theoretical study on structures of small nonstoichiometric aln clusters

Theoretical study on structures of small nonstoichiometric AlN clusters

Li Ling, Bin Song*, Pei-Lin Cao

Department of Physics and State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310027, People’s Republic of China

Received 8 September 2004; revised 3 May 2005; accepted 31 May 2005

Available online 11 July 2005

Abstract

Using the full-potential linear-muffin-tin-orbital molecular-dynamics method, we have performed calculations on the structures of

nonstoichoimetric AlxNy (xCyZ6–9) clusters. The lowest energy structures of these clusters are obtained. The stabilities and bond trends are

discussed. For N-rich cluster, the structure is dominated by N–N bonds. For Al-rich cluster, it appears that, to have a N bonded only to Al

atoms, its coordination needs to be at least three. The binding energy of the cluster increases as the size of cluster composition increases. The

HOMO–LUMO gaps of these clusters have been evaluated.

q 2005 Elsevier B.V. All rights reserved.

Keywords: AlN clusters; Lowest energy structure; FP-LMTO-MD method

1. Introduction

Atomic clusters represent an interesting phase of matter

between molecules and solids, and show increasing

potential for technological applications; for this reason,

their structures and energetics have been the focus of

many experimental and theoretical studies in the past few

years [1,2]. Among mixed clusters, AlN clusters have drawn

increasing attention due to their importance in the

microelectronics industry as a dielectric for semiconductor

devices [3]. Experimentally, AlN clusters can be produced

by nitrogen ion beam bombardment of aluminium target [4]

or by reactions of laser-ablated Al atoms with N atoms [5].

Theoretically, there are some studies, which are mainly

concerned with small stoichiometric AlN clusters [6–15]. A

notable theoretical contribution came from two groups

(e.g. Kandalam group and Wu group). Kandalam et al. [6–8]

calculated the structures of AlnNn (nZ2–6) using the first

principles calculations based on the nonlocal density

approximation to the density functional theory. They

pointed out that Al–N bond is found to dominate the lowest

energy structures of AlnNn (nO2). As the cluster size

0166-1280/$ - see front matter q 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.theochem.2005.05.026

* Corresponding author. Tel.: C86 57187951972; fax: C86

57187951328.

E-mail address: [email protected] (B. Song).

increases from Al4N4 to Al6N6, they also found that there is

a transition from planar to bulklike three-dimensional

structures. Wu et al. [11,12] studied the AlnNn cages

nZ2–41 at the B3LYP density functional level of theory,

and indicated that there do not exist Al–Al and N–N bonds

and only exist Al–N bonds in the ground state structures of

AlnNn clusters except in the case of Al2N2 Compared with

the stoichiometric AlN clusters, researches for nonstoichio-

metric AlN clusters are scare. The existing calculations on

nonstoichiometric AlxNy (xsy) clusters mostly include up

to five atoms, to the best of our knowledge. These include a

study of AlN2 and Al2N [6] noted earlier, a calculation of

AlN3 and Al3N using the second-order Moller–Plesset

(Mp2) perturbation theory [14], a study of Al2N3 and Al3N2

using full-potential linear-muffin-tin-orbital molecular-

dynamics (FP-LMTO-MD) method [16], a first principles

study of AlnN (n%6) clusters [17], and a calculation of

Al7N using ab initio molecular dynamics [18]. In order to

understand the structural properties of nonstoichiometric

AlN clusters, especially for size-dependent properties, we

think it is worthwhile doing more calculations for

nonstoichiometric AlN clusters.

Since the AlxNy clusters of nonstoichiometric compo-

sitions (xsy) within the size range, xCyZ3–5, almost have

been reported, in this paper we will carry out an ab initio

investigation for the larger AlxNy clusters within the size

range, xCyZ6–9. We performed calculations for AlxNy

clusters using FP-LMTO-MD. The accuracy of the method

has been confirmed by our previous studies on small Sin,

Journal of Molecular Structure: THEOCHEM 728 (2005) 215–223

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L. Ling et al. / Journal of Molecular Structure: THEOCHEM 728 (2005) 215–223216

Gen, GanAsn, GanNn clusters [19–24]. The results are

presented in Section 3.

