Dec. 2004
Journal of Electronic Science and Technology of China
Vol.2 No.4
Electronic and Magnetic Properties of Small Iridium Clusters *
KUANG Xiang-jun
(School of Science, Southwest University of Science and Technology Sichuan Mianyang 621002 China)
Abstract
The electronic and magnetic properties of small IrN clusters (N=5, 6, 9, 13, and 19 ) are
studied by using the discrete-variational local-spin-density-functional method. The equilibrium bond
length in the chosen geometry for IrN clusters are determined and show bond contraction compared with
the bulk interatomic spacing. The clusters with magnetic ground state have ferromagnetic interaction
and their average magnetic moment per atom has a complex size dependence. At last, the reactivity of
IrN clusters toward H2, N2 and CO molecules is predicted.
Key words small iridium clusters; electronic structure; magnetic properties; contraction effect
Small clusters have been a subject of intensive
investigation in recent years. With quite large
surface-to-volume ratio, cluster may have intrinsic
electronic, optical, magnetic and structural properties
different from those of its bulk phase. Exploring these
uncommon properties is of great importance in
developing
new
cluster-based
material
for
technological applications and may serve as models for
understanding localized effects in solids.
Transition-metal (TM) clusters are of significant
interest due to their promising practical application in
developing new magnetic materials with large
moments and new catalysts with high reactivity. Many
theoretical calculations and experiment measurements
have been done for 3d and some 4d TM clusters[1~8],
both the theoretical and experimental studies of small
3d clusters have indicated that these cluster atoms have
large average magnetic moment per atom than atoms in
bulk phase and the average magnetic moment per atom
is almost independent of cluster size. For 4d TM
clusters, by using the local-spin-density-function (LSD)
theory, only a few studies from theory and experiment
proposed that 13-atom clusters of Pd, Rh and Ru will
be magnetic, but no study on 5 d TM clusters[5~8].
In this paper, we perform a comprehensive firstprinciple study on IrN clusters with N = 6, 9, 12, 13 and
19 by using the discrete-variational local-spin-densityfunctional (DV-LSD) method. From our study, we
hope it can help us understand the size dependence of
the structural, electronic and magnetic properties of
iridium clusters.
1
Method and Model
The electronic structures and magnetic properties
of the Ir clusters are calculated with the first-principles
discrete variational local-spin-density-function method
(DV-LSD), the same method has already been
employed in several other studies on metal clusters and
described in detail elsewhere[3,7,9]. In brief, the
numerical atomic orbitals are used in construction of
molecular orbitals, in the present work, atom-orbital
configurations composed of 5d7, 6s1.9 and 6p0.1 for Ir
atoms are employed to generate the valence orbitals.
The secular equation is then solved self-consistently
using
the
matrix
elements
obtained
by
three-dimensional numerical integrations on a grid of
random points by the Diophantine method. About 1 000
sampling points around each Ir atom are employed,
these points were found to be sufficient for
convergence of the electronic spectrum within 0.01 ev.
The self-consistent-charge (SCC) scheme and Von
Barth-Hedin exchange-correlation function are used in
the calculations.
Since the exact structure of IrN clusters is not
available experimentally, we assume the structure
models of the cluster as shown in Fig.1. Ir5 is a trigonal
Received 2004-06-29
* Supported by theYouth Natural Science Foundation of Educational Bureau of Sichuan Province (No.212-114879)
80
Journal of Electronic Science and Technology of China
(a) N=5, trigonal
dipyramid (D3h)
(b) N=6, octahedron (Oh)
(d) N=13, icosahedron (Ih)
Fig.1
2
(c) N=9, cube (Oh)
(e) N=19, double icosahedron (D5h)
Structure of iridium clusters
Results and Discussion
We discuss the results in three different steps.
First, we optimize the bond lengths for all clusters by
minimizing the binding energy. Second, the electronic
configuration and the magnetic moments of each
cluster calculated at the optimized bond length are
presented, this is followed by the discussion on the
density of state (DOS).
