heusler Alloys 971

Fig. 2. Total spin resolved density of states (DOS) and the partial density of states (PDOS) ofCo2MnSi by GGA and GGA + U . The upper part of the picture is for spin majority state (solidline) and lower part for spin minority state (dotted line).

Si s electrons and is separated with respect to the other hybridized bands, being

basically unaffected by the Mn or Fe and Co exchange interaction. The majority

spin DOS is strongly metallic, while the minority spin DOS shows a semiconducting

gap around the Fermi level, EF . The high DOS below EF for the minority states

is dominated by d-states being located at Co and Fe or Mn sites. Inspecting the

majority DOS, one finds a small DOS near EF . We can also see that the contribution

of Si is very small near the Fermi level EF .

For example, for Co2FeSi in the majority spin component, Fe 3d states are

occupied and hybridized with Co 3d electrons; however, in the minority spin part,

local and mostly unoccupied Fe and Co 3d states are found at about 0.5 eV above

EF . Particularly, we point out that features of the Fe projected density of states

(PDOS) can be traced back to the eg − t2g splitting of Fe 3d levels in a cubic

crystal field: the majority (minority) spin states show two well-separated peaks at

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−4 and −2 eV (−2.5 and 0.5 eV). On the other hand, the eg− t2g splitting is not so

evident in the Co majority PDOS, which does not reveal any prominent features.

Similarity is also seen from Fig. 2 for Co2MnSi.31 By GGA, the fact is that for

Co2MnSi, the half-metallicity occurs at the equilibrium lattice constant, however,

not half-metallicity for Co2FeSi because the Fermi energy cuts the minority DOS

resulting in decreasing and not integer of the magnetic moments.

By GGA + U , the main difference is in the majority spin component, DOS move

to the lower energy and in the minority spin component, DOS moves to the higher

energy relative to EF , which is the reason of occurrence of the half-metallicity

for Co2FeSi at the equilibrium lattice constant. At the same time, the energy gap

widens compared to the GGA scheme as known to us for GGA + U .

We know that Fe has six 3d electrons and Mn has five 3d electrons, however,

the atom radius 1.72 A of Fe is shorter than Mn 1.79 A. By comparing Fe 3d DOS

in Fig. 1 with Mn 3d DOS in Fig. 2 by GGA, we can see that Fe 3d DOS is more

local near EF . This indicates that the on-site correlation of Fe is stronger than Mn.

Especially as the difference of Fe and Mn element Co 3d DOS near EF in Co2FeSi in

Fig. 1 is also more local than in Co2MnSi in Fig. 2 by GGA, which also foretells that

the correlation in Co2FeSi is stronger than in Co2MnSi, so GGA+U scheme should

be necessary, especially for Co2FeSi. By GGA+U , we calculated and obtained the

measured magnetic moments 6 µB in experiments, which were not obtained by

GGA for Co2FeSi. Because the correlation is relatively weak in Co2MnSi, the on-

site Coulomb interaction will destroy the half-metallicity of Co2MnSi when using

slightly larger U value. Using the same U value which was used for Co2FeSi, we got

5.2 µB magnetic moments (corresponding to ∆M/Mexp = +4%) for Co2MnSi.

Although to the best of our knowledge, no relevant experimental study exists,

the homogeneous change of the lattice constant under hydrostatic pressure can

influence the electronic and magnetic properties of L21 Heusler alloys as Co2MnSi31

and Co2FeSi. This can also help us to understand the stabilization of half-metallicity

upon pressure. So we compressed and expanded by ±15% (∆a/aexp×100%) for the

experimental lattice parameter and investigate the dependence of half-metallicity

and the magnetic moments on the lattice parameter. Figure 3 shows the dependence

of the external energies of the lower (valence) band and the upper (conduction) band

of the minority states enveloping the gap on the lattice constant for two materials

by GGA and GGA+ U schemes, respectively.

Figure 3(a) shows the dependence of the gap on the lattice parameter for

Co2FeSi by GGA method. There is not half-metallicity at the optimized lattice

parameter (or the experimental lattice parameter) by GGA method. The minor-

ity band gap shrinks with compression and vanishes when the lattice constant is

compressed by −4%. With expansion, the minority band gap increases and half-

metallicity will appear at +10% and disappear over +14%. Such a large expansion

of the crystal volume by about 1/3 is rather unrealistic and falls far out of the ex-

pected uncertainties for the experimental measurement of a. However, it may help

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First Principles Study and Heusler Alloys 973

Fig. 3. Dependence of the minority band gap and Fermi energy on the lattice parameter. Shownare the external energies of the upper (conduction) band and the lower (valence) band of theminority states enveloping the gap and Fermi energy depending on the lattice parameter forCo2FeSi and Co2MnSi by GGA and GGA + U , respectively. Line is drawn to guide the eye.

us to inspect which changes in the electronic structure are caused by such an expan-

sion. If GGA+ U method is used, from Fig. 3(b) it is seen that at the equilibrium

lattice constant of 5.65 A (or the experimental lattice constant) half-metallicity

exists and the gap encloses the Fermi energy between −3% and +9% change of the

lattice parameter. Obviously, the gap (1.0 eV at +5%) is larger compared by GGA

scheme (0.22 eV at +11%). This particular gap value was chosen, as it is the case

where the Fermi energy lies just in the middle of the gap of the minority states and

thus the material would securely be in a HMF state. At the same time the minority

band gap will shrink when the lattice constant is expanded, and will increase with

compression and again shrink beyond −10%. So the on-site correlation should be

taken into account to explain the realistic complexion. We also saw that the effect of

expanding volume is seemingly the same as counting on-site correlation interaction

for presenting material half-metallicity.

