The size dependence of cluster average binding energies per atom Eb/atom, second-order differences of total energies D2E, fragmentation energies Ef and HOMO–LUMO gaps of Gen+1and GenAl n
Trang 1A computational investigation of aluminum-doped germanium clusters
by density functional theory study
Shunping Shia,⇑, Yiliang Liub, Chuanyu Zhanga, Banglin Denga, Gang Jiangc
a
Department of Applied Physics, Chengdu University of Technology, Chengdu 610059, China
b
College of Electrical and Information Engineering, Southwest University for Nationalities, Chengdu 610041, China
c
Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
a r t i c l e i n f o
Article history:
Received 9 September 2014
Received in revised form 27 October 2014
Accepted 4 December 2014
Available online 11 December 2014
Keywords:
Density functional theory
Ge n+1 cluster
Ge n Al clusters
Structure of clusters
a b s t r a c t
We report a computational study of the aluminum doped germanium clusters GenAl (n = 1–9) The molecular geometries and electronic structures of the GenAl clusters are investigated systematically using quantum calculations at the B3LYP level with the 6-311G(d) basis sets The growth pattern behaviors, sta-bilities, electronic properties, and magnetic moments of these clusters are discussed in detail Obviously different growth patterns appear between small and larger Al-doped germanium clusters, the optimized equilibrium geometries trend to prefer the close-packed configurations for Al-doped germanium clusters
up to n = 9 The size dependence of cluster average binding energies per atom (Eb/atom), second-order differences of total energies (D2E), fragmentation energies (Ef) and HOMO–LUMO gaps of Gen+1and GenAl (n = 1–9) clusters are studied The stability results show that Gen+1cluster possess relatively higher stability than GenAl cluster Furthermore, the investigated highest occupied molecular orbital-lowest unoccupied molecular orbital gaps indicate that the Gen+1and GenAl clusters have different HOMO– LUMO gap In addition, the calculated vertical ionization potentials and vertical electron affinities confirm the electric properties of Gen+1and GenAl clusters Besides, the doping of Al atom also brings the decrease
as the cluster sizes increase for atomic magnetic moments (lb)
Ó 2014 Elsevier B.V All rights reserved
1 Introduction
The semiconductor clusters with transition metal have
attracted great interest for optoelectronic materials, catalyst, and
the development of new species in nanoscale applications
Germa-nium clusters have also widely been studied because they are
important for the fine processing of semiconductors and the
syn-thesis of novel materials The studies have shown that the
struc-ture and the bonding of bulk germanium are very similar to that
of bulk silicon, and the bulk surfaces show similar reconstruction
[1] However, although small silicon and germanium clusters
appear to have similar geometries, the larger ones are
fundamen-tally different[2] During the past two decades, Genclusters have
been intensively studied both experimentally[3–9]and
theoreti-cally[10–21]because of their fundamental importance and
poten-tial applications in nanoelectronics The photoionization study has
been investigated by Yoshida and Fuke to characterize the
elec-tronic structures of germanium cluster, they found a rapid
decrease in the ionization potentials (IPs) for Genbetween n = 15
and 26, which was very similar to that for silicon clusters [4] The low-lying stages of Ge2and Ge2have also been probed using negative ion zero electron kinetic energy spectroscopy[7] Because
of the lack of experimental method to characterize the structure of germanium clusters, most of the geometrical data come from the-oretical calculations Geometrical and electronic properties of Gen
(n = 5–10) neutrals, cations, and anions have been investigated using the density functional method of Becke’s three-parameter hybrid functional with the Perdew/Wang 91 expression by Li
et al.[12] Yoo and Zeng performed a constrained search for the geometries of low-lying neutral germanium clusters in the size range of 21 6 n 6 29[14] Wang et al calculated dipole polarizabil-ities of Genclusters at FF level of density functional theory, which show the dipole moment and polarizabilities of Genclusters are sensitively dependent on the cluster geometries and electronic structures[15] King et al reported the effect of electron count
on cluster geometry of nine and ten atom germanium clusters using B3LYP level of DFT[20]
The pure germanium clusters are chemically reactive and thus not suitable as a building block of self assembly materials By an appropriate choice of the metal dopant, it is possible to design metallic as well as semiconducting nanotubes using Gen as
http://dx.doi.org/10.1016/j.comptc.2014.12.004
2210-271X/Ó 2014 Elsevier B.V All rights reserved.
