Journal of Magnetismand Magnetic Materials 3 (1976) 354-360 © North-Holland PublishingCompany
MAGNETIC MOMENTS IN HEUSLER ALLOYS C.C.M. CAMPBELL Physics Department, McMaster University, Hamilton, Ontario, Canada Received 8 July 1976; in revised form 29 October 1976 The lteusler alloys are a series of local moment ferromagnets of composition X2MnY, with a magnetization of ~4~ B per Mn atom. Magnetic measurementson the alloy series Ni2MnxTl .xSn, where T is Ti, V or Cr, indicate that the T site moment changes from -~B (for Ti), through zero (for V), to +I~B(f,lr CR). A simple physical interpretation is proposed for this sign change in the present work. This approach facilitates the interpretation of several other features observed in Heusler alloys.
An alternative approach, based on virtual double exchange, has recently been suggested by Kasuya . On the experimental side, the hyperfine fields in Heusler alloys have been the subject of a great deal of investigation [10-19]. A recent study of the magnetic interactions has been undertaken by lshikawa et al. [20 -22] by spin wave measurement. They reported several interesting effects. Firstly, the magnetic interactions appeared to be long range, extending further than the eight near-neighbour shell. They found that interactions, further than the third nearest neighbour shell, could be reasonably well-explained by the s - d interaction based on a nearly free electron model. However, the first and second near-neighbour interactions could not be so explained . They concluded that the strength of the magnetic interaction and the Curie point were mainly determined by the first and second near-neighbour interactions. Secondly, the magnetic form fa< tors agreed remarkably well  with those calculat.~d for a Mn 2+ ion by Watson and Freeman. In order to mterpret the size of the Mn moment of 4t~B, it has usually been assumed (e.g. ) that the Mn ion is in a 3d 6 configuration. Both the 3d 6 :~nd 3d 4 form factors are in disagreement with the results of Ishikawa et al. . Thirdly, they have shown  that the local Mn moments behave like the localized moments in an insulator. In particular, their results can be interpreted in terms of spin wave theory, includi~:g magnon-magnon interactions developed for a Heisenberg system.
1. Introduction There has been considerable interest recently in the magnetic behaviour of ordered, intermetallic alloys of the Heusler structure, X2MnY. Their crystal structure and spontaneous magnetization have been well-established [1-6]. They order in the L21 structure shown in fig. 1. The X sites form a simple cubic matrix with the Mn and Y sites in alternate body-centre positions. Typical X site elements are Cu, Co, Ni or Pd. Common Y site elements are In, Sn, Sb, A1, or Ga. Magnetization and neutron diffraction studies [1,5] indicate that the magnetic moment per molecule is ~4/a B and is confined to the Mn site. In view of the large separation of the Mn ions C>4 A,) the ferromagnetic coupling between their spins is thought to arise from an indirect interaction, rather than by direct d-d overlap. A survey of the magnetic properties of Heusler alloys has been reported elsewhere [ 10]. Theoretical treatments have considered the magnetic coupling between the localized Mn moments, in the free electron limit. Shinohara  has interpreted the coupling in terms of an s - d interaction of RKKY form. Caroli and Blandin  have used the virtual bound-state model and associated double-resonance coupling. Both models give rise to long-range spin density (or Friedel) oscillations, at the Fermi surface of the conduction band, which, in this free electron approach, are considered to be responsible for the ferromagnetic coupling between the Mn local moments. 354
C.C.M. Campbell /Magnetic moments in Heusler alloys
.9/-L---" ,9, H"'~ *'i ....... Lz"r
2. Experimental results
. .o- - - ' , - - . . 6
Fig. 1. The L2 t Heusler structure, X2MnY.
