Half-metallic ferromagnetism in hexagonal ... - APS Link Manager

PHYSICAL REVIEW B 76, 115201 共2007兲

Half-metallic ferromagnetism in hexagonal MAl7N8 and cubic MAl3N4 (M = Cr and Mn) from first principles Li-Jie Shi and Bang-Gui Liu Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China and Beijing National Laboratory for Condensed Matter Physics, Beijing 100080, China 共Received 16 April 2007; revised manuscript received 26 June 2007; published 6 September 2007兲 Motivated by recent reports on high Curie temperatures in Cr- and Mn-doped aluminum nitrides and the fabrication of heavily transition-metal-doped ones with room-temperature ferromagnetism, we study systematically 12.5% transition-metal substituted aluminium nitrides, hexagonal CrAl7N8 and MnAl7N8, using a full-potential density-functional method. We optimize fully the crystal structures and prove the stability of their ferromagnetism against possible antiferromagnetic orders. Their volumes are expanded only by about 1% compared with that of AlN. Every d electron of +3 cations of Cr and Mn contributes one Bohr magneton to the moment. Our calculations show that both of them are half-metallic ferromagnets with half-metallic gaps larger than 1 eV. The transition-metal substitution creates 10 transition-metal d-dominated impurity bands in the semiconductor gap of AlN. The hybridization of the d states with the N p ones yields a large spin exchange splitting of the d states which drives the ferromagnetism. The half-metallicity is attributed to the wide minority-spin gap across the Fermi level and the favorable majority-spin density of states of the d-dominated bands in addition to the above factors. 25% Cr and Mn substituted cubic aluminum nitrides, cubic CrAl3N4 and MnAl3N4, are studied in the same way and proved to share almost all the properties of the two hexagonal ones. The mechanism for the ferromagnetism should be useful for understanding nitride-based diluted magnetic semiconductors. These half-metallic transition-metal aluminum nitrides, at least some of them, could be useful in spintronics. DOI: 10.1103/PhysRevB.76.115201

PACS number共s兲: 75.90.⫹w, 71.20.⫺b, 75.50.⫺y, 75.75.⫹a

I. INTRODUCTION

The half-metallic ferromagnet could be useful for spintronics,1,2 because its electronic states at the Fermi level have only one spin channel.3 This feature makes high spin polarization 共100% in the ideal case3,4兲 and on the other hand avoids some spin-related scattering processes that should exist otherwise, which is important to spintronic applications. The half-metallic ferromagnetism has been found in half-Heusler alloys,3 full-Heusler alloys,5,6 transitionmetal oxides,7–10 binary transition-metal pnictides and chalcogenides,11–19 ternary transition-metal compounds based on conventional semiconductors,20,19 and even in graphene nanoribbons.21 It is still highly desirable to seek more useful half-metallic ferromagnets. AlN is very interesting because, on the one hand, its hexagonal and cubic phases, with wide semiconductor gaps, have been fabricated,22 and on the other hand, diluted magnetic semiconductors with high Curie temperatures have been realized with Cr and Mn doping in it.23–32 The Curie phase transition temperature reportedly can be above 900 K,23–25 which makes diluted magnetic semiconductors based on AlN promising. The ground state phase of AlN has wurtzite structure. The zinc-blende phase of it is higher in total energy by 48 meV per formula unit than the wurtzite one. It is reported that up to 35.7% Cr and 13.6% Mn can be doped into the hexagonal AlN 共Refs. 23–41 兲 and the Curie temperature can be 300– 340 K in the thin films with Cr doping concentrations 0.15 共Ref. 26兲 and 0.357 共Ref. 27兲 and Mn concentration 0.136,28 respectively. Because of these encouraging experiments, it is highly desirable to study the magnetic properties of transition-metal-doped AlN with high transition-metal concentration. 1098-0121/2007/76共11兲/115201共9兲

In this paper we substitute 12.5% and 25% transitionmetal atoms 共Cr and Mn兲 for Al in wurtzite and zinc-blende AlN, respectively. We optimize fully the crystal structures of the resulting hexagonal and cubic ternary nitrides using a full-potential linear-augumented-plane-wave method within the density-functional theory. The electronic structures and properties are studied with the optimized structures. Our study shows that all four ternary nitrides, the hexagonal CrAl7N8 and MnAl7N8 with space group 156, and the cubic CrAl3N4 and MnAl3N4 with space group 215, are typical half-metallic ferromagnets. Each Cr 共Mn兲 atom contributes 3␮B 共4␮B兲 to the total magnetic moment. There are 10 d-dominated bands in the semiconductor gap of AlN. They can be divided into three sets, two majority-spin sets and one minority-spin one. There are three and four 3d electrons in the 10 bands for Cr and Mn cases, respectively. The ferromagnetic interaction between the d spins is established through a hybridization of the d electrons with the lower p electrons. The half-metallicity can be attributed to the favorable majority-spin density of states of the d-dominated bands and the wide minority-spin gap across the Fermi level thanks to the large exchange splitting and the strong pd hybridization. The remaining part of the paper is organized as follows. We present our computational details in the next section. In Sec. III we perform geometric optimization with the internal atomic positions relaxed, and present the important parameters of the optimized structures and magnetic properties. In Sec. IV we present the spin-dependent density of states and bands of the optimized structures, show the half-metallic ferromagnetism in the four hexagonal and cubic ternary transition-metal aluminum nitrides, and deduce basic param-