Tab.1
The equilibrium bond length and bind
energy for the iridium clusters
Cluster
Ir5
Ir6
Symmetry
D3h
Oh
Ir9
Oh
Ir13
Ih
Ir19
D5h
Spin
5/2
0
3/2
0
5/2
7
3/2
15/2
19/2
5/2
17/2
25/2
re / 0.1nm
2.72
2.79
2.77
2.78
2.75
2.77
2.73
2.75
2.75
2.74
2.75
2.75
Eb / ev
16.54
25.56
25.49
41.89
42.01
41.92
61.17
61.02
59.99
90.12
90.14
90.17
5.5
E
__b
/ ev·(atom)−1
N
dipyramid, Ir6 is an octahedron, Ir9 is a cube, Ir13 is an
icosahedron, Ir19 is a double icosahedron, the structures
of Ir6, Ir9 are portions of FCC crystal of iridium.
However, the icosahedral growth sequence is
suggested for the transition-metal clusters, so we use
the icosahedral structure for the Ir13 cluster and Ir19
cluster.
Vol.2
5.0
4.5
4.0
3.5
3.0
0
5
10
15
20
N
Fig.2
Size dependence of the binding energy per atom
We optimize the structure of IrN clusters while
maintaining the specific symmetry of these clusters.
The results are shown in Tab.1 and Fig.2. It is clear
that the binding energy per atom increases gradually as
the cluster size increases, however, the binding energy
of all iridium clusters is also smaller than the bulk
cohesive energy of 5.97 ev, because the surface atoms
are more weakly bonded than the atoms in the bulk.
Comparing with the bulk interatomic spacing of
0.287 nm, the bond length for all iridium clusters is a
little shorter. Such a contraction effect was observed by
extended X-ray-absorption fine structure in Cu and Ni
clusters and the contraction ratio was found to be
proportional to the surface-to-volume ratio[10], so this
effect is believed to be a reflection of surface effect. It
should be pointed out that the binding energy
calculated by the DVM method depends on the
variational basis set, whether include the 6p atomic
orbit in the basis set may lead remarkable affection in
binding energy calculation of iridium clusters.
The results for Mulliken orbital and spin
populations evaluated for IrN clusters are given in
Tab.2. With reference to the atomic configuration 5d7
6s1.9 6p0.1, one can easily see how the electron of the 6s
is redistributed to the 5d and 6p with the increase of
cluster size. Spin populations show some common
magnetic features for IrN clusters, for example, all the
clusters with magnetic ground states have FM
interactions, the cluster moment mainly comes from
the 5d local moment and the local moments of 6s and
6p often align antiferromagnetically with that of 5d.
The size dependence of magnetic moment per atom for
IrN clusters is shown in Fig.3, here we have a complex
size dependence of the moment in contrast to the
nearly size-independence relationship for the moment
in Fe, Co and Ni clusters, Cox et al measured the
magnetic moments per atom for RhN clusters and also
No.4
Tab.2 Mulliken orbital and spin populations for the IrN
clusters, a, b, c are the types of inequivalent atoms
within the cluster point group and the number of
atoms of each inequivalent type is given in
parentheses
Cluster
a (2)
b (3)
Charge
6s 6p
1.55 0.26
1.59 0.23
1.51 0.28
1.45 0.44
1.59 0.28
1.21 0.25
1.55 0.30
1.45 0.33
1.36 0.37
1.43 0.45
1.44 0.51
5d
0.89
0.71
0.12
1.08
1.25
1.44
1.06
0.79
0.61
0.68
0.81
Net Spin
6s 6p
−0.08 0.17
−0.02 0.09
−0.03 0.01
0.09 −0.11
0.09 0.07
0.12 −0.19
0.11 0.07
0.00 −0.01
−0.03 0.01
0.00 −0.04
0.07 0.01
Total
0.98
0.78
0.10
1.06
1.41
1.37
1.24
0.78
0.59
0.64
0.89
EF
N=19
N=13
N=9
N=6
N=5
−16
−12
−8
−4
0
Energy /ev
4
8
(a) Majority spin
EF
1.6
1.2
DOS
Magnetic moment
/uB⋅(atom)−1
Ir6
Ir9 a (1)
b (8)
Ir13 a (1)
b(12)
Ir19 a (2)
b (2)
c (5)
d (10)
5d
7.19
7.18
7.21
6.87
7.16
6.46
7.24
6.92
7.19
7.13
7.12
of cluster size is somewhat complex. The Ir13 cluster
has the largest VBW, and the VBW of Ir5 cluster is
found to be smaller than those of adjacent cluster size.