A recent experimental study  examined the magnetic properties of the Heusler alloy series, Ni2MnxTl-xSn, where T is an element from the first transition series. The original motivation underlying the study was to examine the magnetic properties of Ni2MnSn, as a function of Mn concentration, by substituting a nonmagnetic atom for Mn. Since the alloys Ni2TiSn and Ni2VSn are nonmagnetic and wellordered [25,26], Ti and V were chosen as nonmagnetic solutes in Ni2MnSn. Interpretation of the data suggests, however, that a Ti atom contributes ""P-B to the magnetic moment of the mixed system, NizMn x Til _xSn. There appears to be zero moment associated with a V atom in the series, Ni2MnxV t _xSn and a Cr atom contributes "+/~a to the magnetic moment of the alloys, Ni2MnxCr t _xSn. Thus, the magnetic moment of a substituted atom, T, changes sign from negative to positive, as T changes from Ti to Cr, through V. The present work will review the evidence for this effect and present a simple, physical explanation of it. This interpretation of the sign change of the T site m~an~tit,
nra~ricl~e ~n inrl;t, ntia.
chanism underlying the magnetic coupling at nearneighbour and next-near-neighbour Mn separations. It also predicts that the Mn ion is in a Mn 2+ configuration, in agreement with the results of Ishikawa et al. and explains the constancy of the Mn moment of 4/~B, in the majority of Heusler alloy systems. Several predictions are made on the basis of the proposed model.
Magnetic measurements were carried out on the alloy series, Ni2Mnx'rl _xSn, where T is Ti, V or Cr. AJloys containing Ti or V were studied at compositions corresponding t o x -- 1, 0.8, 0.6, 0.4, 0.2 and 0.0. The Cr series was studied for x = 1,0.8, 0.6 and 0.4; beyond this composition, the samples were polyphase. The magnetic properties of the samples were examined above and below their Curie points, using a vibrating sample magnetometer. Measurements were taken between 4 K and 650 K and the results are reported in table 1. The following featu-es were observed. Firstly, it was found that the local moment feature associated with the Mn atom persists throughout the concentration range of all three series. The effective paramagnetic moment per Mn atom, shown in table 1, remains constant at ~5/~B, in the Ti, V and C~rseries. This corresponds to a saturation moment of 4/~B per Mn atom, assuming an electron g-factor of 2. Secondly, it may be seen from fig. 2 that the Curie points of the Ti and Cr series seem anomalously high. The dilution of a ferromagnet by a nonmag~etic ion has been studied using percolation theory. Duff and CanneUa  have calculated the percolation limit using two different interaction models. Model A assumes ferromagnetic interactions between first-r, ear
Ni*T"x MnxSno / : / ~ "
y / "
Fig. 2. Plot of Curie temperature, Tc (in K), vs Mn cencentration, x, for the series, Ni2MnxTl_xSn, where T is Ti, V or Cr.
C CM. Campbell / Magnetic moments in Heusler alloys
Table 1 Magnetic data for the series Ni2MnxT l_xSn, where T is Ti, V or Cr. ta is the magnetic moment/molecule at 4 K; p is the paramagnetic moment/Mn atom; Tc and 0 c are the ferromagnetic and paramagnetie Curie temperatures; ~tTi, tzv and taCr are the calculated magnetic moments of the Ti, V and Cr atoms, assuming a moment of 4.04 ± 0.10 ~B on the Mn site Ti series
Mn concentration, X 0.8 3.18 5.0 334 311 -0.25
ta (in taB) p (in taB) Te (in K~ 0c (in K) taXi
4.04 5.0 345 359
± 0.1 +-0.2 ±2 ±2
Mn concentration, X
I 4.04 ± 0.I
p (in taB) Tc (inK) 0 c (in K) (1-x)ta v
5.0 ± 0.2 345 +, 2 359 *2 -
Mn concentration, X 1
4.04 5.0 345 359 .
± 0.1 t0.3 -+ 2 +- 2 .