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©2007 The American Physical Society

PHYSICAL REVIEW B 76, 115201 共2007兲

LI-JIE SHI AND BANG-GUI LIU

atoms. For each of the four structures, we determine the stable magnetic order by comparing the total energy of the ferromagnetic spin order and those of various antiferromagnetic ones. III. STRUCTURAL OPTIMIZATION

FIG. 1. 共Color online兲 Schematic structures of hexagonal XAl7N8 in a 2 ⫻ 2 ⫻ 1 wurtzite supercell 共a兲 and cubic XAl3N4 in the Lazarevicite structure 共b兲, where X is Cr or Mn. The biggest atom 共atom 2 on the left-hand side, or the corner atom on the right-hand side兲 represents an X atom, the smallest ones are N atoms and the others are Al ones.

eters concerned. In Sec. V we present energy resolved electron density distributions in some important planes. In Sec. VI we discuss the elementary bonding properties and the mechanism for the half-metallicity. Finally, we present our conclusion in Sec. VII.

II. COMPUTATIONAL DETAILS

We use a full-potential linear-augmented-plane-wave 共FLAPW兲 method within the density-functional theory 共DFT兲,42 as implemented in the Vienna WIEN2k package.43 The generalized gradient approximation 共GGA兲44 is used as the exchange-correlation potential for all of our results presented in the following. Local spin-density approximation 共LSDA兲 is also used for some comparison with GGA.45 Full relativistic calculations are done for core states and the scalar approximation is used for the others, with the spin-orbit coupling being neglected.46 We use the parameters RmtKmax = 8.0 and lmax = 10. We use 200 k points in the first Brillouin zone 共24 k points in the reduced wedge兲 for the hexagonal structures with space group 156, and 3000 k points 共84 k points in the reduced wedge兲 for the cubic structures with space group 215. The radii of muffin tins are adjusted so as to achieve the best accuracy of self-consistent calculations. A strict convergence standard is used for all our self-consistent calculations. We use a 2 ⫻ 2 ⫻ 1 supercell approach to model the 12.5% substitution of transition metals for Al in the hexagonal AlN 共h-AlN兲. With the substitution of Cr and Mn, we obtain two ternary hexagonal compounds: CrAl7N8 共h-CrAl7N8兲 and MnAl7N8 共h-MnAl7N8兲. They have the structure shown in Fig. 1共a兲, having space group 156. As for the 25% substitution of transition-metal atoms for Al in the zinc-blende AlN 共c-AlN兲, we simply replace the Al atom at the vertex of the unit cell. The resulting cubic CrAl3N4 共c-CrAl3N4兲 and MnAl3N4 共c-MnAl3N4兲 both have the crystal structure 共with space group 215兲 shown in Fig. 1共b兲. The geometric structures are optimized in terms of total energy and the internal atomic positions are optimized in terms of forces applied on

AlN assumes wurtzite structure with space group 186 in its ground state phase, and its zinc-blende phase with space group 216, although metastable, has been fabricated because its total energy is only slightly 共48 meV per formula unit in a full-potential DFT calculation兲 higher than that of the wurtzite phase. The h-AlN has 6.21 eV as its semiconductor gap, and the c-AlN has 5.34 eV. Their Kohn-Sham gaps are 4.01 and 3.31 eV, respectively, as are calculated with the fullpotential DFT method. We substitute transition-metal atoms such as Cr and Mn for Al atoms in these two structures, and obtain two hexagonal ternary nitrides, h-CrAl7N8 and h-MnAl7N8, and two cubic ternary nitrides: c-CrAl3N4 and c-MnAl3N4. We optimize fully their geometric structures and internal atomic positions by combining total energy method and force optimization. All of their electronic structures, including density of states 共DOS, in units of state/eV per formula兲, energy bands, and electron density, and their magnetic properties are calculated and studied with the optimized lattice constants and atomic configurations in the following. In Table I we list the optimized lattice constants, magnetic moments, half-metallic gaps, and energy differences per magnetic atom between the lowest antiferromagnetic structures and the ferromagnetic ones. The h-AlN and c-AlN are also presented for comparison. The substitution of Cr and Mn for Al atoms makes the cells of the hexagonal structures expand slightly 共0.4%/0.3% and 0.5%/0.4%兲. The lattice constants of c-CrAl3N4 and c-MnAl3N4 are larger approximately by 0.7% than that of c-AlN. This difference can be attributed to different crystalline symmetry and different percentage of the transition-metal substitution. The ferromagnetic order is more stable for the hexagonal structures than for the cubic ones. Each Cr atom contributes 3␮B to the total moment per magnetic atom, and each Mn contributes 4␮B. The Curie temperatures for the hexagonal cases could be above room temperature because the ferromagnetic phases are lower by at least 78 meV per magnetic atom than corresponding antiferromagnetic structures. We present in Table II optimized bond angles 共␣h, ␤h, and h ␥ 兲 and bond lengths 共bh1, bh2, and bh3兲 of the hexagonal structures 共h-CrAl7N8 and h-MnAl7N8兲 in comparison with those of h-AlN. ␣h is defined as the angle formed by atoms 1 and 3 with respect to atom 2, ␤h that by atoms 3 and 4 with respect to atom 2, and ␥h that by atoms 5 and 2 with respect to atom 1, as shown in Fig. 1共a兲. The bh1 is defined as the length of the bond between atoms 1 and 2, bh2 that between atoms 2 and 3, and bh3 that between atoms 1 and 5. An Al atom in the h-AlN has four nearest N atoms which form an approximate tetrahedron, because bh1 共1.918 Å兲 is larger than bh2 共1.905 Å兲 and ␣h 共108.1° 兲 is less than ␤h 共110.8° 兲. The increasing of the bh1 and bh2 with the Cr and Mn substituted for 12.5% Al atoms reflects the fact that the radii of Cr+3 and Mn+3 ions are larger than that of Al+3. The decreasing 共1.9°