2. Method

The FP-LMTO method [25–28] is a self-consistent

implementation of density functional theory within the

local-density approximation. Here we give just a brief

description of the main features of the method. The real

space is divided in two regions: the muffin-tin spheres,

where the charge density is described by a spherical

harmonic expansion and the interstitial region, where the

charge is presented by a linear combination of Hankel

envelope functions having negative energies (K1.0, K3.0,

K4.0 Ry, in our case). The Hankel functions are solutions to

the Helmholtz equation for a point source of spherical

waves. The full-charge density, which is fitted to these

functions, is used to solve Poisson’s equation in the whole

space and to evaluate the matrix elements of the full-

potential. The fitting is tabulated as a function of the

distance between muffin-tin spheres of two atoms from zero

up to infinity. The tabulated density is used for the

evaluation of the Harris functional [29] and its derivatives

relative to the atomic displacement are used for calculating

the forces [27,28]. Geometry optimization is carried out by

using the forces on the nuclei. To minimize the electronic

energy function (Harris energy function), we use a

dynamical simulated annealing technique in which friction

term is used to cool off. Each calculation is considered to be

converged when the maximum of the forces is less than

0.001 (Ry/Bohr), and variation in total energy is less than

10K4 Ry. The details of how the molecular dynamics

simulation can be performed are given in references [27,28].

In the present calculations, the muffin tin radii for Al and N

were taken as 2.10 and 1.00 a.u., respectively. The LMTO

basis sets included s, p, and d functions on all spheres.

The number of possible atomic arrangements of a cluster

increases rapidly with cluster size. In order to find the global

minimum energy structure (lowest energy structure), many

different seed structures were relaxed until the local

minimum of the total energy was found. To confirm a

structure corresponds to a local minimum, we have altered

the optimized geometry slightly and then the optimization

process was again carried out. The main seed structures were

set up by random selections of atomic positions in three-

dimensional space. The separation of Al–Al, Al–N and N–N

atoms was confined in a range. The range for Al–Al, Al–N

and N–N was 2.38–2.86, 1.86–2.23, and 1.11–1.32 A,

respectively. The separation of any pair of atoms was set

randomly when the seed structures were carried out. We

selected 40 three-dimensional and 40 quasi-planar structures

as initial structures for geometrical optimization. In addition,

we considered several other, usually high-symmetry,

structures as our starting structures for straightforward

structural optimization. In the view of the large number of

different seed structures used, we feel confident that lowest

energy structures are accurate. However, we cannot exclude

the possibility of more stable structures.

3. Results and discussion

3.1. Structures and energies

The obtained lowest-energy structures and some

metastable isomers for AlxNy (xCyZ6–9) are shown in

Figs. 1–4, along with their symmetries and binding energies.

We calculated the binding energy of each cluster of AlxNy,

given by

Eb Z xEðAlÞCyEðNÞKEðAlxNyÞ (1)

where E(Al), E(N) and E(AlxNy) are the energies of an

isolated Al atom, a N atom and the AlxNy cluster,

respectively. In some cases, the geometrical optimization

of several different starting configurations is found to give

the same geometrical parameters upon optimization.

However, it should be pointed out that the structure

dissociates into smaller AlN cluster and N2 molecule(s) in

many cases, especially for N-rich clusters. To explain the

relative stability of the various isomers of AlxNy (xCyZ6–9) clusters, we should bear in mind the binding

energies of the bonds occurring in a given configuration.

Because the bond energy decreases in going from N–N to

Al–N and Al–Al, it is expected that skeletons of the stable

isomers of AlxNy clusters are governed by the relative

strengths of the bond energies, favoring, in this order, the

appearance of N–N, Al–N and Al–Al sequences.

