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Work functions, ionization potentials, and in-between: Scaling relations based on the image charge model

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a r X i v : p h y s i c s / 0 2 0 7 1 1 6 v 2 [ p h y s i c s . a t m - c l u s ] 1 9 N o v 2 0 0 2
Work functions, ionization potentials, and in-between:Scaling relations based on the image charge model
Kin Wong, Sascha Vongehr, and Vitaly V. Kresin
Department of Physics and Astronomy,University of Southern California, Los Angeles, CA 90089-0484, USA
(Dated: February 2, 2008)
Abstract
We revisit a model in which the ionization energy of a metal particle is associated with the workdone by the image charge force in moving the electron from inﬁnity to a small cut-oﬀ distance justoutside the surface. We show that this model can be compactly, and productively, employed tostudy the size dependence of electron removal energies over the range encompassing bulk surfaces,ﬁnite clusters, and individual atoms. It accounts in a straightforward manner for the empiri-cally known correlation between the atomic ionization potential (IP) and the metal work function(WF), IP/WF
∼
2. We formulate simple expressions for the model parameters, requiring only asingle property (the atomic polarizability or the nearest neighbor distance) as input. Without anyadditional adjustable parameters, the model yields both the IP and the WF within
∼
10% for allmetallic elements, simulates the concentration dependence of the WF of regular binary bulk alloys,and matches the size evolution of the IP of ﬁnite metal clusters for a large fraction of the experi-mental data. The parametrization takes advantage of a remarkably constant numerical correlationbetween the nearest-neighbor distance in a crystal, the cube root of the atomic polarizability, andthe image force cutoﬀ length. The paper also includes an analytical derivation of the relation of the outer radius of a cluster of close-packed spheres to its geometric structure.
PACS numbers: 79.60.Jv, 78.67.-n, 61.46.+w
1
I. INTRODUCTION
While the good agreement between theoretical and experimental atomic ionization po-tentials (IP) is a major triumph for quantum mechanics, it is prohibitively more diﬃcultto rigourously solve the polyatomic quantum problem, not to mention extrapolation to aninﬁnite bulk metal. The IP for an atom is a well understood quantity. The same cannot,however, be said about the work function (WF) for a metal or even for a ﬁnite cluster. Onthe other hand, it is an experimentally realistic task to produce clusters ranging in size fromtwo atoms up to tens of thousands of atoms and to measure the size dependence of the elec-tron removal energy. The clusters can be made big enough that they can be considered closeto bulk metals, hence the evolution from the atomic IP to the metal WF can be mappedout. Experiments over such a wide range have been performed, e.g., for sodium
1
. For eachelement in the periodic table one would therefore expect that there exists a function whichcan predict the electron removal energy for a particle of arbitrary size.The exact derivation of such a scaling law is a daunting task. However, the availableexperimental data on the IP, the WF, and on clusters reveal some characteristic features.For example, it has been noted a long time ago that the IP and the WF of metallic elementsare approximately correlated to each other via the factor
2,3
I/W
≈
2
.
(1)To give another example, for many metal clusters of intermediate sizes the electron removalenergy has been found to scale as
φ
≈
W
+
γ e
2
R,
(2)where
R
is the radius of the cluster and
γ
is a constant factor
4,5
. Henceforth,
I
will denotethe atomic ﬁrst ionization potential,
W
will denote the polycrystalline bulk surface workfunction, and for a ﬁnite metal particle the term ”electron removal energy” will be employed(denoted by
φ
).A signiﬁcant number of ﬁrst-principles calculations have been performed for work func-tions of metallic systems
6,7,8
. In addition, simple models based on semi-empirical approachescombined with classical electrostatics have been rather successful at reproducing WF trendsand values
9
. This suggests that some features of the desired scaling law may be found byemploying such a model. In this paper we demonstrate that by combining the image-charge2
potential function for a ﬁnite particle with just a single material-dependent scale parameter(the atomic polarizability
α
or the crystalline nearest-neighbor distance
r
nn
) one can obtainan interpolation formula covering the full size range from the atom through the cluster tothe bulk. This formula estimates both the IP and the WF within
∼
10% for all metallicelements in the periodic table, yields values in reasonable agreement with experiment formany intermediate sized clusters, and provides a natural justiﬁcation for the aforementionedIP/WF
≈
2 ratio.The plan of the paper is as follows. In Sec. II we consider the image-charge expressionwhich describes the removal of an electron from an isolated sphere. By focusing on the limitsof a sphere of atomic radius and one of inﬁnite radius, we show that the relation (1) follows asnaturally from the image-charge consideration as does Eq. (2). In Sec. III we demonstrate
some striking parallels between the variation of atomic sizes and image-force cutoﬀ distancesacross the periodic table and use these observations to formulate compact expressions forestimating both WF and IP. An extension to the case of binary alloys is discussed in Sec.IV. Finally, we show that by using an appropriately interpolated expression for the clusterradius a good description of the size evolution of cluster ionization potentials can be obtained(Sec. V). Some rigorous formulas on close-packed cluster radii are derived in the Appendix.