⇑Corresponding author Tel.: +86 2884078267; fax: +86 28 85415508.
E-mail address: shishunping13@cdut.cn (S Shi).
Contents lists available atScienceDirect
Computational and Theoretical Chemistry
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p t c
Trang 2building blocks Doped germanium clusters have been performed
the focus of a few experimental and theoretical studies[22–35],
which exhibit many novel properties such as the sizes selectivity,
the highest occupied molecular orbital-lowest unoccupied
molecu-lar orbital gap, different charge transfer direction and the magnetic
property Tai and Nguyen[22]found the structure and stability of
the Ge12Mxclusters with M = Li, Na, Be, Mg, B, Al, and x from 1 to
+1, they obtained the high thermodynamic stability of the
icosahe-dra arises from a combination of their closed crystal field shells,
spherical aromaticity and electrostatic attraction force Electronic
properties of germanium–fluorine cluster anions (GenFm; n = 1–11,
m = 1–3) were studied by Negishi et al using photoelectron
spec-troscopy with a magnetic-bottle type electron spectrometer, which
showed that the doped F atom in GenFdeprives each Gencluster
of the excess electron without any serious rearrangement of the
Gen framework [23] In addition, the geometries, stability, and
electronic properties of TM-doped germanium clusters (TM = Zn,
Fe, Mn, Si, Ni, W, Cr, Cu, Au)[25–35]have also been systematically
investigated by using different method The remarkable features of
Zn-doped Genclusters are distinctly different from other TM-Gen
clusters, indication that the growth pattern of the TM-Genclusters
depends on the kind of doped TM impurity
Although many studies have been taken on pure germanium
clusters and doped germanium clusters, to our knowledge, surely
systematic and theoretical investigated on aluminum-doped
ger-manium clusters have not been reported so far In this work, an
investigation on the structures, stabilities, magnetism, and electronic properties of the Al-doped germanium clusters were cal-culated using density functional theory by considering a consider-able number of structural isomers In order to reveal the effect of the doped Al atom to the germanium clusters, in this paper, we optimize the geometrical structures of GenAl (n = 1–9) clusters by employing DFT approach to find the structural and stability, and combined with pure germanium clusters for comparison by using identical methods and basis sets
2 Computational details The geometry optimizations of the Gen+1 and GenAl (n = 1–9) clusters with spin configurations considered are performed by using density functional theory (DFT) with the B3LYP exchange– correlation potential and 6-311G(d) basis sets The B3LYP method,
it is based on the Becke three-parameter exchange functional and the Lee, Yang and Parr correlation functional[36,37] In order to test the reliability of our calculations, some test calculations are carried out on Ge2 and Al2 using B3LYP, B3P86, PBE1PBE, and B3PW91 method with LANL2DZ, Def2-TZVP, and 6-311G(d) basis sets The computed spin multiplicities, bond lengths (Re), vibra-tional frequencies (xe), and dissociation energies (De) of dimers (Ge2, and Al2) and available experimental and previous theoretical data are summarized inTable 1 Comparing with the experimental data, we can find that the B3LYP method with 6-311G(d) basis sets
Table 1
The computed spin multiplicities, bond lengths (R e ), vibrational frequencies (xe ), and dissociation energies (D e ) of dimers (Ge 2 , and Al 2 ) and available experimental and previous theoretical data.