3.1.1. 6-atom clusters

Some low-lying isomers of these clusters are shown in

Fig 1. For AlN5, a planar C2v structure (Fig. 1(a)) with an Al

atom locating at the side of pentagonal N5 ring is the lowest

energy structure. The Al atom binds to N atoms with the

bond length 2.10 A. The bond lengths of N(2)–N(3),

N(3)–N(5) and N(5)–N(6) in N5 ring are 1.32, 1.31 and

1.32 A, respectively. If one of the Al–N bonds in Fig. 1(a) is

broken, the structure changes to Fig. 1(b), however, it lies

0.13 eV above the lowest energy structure. Structure 1C

includes a N2 and N3 subunit, however, it is 1.50 eV higher

in energy than the lowest energy structure.

For Al2N4, two degenerate isomers (Fig. 1(d) and (e)) are

obtained as the lowest energy structure. The bond lengths of

N(4)–N(6) and N(3)–N(5) are 1.12 and 1.19 A in structure

1D and those are 1.16 and 1.17 A in structure 1E,

respectively. The N–N bonds of these structures are only

slightly elongated from that of N2 dimmer (1.11 A) obtained

by us. The average bond length of Al–N in structure 1D and

1E is 2.07 and 2.23 A, respectively.

For Al4N2, the lowest energy structure is a fan-shaped

structure shown in Fig. 1(f). This structure is composed of

C2v 43.87eV C2v 43.74eV C1 42.37eV

1

6 5

4 3

2

1

4

5 3

26

4

6

3

1

2

5

C1 38.20eV C1 38.20eV Cs 27.83eV

531

6

4

2

12

4 6

3 5

4

5

2 3

61

C4v 20.65eV C2v 20.19eV

52

6

1

43

42

5

6

1

3

(a) (b) (c)

(d) (e) (f)

(g) (h)

Fig. 1. Geometries of the lowest energy structures of nonstoichiometric AlxNy (xCyZ6) clusters, along with their symmetries and binding energies. Small

circles represent N atoms, and large circles represent Al atoms.

L. Ling et al. / Journal of Molecular Structure: THEOCHEM 728 (2005) 215–223 217

a quasi-planar quadrangle of 2, 3, 5, 6 atoms with the atom 1

and 4 a little bit higher than them. In this structure every N

atom binds to three Al atoms with the same bond length

1.87 A. It is interesting to note that this bonding pattern is

presented in the bulk wurtzite structure where N is threefold

coordinated to Al atoms. Furthermore, the Al–N bond

distance in the wurtzite structure is 1.89 A, which is very

close to the bond length seen in this cluster. Nayak et al. [17]

reported an unusually stable planar structure of Al3N. It is

an equilateral triangle in which a central N bonds to three Al

atoms with bond length 1.86 A. Structure 1F has not only

the same bonding pattern to bulk wurtzite and planar Al3N

structures, but also almost the same bond length to them,

hence, it is not surprised that structure 1F appears to be the

lowest energy structure of Al4N2.

For Al5N, a capped square pyramid C4v structure

(Fig. 1(g)) is the lowest energy structure. The calculated

bond lengths are 2.67 [Al(1)–Al(2)], 2.55 [Al(2)–Al(3)] and

1.98 [Al(2)–N(6)] A, and the bond-length ratio of which is

1.35:1.29:1.00. Structure 1H, including a Al3N subunit, lies

0.46 eV above the structure 1G. These structures (Fig. 1(g)

and (h)) were also reported by Nayak et al. [17]. However,

they considered 1H was the lowest-energy structure of Al5N

clusters, though the energy difference of the two structures

was only 0.24 eV. We carefully compared these structures

and found the corresponding bond-length ratio of the C4v

structure of Nayak et al. is 1.41:1.32:1.00. Therefore, the

lowest energy structure we obtained (Fig. 1(g)) is more

compact than that Nayak obtained, though the symmetry of

the structure is the same.