In FM clusters, it has been shown that a narrow VBW
is one of the favorable conditions for enhancing the
energy gain for ferromagnetism[11]. Here we could not
find any explicit correlation between the VBW and the
cluster size.
DOS
found that they depend significantly on cluster size,
our conclusion is consistent with their founding[2, 8]. As
we have seen above, the average magnetic moment of
Ir6 cluster is small because it has PM ground state.
Ir5
81
KUANG Xiang-jun: Electronic and Magnetic Properties of Small Iridium Clusters
0.8
N=19
N=13
0.4
N=9
N=6
0
4
8
12
16
N
Fig.3
N=5
20
Size dependence of magnetic moment per atom
−16
−12
−8
−4
0
Energy /ev
4
8
(b) Minority spin
Fig.4a and 4b show the density of states (DOS)
for the majority spin and minority spin electrons in the
IrN clusters. The DOS is obtained by a Lorentzian
extension of discrete energy levels and a summation
over them, the broadening width parameter is chosen
to be 0.35 ev. From Fig.4, we can see that all the DOS’s
show a large peak near the top of the valence band，and
that EF lies in the minority peak for Ir19, Ir9 and Ir6
cluster.
With theses figures, we can determine the
exchange splitting (ΔE1) and the valence bandwidth
(VBW) for IrN clusters, as listed in Tab.3. Comparing
Tab.3 with Fig.3, one can find that ΔE1 correlates in a
striking way with the cluster moment: the larger the
cluster moment, the larger ΔE1. All the VBW of
clusters are found to be smaller than the bulk value
(7.9 ev), however, the variation of VBW as a function
Fig.4
Density of state for IrN clusters
The results for the highest occupied molecular
orbit (HOMO) and the lowest unoccupied molecular
orbit (LUMO) of IrN cluster are presented in Tab.3. The
gap between the HOMO and LUMO is found to be
rather small for Ir13 and Ir19 cluster, the HOMO as a
function of cluster size has a local minima at Ir13
cluster and a local maximum at Ir6 cluster. It is
interesting to link the variation of the HOMO with the
cluster size with the reactivity of IrN clusters toward H2,
N2 and CO molecules, following the method of Rosen
and Rantala[12], we can predict that Ir6 cluster has
substantial reactivity, while Ir13 cluster has high
stability toward H2, N2 and CO molecules.
For a cluster, the number of electrons in the
HOMO determines its ground-state electronic
82
Journal of Electronic Science and Technology of China
configuration. From Tab.3, we can see that the HOMO
is occupied by the minority-spin electrons for Ir5, Ir9
and Ir19, and occupied by the majority-spin electrons
for the Ir6, Ir13 cluster. The HOMO of all clusters
except Ir13 are fully occupied, which lead to ground
state with closed electronic shell, thus these clusters
are expected to be remarkably stable. The Ir13 cluster
has degenerate ground state because its HOMO is
partially occupied, according to the John-Teller
theorem, Ir13 cluster tend to distort further toward
lower symmetry so as to lift the degeneracy of its
ground state and lower the energy.