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HALF-METALLIC FERROMAGNETISM IN HEXAGONAL…

TABLE I. The calculated equilibrium lattice constants 共a / c or a兲, relative volume change 共␦v兲 with respect to the pure AlN, magnetic moment per magnetic atom 共m兲, and relative energy per magnetic atom 共⌬E兲 of the lowest antiferromagnetic structure with respect to the ferromagnetic one. The AlN phases are presented for comparison. a / c 共Å兲

␦v 共%兲

m 共 ␮ B兲

⌬E 共meV兲

h-CrAl7N8 h-MnAl7N8 h-AlN Name

3.147/5.036 3.151/5.042 3.136/5.022 a 共Å兲

1.0 1.4

3 4 0 m 共 ␮ B兲

78 108

c-CrAl3N4 c-MnAl3N4 c-AlN

4.440 4.437 4.407

Name

␦v 共%兲 2.3 2.1

⌬E 共meV兲

3 4 0

38 12

or 1.6°兲 of ␣h reflects that bh1 and bh2 becomes larger at different rates 共5.8% and 2.6% for the h-CrAl7N8, or 5.4% and 3.3% for the h-MnAl7N8兲. As a result, the ␤h angles increase by approximately 1.5% with respect to that of h-AlN. The substitution-driven change of atomic positions is limited to the nearest N atoms only, and the next-nearest atoms and beyond remain almost unchanged. The c-AlN lattice consists of an Al face-centered-cubic 共fcc兲 sublattice and an N fcc sublattice. Letting the four Al atoms occupy the vertex and the three fcc sites of the unit cell, the four N atoms occupy 共u , u , u兲, with u = 0.25, and the ¯ , ¯u , u兲, 共u ¯ , u , ¯u兲, and 共u , ¯u , ¯u兲. other three equivalent sites, 共u An Al 共or N兲 atom has four equivalent N 共or Al兲 atoms as the nearest neighbors and the four N 共or Al兲 atoms form a standard tetrahedron. The angle formed by any two of the four N atoms with respect to the centering Al atom is equivalent to the standard 109.5°. Substituting Cr 共or Mn兲 for the Al at the vertex, we obtain the c-CrAl3N4 共or c-MnAl3N4兲 which has ¯ 3m 共No. 215兲. The full optimization changes space group P4 the lattice constants and the u parameters of the N positions, keeping the Al sites unchanged. The bond angles and lengths are summarized in Table III. ␣c is defined as the angle formed by atoms 1 and 3 with respect to atom 2, and ␤c that by atoms 3 and 4 with respect to atom 2, as shown in Fig. 1共b兲. bc1 is the bond length between atom 1 and atom 2, and b2c that between atom 2 and atom 3. The Cr or Mn substitution makes the lattice constant a increase approximately by 0.7%. The u parameter is equivalent to 0.2543 and 0.256 for the c-CrAl3N4 and c-MnAl3N4, respectively, instead of 0.25 for the pure c-AlN. The bond angles ␣c become smaller by −0.9% and −1.2%, respectively, for the c-CrAl3N4 and

The spin-dependent DOS and energy bands of h-CrAl7N8 are presented in Fig. 2. The energy bands labeled with Ah1 between −8.26 eV and −2.95 eV, including both spin channels, are similar to those of h-AlN. Although the partial DOS of the Cr d is small in this energy region, the total Cr d weight between −8.26 eV and −2.95 eV is not very small because the energy region is quite broad. The most distinguishing feature of the DOS in Fig. 2 is that there exist 10 d-dominated spin-split bands, divided into three sets: Bh1 共two majority-spin bands兲, Ch1 共three majority-spin bands兲, and Dh1 共five minority-spin bands兲, in the semiconductor gap of h-AlN. The crystalline field divided the d quintet states into a doublet 共Bh1兲 and a triplet 共Ch1兲. There is a gap of 1.6 eV between the Ah1 and Bh1. The doublet and triplet merge into the quintet at the higher energy. The two Bh1 bands are filled. The three Ch1 bands are partly filled with one electron. The five Dh1 bands are completely empty. The magnetic moment per Cr atom is 3␮B, coming from the two Bh1 electrons and one electron in the Ch1 bands. Taking −0.4 eV as the weight center of the five majority-spin bands in Bh1 and Ch1, we esti-

TABLE II. The calculated results of the three bond angles 共␣h, ␤ , and ␥h兲, and the three bond lengths 共bh1, bh2, and bh3兲 of the two hexagonal structures around the transition-metal atom. The h-AlN is presented for comparison.