3.1.2. 7-Atom clusters

Some low-lying isomers of 7-atom clusters are shown in

Fig. 2. For AlN6, a three-dimensional Cs structure with three

N2 subunits (Fig. 2(a)) is the lowest energy structure. The

three N2 bind to Al atom with the bond length 2.12 [Al(1)–

N(2)], 2.25 [Al(1)–N(7)], 2.28 [Al(1)–N(5)] A, respect-

ively. The bond lengths of N–N in N2 subunit are all 1.12 A,

which are quite close to the bond length of N2 dimer

(1.11 A) obtained by us. The structure with a N6 ring

(Fig. 2(b)) is also found in our calculation; however, it is

4.28 eV above the lowest-energy structure 2A.

1

7

2

5

3

4

6

Cs 53.64eV C1 49.36eV C1 46.58eV

Cs 46.34eV C2v 41.53eV C1 35.56eV

C1 30.34eV C2v 30.33eV C1 23.73eV

Cs 22.88eV

1

35

42

6

72

4

3

1

75 6

17

65

2

4

3

2

531

76 4

3

5

4 2

761

2

6

54

7 13

2 15

76

34

3

5

7

6

1

4

2

6 2

1

3

4

7

5

(a) (b) (c)

(d) (e) (f)

(g) (h)

(j)

(i)




For Al2N5, the lowest energy structure we found is a

three-dimensional structure with two N2 subunits

(Fig. 2(c)). The two N2 subunits bind to the Al atom with

the bond lengths 2.16 [Al(2)–N(3)] and 2.18

[Al(2)–N(5)] A, respectively. The bond length of N–N in

the two N2 subunits is both 1.12 A. Structure 2D with a N2

and N3 subunit is the next high-energy structure. It lies

0.24 eV above the lowest energy structure 2C. For Al3N4,

the lowest energy structure is shown in Fig. 2(e).

This structure is a planar structure with a N3 subunit. It

can be viewed as a planar Al3N structure [17] binds to a N3

subunit with Al–N average bond length 2.05 A. For Al4N3,

the lowest energy structure, as shown in Fig. 2(f), is also a

planar structure. In this structure, atom 5 (a N atom) binds to

three Al atoms with the average bond length 1.86 A (similar

to that in planar Al3N structure [17]). The N–N bond length

of the structure is 1.20 A. The N2 subunit binds to three Al

atoms with the average bond length 2.08 A.

15

8

7

6

23

4

C1 59.64eV C1 56.41eV C1 55.97eV

C1 55.32eV C1 49.39eV C2v 40.08eV

78

5

46

1

3

2

6

1

3

2

7

5

48

2

4

7

6

1

3

8 5

3

6

21

8

475

31

68 4

52 7

Cs 34.79eV C3v 28.27eV

61

8

3

4

7

25

7

6

3

2

8

1

4 5

(a) (b) (c)

(d) (e) (f)

(g) (h)




For Al5N2, two degenerated structures (Fig. 2(g) and (h))

are obtained as the lowest energy structures. The energy

difference between them is only 0.01 eV. Structure 2G has a

threefold coordinated and a fourfold coordinated N atom,

while structure 2H only has threefold coordinated N atoms.

For fourfold coordination N atom, the average bond length

of Al–N is 1.93 A because of larger coordination of N atom.

Structure 2H can be viewed as two planar Al3N structures

by sharing a common Al atom. Two isomers of Al6N

clusters are shown in Fig. 2(i) and (j). The lowest energy

structure (Fig. 2(i)) of Al6N is a three-dimensional structure.

The structure contents the maximum number of Al–N bond.

The Al–N average bond length of the structure is 2.14 A.

Structure 2J is a N atom capping the triangular face of

octahedron. This structure was presented as the lowest

energy structure of Al6N by Nayak et al. [17], however, it

lies 0.85 eV above the structure 2I. It is obvious that

structure 2I is more compact than the structure 2J.


