ARTICLE IN PRESS

0921-4526/$ - see

doi:10.1016/j.ph

�CorrespondiE-mail addre

Physica B 403 (2008) 200–206

www.elsevier.com/locate/physb

Ab-initio investigation of electronic properties andmagnetism of half-Heusler alloys XCrAl (X ¼ Fe, Co, Ni)

and NiCrZ (Z ¼ Al, Ga, In)

Hongzhi Luoa,�, Zhiyong Zhua, Guodong Liua, Shifeng Xua, Guangheng Wua,Heyan Liub, Jingping Qub, Yangxian Lib

aBeijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100080, PR ChinabSchool of Material Science and Engineering, Hebei University of Technology, Tianjin 300130, PR China

Received 16 June 2007; received in revised form 27 August 2007; accepted 27 August 2007

Abstract

The electronic structures and magnetism of the half-Heusler alloys XCrAl (X ¼ Fe, Co, Ni) and NiCrZ (Z ¼ Al, Ga, In) have been

investigated to search for new candidate half-metallic materials. Here, we predict that NiCrAl, and NiCrGa and NiCrIn are possible half-

metals with an energy gap in the minority spin and a completely spin polarization at the Fermi level. The energy gap can be attributed to

the covalent hybridization between the d states of the Ni and Cr atoms, which leads to the formation of bonding and antibonding peaks

with a gap in between them. Their total magnetic moments are 1mB per unit cell; agree with the Slater–Pauling rule. The partial moment

of Cr is largest in NiCrZ alloys and moments of Ni and Al are in antiferromagnetic alignment with Cr. Meanwhile, it is also found that

FeCrAl is a normal ferromagnetic metal with a magnetic moment of 0.25mB per unit cell and CoCrAl is a semi-metal and non-magnetic.

r 2007 Elsevier B.V. All rights reserved.

PACS: 71.20.Be; 71.20.Lp; 75.50.Cc

Keywords: Half-Heusler alloy; Band structure; Half-metallicity

1. Introduction

The rapid development of spintronics [1,2] gets muchattention in recent years. Widely possible applications suchas single spin electron sources and spin injectors have beenenvisaged [3,4]. One key ingredient for spintronics is thesource of the spin-polarized charge carriers. An idealchoice is the half-metallic materials, including half-metallicferromagnets (HMFs) and half-metallic antiferromagnets.All the half-metallic materials have a common character intheir energy bands, they are semiconductor-like in one spindirection at the Fermi level whereas the other spin isstrongly metallic, which results in a complete (100%) spinpolarization of the conduction electrons at the Fermi level.The HMFs were first discovered by theoretical band

front matter r 2007 Elsevier B.V. All rights reserved.

ysb.2007.08.214

ng author. Tel.: +86 10 8264 9247; fax: +86 10 6256 9068.

ss: luohz@aphy.iphy.ac.cn (H. Luo).

structure calculations for half-Heusler alloy NiMnSb [5].Then much investigation has been put on both half- andfull-Heusler alloys [5–14]. For the so-called half-metallicantiferromagnet, its magnetic properties are unique. It hasferrimagnetic coupling with completely compensated mag-netic moment and 100% spin polarization at the Fermilevel. So, it has an advantage for some technical applica-tions, for it is non-magnetic and not easy to be affected byexternal magnetic fields [15–17]. In 1991, de Groot [15]predicted MnCrSb with half-Heusler alloy structure to be ahalf-metallic antiferromagnet. Later, similar Heusler alloyFe8MnV7Sb7In [17] and Mn3Ga [16] has also beenpredicted by band structure calculation. However, thesealloys are difficult to be synthesized with its stoichiometricform.Recently, Wurmehl et al. [16] proposed the rule for

designing half-metallic antiferromagnets that is this kind ofmaterial might be expected in half-Heusler alloys with 18