TABLE III. The calculated results of the internal structural parameter 共u兲, the two bond angles 共␣c and ␤c兲, and the two bond lengths 共bc1 and bc2兲 of the two cubic structures around the transitionmetal atom. The c-AlN is presented for comparison.

h

Name h-CrAl7N8 h-MnAl7N8 h-AlN

␣h 共deg兲 ␤h 共deg兲 ␥h 共deg兲 bh1 共Å兲 bh2 共Å兲 bh3 共Å兲 106.2 106.5 108.1

112.5 112.3 110.8

106.6 106.6 108.1

2.030 2.021 1.918

1.954 1.969 1.905

1.909 1.916 1.905

c-MnAl3N4, and ␤c larger by 0.8% and 1.1%. The lengths bc1 of Cr-N and Mn-N bonds are enlarged by 2.4% and 3.1%, respectively, with respect to the Al-N bond length in the c-AlN, but the bond lengths bc2 change only slightly 共by 0.1% and −0.1%兲. The change of atomic positions due to the substitution is limited to the nearest N atoms of the Cr or Mn atom. IV. ELECTRONIC STRUCTURES AND HALF-METALLIC FERROMAGNETISM

Name c-CrAl3N4 c-MnAl3N4 c-AlN

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u

␣c 共deg兲

␤c 共deg兲

bc1 共Å兲

bc2 共Å兲

0.2543 0.2560 0.25

108.5 108.2 109.5

110.4 110.7 109.5

1.954 1.967 1.908

1.910 1.906 1.908

PHYSICAL REVIEW B 76, 115201 共2007兲

LI-JIE SHI AND BANG-GUI LIU

-8

Density of States

8

-2

-4

-6

2

4 total Crtot Altot Ntot Inter

B1h

A1h

4

0

C1h

12

12

8

8

4

0

0

4

4

8 12

D1h -8

-6

-4

-2

0

2

Density of States

12

4

-6

-8

-4

A2h

4

Γ

Σ

Μ

Κ

Λ

Γ

∆

Α

Σ

Μ

4

12

total Mntot Altot Ntot Inter

C 2h

8 4 0

4

4

8

12

12

D2h -8

-6

-4

-2

0

8

2

4

12

Energy (eV)

EF

Γ

2

0

8

Κ

Λ

Γ

∆

Energy (eV)

Energy (eV)

6 5 4 3 2 1 EF 0 -1 -2 -3 -4 -5 -6 -7 -8 -9

0

B2h

Energy (eV) 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9

-2

Α

FIG. 2. 共Color online兲 Upper part: Spin-dependent total 共thick line兲 and partial DOS of the h-CrAl7N8. The thin, dotted, dashed, and dashed-dotted lines represent the partial DOS in the muffin-tin sphere of Cr, Al, and N, and in the interstitial region, respectively. The upper-half panel is for majority-spin channel and the lower-half panel for minority-spin channel. The Ah1 part, including both majority-spin and minority-spin DOS, corresponds to that of pure h-AlN, The Bh1, Ch1, and Dh1 parts are mainly attributed to Cr. Lower part: Spin-dependent energy bands of the h-CrAl7N8. The left-hand panel is majority-spin bands and the right-hand panel is minorityspin bands. There is a little overlap between the Ch1 bands and the Dh1 bands.

mate the spin-splitting energy at 2.0 eV. Because the Fermi energy is in the minority-spin gap 共3.8 eV兲 between the Dh1 and Ah1 bands, the h-CrAl7N8 is a half-metallic ferromagnet.3,14,16 The half-metallic gap is 1.02 eV. It is the minimal energy at which an electron at the Fermi level has its spin reversed.16,19 In Fig. 3 we present the spin-dependent DOS and bands of the h-MnAl7N8. The Ah2 part is similar to the Ah1 one of the h-CrAl7N8. The minority-spin gap between the Dh2 bands and the Ah2 ones is 3.2 eV, being smaller than that of the h-CrAl7N8. The Bh2 and Ch2 bands move toward the low energy end. As a result, the gap between the bands Bh2 and Ah2 is only 0.2 eV, in contrast to 1.6 eV in the case of h-CrAl7N8. We can define a virtual semiconductor gap for each of the two hexagonal ternary nitrides by removing the in-gap bands of Bh2, Ch2, and Dh2. It is 4.7 eV for the h-CrAl7N8 and 4.6 eV for the h-MnAl7N8. The total magnetic moment per Mn atom

6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8

Γ

Σ

Μ

Κ

Λ

Γ

∆

6 5 4 3 2 1 EF 0 -1 -2 -3 -4 -5 -6 -7 -8

Α

EF

Γ

Σ

Μ

Κ

Λ

Γ

∆

Α

FIG. 3. 共Color online兲 Upper part: Spin-dependent total 共thick line兲 and partial DOS of the h-MnAl7N8. The thin, dotted, dashed, and dashed-dotted lines represent the partial DOS in the muffin-tin sphere of Mn, Al, and N, and in the interstitial region, respectively. The upper-half panel is for majority-spin channel and the lower-half panel is for minority-spin channel. The Ah2 part, including both majority-spin and minority-spin DOS, corresponds to that of pure h-AlN, The Bh2, Ch2, and Dh2 parts are mainly attributed to Mn. Lower part: Spin-dependent energy bands of the h-MnAl7N8. The lefthand panel is majority-spin bands and the right-hand panel is minority-spin bands.