Some low-lying isomers of 8-atom clusters are shown in

Fig. 3. For AlN7, the lowest energy structure is shown in

Fig. 3(a). The N–N average bond length is 1.20 A. The

Al–N bond lengths are 2.19 [Al(1)–N(3)], 2.24 [Al(1)–N(8)]

and 1.96 [Al(1)–N(5)] A, respectively. A stable structure

with a N7 ring (Fig. 3(b)) is 3.23 eV higher in energy than

the lowest energy structure 3A. For Al2N6, the structure

containing three N2 subunits is the lowest energy structure

(Fig. 3(c)). The calculated bond lengths of N–N are 1.14

[N(3)–N(6)], 1.13 [N(5)–N(7)] and 1.12 [N(4)–N(8)] A,

respectively. The three N2 bind to two Al atoms with bond

lengths 1.99 [Al(1)–N(6)], 1.95 [Al(2)–N(7)] and 1.99

[Al(2)–N(4)] A, respectively. If three N2 subunits of the

structure are replaced by two N3 subunits, the binding

energy decreases 0.65 eV (Fig. 3(d)).

For Al3N5, the lowest energy structure is a three-

dimensional structure with a N2 (1.21 A) and N3 subunit


(1.15, 1.21 A) (Fig. 3(e)). It can be obtained from the isomer

of Al2N5 cluster (Fig. 2(d)) by adding another Al atom. The

Al–N average bond length in this structure is 2.09 A. For

Al5N3, the lowest energy structure is shown in Fig. 3(f). It

can be viewed as Al5N2 clusters (Fig. 2(h)) by adding

another N atom. In this structure, the average bond length of

Al–N is 1.88 A and that of Al–Al is 2.72 A, respectively.

For Al6N2, the N atom capping the triangular face of the

octahedron is the lowest energy structure (Fig. 3(g)). The N

atom binds to three Al atoms with bond length 1.89 A. For

Al7N, the lowest energy structure, shown in Fig. 3(h), is

composed of a Al6N structure attached to a Al atom. The

structure also contents the maximum number of Al–N bond.

The N atom binds to each Al atoms with average bond

length of 2.07 A. The structure we obtained is in good

agreement with the result of Sun [18].


The lowest energy structures of 9-atom clusters are

shown in Fig. 4. For AlN8, we relaxed 86 initial structures

C1 71.27eV C1 64.51

Cs 53.49eV C1 48.25

8

1

6

7

9

4

5

2

3

76

4

6

3

2 1

5 8

4

7 9

6

1

4

8

C1 36.93eV

67

8

3

9

2

1

5

4

(a) (b)

(d) (e)

(g) (

Fig. 4. Geometries of the lowest energy structures of nonstoichiometric AlxNy (x


and only obtained one stable structure (Fig. 4(a)). The rest

all dissociated into smaller AlN cluster and N2 molecule(s).

Structure 4A contents four N2 subunits, the average N–N

bond length of which is 1.12 A. The Al–N average bond

length is 2.24 A. For Al2N7, the lowest energy structure is

shown in Fig. 4(b). Structure 4B can be derived by structure

of AlN8 by using an Al atom replacing the location of N(4)

atom. For Al3N6, the lowest energy structure is shown in

Fig. 4(c). It can be regarded as a N2 adding to the structure

of Al3N4 cluster (Fig. 2(e)). For Al4N5, the lowest energy

structure, shown in Fig. 4(d), is a three-dimensional

structure with Cs symmetry. The average bond lengths of

Al–Al, Al–N and N–N are 2.71, 2.02, and 1.20 A,

respectively. For Al5N4, the lowest energy structure is

shown in Fig. 4(e). The average bond length of Al–N is

1.98 A. The N2 binds to three Al atoms with the average

bond length 2.21 A. For Al6N3, the lowest energy structures

is shown in Fig. 4(f). In this structure, the average Al–N and

Al–Al distances are 1.97 and 2.75 A, respectively. For

Al7N2, the tricapped octahedral structure is the lowest

eV C1 59.43eV

eV C1 41.43eV

5

2

9

1

83

3

72

9

4

1

865

2

3

9

7

5

5

12

6

78

9

4

3

Cs 29.88eV

47

6 1

9

53

8 2

(c)