2.548 b
281 c
2.34 d
274 f
2.65 f
286 + 5 g , h
2.70 ± 0.07 g , h
2.7 k
284.2 k
1.34 l a
Ref [32]
b
Ref [33]
c
Ref [34]
d
Ref [35]
e
Ref [6]
f Ref [7]
g Ref [8]
h Ref [9]
i
Ref [39]
j
Ref [40]
k
Ref [41]
l
Trang 3is more optimal than others Therefore, the B3LYP/6-311G(d)
scheme is reliable and accurate enough for describing the systems
involving Ge and Al atoms The B3LYP method or 6-311G(d) basis
sets was successfully used for pure Gen clusters[12,17] and for
GenLi clusters[22], GenCr clusters[31]
In this paper, the conformations of the pure Gen+1clusters are
obtained firstly by reference to the configurations in Refs
the GenAl clusters, we have considered possible isomeric structures
by placing the Al atom on each possible site of the Gen+1clusters as
well as by substituting one Ge atom by the Al atom from Gen+1
cluster Furthermore, different spin states of GenAl clusters are
con-sidered and calculated by using the Gaussian 03W package[38],
the optimized results are obtained that the most stable structures
of the Al-doped germanium clusters The detailed calculated
results and discussions are followed
3 Results and discussions
3.1 Lowest-energy structures
Using the computation scheme described in Section2, we have
explored a number of lying isomers and determined the
low-est-energy of GenAl clusters up to n = 9 The obtained ground state
geometries and some low-lying metastable isomers are shown in
state geometries of pure Gen+1(n = 2–9) clusters The point group
symmetries (PG), the spin multiplicities, the electronic states,
geometry property, and relative energy DE (relative to
lowest-energy structure) of the most stable and low-lying GenAl (n = 1–9)
clusters are summarized in Table 2 For Ge2 dimer, the
dissociation energy and vibrational frequency are obtained as
2.87 eV and 276.6 cm1 Our current results are in satisfactory
agreement with the experimental data (dissociation energy
De= 2.7 eV and x= 286 cm1) [8,9] The Ge–Ge bond length for
Ge2dimer is predicted to be 2.413 Å, this is also consistent with
results 2.44 Å of experimental [6] The electronic state and spin
multiplicity are 3
Rg and triplet spin state, respectively, which agrees well with the results of Deutsch et al [16] and
Bandyopadhyay and Sen [32] For the GeAl monomer with C1v
symmetries, the optimized results indicate that the quadruple
spin state is lowest energy Therefore, the quadruple GeAl
mono-mer with a bond length of 2.491 Å is most stable structure, the
corresponding electronic state is4R
For Ge3, the isosceles triangle structure is suggested as the
low-est energy structure with an apex angle of 83.8° corresponding to
the1A0 Our result is excellent with previous theory[19], in which
the isosceles triangle structure with an apex angle 84.9° The most
stable structure of Ge2Al cluster is also an isosceles triangle
struc-ture (3-a) This configuration presents the low spin state of1A0 The
Ge–Ge bond length, the Ge–Al bond length, and the vibrational
fre-quencies of ground1A0 state of Ge2Al cluster is 2.585 Å, 2.408 Å,
and 167.4 cm1, respectively The linear C1v(Ge–Ge–Al)
configura-tion is also considered, corresponding to 3-b isomer with1A00state,
the linear C1vstructure is lower in energy than the linear Cs
iso-mers The linear Ge–Al–Ge geometry with quartet spin multiplicity
is also found to be stable However, the configuration corresponds
to very high relative energies of 1.120 eV
The Ge4is a rhombus structures with C2Vsymmetry, the
corre-sponding to bond length is 2.475 Å and electronic state is1A1 Five
kinds of Ge3Al clusters can be optimized to the minima When
n = 3, the planar structures (3-a) are proved to be the
lowest-energy structures, but three-dimensional (3D) structures (3-b, 3-c,
3-d, and 3-e) are not the most stable structures in our calculated
clusters The Ge–Ge–Ge bond angle (104.3°) of the 3-a isomer,
generated from substitution of Ge4 3-a0rhombus by Al, is much larger than that of the Ge3cluster The 3-b isomer is a distorted Y-type structure, which can be described as one Al atom being bonded on the apical Ge atom in the lowest energy Ge3cluster If one Al or Ge atom is capped on the lowest energy 2-a0or 2-a, the 3-c or 3-d isomer may be formed 3-e is the highest energy between planar structures and 3D structures The most stable structure (3-a) have CSsymmetry and2A0electronic state The most favorable geometry of Ge5cluster is a distorted trigo-nal bipyramid structure with D3Hsymmetry, corresponding to the