II. THE IMAGE POTENTIAL MODEL
It is stated in the literature that the earliest attempt to explain the work function of ametallic surface using classical electrostatics was due to Debye
10
. He proposed that it isequal to the energy required to pull an electron out to inﬁnity against its image charge.Since the image force diverges at the surface, Schottky
11
suggested that one may be able todeﬁne a microscopic cutoﬀ distance
d
at which the image force starts to act. Despite thesimplicity of this model
12
, there have been numerous attempts to estimate the parameter
d
in order to ﬁt the experimental values of the WF
2,3,9,13,14,15
. Although the particular choicesof
d
were supported only by plausibility arguments, they were frequently able to oﬀer rathernice agreement with the experimental data.In a similar spirit, let us now consider the image-force expression for the energy requiredto remove an electron from an isolated ﬁnite metal particle, modelled as a conducting sphereof radius
R
. The particle is assumed initially neutral, i.e., after the removal of the electron it3
acquires an unit positive charge. A calculation of the work required to move the electron froma distance
d
outside the metal surface to inﬁnity against its image charge is a straightforwardexercise
16
:
φ
(
R
) =
e
2
4
d
1 + 4(
d/R
) + 6(
d/R
)
2
+ 2(
d/R
)
3
(1 +
d/
2
R
)(1 +
d/R
)
2
.
(3)The cutoﬀ parameter
d
is assumed to be a material-dependent constant. The ﬁrst factor onthe right-hand side represents the bulk (
R
→∞
) work function:
W
=
e
2
/
(4
d
). This formulahas been applied to yttrium and lanthanide clusters in Ref. [17]. However, the authors didnot extend it to the bulk or the atomic limit; they used the equation as an extrapolationformula for small clusters with the WF as a given boundary condition.It is convenient to rewrite the above equation in the following form:
φ
(
R
)
W
=
η
(
d/R
)
,
(4)where the dimensionless scaling function
η
(
d/R
) is deﬁned by the second factor in Eq. (3)and plotted in Fig. 1. If Eq. (4) were applied all the way down to the atomic limit, it would
give an estimate of the ratio IP/WF as the value of
η
(
d/R
at
), where
R
at
is a quantity charac-terizing the atomic size. On the other hand, we know from numerous investigations
2,3,9,13,14,15
that the cutoﬀ parameter
d
is, sensibly enough, also of the same order of magnitude. In theatomic limit, therefore, the ratio
d/R
at
should be on the order of unity. In other words, if one assumes that
d
∼
R
at
, then, independent of the exact expression for either parameter,the scaling function predicts that
I/W
∼
η
(1)
≈
2
.
(5)This is a new explanation of the well known empirical result mentioned in the Introduction,Eq. (1). In the next section we suggest some speciﬁc parametrizations of the variables
d
and
R
at
and show that these can give an even more accurate value of the IP/WF ratio.In the large
R
limit, the scaling function can be expanded to ﬁrst order in
d/R
. Theresult is:
φ
(
R
) =
W
+ 38
e
2
R
+
O
(
dR
)
2
.
(6)This is the well known ﬁnite size correction for the ionization potential of metallicclusters
4,5,18,19,20
. This scaling law has been experimentally veriﬁed for many metalclusters
4,5,20,21
. Although there is still some controversy whether the 3/8 factor is suﬃcientlyrigorous
22,23,24,25,26
, Eq. (6) does ﬁt the experimental data relatively well.4
It appears, therefore, that Eq. (3) oﬀers a consistent estimate for the scaling of electronremoval energy from the atomic IP to ﬁnite particle sizes to the bulk WF. It is interestingto ask whether some simple parametrizations for the image force cutoﬀ distance and atomicand cluster radii may be proposed so as to enable more quantitative applications of Eq. (3)to experimental data. This is the subject of the sections that follow.
III. LENGTH SCALE PARAMETERS
For guidance with length scaling, let us begin by plotting the ”experimental” image forcecutoﬀ distances as deﬁned by equating
e
2
/
(4
d
) to the experimentally measured bulk surfacepolycrystalline work functions
27
. These values are shown in Fig. 2(a). The same ﬁguredisplays two quantities reﬂecting the size of individual atoms: the cube root of the atomicpolarizability
27
(
r
α
) and one-half of the nearest neighbor distance in a crystal
28
(
r
nn
/
2).Although
d
shows strong correlation with both the polarizability radius and the nearestneighbor distance, it’s consistently lower than either one. However, by forming the inversesum of the two, a quantity emerges which follows the experimental values of
d
beautifully.This parametrization has the form1
/d
≈
1
/r
α
+ 2
/r
nn
(7)and agrees with the empirical value of the cutoﬀ parameter to within
∼
10% for mostmetallic elements, see Fig. 2(b). Put another way, the WF for most metallic elements canbe estimated to better than 10% by using this parametrization of
d
.The coexistence of an atomic part and a bulk part in a work function model resemblesqualitatively the elegant argument in Ref. [29] in which the WF value is calculated as arisingfrom the IP of a neutral atom reduced by the work done by the image force in bringing theresulting ion back to the crystal surface.As a matter of fact, Fig. 2(a) reveals that all the size parameters turn out to haveessentially the same trend across the periodic table, diﬀering only in overall magnitude.We are not aware of a quantitative theory explaining this observation for the crystallinestate, but, roughly speaking, one does expect bigger atoms to have higher polarizabilities,larger lattice spacings in crystal form, and lower IP together with lower WF [via Eqs.(1,5)] and therefore greater
d
values. In fact, remarkably linear correlations between the5

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