(f)

h)

CyZ9) clusters, along with their symmetries and binding energies. Small

Table 1

Binding energies (Ea in eV) per atom and HOMO–LUMO gaps (Eg in eV) for the lowest energy structures of AlxNy (xCyZ6–9) clusters

Species AlN5 Al2N4 Al4N2 Al5N

Ea 7.31 6.37 4.64 3.44

Eg 3.83 1.46 1.06 0.95

Species AlN6 Al2N5 Al3N4 Al4N3 Al5N2 Al6N

Ea 7.40 6.65 5.93 5.08 4.33 3.39

Eg 3.07 1.28 2.17 0.83 1.02 1.16

Species AlN7 Al2N6 Al3N5 Al5N3 Al6N2 Al7N

Ea 7.08 7.00 6.17 5.01 4.35 3.53

Eg 1.87 1.17 1.16 0.49 1.90 2.22

Species AlN8 Al2N7 Al3N6 Al4N5 Al5N4 Al6N3 Al7N2 Al8N

Ea 7.92 7.17 6.60 5.94 5.36 4.60 4.10 3.32

Eg 1.06 1.25 1.40 0.57 0.38 0.72 0.36 0.29


energy structure (Fig. 4(g)), the capping atoms of which are

an Al and two N atoms. It can be derived by an Al atom

capping the lowest energy structure of Al6N2 (Fig. 3(g)). For

Al8N, the lowest energy structure, shown in Fig. 4(h), can be

viewed as another Al atom attaching to the structure of Al7N

(Fig. 3(h)). The average Al–N bond length in this structure

is 2.09 A, which is 0.02 A deviated from that of Al7N.

3.2. Binding energies and HOMO–LUMO gaps

The binding energies per atom and the HOMO–LUMO

gaps of the lowest energy structure calculated for AlxNy

(xCyZ6–9) clusters are shown in Table 1. Table 1 shows

that the binding energy per atom changes as the N content of

the cluster changes. The binding energy per atom increases

as the N content of the cluster increases. On the other hand,

the binding energy per atom decreases as the Al content

increases. To explain these energy trends, we should bear in

mind that the binding energies of the bonds occur in given

configuration. Because the bond energy decreases in going

from N–N to Al–N and Al–Al, it is expected that the binding

energy per atom of AlxNy clusters are governed by the

relative strengths of the bond energies, favoring, in this

order, the appearance of N–N, Al–N and Al–Al sequences.

The energy difference between the highest occupied

molecular orbital (HOMO) and the lowest unoccupied

molecular orbital (LUMO), or the HOMO–LUMO gap is

the equivalent of the energy gap in a bulk semiconductor

and it is of considerable importance in the study of

semiconductor materials. For small clusters, the trend

followed by the HOMO–LUMO gap is still of considerable

interest, although a comparison with bulk values should

require much bigger clusters. The HOMO–LUMO gap of

the AlxNy clusters, as shown in Table 1, depends on cluster

composition. Table 1 show that the magnitude of the gaps

changes from 0.29 to 3.83 eV. The smallest HOMO–LUMO

gap is 0.29 eV for the lowest energy structure of Al8N. This

is quite small compared to the band gap of AlN bulk

wurtzite structure (6.28 eV) [30], but could be expected

for clusters.