0.6 2.04 5.0 308 270 -0.95
0.6 2.31 + 0.I
0.4 1.61 + 0.I
5.1 ± 0.3 266 ±2 283 ±2 0.0 ± 0.2
5.3 197 198 -0.1
* 0.3 ±2 ±2 ± 0.1
5.4 110 128 0.0
± 0.3 ~3 ±2 ± 0.1
0.8 3.7 5.2 308 320 +2.2
0.6 2.9 5.4 277 291 +1.2
+, 0.1 ±0.3 ±2 ±2 +-0.3
0.4 2.1 5.8 220 233 +0.8
± 0.05 +-0~3 ±2 ±2 +-0.2
ta (in taB)
ja (in taB) p (in taB) Tc (in K) 0c (in K) ~Cr
± 0.1 ±0.2 ±2 ±2 ± 0.8
+ 0.1 +0.3 +- 2 +, 2 +-0.9
± 0.1 +0.2 ±2 ±5 ± 0.3
0.4 1.20 4.9 297 264 -0.7
± 0.1 ±0.2 ±2 ±4 ± 0.1
0.2 0.47 4.8 261 233 -0.43
± 0.1 ±0.2 ±2 ±4 ± 0.04
0.2 0.60 ± 0.05
5.3 45 70 -0.26
* 0.3 t 5 ±5 ± 0.1
neighbour magnetic atoms only and gives a percolation limit at 20% magnetic atom concentration. Model B assumes ferromagnetic interactions between firs' and second near-neighbour magnetic atoms. The limit of o~ rcolation for model B was 13%. The decrease of Tc with increasing V concentration is very similar to that predicted by Duff and Cannella for model B. It is assumed, therefme, that the magnetic interaction, in the V series, is Cominated by the first and second nearneighbour Mn atoms. This is in complete agreement with the conclusions of Ishikawa et al., commented on previously. Thirdly, it may be seen from table 1, that for the Ti series, the ferromagnetic Curie temperature, Tc, is higher than the param~ gnetic Curie temperature Oc. This result is usutlly associated with antiferromagnetic effects. It is also seen that, for the Cr series, Tc is lower than Oc, an effect usually associated with ferromagr, elic order. These three features can be interpreted by con-
sidering the magnetization results in the following w a y It is assumed that there is a moment of 4ta 8 associated with the Mn atom in all alloys investigated in the present study. The constancy of the paramagnetic moment per Mn atom indicates this is a reasonable assumption. The Ti, V or Cr contribution to the magnetic moment, implied by this assumption, was cal. culated from the saturation magnetization, using a val. ue o f 4.04 -+ 0.1 taB for the Mn moment, obtained from the data for Ni2MnSn. As shown in table 2, it was estimated that a Ti atom contributes ~tab to the magnetic moment, the ~ contribution is zero, and a Cr atom contributes "+taB. The paramagnetic susceptibility depends on the square of the effective moment. Hence, the small Ti and Cr moments would make a minor contribution to the susceptibility. Since this interpretation rests on an assumption for the size of the Mn moment, it is proposed to undertake a neutron diffraction study  of all samples reported in the present work. Preliminary results are in agree-
C.C.M. Campbell/Magnetic moments in Heusler alloys Table 2 Magnetic data for the series, Ni2Mnl+xSnl_ x. taexp is the
magnetic moment/molecule at 4 K; lUcalcis the magnetic moment/molecule, assuming the fraction of Mn atoms X are coupled antiferrornagnetically to the others. (1 +x) 1.1 1.2 1.3 1.4
3.66±0.2 3.24±0.2 3.19±0.2 2.34±0.2
3.6 3.2 2.8 2.4
ment with the above assumptions and the final analysis will be reported elsewhere. This interpretation of the Ti, V and Cr moments is consistent with the dependence of the Curie point on Mn concentration, shown in fig. 2. The observed decrease of Tc with increasing V concentration is consistent with dilution of a ferromagnet by a nonmagnetic impurity. The less rapid decrease for the Ti and Cr series can be explained if these cases do not represent dilution by nonmagnetic atoms. Finally, the estimates are consistent with the observation that Tc > 0c, for the Ti series, and T¢ < 0 for the Cr series.