ARTICLE IN PRESSH. Luo et al. / Physica B 403 (2008) 200–206 201

valence electrons or full-Heusler alloys with 24 electronsand with Mn on the Y site. The Mn at Y site in Heusleralloys tends to have a high, localized magnetic moment andcancels the moments at X sites. In fact, a large magneticmoment has also been found in Cr at Y site [18]. So, it ispossible to design a half-metallic antiferromagnet in half-Heusler alloys XYZ with 18 valence electrons and Cr atomat Y site. Here, we select half-Heusler alloy CoCrAl to findif half-metallicity with compensated magnetic moment canbe expected in it as has been found in MnCrSb [15].Meanwhile, it is known that the magnetic and electronicproperties of half-Heusler alloy strongly depend on thenumber of valence electrons [12]. So, in this paper weinvestigate the electronic structure and magnetism of half-Heusler alloys with 17–19 electrons namely XCrAl(X ¼ Fe, Co, Ni) and NiCrZ (Z ¼ Al, Ga, In) with first-principles band structure calculations. Three possible half-metals are predicted as NiCrAl, NiCrGa and NiCrIn. Alarge magnetic moment on Cr site and antiferromagnetismcoupling between the moments Ni and Cr is observed.

2. Computational methods

We carried out the electronic structure calculation usingthe self-consistent full-potential linearized-augmentedplane wave (FLAPW) method based on the local spin-density approximation within the density functional theory[19], where the potential and/or the charge density in thecrystal are treated with no shape approximation. Onehundred and eighty-two k points are employed in theirreducible Brillouin zone. The self-consistent calculationstops as the charge density deviation is less than 0.01me/a.u. and the total energy deviation is better than 0.1mRyper cell. The density plane-wave cutoff is Rkmax ¼ 8.0. Theelectron states were treated in a scalar relativistic approx-imation. Using the energy eigenvalues and eigenvectors atthese points, the density of states (DOS) was determined bythe tetrahedral integration method [20].

The half-Heusler alloy XYZ crystallizes in the face-centered cubic (fcc) structure with one formula unit perprimitive cell. The space group is F4-3m. In this study, Fe,Co and Ni represent the X, while Cr and (Al, Ga, In)represent the Y and Z atom, respectively. The sitepreference in half-Heusler alloys has been studied

Table 1

The equilibrium lattice constants, total energy differences DE, calculated total a

spin polarization ratio P for the XCrAl (X ¼ Fe, Co, Ni) and NiCrZ (Z ¼ A

Compound VEN Lattice constant (A) DE (eV) Mt (mB) MX (mB) MY (m

FeCrAl 17 5.57 �0.04 0.25 0.3 �0.04

CoCrAl 18 5.52 �0.002 0.00 0.00 0.00

NiCrAl 19 5.49 �0.20 1.00 �0.36 1.60

NiCrGa 19 5.51 �0.19 1.00 �0.26 1.46

NiCrIn 19 5.70 �0.30 1.01 �0.32 1.54

VEN, valence electrons number; SM, semi-metal; HMF, half-metallic ferroma

[13,21,22]. It is found that the conventional stable structureis that the Y and Z atoms locate at (0, 0, 0) and (1

2; 12; 12) sites

and form the rock salt structure while the X atom locates inthe octahedrally coordinated pocket, at one of the cubecenter site (1

4; 14; 14) leaving the other site (3

4; 34; 34) unoccupied.

The crystal with different configurations as XYZ, ZYX andYXZ has been investigated and the configuration withmore valence electrons at X site is found to be lowestin energy [13]. In XYZ structure, the X atom hasfour Y and four Z as nearest neighbors whereas Y and Zatoms only have four X atoms in their nearest neighborscoordinations.

3. Results and discussion

To determine the theoretical lattice parameter, weperform total energy calculations on XCrAl and NiCrZalloys both for the non-magnetic (PM) and the ferromag-netic (FM) states at different lattice parameters.The calculated equilibrium lattice parameters are listed inTable 1.For XCrAl, it is found that when the X atoms belong to

the different column of the periodic table their magneticbehavior is different. For FeCrAl or NiCrAl, it can be seenthat the ferromagnetic state is more stable in energy thanthe paramagnetic one. Whereas in CoCrAl, the total energydifference between the paramagnetic state and correspond-ing ferromagnetic one is very small and can be neglected.Both of them give a zero total and partial magneticmoment indicating paramagnetic configuration is stable.The energy difference DE between the FM and PM statesare also listed in Table 1. In NiCrZ alloys, calculations alsoindicate the ferromagnetic state is more stable.In order to get a deep study on the electronic structure