3.3. Stability

We now present the results on the relative stability of

AlxNy (xCyZ6–9) with respect to their fragmentation in

atoms and/or clusters. We will only consider the lowest-

energy structure for each of the clusters involved, neglecting

contributions from the zero-point vibration energy. Table 2

is a collection of the fragmentation energies of the AlxNy

(xCyZ6–9) clusters dissociating via several possible paths.

It is noteworthy that most of these clusters are stable with

respect to their fragmentation into atoms or smaller clusters.

As is readily seen, the relative dissociation energies for the

various fragmentation paths can be explained by the

different strength of the N–N, Al–N, and Al–Al bonds.

Therefore, products containing N2 are obviously favored,

and this explains why the N-rich clusters are relatively less

stable than the Al-rich clusters. The fragmentation leading

to a N2 molecule is the preferred path for the clusters

containing more than two N atoms. However, for AlxN

clusters, the preferred fragmentation path always leads to a

smaller AlxK1N cluster and a Al atom, not to a Al2molecule.

3.4. Binding trends

For AlnNn (nO2) clusters, Kandalam et al. [6–8] and Wu

et al. [11,12] have pointed out that Al–N bonds dominated

the lowest-energy structures. However, for AlxNy clusters,

the structures found in the present works show some

interesting trends. For AlxN clusters, the lowest energy

structure corresponds to the structure that would maximize

the number of Al–N bonds. This may explain as that the

binding energy of Al–N is larger than that of Al–Al. For

AlxNy (x!y) clusters the structures are dominated by the

N–N bonds. It is shown that N2 and N3 subunits are

preferred, except for the cases of AlN5. For the structures of

AlxNy (xOy) clusters, we should note that in the cases of

Al4N2, Al5N2, Al5N3 and Al7N2, N atoms are threefold

coordinated to Al atoms and Al–N bond length resumes its

1.86 A value seen in Al3N. It appears that, to have a N

bonded only to Al atoms, its coordination needs to be at

Table 2

Dissociation energies (eV) for the lowest-energy structures of AlxNy (xCyZ6–9) clusters via various fragmentation paths.

Reaction Dissociation energy (eV) Reaction Dissociation energy (eV)