is 4␮B, which means that all four d electrons of Mn+3 contribute to the moment. The two Bh2, three Ch2, and five Dh2 bands in the semiconductor gap of h-AlN are d dominated too, although some of the spectrum weight is transferred to the Ah2 bands. The electronic structures show that the h-MnAl7N8 is a half-metallic ferromagnet. The half-metallic gap is 1.29 eV, a little larger than that of the h-CrAl7N8. The spin-dependent DOSs and energy bands of the c-CrAl3N4 and c-MnAl3N4 are presented in Figs. 4 and 5, respectively. The DOS between −8 and 3 eV can still be divided into four sets: Aci , Bci , Cci , and Dci 共i = 1 and 2兲, as those of the two hexagonal nitrides do accordingly. There are less bands in the Ac part than in the Ah part of the hexagonal nitrides because of less atoms in the unit cells of the cubic structures. The c-CrAl3N4 has a little larger gap between the Ac bands and the Bc ones than the h-CrAl7N8, and so does the c-MnAl3N4 than the h-MnAl7N8. The widths of the Bc, Cc, and Dc bands are clearly larger than those of the Bh, Ch,

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HALF-METALLIC FERROMAGNETISM IN HEXAGONAL…

-4

-6

Density of States

6

B1c

c 1

A

4

0

-2

2

4

8

8

6

6

4

C1c

2

2

0

0

-2

-2

total Crtot Altot Ntot Inter

-4 -6 -8

-8

-6

-4

-2

-4

D1c

0

2

4

Density of States

-8

8

-8

-6

-4

4

R

∆

Γ ∆ Χ Ζ Μ

Σ

Γ

∆

4

8

C

B2c

c 2

4 2

0

0

-2

total Mn-tot Al-tot N-tot Inter

-4 -6

-8

-8

-8

-6

-4

-2

0

-2 -4

D2c

-6

2

4

-8

Energy (eV)

EF

R

2

2

-6

Γ ∆ Χ Ζ Μ

Σ

Energy (eV)

Energy (eV)

4 3 2 1 EF 0 -1 -2 -3 -4 -5 -6 -7 -8

0

6

A2c

Energy (eV) 4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8

-2

Γ

FIG. 4. 共Color online兲 Upper part: Spin-dependent total 共thick line兲 and partial DOS of the c-CrAl3N4. The thin, dotted, dashed, and dashed-dotted lines represent the partial DOS in the muffin-tin sphere of Cr, Al, and N, and in the interstitial region, respectively. The upper-half panel is for majority-spin channel and the lower-half panel is for minority-spin channel. The Ac1 part, including both majority-spin and minority-spin DOS, corresponds to that of pure c-AlN, The Bc1, Cc1, and Dc1 parts are mainly attributed to Cr. Lower part: Spin-dependent energy bands of the c-CrAl3N4. The left-hand panel is majority-spin bands and the right-hand panel is minorityspin bands. There is some overlap between the Cc1 bands and the Dc1 ones.

and Dh bands of the h-CrAl7N8 and h-MnAl7N8. The crystalline splitting reflected by the gaps between the Bc bands and the Cc bands is smaller than that of the hexagonal nitrides. In the case of c-MnAl3N4, the gap is closed. We also can define the virtual gaps for the cubic nitrides by removing the in-gap bands: Bci , Cc1, and Dci 共i = 1 and 2兲. They are approximately equivalent to the corresponding virtual gaps of the hexagonal cases. The half-metallic gaps of the two cubic nitrides are 0.49 and 0.95 eV, smaller than those of the hexagonal ones. There are 24 bands 共12 majority spin and 12 minority spin兲 in the Ac part of the c-CrAl3N4 or c-MnAl3N4 because we have 6 ⫻ 4 p electrons. As for the h-CrAl7N8 or h-MnAl7N8, we have 48 bands in the Ah part of the bands because we have 6 ⫻ 8 p electrons. The width of the Ah bands is 5.3 eV and that of the Ac bands is 4.9 eV. The h-AlN and c-AlN phases have semiconductor gaps of 6.21

4 3 2 1 0 -1 -2 -3 -4 -5 -6 -7 -8

R

∆

Γ ∆ Χ Ζ Μ

Σ

4 3 2 1 EF 0 -1 -2 -3 -4 -5 -6 -7 -8

Γ

EF

R

∆

Γ ∆ Χ Ζ Μ

Σ

Γ

FIG. 5. 共Color online兲 Upper part: Spin-dependent total 共thick line兲 and partial DOS of the c-MnAl3N4. The thin, dotted, dashed, and dashed-dotted lines represent the partial DOS in the muffin-tin sphere of Mn, Al, and N, and in the interstitial region, respectively. The upper-half panel is for majority-spin channel and the lower-half panel is for minority-spin channel. The Ac2 part, including both majority-spin and minority-spin DOS, corresponds to that of pure c-AlN, The Bc2, Cc2, and Dc2 parts are mainly attributed to Mn. Lower part: Spin-dependent energy bands of the c-MnAl3N4. The left-hand panel is majority-spin bands and the right-hand panel is minorityspin bands. There is a little overlap between the Bc2 bands and the Cc2 ones.