1A0state The lowest energy structure of Ge4Al is 4-a in C1 symme-try with2A electronic state, which is formed by capping one Al atom on the top of Ge4cluster 4-b and 4-c can be viewed as Al atom substituted a Ge atom from the apical and middle in the
Ge5isomer, which are obvious higher in energy than the lowest energy structure 4-a by 0.548 and 0.748 eV, respectively One pla-nar isomer (4-d) with C1 symmetry behaves the highest energy (DE = 1.076 eV) among all isomers of Ge4Al clusters Therefore, from n = 4, the 3D structures are more stable than the planar struc-tures (4-a > 4-b > 4-c > 4-d)
The distorted octahedron is obtained for Ge6, which has D3 sym-metry, corresponding to the electronic is 3A1 Four structures obtained for Ge5Al clusters have C1 symmetry The most stable (5-a) with2A electronic state, corresponding to the Ge–Ge bond length, and the Ge–Al bond length are 2.612 Å and 2.802 Å The prism structure (5-b) and the distorted octahedron structure (5-c) usually are considered the most stable, but in our calculation, their energies higher than the 5-a, which are 0.441 and 0.603 eV, respec-tively The structure of 5-d isomer with C1symmetry and spin mul-tiplicity (PG = 2), which relation energy is 1.144 eV
In the case of n = 7, the pure Ge7adopted the pentagonal bipyr-amid structure with C1symmetry, corresponding to the electronic state and Ge–Ge bond length are 1A and 2.681 Å, respectively Although geometries structure of Ge7 cluster is same as Refs
for Ge6Al The most stable structure (6-a) is obtained by one Al atom substitute one Ge atom from the waist of pure Ge7cluster (6-a0), which has C1 symmetry, and its the Ge–Ge bond length, and the Ge–Al bond length are 2.697 Å and 2.636 Å The other pen-tagonal bipyramid structure (6-b) is one Al atom substitute one Ge atom from the top of pure Ge7cluster (6-a0) The relation energy of 6-c and 6-d are 0.607 and 0.827 eV, respectively
When the size of Genclusters is up to 8, one structure, which is obtained from the pentagonal bipyramid Ge7(6-a0), is proven to be stable structure The Ge–Ge bond length, and the vibrational fre-quencies of ground 1A state of Ge8 cluster are 2.681 Å, and 86.5 cm1, respectively The most stable Ge7Al (7-a) cluster can
be generated from one Al atom substitute one Ge atom on the low-est energy Ge8cluster It displays C1symmetry with2A electronic state The 7-b, 7-c, and 7-d isomers have higher energies compared with the 7-a structure in its ground state by 0.051, 0.249, and 0.641 eV, respectively
As for Ge9, the lowest-energy structure is a bicapped pentagonal bipyramid structure with C1symmetry, corresponding to the spin multiplicity is 1 The configuration of Ge9can be easily understood
as growth on the basis of Ge8 Five kinds of stable structures can be verified to be the minima in Ge8Al The most stable structure 8-a with C1symmetry, it is seen that the Al atom substitutes one Ge atom of the Ge9 cluster Other possible isomers (8-b–8-e) have energies higher than 8-a by 0.202, 0.260, 0.276, 0.905 eV Especially, the 8-e isomer is much higher in energy than 8-a by 0.905 eV Although both Ge8Al 8-c and 8-d isomers are close in energy, they have different structures
The ground-state structure obtained for Ge10has C1symmetry and it can be built from Ge9wedges The electronic state and the Ge–Ge bond length are 1A and 2.728 Å, respectively For Ge Al,
Trang 4the four most stable isomers are listed inFig 1(9-a–9-d), although
from 9-a to 9-d isomers are different in structure, they have same
symmetry (C1) and electronic state (2A) The lowest energy isomer
is in C symmetry, a multirhombus prism with one side capped Ge
3-e
5-a0 4-d
6-a0 5-d
5-c
6-d
7-d
9-a0 8-e
8-d
9-d 9-c
9-b
3-e
5-a0 4-d
6-a0 5-d
5-c
6-d
7-d
9-a0 8-e
8-d
9-d 9-c
9-b Fig 1 Ground-state configurations and low-lying isomers of Ge n Al (n = 1–9) clusters and the lowest-energy structures of pure Ge n+1 (n = 1–9) clusters The first Ge n Al structure is the lowest-energy one for Ge n Al (n = 2–9).