AlN5/AlN3CN2 K0.47 Al5N2/Al3CAl2N2 6.27

AlN5/AlNC2N2 3.87 Al5N2/Al4CAlN2 6.60

AlN5/AlCN5 5.46 Al5N2/Al4NCAlN 7.43

AlN5/AlN2CN3 5.76 Al6N/Al5NC Al 3.07

Al2N4/Al2N2CN2 0.82 Al6N/Al3CAl3N 4.34

Al2N4/Al2C2N2 2.86 Al6N/Al4NCAl2 5.47

Al2N4/2AlN2 2.98 AlN7/AlN5CN2 K1.60

Al4N2/Al4CN2 4.34 AlN7/AlN3C2N2 K2.07

Al4N2/AlCAl3N2 5.15 AlN7/AlN4CN3 3.38

Al4N2/AlN2CAl3 6.38 AlN7/AlCN3C2N2 4.41

Al4N2/Al2N2CAl2 6.98 Al2N6/Al2N4CN2 0.40

Al4N2/Al3NCAlN 7.03 Al2N6/Al2N2C2N2 1.01

Al5N/AlC Al4N 2.86 Al2N6/AlN4CAlN2 2.60

Al5N/Al2C Al3N 4.51 Al2N6/Al2C3N2 3.26

Al5N/Al3CAl2N 5.62 Al3N5/Al3N3CN2 0.32

AlN6/AlN4CN2 0.52 Al3N5/Al2N2C AlN3 2.16

AlN6/AlN2C2N2 1.30 Al3N5/Al3N2CN3 6.18

AlN6/AlC3N2 1.55 Al3N5/Al3CN2CN3 7.67

AlN6/AlN3CN3 6.16 Al5N3/Al5NCN2 2.14

Al2N5/Al2NC2N2 0.65 Al5N3/Al3NC Al2N2 4.37

Al2N5/Al2N3CN2 0.93 Al5N3/AlC Al4N3 4.60

Al2N5/AlN4CAlN 5.56 Al5N3/Al4N2CAlN 7.06

Al2N5/AlNC AlN2CN2 6.34 Al5N3/Al2C Al3N3 7.86

Al3N4/Al3N2CN2 1.40 Al6N2/2Al3N 3.70

Al3N4/Al2N2CAlN2 3.60 Al6N2/Al6CN2 4.08

Al3N4/Al3NCN3 5.39 Al6N2/AlC Al5N2 4.48

Al3N4/AlNCAl2N3 7.88 Al6N2/Al4NCAl2N 5.97

Al4N3/Al4NCN2 1.00 Al7N/Al6NCAl 4.55

Al4N3/Al3N3CAl 2.11 Al7N/Al3NC Al4 6.59

Al4N3/Al3NCAlN2 2.48 Al7N/Al5NCAl2 7.01

Al4N3/Al3N2CAlN 7.60 Al5N4/Al5N2CN2 0.55

AlN8/AlN6CN2 0.26 Al5N4/Al4N2CAlN2 2.81

AlN8/AlN4C2N2 0.78 Al5N4/Al5C2N2 4.10

AlN8/AlN2C3N2 1.57 Al6N3/Al6NCN2 0.34

AlN8/AlC4N2 1.81 Al6N3/Al5N3CAl 1.27

Al2N7/Al2N3C2N2 0.11 Al6N3/Al4N2CAl2N 2.40

Al2N5/Al2N5CN2 0.56 Al6N3/Al3NCAl3N2 3.19

Al2N7/Al2NC3N2 1.21 Al6N3/Al4NCAl2N2 3.56

Al2N7/AlN4CAlN3 1.77 Al6N3/Al4N3CAl2 5.26

Al3N6/Al3N4CN2 0.64 Al7N2/Al6N2C Al 2.11

Al3N6/Al3NCN2CN3 6.02 Al7N2/Al7CN2 2.40

Al3N6/Al3N3CN3 7.26 Al7N2/2Al3NCAl 5.82

Al4N5/Al4N3CN2 0.57 Al8N/Al7NCAl 1.63

Al4N5/Al4NC2N2 1.12 Al8N/Al3NC Al5 4.93

Al4N5/Al3N5CAl 4.12 Al8N/Al5NC Al3 5.42

Al4N5/AlN2C Al3N3 4.20 Al8N/Al6NC Al2 5.57

Al5N2/AlCAl4N2 2.51

Al5N2/Al5CN2 3.55

Al5N2/Al3NCAl2N 3.58


least three. This supports the results of Castales [31]. Hence,

to establish the structures without a N–N bond, we should

increase the coordination of the N atom by having a higher

local concentration of Al atoms.

4. Conclusion

We have performed a study of AlxNy (xCyZ6–9)

clusters, focusing on the geometries, stabilities and their

binding trends. Through relaxing a larger number of initial

structures, we have obtained the lowest energy structures of

these clusters. The lowest energy structure depends on its

size and composition. For N-rich cluster, N–N bonds

dominate the structure. For Al-rich cluster, it appears that,

to have a N bonded only to Al atoms, its coordination needs

to be at least three. The binding energy increases as the N

content of the cluster increases and decreases as the Al

content increases. The most of the clusters are stable

against the various dissociation products considered here.


For the clusters containing more than two N atoms,

generally dissociation leading to a N2 molecule is the one

requiring the least energy. For AlxN clusters, dissociation

into a Al atom and a smaller AlxK1N clusters is favored.

This can be explained by the relative strengths of the bond

of Al–Al, Al–N, and N–N. The HOMO–LUMO gaps of

these clusters are evaluated, and they also depend on cluster

composition.

Acknowledgements

This work was supported by the National Natural Science

Foundation of China (No.10174062).

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theoretical study on structures of small nonstoichiometric aln clusters

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