3. Interpretation It has been observed that in the alloy series, Ni2MnxTt _xSn, the magnetic moment at the T site changes sign from negative to positive, as T changes from Ti to Cr, through V. It was also concluded that the Mn spins in Ni2MnSn are ferromagneticaUy aligned by their first and second near-neighbour Mn atoms. At first sight, the sign change of the T site moment is reminiscent of the situation where elements of the first transition series are substituted as dilute impuln(gtal~i, Fe, LO rities in the ierromagneuc ............. " alltd . . . . . ~,~ . L,:O'"" 30]. It has been observed that the magnetic moment per added impurity changes sign in Fe, Co and Ni. Friedel  has interpreted the sign change at impurity sites in Ni, in terms of a virtual bound state being subtracted from the Ni d-band as the potential of the 3d impurity is decreased. It is unlikely that the sign change of moments observed in Heusler aUoys is due to the same effect, since their Ni atoms carry no moment and the Mn moments are localized [ 1 ]. Both
Ni2TiSn and Ni2VSn are nonmagnetic [25,26]. Since Friedel's explanation of the sign change depends on an unf'dled Ni d-band, his model is not applicable to the present case. For example, in the case of Ni2MnxTil _xSn, it is clea that is is the Mn atom (not the Ni) that pulls o t t a magnetic moment on the Ti, since Ni2TiSn is nonmagnetic. Since there is little overlap between the d-wave functions at the Mn and Ti sites, this is probably due to an indirect interaction. Heusler alloys are metals. They have a large number of conduction electrons and hence it seems reasonable to try to interpret their magnetic properties using the s - d interaction. This approach has been justified, to some extent, by the results of Ishikawa ,~t al. in which the RKKY oscillations have probably been observed. On the other hand, the magnetic moment on the Mn atom, in Heusler alloys, remains constant at 4/aB. This seems rather surprising in view of the range of outer s - p electron structure exhibited by these alloys. In order to account for the size of the Mn moment, the Mn atom is usually considered to be in a 3d 6 configuration. If a rigid band model were applicable to Heusler alloys, it should be possible to raise or lower the value of the Mn moment by raising or lowering the Fermi level. By changing the element at the X or Y site, .he number of electrons in the conduction band would be expected to change. However, such a change in X or Y site element has no significant effect on the size of the Mn moment. Except for the Co series where there is a moment on the Co atom, it remains constant at 4/a B. It is unaffected by the char.oes io the Fermi level indicated, in disagreement with t~e u'~lal rigid band effects in a metal. ~tral features observed in Heusler alloys can be interpreted by taking a localized molecular orbital approach. For example, the observed sign change of the impurity site moment can be explained if the magnetic cou~A'ng between the Mn spins takes place via the Y site atom. A similar . . . . . . . .mechanism . . . . . . . . . . wa~ ~~,~t . . propo~u . . . 1 many years ago by Goodenough  and has been largely neglected, in general, such an approach is open to criticism that it may be inappropriate to use overlapping atomic, orbitals in metals, instead of hybridized bands. Despite this fact, it remains that certain characteristics of Heusler alloys can be extremely well-interpreted in terms of this basic mechanism. An exact calculation of this interaction would be extremely difficult, but the main features can be seen quite clearly
C CM. Campbell /Mag,~etic moments in Heusler alloys