for the XCrAl (X ¼ Fe, Co, Ni) alloys, which have 17, 18and 19 valence electrons, respectively, we study theirtotal DOS in paramagnetic configuration first. As shownin Fig. 1, their shapes are similar and the characters can bedescribed as follows: the states below �6 eV are mainly selectrons of Al atom, which are relatively small andseparated from the d states by a dip in DOS. The lowenergy part around �5 eV are mainly the p states of Alatom in the occupied valence states, which hybridize withp and d electrons of the X atoms and determine the

nd partial magnetic moments, and band gap, HM gap width together with

l, Ga, In) alloys

B) MZ (mB) Gap width (eV) HM gap (eV) P (%) Physical property

�0.00 – – 11 Metal

0.00 – – – SM

�0.24 0.45 0.02 100 HMF

�0.20 0.52 0.17 100 HMF

�0.20 0.51 0.07 100 HMF

gnet.

ARTICLE IN PRESS

0

10

20

(17)FeCrAl

DO

S (

ele

ctr

ons/e

V)

0

10

(18)CoCrAl

0

10

(19)NiCrAl

0

10

20 (19)NiCrGa

-8 -6 -4 -2 0 2 4 60

10

20(19)

Energy (eV)

NiCrIn

Fig. 1. Total DOS in paramagnetic state for FeCrAl, CoCrAl, NiCrAl, NiCrGa and NiCrIn alloys. The number of valence electrons is shown in the

figure.

H. Luo et al. / Physica B 403 (2008) 200–206202

occupation degree of the p–d orbitals. The states of 3dmetal atoms extend from �4 to +2 eV and hybridize witheach other. It has been reported that the covalenthybridization between the lower-energy d states of thehigh-valent transition metal atom X and the higher-energyd states of the lower-valent transition metal Cr can lead tothe formation of bonding and antibonding bands. Thebonding hybrids are localized mainly at the high-valenttransition metal atom site while the unoccupied antibond-ing states mainly at the lower-valent transition metal site[23]. So, a d–d band gap is formed near the Fermi level.There are nine bands below the d–d gap and canaccommodate 18 valence electrons in paramagnetic state[13]. So, for a half-Heusler alloy with 18 valence electronsthe Fermi level will locate in the energy gap and usuallymake it a semiconductor or a semi-metal. When the alloyhas more or fewer than 18 valence electrons, the Fermilevel will be shifted and locate at the antibonding orbonding peak. The paramagnetic state is no longer stable inenergy when EF lies in the antibonding peak. Spinpolarization and forming magnetic moment can releasethis instability [13]. It has been reported that if theparamagnetic electronic structure supports an energy gap,then in the process of spin polarization, the half-metallicferromagnetism may be stabilized depending on the relativeposition of the gap and the Fermi level [24]. Theparamagnetic electronic structure of other HMF systemslike Fe2MnSi [25], NiMnSb, FeVSb [24] also sustains a gapclose to the EF. So, here, we will discuss their spin-polarized DOS below.

The total and partial DOS in ferromagnetic states forXCrAl alloys are given in Figs. 2–4. It is clear that the threealloys are different in their magnetic properties. CoCrAl

has 18 valence electrons and in both spin directions thenine bands are occupied. So, it has a symmetrical DOS inmajority and minority spins and a zero magnetic moment.In this calculation, CoCrAl is a semi-metal rather than thewanted half-metallic antiferromagnet. The situation inNiCrAl and FeCrAl is different; they show ferromagneticbehavior in their electronic structures.For NiCrAl, it can be seen from the total DOS that there

is an energy gap in the minority spin at the Fermi level,whereas there is a high DOS peak in the majority spin.So, a 100% spin polarization is obtained and NiCrAlis a half-metal. From the partial DOS, it can be seenthat the Cr atom gains a large exchange energy during thespin polarization. In the majority spin, the Cr d statesbasically lie below the Fermi level and are occupied, whichform a common d band with the Ni d states. While in theminority spin, the antibonding peak is shifted 1.3 eV abovethe Fermi level and unoccupied. The exchange splittinginduces a large magnetic moment on Cr site. The calculatedpartial spin moment of Cr is 1.60mB, which is larger thanthe calculated total magnetic moment of 1mB and iscompensated by the antiferromagnetic aligned Ni and Almoments. The exchange splitting is relatively small for Ni dstates and most of the Ni d states are below EF andoccupied. So, the majority and minority spins of Ni d statesare equally populated and form a moment of �0.36mB. TheDOS of Al is almost symmetrical in both spin directionsand only has a small moment of �0.24mB.The total and partial DOS for FeCrAl alloy are shown in