Trang 5atom and the other side capped Al atom, which can be view as one
Ge atom capped on the 8-d cluster Other possible isomers
(8-b–8-d) have energies higher than 8-a by 0.276, 0.361, and 1.043 eV as
shown inTable 2
3.2 Relative stability of different sized GenAl clusters
The understanding of the relative stability of different sized
GenAl (1–9) clusters is important for novel cluster-assembled
optoelectronic materials and can provide a good way to show the
relative local stability of small clusters So the relative stability of
different GenAl clusters can be represented with the average
bind-ing energies (Eb), second-order differences of total energies (D2E)
and fragmentation energies (Ef) Firstly, we consider the
corre-sponding Eb,D2E, and Efof Gen+1(n = 1–9) clusters to provide an
interpretation They are expressed as
Eb½Genþ1 ¼ ððn þ 1ÞE½Ge E½Genþ1Þ=ðn þ 1Þ ð1Þ
D2E½Genþ1 ¼ E½Genþ2 þ E½Gen 2E½Genþ1 ð2Þ
Ef½Genþ1 ¼ E½Gen þ E½Ge E½Genþ1 ð3Þ
where E[Ge], E[Gen], E[Gen+1], and E[Gen+2]denote the total energies
of the lowest energy Ge, Gen, Gen+1, and Gen+2clusters, respectively
For GeAl (n = 1–9) clusters, the average binding energies (E),
second-order differences of total energies (D2E) and fragmentation energies (Ef) can be calculated by following formulas:
Eb½GenAl ¼ ðnE½Ge þ E½Al E½GenAlÞ=ðn þ 1Þ ð4Þ
D2E½GenAl ¼ E½Genþ1Al þ E½Gen1Al 2E½GenAl ð5Þ
EfðGenAlÞ ¼ EðGen1AlÞ þ EðGeÞ EðGenAlÞ ð6Þ
where E[Ge], E[Al], E[Gen1Al], E[GenAl], and E[Gen+1Al], respec-tively, are the total energies of the stable atoms or clusters for Ge,
Al, Gen1Al, GenAl, and Gen+1Al
Based on the above formulas, the calculated results of average binding energies, second-order differences of total energies and fragmentation energies are shown inFigs 2–4 FromFig 2, in gen-eral, it can be seen that binding energies of Gen+1clusters increase with cluster size up to n = 9 and contain one minor bump at n = 6 implying that the cluster for n = 6 is more stable than their neigh-bors, our results are excellent with previous theory[19,33] When
Al is doped on the pure germanium clusters, the averaged binding energy increase smoothly with the size of GenAl clusters increases from 1 to 9, the tendency is almost consistent with the binding energy of Gen+1 cluster Although, the average binding energy increase unceasingly, the average binding energy’s increment speed slows down gradually for n = 1–3, 3–5, 5–7, 7–9 In general, the average binding energy grows gradually as clusters size n and
Table 2
The point group symmetries (PG), spin multiplicity, electronic state, bond lengths and vibrational frequencies, and relative energies of Ge n+1 and Ge n Al (1–9) clusters R av1 and R av2 denote the average bond lengths of Ge–Ge and Ge–Al, respectively; Freq denotes the lowest vibrational frequency of the Ge n+1 and Ge n Al equilibrium geometry.
Trang 6present local maximum value at n = 1, 3, 5 and 7, implying that the clusters are more stable than their neighbors However, as shown
that of the Gen+1cluster.Fig 3shows the second-order difference
in cluster total energies, as a function of the cluster size The
Gen+1 clusters stabilities exhibit pronounced odd–even alterna-tions, but the phenomenon is changed when n = 4, so maxima are found at n = 1, 3, 6, 8, indicating these clusters possess relatively higher stability For GenAl cluster, it is found that the sec-ond-order difference in cluster total energies exhibits odd–even oscillations from 1 to 9 Odd n gives high stability while even n gives low stability, but the stability of Ge6Al, Ge7Al and Ge8Al cluster are inverse A distinct characteristic for the stability of the
GenAl clusters can also be observed, around n = 6, this behavior is exceptional As shown inFig 4, it is clearly found that the doping impurity Al atom makes the thermodynamic stability pattern of the host germanium cluster same from n = 1 to n = 9, except for
n = 6 And the Gen+1cluster with higher fragmentation energy than the GenAl clusters, suggesting Gen+1cluster have higher stability, except for Ge4, Ge5, and Ge7clusters The local maxima of the frag-mentation energy of Gen+1and GenAl clusters appear at 3, 6, 9 and
3, 5, 9 That is to say the Ge4, Ge7, Ge10and Ge3Al, Ge5Al, Ge9Al clusters have higher fragmentation energy, indicating that these clusters are more stable than their neighboring ones
3.3 Homo–Lumo gap
The calculations of molecular orbital were based on the Hückel method proposed by Erich Hückel in 1930, which is a linear com-bination of atomic orbital (LCAO) method If the closed electronic configuration of a cluster has a large highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) gaps, which show the cluster contain high chemical stability As seen fromFig 5, the Ge2and Ge6clusters have a larger HOMO– LUMO gap while the Ge5and Ge8clusters have a smaller HOMO– LUMO gap However, when Al is doped on the pure germanium clusters, the GeAl, Ge4Al and Ge8Al clusters have a larger HOMO– LUMO gap while the Ge5Al, Ge7Al and Ge9Al clusters have a smaller HOMO–LUMO gap As shown inFig 5, for GeAl, Ge4Al and Ge7Al clusters, the HOMO–LUMO gaps of GenAl clusters are usually larger than those of Gen+1 clusters, while for the other clusters, the HOMO–LUMO gaps of GenAl clusters are usually smaller than those
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
Cluster size n
Ge n+1
Ge n Al
Fig 2 Calculated binding energy per atom of germanium clusters and Al-doped Ge
clusters (n = 1–9) plotted as function of number of Ge atoms.