Tab. 3
The electronic structure and ground-state electronic
configuration for IrN clusters (ev)
Cluster HOMO LUMO ΔE1 ΔE2 VBW S* E** EC***
Ir5
Ir6
Ir9
Ir13
Ir19
−3.32
−3.19
−3.89
−4.46
−3.29
−3.17 0.47 0.13
−3.02 0.05 0.33
−3.55 0.53 0.10
−4.43 0.71 0.01
−3.23 0.31 0.03
5.19
5.34
5.67
5.89
5.27
a2′ ↓
t2u ↑
t1 ↓
hu ↑
e1′ ↓
2 closed
1 closed
3 closed
5 open
4 closed
* S: symbol; ** E: electronics; *** EC: Electronic configuration
3
Summary
In this paper, we report a comprehensive study on
the electronic and magnetic properties of small Ir
clusters by using the first-principles DV-LSD method.
The results we have obtained can be summarized as
follows.
The binding energy per atom increases gradually
as the cluster size increases, but is still smaller than the
bulk cohesive energy. All iridium clusters have bond
length contraction effect.
All the clusters with magnetic ground states have
FM interactions, the cluster moment mainly comes
from the 5d local moment and the local moments of 6s
and 6p often align antiferromagnetically with that of
5d. The size dependence of magnetic moment per atom
for IrN clusters is complex.
We can predict that Ir6 cluster has substantial
reactivity，while Ir13 cluster has high stability toward
H2, N2 and CO molecules, Ir13 cluster tend to distort
Vol.2
further toward lower symmetry so as to lift the
degeneracy of its ground state and lower the energy.
References
[1] Liu F, Khanna S N, Jena P. Magnetism in small vanadium[J].
Phys. Rev., 1991, B43: 8 179-8 182
[2] Pastor G M, Dorantes J, Benneman K H. Magnetic
anisotropy of 3d transition-metal cluster[J]. Phys. Rev.
Letters, 1995, 75: 326-329
[3] Lee K, Callaway J. Electronic structure and magnetism of
small V and Cr cluster[J]. Phys. Rev., 1993, B48: 15 35815 364
[4] Chen J P, Sorensen C M, Klabunde K J. Enhanced
magnetization of nanoscale colloidal cobalt particles[J]. Phys.
Rev., 1995, B51: 11 527-11 532
[5] Reddy B V, Khanna S N. Giant magnetic moment in 4d
clusters[J]. Phys. Rev. Letters, 1993, 70: 3 323-3 326
[6] Blugel S. Two-dimensional ferromagnetism of 3d, 4d and 5d
transition metal monolayers on nobel metal substrates[J].
Phys. Rev. Letters, 1992, 68: 851-854
[7] Cox A J, Louderback J G, Apsel S E. Magnetism in
4d-transition metal clusters[J]. Phys. Rev., 1994, B49:
12 295-12 298
[8] Cox A J, Louderback J G, Bloomfield L A. Experiment
observation of magnetism in R hodium cluster[J]. Phys. Rev.
Letters, 1993, 71: 923-926
[9] Delly B, Ellis D E. Efficient and accurate expansion methods
for molecules in local density models[J]. J. Chem. Phys.,
1982, 76: 1 949-1 962
[10] Delly B, Ellis D E, Freeman A J. Binding energy and
electronic structure of small copper particles[J]. Phys. Rev.,
1983, B27: 2 132-2 144
[11] Li Z Q, Yu J Z, Kaoru O. Calculation on the magnetic
properties of rhodium[J]. J. Phys: Condense matter, 1995, 7:
47-53
[12] Rosen A, Rantala T T. Magnetism of small TM clusters[J]. Z.
Phys., 1986, D3: 205
Brief Introduction to Author(s)
KUANG Xiang-jun (邝向军) was born in 1967 and
received doctor degree from USTC in 1999. He is now is an
associate Professor in the School of science of SWUST. His
research interests are in the study on the structure, electronic
and magnetic properties of TM atom clusters.