Fig. 2. It is clear that a magnetic moment is formed in thespin-polarized calculation. However, there are states at theFermi level in both spin directions, so FeCrAl is only anormal ferromagnetic metal. It can be seen that both the

ARTICLE IN PRESS

-10

-5

0

5

10

TotalFeCrAl

Energy (eV)

DO

S (

ele

ctr

ons / e

V)

-4-2

0

24

Fe

-4-2

0

24 Cr

-0.8

-0.4

0.0

0.4

0.8Al (p)

-8 -6 -4 -2 0 2 4 6

-0.4

-0.2

0.0

0.20.4

Al (s)

Fig. 2. The total and partial DOS in ferromagnetic state for FeCrAl alloy. Minority states are on negative scale.

-10

-5

0

5

10

Energy (eV)

TotalCoCrAl

DO

S (

ele

ctr

on

s /

eV

)

-6

-3

0

3 Co

-3

0

3 Cr

-0.8-0.40.00.40.8

Al (p)

-8 -6 -4 -2 0 4 6

-0.4-0.20.00.20.4 Al (s)

2

Fig. 3. The total and partial DOS in ferromagnetic state for CoCrAl alloy. Minority states are on negative scale.

H. Luo et al. / Physica B 403 (2008) 200–206 203

majority and minority spins are partly occupied. Theexchange splitting of Cr is weaker in FeCrAl than that inNiCrAl, which introduces a relatively small magneticmoment of 0.30mB for Cr compared with 1.60mB in NiCrAl.Nanda and Dasgupta [13] studied the interaction in somehalf-Heusler alloys with COHP method. They found thatthe X–Y (X ¼ Fe, Co, Ni; Y ¼Mn, V) interaction isstrongest for Fe and weakest for Ni. This implied a largeadmixture of Y atom for Fe in the minority valence bandand a reduction of the magnetic moment at Y site. The Featom has four Cr and four Al as nearest neighbors and

shows a three-peak structure in both spin directions.The exchange splitting is also weak in Fe d states andleads to an almost symmetrical DOS in the two spindirections. So, Fe only has a small moment of 0.04mBantiparallel to that of Cr.The half-metallic behavior in NiCrAl and normal ferro-

magnetism in FeCrAl relate to the different degree of exchangesplitting at Cr site. According to Nanda and Dasgupta [13] alarge exchange splitting in the minority spin band is preferredin the formation of the half-metallic properties, which influencethe formation of the energy gap at the Fermi level.

ARTICLE IN PRESS

-10

-5

0

5

10TotalNiCrAl

Energy (eV)

DO

S (

ele

ctr

ons / e

V)

-4

0

4Ni

-3

0

3

6Cr

-0.6

0.0

0.6

1.2Al(p)

-8 -6 -4 -2 0 2 4 6

-0.4-0.20.00.20.4 Al(s)

Fig. 4. The total and partial DOS in ferromagnetic state for NiCrAl alloy. Minority states are on negative scale.

H. Luo et al. / Physica B 403 (2008) 200–206204

In order to know more about the half-metallicity inCr-based half-Heusler alloys, we calculated the electronicstructure of the NiCrZ (Z ¼ Al, Ga, In) alloys usingtheoretical lattice constants. Their total and partial DOSare given in Figs. 4–6, respectively. It can be seen that theirDOS structures are quite similar. As has been discussedabove, in the majority spin the Cr 3d states are basicallyoccupied with the high antibonding peak lying at the Fermilevel. It strongly hybridizes with Ni d states and forms awildly spread d band. While in the minority spin theantibonding peak of Cr states are shifted to higher energyabove EF by the exchange splitting and cause an energy gapat the Fermi level. So, the half-metallic properties areobtained in the three NiCrZ half-Heusler alloys. The originof the gap in half-Heusler alloys NiCrZ can be attributedto the strong hybridization between the d states of the Niand Cr atoms. The Fermi level lies in the bonding t2gorbitals and antibonding eg orbitals. This is quite similar tothe formation of the gap in semiconductors such as thewell-known GaAs [23].