-2
-1
0
1
2
3
Cluster size z
Gen+1 GenAl
Fig 3 Calculated second difference in energies for the germanium clusters and
Al-doped Ge clusters as a function of number of Ge atoms.
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Cluster size n
Gen+1 GenAl
Fig 4 Calculated fragmentation energies for the germanium clusters and Al-doped
Ge clusters as a function of number of Ge atoms.
1.0 1.5 2.0 2.5 3.0 3.5
Cluster size n
Ge n+1
Ge n Al
Fig 5 Calculated HOMO–LUMO gaps for the germanium clusters and Al-doped Ge clusters as a function of number of Ge atoms.
Trang 7of Gen+1clusters FromFig 5, maxima of GenAl cluster is found at
n = 1, indication GeAl cluster possesses the highest stability
3.4 Electronic properties
In cluster science, electron affinity (EA) and ionization potential
(IP) are used as important properties to study the change in
elec-tronic structure of the cluster size Vertical electron affinity (VEA)
is defined as the energy difference between the anionic and neutral
clusters with both at the optimized geometry of the anionic cluster,
while the vertical ionization potential (VIP) is defined as the energy
difference between the cationic and neutral clusters with both at
the optimized geometry of the neutral cluster Chemical hardness
(g) is expressed as g= VIP–VEA based on the basis of a
finite-difference approximation and the Koopmans theorem[43], and is
established as an electronic quantity which may be applied in
characterized the relative stability of molecules and aggregate
through the principle of maximum hardness (PMH) proposed by
Pearson[44] The VIP, VEA andgof the most stable Gen+1and GenAl
(n = 1–9) clusters are calculated and listed in Table 3 The VIP
shows an oscillating behavior from Ge2 to Ge10, except for Ge9,
the EVA shows an increase from Ge2 to Ge6 and an oscillating
behavior from Ge6to Ge10, therefore, the Ge3 has the maximum
hardness while Ge6has the minimum hardness When Al atom is
doped Gen+1 cluster, it clearly sees that doping with Al atom
reduces the vertical ionization potential and chemical hardness
of germanium clusters, but doped Al atom raise the vertical
elec-tron affinity of germanium clusters For GenAl clusters, The VIP
shows a decrease from Ge2Al to Ge9Al, except for Ge8Al, while
VEA increase with n In addition, the chemical hardness decreases
with cluster size
3.5 Magnetisms
Finally, we comment on the magnetic properties of the Gen+1
and GenAl clusters InFig 6, we compare the magnetic moments
of the Gen+1 clusters with computed values for size of the GenAl
clusters It is interesting that the atomic averaged magnetic
moments are clearly different between the Gen+1clusters and the
GenAl clusters, indicating that the magnetic moments for the GenAl
clusters are relatively more larger than the Gen+1clusters, except
Ge6cluster In Gen+1system, it should be mentioned that the most
stable Ge2and Ge6clusters exhibit magnetic moments, but Ge3,
Ge4, Ge5, Ge7, Ge8, Ge9, and Ge10 clusters exhibit nonmagnetic
ground state, it means that 1-a0and 5-a0isomers are the magnetic
structures, the other isomers are nonmagnetic structures
However, when Al atom is doped germanium clusters, the
mag-netic moment changes discontinuously with the cluster size and
the magnetic moments are decreased with increasing cluster size
For the GeAl dimmer, it is 1.50lb, it decreases up to 0.1lbfor n = 9
There is sharply decrease from n = 1 (1.50lb/atom) to n = 2 (0.33lb/atom) in the curve of magnetic moments, staring from
n = 2, the moments gradually decrease as the cluster sizes increase
4 Conclusions