from the following qualitative discussion. In Heusler alloys, the Mn atom may be considered to be in a 3d s state, with the t~g and e~ orbitals halfftdl. Hund's rule requires that the electrons be of parallel spin. The p states of the Sn atom are less than halffull and the neutron diffraction results indicate the absence of a moment at this site [1 ]. This merely reflect,, the usual prevalence of chemical bonds for containing electron pairs of opposite spin. It arises from the attractive potential of the Sn atom core being stronger than the surrounding electron-electron repulsions. Admixture of the Po-orbitals of a Sn anion with ~:he %crbitals of the adjacent Mn cations would result in a bonding intera'tion between these atoms. The observed ferromagnetic coupling arises from admb' ture of the po~-orbital of the Sn with the Mn egt-orbitals on either side. Sine:. the stoichiometry ensures there is one Sn atom per Mn, the observed Mn moment of 4/aB is consistent with spin-pairing at the Sn site, if a Mn and Sn atom each contribute one electron (of eg and Po character, respectively) to the bonding interaction between them. This mechanism i~ able to interpret the observed Mn moment of 4/aB without requiring the Mn atom to be m either a 3d 6 or 3d 4 configuration. It suggests it is in a 3d s state, in agreement with the resuits of lsh~kawa ct al. It is also able to explain the sTabdity of the size of the Mn moment in the face of substantial changes m tlae number of conduction elec. trons. Since the size of the moment is determined by a localized orbital approach, it is less likely tt, be affected by the kind of changes in the Fermi level menhoned be fore. In addition, this mechanism is able to interpret the local moment siga change in Ni2MnxT 1 _xSn. The fact that both Ni2TiSn and Ni2VSn are nonmagnetic can be attributed to the weakness of into -tomic exchange at the Ti and V sites. If there is a spin-paired 3d 2 state associated with the Ti site in Ni2TiSn, then the negative mome.nt on the Ti site, in the mixed system. Ni2MnxTi ! _xSn, can be explained by a straiglu%rward extension of the inte action described above. I h e effect of an adjacent Mn atom is to pull out a spm t electron from the 3d 2 state at the Ti site into the spin-paired Po-eg orbit~ I'his leaves a spin ~ electron and an associated mort "nt of -/ai3 at the Ti site. A ~.imilar mechanism can be considered to occur for the Mn-V iJ~teraction in the alloy series, Ni2Mn x \"1 - , ~;'~. wbe'e the V atom is considered to be in a
3d a state. As before, the effect of an adjacent Mn atom is to pull out a d electron from the V site into the common orbital. Intra-atomic exchange at the V site is insufficiently strong to sustain a moment, and the remaining two electrons are paired with opposite spins, leading to the observed zero moment. This is no longer the case for the Cr atom, however, and intra-atomic exchange at a Cr site, in Ni2MnxCr I _xSn leads to the observed positive moment. It appears, therefore, that certain magnetic features of Heusler alloys discussed above can be interpreted in terms of magnetic coupling between the Mn eg orbitals, through the intermediate Sn ion. It is clear that not all the magnetic features reported for these alloys can be explained by this mechanism. In particular, the role of the X site atom is important in effecting the ferromagnetic-antiferromagnetic transition in the alloy series. CuxNi I _xMnSb . The only antiferromagnetic Heusler alloys of L2 t structure are Pd or Pt based alloys, where the Y site element is characterized by 3 s - p electrons (e.g. A], Ga or In). It seems likely there. fore that additional effects depend on the magnetic character of the X site atoms. Kasuya's  model of virtual double exchange may prove helpful in this regard. It is not clear at this stage how his model would interpret the features commented on previously. It is proposed to make three predictions based on the mechanism, discussed in the present work, in order to test it. Firstly, it has been shown that the T site moment in the alloy series, Ni2MnxT l _xSn, can be determined where T is Ti, V or Cr. The proposed model is also able to [ redict the moment where T is Fe. To be consistent with the 3d s Mn configuration, it is anticipated that an Fe atom in these alloys wotdd be in a 3d 6 state. As indicated above, the effect of an adjacent Mn atom is to pull out a d-electron from the Fe site into the common M n - S n - F e orbital. Anti-parallel spin occupancy of this orbital indicates that Hund's rule coupling of the other five F e d electrons will result in a predicteta moment of 5/aB at an Fe site. The Fe moment in Ni 2 Mno. 8 Feo. 2 has been measured and a value of 5.3 +_-0.5/a B obtained  in agreement with the above prediction. A second prediction can be made on the basis of the proposed model. The Mn.Y site separation is ~-3 h. If an Mn atom were on a Y site, then its eg orbital would overlap the e~. orbital on the adjacent regular Mn site. If both Mn ,,rams are in the 3d s configuration