and 5.34 eV and Kohn-Sham gaps of 4.01 and 3.31 eV, respectively. The widths of the p valence bands of the h-AlN and c-AlN, corresponding to those of the Ah and Ac, are 5.9 eV and 5.7 eV, independent of the different crystalline symmetry. The virtual gaps 共GV兲, the energy gaps between the B bands and A 共GAB兲, the minority-spin gaps crossing the Fermi levels 共G MI兲, and the half-metallic gaps 共GH兲 are summarized in Table IV. The virtual gaps of the hexagonal nitrides are larger by 0.6– 0.7 eV than the Kohn-Sham gap of h-AlN, and those of the cubic ones are larger by 1.5 eV than the Kohn-Sham gap of c-AlN. This implies that the presence of Cr or Mn atoms widens the p energy gaps crossing the Fermi levels. The GAB of the two Cr substituted nitrides are much larger than those of the two Mn substituted nitrides. This means that the Mn d states are much lower than the Cr d ones. The two hexagonal nitrides have much wider halfmetallic gaps than the cubic ones do.

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PHYSICAL REVIEW B 76, 115201 共2007兲

LI-JIE SHI AND BANG-GUI LIU TABLE IV. The calculated virtual gaps 共GV兲, A-B gaps 共GAB兲, minority-spin gaps crossing the Fermi energy 共G MI兲, and halfmetallic gaps 共GH兲 of the four optimized ternary structures. Name h-CrAl7N8 h-MnAl7N8 c-CrAl3N4 c-MnAl3N4

GV 共eV兲

GAB 共eV兲

G MI 共eV兲

GH 共eV兲

4.7 4.6 4.8 4.8

1.6 0.2 1.7 0.5

3.8 3.2 3.6 3.5

1.02 1.29 0.49 0.95

V. ELECTRON DENSITY DISTRIBUTION

The energy resolved electron density distribution is very important to investigating the bonding and magnetic properties. We shall divide the electron density between −9 eV and 0 eV into three parts: A, B, and C. The A part consists of 24 majority-spin and 24 minority-spin bands for the h-CrAl7N8 and h-MnAl7N8, and 12 majority-spin and 12 minority-spin bands for the c-CrAl3N4 and c-MnAl3N4. We call the corresponding part of the bands Ah0 for the h-AlN, and Ac0 for the c-AlN, with the number of the bands being normalized in terms of the h-CrAl7N8 or the c-CrAl3N4, respectively. We have two electrons in the Bhi or Bci 共i = 1 , 2兲 part. Either Chi or Cci 共i = l , 2兲, is partly filled. We have only one electron in each of them. In the upper three panels of Fig. 6, we present the charge 共both majority-spin electron and minority-spin electron兲 density distributions of the Ahi 共i = 0 , 1 , 2兲 bands of the h-AlN, h-CrAl7N8, and c-MnAl7N8, respectively, in the 1-2-3 plane 共the H1 plane兲 of the hexagonal structure shown in Fig. 1共a兲, and in the lower three panels of Fig. 6, we present those of the Aci 共i = 0 , 1 , 2兲 bands of the c-AlN, h-CrAl7N8, and c-MnAl7N8, respectively, in the 1-2-3 plane 共the C1 plane兲 of the cubic structure shown in Fig. 1共b兲. The charge density can be large only in the neighborhood of the N atom. The charge density in the inner part of the N atom, larger than 0.05e / a.u.3 共e is the absolute value of the electron charge兲, is

approximately spherically symmetric. In contrast, it is very small near the Al atom. The minimal charge density, smaller than 0.004e / a.u.3, is reached only near the Al atom. The presence of the Cr or Mn atom changes the charge density only locally around it, which is independent of crystalline symmetry. Presented in the upper eight panels of Fig. 7 are the density distributions of the Chi and Bhi electrons, i = 1 and 2, in the H1 plane and the 4-2-3 plane 共the H2 plane兲 defined in terms of the hexagonal structure shown in Fig. 1共a兲. These are all the electron density distributions of the Bh and Ch parts of the h-CrAl7N8 and h-MnAl7N8 in the H1 and H2 planes. We present in panels 共c1兲–共c4兲 of Fig. 7 the electron density distributions of the Cc1, Bc1, Bc1 + Cc1, and Bc2 + Cc2 bands in the C1 plane. The C electron density in the neighborhood of the Cr or Mn atom in all three planes 共H1, H2, and C1兲, looking like a butterfly, is dominant, and that of the N atom in the same planes, looking like an encapsulated dumbbell, is much smaller. The B electron density in the neighborhood of the Cr or Mn atom in the same three planes is dominant too, and its shape looks like a quatrefoil. The B electron density near the N atom, being very small, looks like a deformed butterfly. The electron density near the Al atom, from the B or C bands, is very small, which does not change even after Cr or Mn is substituted for Al. The minimal charge density, smaller than 0.0005e / a.u.3, is reached only near the Al atom. In the case of the C1 plane for the cubic nitrides, the electron density distribution of the Bci + Cci 共i = 1 and 2兲 bands in the neighborhood of the Cr, Mn, or N atom becomes less anisotropic. Investigating the electron density distributions of the B and C bands, we find that high density, defined to be larger than 0.03e / a.u.3, appears only in the near neighborhood of the Cr and Mn atoms. We observe different shapes between the B electron density and the C electron density for both Cr and Mn cases. The high density distributions are anisotropic, having tetrahedral symmetry. Because the Cr 共or Mn兲 site is the center 共OT兲 of the tetrahedron of the four N atoms, we let the vertices be VT points, and define LT as the middle point of an edge of the tetrahedron and FT as the center of a face. The

