Ionization potential of aluminum clusters
J. Akola, H. Ha
¨
kkinen,* and M. Manninen
Department of Physics, University of Jyva ¨skyla ¨, P.O. Box 35, FIN-40351 Jyva ¨skyla ¨, Finland
Received 2 February 1998
Structure, electronic structure, and ionization potential of aluminum clusters of 2–23 atoms are studied with
a total energy method based on the density-functional theory. The calculated adiabatic ionization potentials
agree remarkably well with the data from threshold photoionization measurements. The analysis of results
gives insight into hybridization effects in the smallest clusters as well as reveals certain clusters that exhibit a
clear jellium-type shell structure. An explanation of the experimental results in the size region of 12–23 atoms
is given in terms of coexisting, competing icosahedral, decahedral, and fcc-based clusters.
S0163-18299800228-8
Measurements of the ionization potential IP of small
metal clusters, i.e., the energy needed for removal of an elec-
tron from the cluster, yield valuable information on the elec-
tronic structure.
1
In the crudest level, the observed trends can
often be understood by considering a simple model of the
cluster as a conducting sphere, in which there are two con-
tributions to the ionization potential: the binding energy of
electron to the sphere analogous to the work function W of
a metal surface and the electrostatic contribution due to the
charging energy of the cluster ion. In fact, the size-evolution
apart from the well-known shell effects
1,2
of measured ion-
ization potentials of simple metal clusters seem to follow
nicely an average trend given by V
i
( R ) =W+ e
2
/ R , where
R is the cluster radius and =0.5 comes from the classical
charging energy of a sphere of radius R . Similarly, the elec-
tron affinity, the energy gained by attaching an electron to
the cluster, is seen to follow a trend A
e
( R ) =W- e
2
/ R ,
with = in the classical consideration. The model fulfills
the obvious limit V
i
=A
e
→W as R → . With quantum cor-
rections to the parameters and the measured difference
V
i
-A
e
is reproduced reasonably well.
3
Small aluminum clusters seem however to behave in a
way that is not consistent with the above model of a metallic
sphere.
4,5
There is an initial rise of IP up to N =4, i.e., N
e
=12, N
e
being the number of valence electrons. This is not
explainable by the jellium model. Furthermore, strong devia-
tions from the e
2
/ R behavior are seen up to N 20, and
even beyond. The probable explanation for the behavior of
IP for small N is an incomplete s - p hybridization, whence
for larger clusters there certainly are strong shell effects aris-
ing from electronic or atomic structure.
6
In this context it is
interesting to note that the mass spectra in the relatively
small size range a few hundred atoms can be explained by
octahedral growth pattern, indicating that already in this size
range the cluster prefers the bulk fcc symmetry.
7,8
In this work we have studied systematically the ionization
potential of small Al clusters in the size range of 2–23 atoms
by an ab initio total energy method.
9
By analyzing the degree
of s - p hybridization for the smallest clusters and the com-
patibility of the jellium-type shell structure for the larger
ones this work is complementary to the previous semiempir-
ical or ab initio calculations.
10–13
Particularly regarding the
ionization potential this calculation is the most systematic
one reported to date, to our knowledge. As a by-product the
calculations also yield information on the geometry of the
ground states of neutral and charged clusters, and some of
the isomers of the neutral ones. Their effect on the measured
IP will be discussed. Our results suggest an aspect that the
oscillation of the measured threshold ionization in the size
range of 12–23 atoms
4
is due to competition and coexistence
of icosahedral, decahedral, and fcc-based structures.
The calculations are done using the BO-LSD-MD Born-
Oppenheimer local-spin-density molecular dynamics
method devised by Barnett and Landman, fully documented
in Ref. 9. In the BO-LSD-MD method one solves for the
Kohn-Sham KS one-electron equations using a suitable
parametrization for the local spin-density approximation to
calculate the exchange-correlation part for the valence elec-
trons of the system corresponding to a given nuclear configu-
ration of the classical ions. From the converged solution the
Hellmann-Feynmann forces on ions can be calculated, which
together with the classical Coulomb repulsion between the
positive ion cores determine the total forces on ions, accord-
ing to which one can perform structural optimizations or
classical molecular dynamics for the ions. The current imple-
mentation uses plane waves combined with fast Fourier
transform techniques as the basis for the one-electron wave
functions and norm-conserving, nonlocal, separable
14
pseudopotentials by Troullier and Martins
15,16
to describe the
valence-electron–ion interaction, and the LSD parametriza-
tion by Vosko, Wilk, and Nusair.
17
Here we wish to stress
that the method does not apply any of the standard supercell
techniques in calculating the total energy of a finite system.
This is an important aspect pertinent to this study: the ion-
ization potential is a straightforward difference in the total
energies of the neutral and charged cluster, without or with
the relaxation of the charged cluster to evaluate the vertical
or adiabatic IP, respectively.
We give the ground-state structures and ionization poten-
tials for Al–Al
7
in Table I. Table II shows IP’s for
Al
12
– Al
23
. Both tables show the experimental data by
Schriver et al.,
4
and our adiabatic values are compared to the
experimental threshold ionization data as well as to some
earlier calculations in Fig. 1. For each cluster both neutral
and ion we found the ground state to have the minimum
total spin i.e., S =0 and S =1/2 for a cluster with even and
PHYSICAL REVIEW B 15 AUGUST 1998-I VOLUME 58, NUMBER 7
PRB 58 0163-1829/98/587/36014/$15.00 3601 © 1998 The American Physical Society