C.CM. Campbell /Magnetic moments in Heusler alloys
proposed and their t2g and eg states are half-full, the Pauli exclusion principle will require that a Mn pair at 3 A separation will couple antiferromagnetically. Alloys of the series, Ni2Mnl +xSnl - x , were prepared for x = 0.1,0.2, 0.3 and 0.4. Their saturation magnetization was measured  and the results are shown in table 2. They are in good agreement with the hypothesis that the fraction, x, of the Mn atoms that go on a Y site, are coupled antiferromagnetically to those on the regular Mn site. The susceptibility measurements are consistent with a local moment of 4VB on each Mn site. This would x,ot be the case if the Mn atom were in a 3d 6 or 3d 4 state, but is exactly what would be obtained for Mn atoms in the 3d s configuration, with overlapping eg orbitals, proposed above. A third test of the model arises from considering the Ni moment in the series Cu2MnxNi I _xSn. Cu2NiSn is reported to be nonmagnetic . The effect of the Cu is presumably to fill the Ni d-shell, so that the electrons are spin-paired at this site. In this case, like that of Ti, the effect of the Mn atoms in Cu2MnxNitoxSn should be to pull out a spin ~ d-electron from the Ni atom, causing an effective moment of - # u at the Ni site. Preliminary results on the series, Cu2 MnxNil - xSn indicate that this is the case .
ment of 4/a a in Heusler alloys a~d in mixed systems such as Ni2MnxT l _xSn. (3) It indicates the Mn atom is in a 3d 5 state, in agreement with the results of Ishikawa et al. Their analysis in terms of a Heisenberg system is in agreement with the spirit of the present work. (4) It predicts an Fe site moment of 5taB, subsequem.ly observed in Ni2 Mno.aFeo.2 Sn. A moment of 5/aB at an Fe site in a metal is extremely unusual and helps to validate the approach used here. (5) It predicts the antiferromagnetic coupling of Mn ions at 3 A separation, subsequently observed in Ni2Mnl +xSnl - x . (6) Lastly, the model predicts the observed negative moment at Ni sites in Cu2MnxNi I _xSn. A completely integrated interpretation of all of the magnetic characteristics of Heusler alloys will not be possible until a band structure calculation is carried out. Such a calculation is very difficult. The above physical arguments indicate a fruitful line of approach.
Acknowledgements It is a pleasure to acknowledge discussions on aspec:.s of this work with C.V. Stager, M.F. Colhns and J.A Cameron.
4. Conclusions The main purpose of the present work is to present a model which interprets various magnetic properties, observed in Heusler alloys, in a unified way. Certain effects can be explained using the s--d interaction; for example, the long-range spin-density oscillations observed by lshikawa et al. However, these workers concluded that this interaction is not the dominant mechanism for the ferromagnetic order. They state that tile principal coupling is due to the first and second Mn near-neighbours. This is in agreement with our resuits on tile V series. "t,,asuya . . . . . . . i~]'"has ..... a~,.t,v,.,~:t'"'~ ,,~o,,,,.,i"-., teraction to virtual do,,~ble-exchange. The present work has ascribed it to a covalent coupling via the Y site. in particular, the proposed mechanism ~s able to interpret the following fea~ ,~es (1) It interprets the local moment sign change from negative to positive in Ni2MnxTi _xSn, where T changes from Ti to Cr through V. (2) It explains the constancy and size of the Mn mo-
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C CM. Campbell /Magnetic moments in Heusler alloys
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