As shown in Fig. 3(c), compared with Co2FeSi for Co2MnSi, half-metallicity

exists at the equilibrium lattice constant or experimental constant by the GGA

method and the minority gap increases first and subsequently shrinks with com-

pression, but when the lattice constant is expanded, the minority gap will decrease

at all times. The range of lattice constant for half-metallicity is from −4% to 3%

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974 J. Qi et al.

change of the lattice parameter. Therefore, a moderate change of the lattice param-

eter does not change the HMF behavior of Co2MnSi. With GGA+U , the difference

is that there does not exist half-metallicity at the equilibrium lattice constant and

the gap-width increases from 0.65 eV at −1% (GGA) to 1.69 eV at −9% (GGA+U),

at the same time the gap encloses the Fermi energy between −13% and −2% change

of the lattice parameter in Fig. 3(d). From that we can see that the on-site correla-

tion destroys the half-metallicity of Co2MnSi at the equilibrium lattice parameter

and also enhances the gap-width.

From the above description, we can see that the general trends are similar:

the minority band gap increases (decreases) with compression (expansion) for the

small change of the lattice constant. Upon compression, the Fermi level moves in

the direction of the conduction band, upon expansion towards the valence band.

Namely, “spin gap” (distance of Fermi level away from the minority valence band

maximum)31 widens upon compression and narrows upon expansion. The “spin

gap” is a very important quantity for the half-metallic ferromagnets because it is

the minimum energy required to flip a minority spin electron from the valence band

maximum to the majority spin Fermi level.

In order to explain those behaviors, we first note that the position of the Fermi

level is determined by the metallic DOS in the majority band. We believe that the

shift of EF is determined by the behavior of the Si p-states, in particular by the

large extension of these states as compared to the d states. With a decrease of a, the

interaction between the atoms becomes stronger and the higher overlap results in a

stronger dc-localization of the electrons. As a result, the p-states are squeezed and

hybridized more strongly, thus pushing the d-states and the Fermi level to higher

energies, i.e., towards the minority conduction band. In addition, the dCo−dCo and

dCo − dMn(Fe) states hybridize more strongly, which tends to increase the size of

the gap while the dMn(Co) states bandwidth increases, which tends to shrink the

gap. As the first of the two effects is stronger, this finally leads to an increase of

the gap width.32 Upon expansion, the opposite effects are observed.

3.3. Magnetic properties

In Table 1, we showed the calculated magnetic moments M and the experimental

values Mexp for Co2MnSi and Co2FeSi, respectively. By GGA, for Co2MnSi, the

calculated magnetic moments agree with the experimental measurement value at

the optimized lattice constant or experimental constant.33 However, for Co2FeSi,

our calculated value of magnetic moments is 5.4 µB, smaller than 6 µB from exper-

imental measurement. Inspecting the spin resolved DOS in Fig. 1, we find that the

minority energy gap is located below the Fermi energy, which is finally one reason

why the magnetic moments are too low and not an integer. With GGA+U , we can

get an improved calculated value for Co2FeSi if we use appropriate U and J value,9

which is a good proof that electron-electron correlation might play an important

role for opening the gap in the minority states and getting the integer magnetic

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First Principles Study and Heusler Alloys 975

Fig. 4. Dependence of the magnetic moments on the lattice parameter. Shown are the total andsite specific magnetic moments of Co2FeSi and Co2MnSi as a function of the lattice parameter byGGA and GGA + U . Line is drawn through the calculated values to guide the eye.

moments. From Fig. 1, we can see that the Fermi energy level is just within the

energy gap, which ensures that the integer value of the magnetic moments is the

same as the experiment.

In order to better understand the difference of GGA and GGA + U for calcu-

lated magnetic moments and position of the gap, the dependence of the magnetic

moments on the lattice parameter was carefully inspected for Co2FeSi and Co2MnSi

separately by GGA and GGA+U in Fig. 4. For Co2FeSi by GGA, in Fig. 4(a) we

can see that the calculated atomic resolved magnetic moments of Co and Fe in-

crease both with lattice constants a, and the overall magnetic moments follows the

same trend, which gets to the experimental value of 6 µB at lattice constants of

6.20 A, which is expanded by +10% with respect to the experimental value. The

site specific moments seem to saturate above +6% changes of the lattice parameter

at about 1.5 µB and 3 µB for Co and Fe, respectively, and total magnetic moments

also have the same change.