In conclusion, we report a systematic study of the geometric structures, relative stabilities, electronic properties and magnetic properties of Gen+1and GenAl (n = 1–9) clusters using density func-tional theory under the generalized gradient approximation scheme Extensive structures and different possible spin states are carefully investigated In order to show the properties of Al atom doped germanium clusters, we also calculate the properties
of pure Gen+1clusters The results can be summarized as follows:
(1) According to the optimized equilibrium geometries of the
Gen+1 and GenAl clusters, the growth pattern of the Gen+1
and GenAl clusters is investigated Theoretical results indi-cate that the low-lying isomers for the Gen+1and GenAl clus-ters become three dimensional structures when the size
n = 4 On the whole, the adopted lowest energy structures
of the GenAl are similar to lowest energy structures of the
Gen+1clusters
(2) The stability analysis in relation to the calculation of the averaged atomic binding energy, the fragmentation energy, and the second order difference of energy shows that Gen+1
cluster possess relatively higher stability than GenAl cluster The average binding energies of the most stable Gen+1 clus-ters are higher than those of the GenAl clusters According
to the fragmentation energy and the second order difference
of energy analysis, it is concluded that the small Ge7 and
Ge5Al isomers are the most stable geometries for Gen+1and
GenAl clusters, respectively
(3) The HOMO–LUMO gaps are extensively analyzed for Gen+1
and GenAl clusters The obtained results reveal that the GeAl,
Ge4Al and Ge8Al clusters have a larger HOMO–LUMO gap while the Ge5Al, Ge7Al and Ge8Al clusters have a smaller HOMO–LUMO gap The VIP of the Gen+1clusters show an oscillating behavior from Ge2 to Ge10, except for Ge9, the EVA of the Gen+1clusters show an increase from Ge2to Ge6 and an oscillating behavior from Ge6 to Ge10 The VIP of the GenAl clusters shows a decrease from Ge2Al to Ge9Al, except for Ge8Al, while VEA of the GenAl clusters increase with n
Table 3
Vertical ionization potential, vertical electron affinity, chemical hardness of the most
stable Ge n+1 and Ge n Al (n = 1–9) clusters (eV).
n = 1 7.627 1.751 5.876 7.599 1.139 6.460
n = 2 7.967 1.702 6.265 7.384 2.033 5.351
n = 3 7.819 1.860 5.959 7.295 1.898 5.395
n = 4 7.956 2.125 5.831 7.444 2.531 4.913
n = 5 6.888 2.615 4.273 7.270 2.265 5.005
n = 6 7.797 1.764 6.033 7.120 2.204 4.916
n = 7 6.875 1.956 4.919 6.239 2.108 4.131
n = 8 7.131 1.482 5.649 7.224 2.876 4.348
n = 9 7.434 1.644 5.790 7.085 3.014 4.071
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Cluster size n
Ge n+1
Ge n Al
Fig 6 Size dependence of the average atomic magnetic moments of the lowest-energy Ge n+1 and Ge n Al clusters.
Trang 8(4) The investigated magnetic moments of the GenAl cluster
indicate that the atomic averaged magnetic moments
decrease with cluster size increasing Moreover, for the
Gen+1 clusters, the Ge2 and Ge6 clusters exhibit magnetic
moments, but Ge3, Ge4, Ge5, Ge7, Ge8, Ge9, and Ge10clusters
exhibit nonmagnetic ground state
Acknowledgements
This research is supported by Cultivating programme of
excel-lent innovation team of Chengdu university of technology (Grant
No JXTD20130) and Cultivating Programme of Middle-aged
backbone teachers of Chengdu University of Technology We
acknowledge Project supported by the Scientific Research
Foundation of the Education Department of Sichuan Province,
China (Grant No 11ZB036 and 11ZB266)
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