FIG. 6. 共Color online兲 The charge density distributions of the electronic states of the A bands 共shown in Figs. 2–5兲 for the h-AlN 共a1兲, h-CrAl7N8 共a2兲, and h-MnAl7N8 共a3兲 in their H1 planes 共see the text for the definition兲, and for the c-AlN 共b1兲, c-CrAl3N4 共b2兲, and c-MnAl3N4 共b3兲 in their C1 planes, respectively. The outermost 共pink兲 line from the N atom represents 0.01e / a.u.3. The contour increment is 0.01e / a.u.3. The charge density takes the minimal value, smaller than 0.004e / a.u.3, only near the Al atom, which remains unchanged after the transition-metal substitution.

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FIG. 7. 共Color online兲 The electron densities of the B and filled C states of the hexagonal nitrides 关共b1兲–共b4兲, 共a1兲–共a4兲兴 and the cubic ones 关共c1兲–共c4兲兴 in different planes 共see the text for their definition兲. For the h-CrAl7N8, we present the electron densities of the Bh1 bands 共bl兲 and the filled part of the Ch1 bands 共al兲 in the H1 plane and those 关共b2兲 and 共a2兲兴 in the H2 plane, respectively. For the h-MnAl7N8, the electron densities of the Bh2 bands 关共b3兲 and 共b4兲兴 and the filled part of Ch2 bands 关共a3兲 and 共a4兲兴 in the corresponding plane are presented, respectively. The electron densities of the Bc1 bands 共c2兲 and the filled part of the Cc1 bands 共cl兲 of the c-CrAl3N4 in the C1 plane of the cubic structure are presented, with the sum of the Bc1 and filled Cc1 densities being presented in panel 共c3兲. The corresponding electron density of the Bc1 and filled Cc1 bands of the c-MnAl3N4 is presented in panel 共c4兲. The outermost 共green兲 line from the Cr or Mn atom represents 0.0005e / a.u.3 共e is the absolute value of the electron charge兲 and the contour increment is 0.001e / a.u.3. The smallest electron density, less than 0.0005e / a.u.3, is reached near the Al atoms in each panel, and in some other regions away from the Cr 共or Mn兲 atom and the N atom in every panel except 共c3兲 and 共c4兲.

largest electron density of the B bands is along the six OTLT directions, and that of the C bands along the four OTVT and OTFT ones. Therefore, the contour surface of the B electron density distributions around the Cr 共or Mn兲 atom has six vertices, and that of the C ones at the same place has eight vertices. VI. BONDING PROPERTIES AND MECHANISM FOR THE HALF-METALLICITY

The charge density distribution in Fig. 6 reveals that for AlN, both hexagonal and cubic, almost all of the Al 3s and 3p electrons are transferred to the N 2p orbitals. As a result, the charge density in regions away from the N atoms can be very small, even down to 0.004e / a.u.3, the minimal charge

density of the p electrons, smaller than 0.004e / a.u.3, is reached only near the Al atom, and almost all the p charge is in the neighborhood of the N atom. Therefore, the bonding between Al and N is almost ionic. When substituted into AlN, Cr or Mn changes the charge density only locally. As shown in Fig. 6, the charge density near the Cr 共or Mn兲 atom is larger than 0.02e / a.u.3 in all four ternary nitrides. This is caused by the fact that Cr or Mn should lose a relatively localized d electron, in addition to the s electrons, to form Cr+3 or Mn+3 and there are still three d electrons in Cr+3 or four d electrons in Mn+3. The charge density on the line between the Cr 共or Mn兲 atom and one of its nearest N atoms always is larger than 0.06e / a.u.3. As a result, there is substantially less ionicity in both Cr-N and Mn-N bonds than in the Al-N bond, and both of the Cr-N and Mn-N bonds are covalent. As shown in Fig. 6, the charge density, except for