In the Fig. 4(b), the total magnetic moments of Co2MnSi also increase slightly

with the lattice parameter, but it stays at 5 µB in the range of ±5% change of a,

which is also the range of half-metallicity occurrence in Fig. 3(c). There is a signif-

icant increase with lattice parameter change in the magnetic moments at Mn sites.

At the same time, the magnetic moments of Co decreases and the magnetic mo-

ments of anti-parallel alignment Si increases slightly. Thus the Co and Si moments

counter balances the Mn moments such that the overall magnetic moments are kept

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976 J. Qi et al.

constant with the lattice parameter. So we can say that a moderate change of the

lattice parameter does not change the overall magnetic moments of Co2MnSi. How-

ever, in the same range of ∆a/a as for Co2MnSi (±5%), an overall change of about

1 µB is observed for the case of Co2FeSi. Because it exhibits some fluctuations about

the integer value at very large lattice parameter, so we can see that the magnetic

moments of Co2FeSi is less stable against variation of the lattice parameter.

With GGA + U scheme, in Fig. 4(c) for Co2MnSi, the difference is that at

about −10% of ∆a/a, the total magnetic moment is 5 µB and slightly increases

with lattice parameter below +2% because, at the same range of the lattice con-

stant, although the magnetic moments of Mn significantly increases from 3 µB to

3.6 µB, the magnetic moments of Co decrease. In particular, over +2% the magnetic

moments of Co begin to increase as Mn, resulting in a rapid increase for total mag-

netic moments. Those also agree with the range of appearance of half-metallicity

in Fig. 3(d). We can see greater difference in the case of Co2FeSi by GGA + U

comparing with GGA scheme. Within the range of half-metallicity, −2 ∼ +9% of

∆a/aexp, the magnetic moments of Co and Fe both slightly increase with increasing

lattice constant, however, the magnetic moments of Si slightly decrease, resulting

in the total magnetic moment kept at about 6 µB, which agrees with the recent

experimental value.33,34 Over +10%, the magnetic moments of Si begin to increase,

particularly for Co, the increase is more in evidence, which results in visible in-

crease for total magnetic moment. At the range of appearance of half-metallicity,

we calculate the ratio MFe/MCo of the magnetic moments of Fe and Co and the

ratio is about 2.2 which agrees very well with the measured value 2.2 at 300 K in

an induction field of 0.4 T.7

Comparing the half-metallicity with the magnetic moments, we can conclude

that there is a close relation between the magnetic moments and the HMF charac-

ter because the formation of the gap and localized magnetic moments in Heusler

compounds is due to hybridization, which was recognized by most researchers. Al-

though an integer value of the magnetic moments may not result automatically

in a real gap in the minority (or majority) density, the appearance of the gap in

the minority density constrains the number of minority electrons to be integers,

consequently resulting in the integer total magnetic moment.

Comparing GGA with GGA+U schemes, we can see that only by GGA+U , the

experimental magnetic moments 6 µB can be obtained for Co2FeSi at equilibrium

lattice constant. So we predicate that electron-electron correlation might play an

important role on the description of half-metallic Heusler compounds with localized

moments not only for Co2FeSi but also for Co2MnSi because two materials have

the same structure and similar on-site Coulomb interaction between d electrons.

The difference is that the correlation in Co2FeSi is stronger than in Co2MnSi. As

GGA scheme considers a part of the correlation interaction, we successfully deal

with Co2MnSi by GGA. For Co2FeSi, as the correlation is stronger, we must use

the GGA + U scheme.

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First Principles Study and Heusler Alloys 977

4. Summary and Conclusion

In this work, we present a comprehensive investigation of the equilibrium structural,

electronic and magnetic properties of Co2MnSi and Co2FeSi, focusing on the depen-

dence of half-metallicity and the magnetic moments on the lattice parameter and

differences between GGA and GGA + U treatments. In the process of relaxation,

the internal parameter and cell shape are not changed, only volumes changed. So we

can investigate the influence of hydrostatic pressure on the electronic and magnetic

properties by only changing the lattice constant. We see that the on-site correla-

tion interaction in Co2FeSi is stronger than in Co2MnSi. So the GGA+ U scheme

is necessary for Co2FeSi. At the same time, we found that a moderate change of

the lattice parameter does not change the HMF behavior for both Co2MnSi and

Co2FeSi. Further, we find that there is a close relation between the magnetic mo-

ments and the HMF character: appearance of half-metallicity is consistent with the

integral magnetic moments, which also agrees with the experiment measurements.

Acknowledgments

We are grateful for support from the National Nature Science Foundation of China

(No. 10372045), program for New Century Excellent Talents in University of China.

X. Jiang was supported in part by the Natural Science Foundation of Jiangsu

Educational Department, China (Grants No. 05KJB140001 and 06KJA43014).

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