However, in half-Heusler alloys XYZ, Z elementsare mainly atoms with large atomic radii, like Sn, Sb orTe. The NiCrAl or NiCrGa compound with C1b structureis not known now due to the smaller atomic radii ofZ atoms. This instability of the bcc structure may beexplained from their electronic structures. According to thestudy of Kandpal et al. [26] in CrCoSb, the high peakin the DOS at the Fermi level may be associated withthe non-existence of CrCoSb alloy. Here, the elec-tronic structure of NiCrZ is quite similar to that ofCrCoSb, so the high peak at EF in their DOS may have thesame effect.

In the calculated energy bands for NiCrZ, both themaximum of the valence band and minimum of the

conduction band are at the X point so the gap is a directgap, which is different from normal half-Heusler alloyswith half-metallic characters in which a G–X indirect gap isformed [23]. It is clear that the Fermi level lies above theminority spin valence band maximum (VBM), which is theminimum energy required to flip a minority spin electronfrom the VBM to the majority spin Fermi level and isoften referred to as the ‘‘spin-flip gap (HM gap)’’ [27].The calculated gap width around EF together with the HMgap for the minority carriers is listed in Table 1.The calculated total and partial magnetic moments for

the XCrAl (X ¼ Fe, Co, Ni) and NiCrZ (Z ¼ Al, Ga, In)alloys are listed in Table 1. In half-Heusler alloys with half-metallic character, their magnetic moment can be predictedby the Slater–Pauling rule that is MH ¼ NV�18, where MH

is the total spin magnetic moment per formula unit and NV

is the total number of valence electrons. This rule comesfrom that way: in half-metals the Fermi energy is pinned inan energy gap in only one spin direction, which leads to thenumber of occupied states being an integer. So, theSlater–Pauling rule for one atom will be expressed asmHMF ¼ nV�6 (nV is the mean number of valence electronsper atom) for the spin magnetic moment per atom. In thecase of three atoms per unit cell, as in half-Heuslercompounds, one should subtract 18 from the total numberof valence electrons NV to get the spin magnetic moment M

per unit cell: MH ¼ NV�18 [28]. It can be seen that boththe magnetic moments of CoCrAl and NiCrZ follow theSlater–Pauling rule. They have 18 and 19 valence electronsand magnetic moments of 0mB, 1mB, respectively. WhileFeCrAl is an exception, according to the Slater–Paulingrule, FeCrAl with 17 valence electrons should have amoment of 1mB. However, it is not a half-metal and has amagnetic moment of only 0.25mB.

ARTICLE IN PRESS

-5

0

5

10

TotalNiCrGa

Energy (eV)

DO

S (

ele

ctr

ons / e

V)

-8-4048

Ni

-8

-4

0

4 Cr

-1.0

-0.5

0.0

0.5

1.0

Ga (p)

-8 -6 -4 -2 0 2 4 6-1.0

-0.5

0.0

0.5

1.0Ga (s)

Fig. 5. The total and partial DOS in ferromagnetic state for NiCrGa alloy. Minority states are on negative scale.

-6

0

6

12

TotalNiCrIn

Energy (eV)

DO

S (

ele

ctr

ons / e

V)

-10

-5

0

5

10Ni

-10

-5

0

5

10

Cr

-0.45

0.00

0.45 In(p)

-8 -6 -4 -2 0 2 4 6-1.2

-0.6

0.0

0.6

1.2In(s)

Fig. 6. The total and partial DOS in ferromagnetic state for NiCrIn alloy. Minority states are on negative scale.

H. Luo et al. / Physica B 403 (2008) 200–206 205

The spin polarization ratio P is calculated as the value of(Nm�Nk)/(Nm+Nk) and listed in Table 1, where Nm andNk are the majority and minority DOS at EF, respectively.It is clear that the P values of FeCrAl and CoCrAl are lowwhereas the NiCrZ alloys predicted to be half-metal allhave a 100% spin polarization at the Fermi level.