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the small circular region near the Al atom, is always larger than 0.004e / a.u.3, and even remains larger than 0.01e / a.u.3 along all the nearest N-N paths which form a network of the N atoms. Because both majority-spin and minority-spin bands in the A part are filled and there is little spin splitting for this part, the A part of the bands is not directly related to the magnetism. It is shown in Figs. 2–5 that there are 10 d-dominated electronic states 共the B, C, and D parts兲 in the semiconductor gap of AlN. They are originated mainly from the Cr or Mn d electrons. These can be proved with the fact that the A-B gaps of the Mn-substituted compounds is substantially narrower than those of the Cr-substituted ones. The d electrons alone are localized, and thus cannot form the ferromagnetism. The way out is through the hybridization of them with the p electrons around the near sites. This pd hybridization explains the d character of the A bands naturally and makes the d-dominated impurity bands in the AlN semiconductor gaps. Furthermore, it causes the ferromagnetic exchange interactions between the magnetic moments of the d spins because the pd hybridization is spin conserved and the p states are delocalized, which results in a spinexchange splitting of the d-dominated impurity bands. The spin-exchange splitting drives the five minority-spin bands in the D part upwards and the five majority-bands in the B and C parts downwards, which means that the ferromagnetism is formed. It is clear from Figs. 2–5 that the d DOS weight in the Ah c 共A 兲 bands and the spin-exchange splitting for the Mn substitution are larger than those for the Cr one. These are because we have stronger pd hybridization for the Mn cases compared to the Cr ones. For the Cr cases, the wide minority-spin gaps between the A parts and the D parts are formed thanks to three factors: the large separation of the Cr d levels from the lower N p ones, the pd hybridization, and the spin-exchange splitting. For the Mn cases, we have much less separation of the Mn d levels from the lower N p ones, but the small separation is favorable to enhance substantially the pd hybridization and the spin-exchange splitting. As a result, although we have narrow A-B gaps 共GAB = 0.2 eV and 0.5 eV兲 for the Mn cases, we have wide minority-spin gaps 共G MI = 3.2 eV and 3.5 eV兲 for the h-MnAl7N8 and c-MnAl3N4, almost the same as those of the h-CrAl7N8 and c-CrAl3N4, as shown in Table IV. In addition, the total DOS of the d-dominated B and C bands are so large that the Fermi level is rightly in the minority-spin gap across the Fermi level. Therefore, there are only majority-spin electronic states at the Fermi level, and this is the half-metallic ferromagnetism. GGA is used as the exchange-correlation potential for calculations of all the results presented above. We also have done some LSDA calculations for comparison.45 For c-MnAl3N4, the LSDA calculations yield the following: a slightly smaller optimized lattice constant 4.371 Å compared to 4.437 Å by GGA, a smaller half-metallic gap 0.61 eV compared to 0.95 eV by GGA, the same magnetic moment 4␮B per Mn atom as that by GGA, and approximately the same GV, GAB, and G MI as those by GGA. LSDA yields

smaller half-metallic gaps than GGA does, with other quantities remaining the same or approximately the same as the GGA ones. All our calculations are done in terms of ordered distributions of substituted Cr and Mn atoms. It could be possible in some real cases that the distributions of the atoms deviate from the ordered structures and assume some randomness. The half-metallicity can be disturbed or even destroyed by too much randomness in distributions of Cr and Mn atoms. Because we have 12.5% Cr or Mn substitution for the hexagonal structures and 25% for the cubic structures, we believe that one can achieve high-quality samples of them, or some of them, in which randomness of Cr and Mn distributions is made small enough and our ordered structures of half-metallic ferromagnetism are realized. VII. CONCLUSION

In summary, we study the four transition-metal substituted aluminium nitrides using a full-potential density-functional method. Our calculations show that the h-CrAl7N8 and h-MnAl7N8 and cubic c-CrAl3N4 and c-MnAl3N4 are halfmetallic ferromagnets. The half-metallic energy gap is quite wide, reaching 1.29 eV for the h-MnAl7N8. The ferromagnetism of the h-CrAl7N8 and h-MnAl7N8 is more stable against antiferromagnetic fluctuations than that of the cubic compounds because the ferromagnetic orders are lower at least by 78 meV and 108 meV per magnetic atom than corresponding antiferromagnetic orders, respectively. The transition-metal substitution creates d-dominated impurity bands in the semiconductor gaps of AlN. The ferromagnetism is attributed to the exchange splitting of the transition-metal d electrons with the help of the pd hybridization. The half-metallicity is formed because of the wide minority-spin gaps across the Fermi level and the favorable distribution of majority-spin DOS up to the Fermi level. These compounds make a type of typical model system for itinerant ferromagnetism. The mechanism for the halfmetallic ferromagnetism, especially for the h-CrAl7N8 and h-MnAl7N8, could be useful in understanding the ferromagnetism in diluted magnetic semiconductors of lightly Cr- 共or Mn-兲 doped aluminum nitrides.47–49 Lattice mismatch between the two hexagonal 共cubic兲 compounds and the h-AlN 共c-AlN兲 is very small, smaller than 0.5% 共approximately 0.7%兲. AlN-based magnetic multilayers could be fabricated with these compounds. Because high Curie temperatures have been achieved in Cr-doped AlN and heavily transitionmetal-doped aluminum nitrides have been fabricated, we believe that high-quality samples of some of the four halfmetallic ferromagnets will be realized experimentally soon, and applied in spintronics. ACKNOWLEDGMENTS

This work was supported by National Natural Science Foundation of China 共Grant No. 904060l0兲, by Chinese Department of Science and Technology under the National Key Projects of Basic Research 共Grant No. 2005CB623602兲, and by Supercomputing Center, Computer Network Information Center 共CNIC兲, Chinese Academy of Sciences.

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