4. Conclusion

We have investigated the electronic structures andmagnetic properties of the half-Heusler alloys XCrAl

(X ¼ Fe, Co, Ni) and NiCrZ (Z ¼ Al, Ga, In) to searchfor new candidate half-metallic materials. Here we predictthat NiCrAl, NiCrGa and NiCrIn are possible half-metalswith an energy gap in the minority spin and a completelyspin polarization at the Fermi level. The energy gap can beattributed to the covalent hybridization between the dstates of the Ni and Cr atoms, which leads to the formationof bonding and antibonding bands with a gap in between.Their total magnetic moments are 1mB per unit cell; agreewith the Slater–Pauling rule. Meanwhile, it is found thatFeCrAl is a normal ferromagnetic metal with a magnetic

ARTICLE IN PRESSH. Luo et al. / Physica B 403 (2008) 200–206206

moment of 0.25mB per unit cell and CoCrAl is a semi-metaland non-magnetic.

Acknowledgments

This work is supported by National Natural ScienceFoundation of China (Grant no. 50531010) and NaturalScience Foundation of Hebei (Grant no. E2006000063).

References

[1] G.A. Prinz, Phys. Today 48 (1995) 58.

[2] K.I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, K. Tokura,

Nature 395 (1998) 677.

[3] J.H. Park, E. Vescovo, H.J. Kim, C. Kwon, R. Ramecsh,

T. Venkatesan, Nature 392 (1998) 794.

[4] S.J. Hashemifar, P. Kratzer, M. Scheffler, Phys. Rev. Lett. 94 (2005)

096402.

[5] R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Buschow,

Phys. Rev. Lett. 50 (1983) 2024.

[6] S. Ishida, T. Masaki, S. Fujii, S. Asano, Physica B 245 (1998) 1.

[7] R. Weht, W.E. Pickett, Phys. Rev. B 60 (1999) 13 006.

[8] S. Picozzi, A. Continenza, A.J. Freeman, Phys. Rev. B 66 (2002)

094421.

[9] S. Ishida, S. Kashiwagi, S. Fujii, S. Asano, Physica B 210 (1995)

140.

[10] A. Kellow, N.E. Fenineche, T. Grosdidier, H. Aourag, C. Coddet,

J. Appl. Phys. 94 (2003) 3292.

[11] S. Wurmehl, G.H. Fecher, H.C. Kandpal, V. Ksenofontov, C. Felser,

Appl. Phys. Lett. 88 (2006) 032503.

[12] J. Tobola, J. Pierre, J. Alloys Compd. 296 (2000) 243.

[13] B.R.K. Nanda, I. Dasgupta, J. Phys.: Condens. Matter 15 (2003)

7307.

[14] M. Zhang, X.F. Dai, H.N. Hu, G.D. Liu, Y.T. Cui, Z.H. Liu,

J.L. Chen, J.L. Wang, G.H. Wu, J. Phys.: Condens. Matter 15 (2003)

7891.

[15] R.A. de Groot, Physica B 172 (1991) 45.

[16] S. Wurmehl, H.C. Kandpal, G.H. Fecher, C. Felser, J. Phys.:

Condens. Matter 18 (2006) 6171.

[17] H. van Leuken, R.A. de Groot, Phys. Rev. Lett. 74 (1995) 1171.

[18] X.Q. Chen, R. Podloucky, P. Rogl, J. Appl. Phys. 100 (2006) 113901.

[19] E. Wimmer, H. Krakauer, M. Weinert, A.J. Freeman, Phys. Rev.

B 24 (1981) 864.

[20] J. Rath, A.J. Freeman, Phys. Rev. B 11 (1975) 2109.

[21] S. Ogut, K.M. Rabe, Phys. Rev. B 51 (1995) 10443.

[22] P. Larson, S.D. Mahanti, M.G. Kanatzidis, Phys. Rev. B 62 (2000)

12754.

[23] I. Galanakis, Ph. Mavropoulos, P.H. Dederichs, J. Phys. D 39 (2006)

765.

[24] B.R.K. Nanda, I. Dasgupta, Comput. Mater. Sci. 36 (2006) 96.

[25] S. Fujii, S. Ishida, S. Asano, J. Phys. Soc. Japan 64 (1995) 185.

[26] H.C. Kandpal, C. Felser, R. Seshadri, J. Phys. D 39 (2006) 776.

[27] K. Capelle, G. Vignale, Phys. Rev. Lett. 86 (2001) 5546.

[28] C.M. Fang, G.A. de Wijs, R.A. de Groot, J. Appl. Phys. 91 (2002)

8340.