Nuclear Instruments and Methods in Physics Research B 230 (2005) 298–304
www.elsevier.com/locate/nimb
Broadening effects in Auger neutralization
of 130–430 eV Ar+ ions at Al surfaces
A. Sindona a,b,*, R.A. Baragiola c, S. Maletta a,
G. Falcone a,b, A. Oliva a,b, P. Riccardi a,b,c
a
Dipartimento di Fisica, Università della Calabria, Via P. Bucci 31C, 87036 Rende (CS), Italy
Unità dell’istituto Nazionale di Fisica della materia (INFM) di Cosenza, Via P. Bucci 30C, 87036 Rende (CS), Italy
Laboratory for Atomic and Surface Physics, University of Virginia, Engineering Physics, Charlottesville, VA 22901, USA
b
c
Available online 26 January 2005
Abstract
We present a model for electron emission from Al surfaces by Auger neutralization of 130–430 eV Ar+ ions, that
includes the singular response of the metal conduction band to the abrupt change of the surface potential caused by
electron capture by the incident ion. This effect, previously identified in X-ray studies, produces a broadening that plays
a significant role in reproducing the higher energy part of the experimental electron kinetic energy distributions.
Ó 2004 Elsevier B.V. All rights reserved.
PACS: 79.20.Rf; 34.50.Dy; 34.70.+e; 71.10.w
Keywords: Ion beam impact and interactions with surfaces; Electron emission and Auger neutralization; Charge transfer; Theories and
models of many-electron systems
1. Introduction
Auger neutralization (AN) of slow ions colliding with metal surfaces is a powerful tool to obtain
information on the electronic structure of the
surface and on the evolution of ionic levels in the
ion–surface interaction. The foundations of both
experimental and theoretical treatments of Auger
*
Corresponding author. Tel.: +39 (0)984 496059; fax: +39
(0)984 496061.
E-mail address: sindona@fis.unical.it (A. Sindona).
processes were established in the 1950s by Hagstrum [1]. Since then, many works of increasing
sophistication have explored the characteristics of
the kinetic energy distributions of the electrons
collected during the collision and the charge state
of the particles reflected from the solid [2–7].
In particular, the study of the experimentally
observed broadenings of the kinetic energy distributions of ejected electrons was used to explain
the basic physical effects taking part in AN [1]. It
was early recognized that the dominant broadening component has a nearly Lorentzian structure,
0168-583X/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.nimb.2004.12.058
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 298–304
which gives rise to a common intersection point
(‘‘magic’’ energy) for spectra acquired at different
ion kinetic energies [3]. The Lorentzian broadening
results from: (i) the asymptotic behavior of the matrix elements of the AN potential, producing a
width scaling linearly with v?, the perpendicular
component of the projectile velocity to the surface;
(ii) the average lifetime of the final state of the metal conduction band, involving two holes below
the Fermi energy. Other sources of broadening
were identified in: (iii) the lifetime of the initial
state arising via the ion survival probability; (iv)
the effect of variation of the energy level of the projectile near the surface of the target; (v) the shift of
the Fermi surface in the rest frame of the projectile; (vi) the effect of the electron–phonon interaction at room temperature [8].
In this paper, we present a theoretical study of
AN that includes the singular response of the conduction band to the sudden change of charge
intrinsic in the neutralization process. This effect,
similar to the Fermi edge singularity predicted by
the Mahan, Nozieres and De Dominicis (MND)
theory of X-ray absorption in metals [9], yields
an additional energy broadening to the kinetic energy distributions of ejected electrons (Section 2).
We apply the model to electron emission from Al
surfaces by 130–430 eV Ar+ ions, showing that,
although the basic theoretical treatment of AN
[1–5] gives a reasonable agreement with the experiment, previously identified broadening components – such as (i)–(vi) – do not adequately
reproduce the high energy behavior of the electron
energy spectra (Section 3).
2. Theory
The interaction between electrons in a Fermi
sea leads to various anomalies in the properties
of the system which involve the energy spectrum
near the Fermi energy. In X-ray absorption, a
transverse photon penetrates deeply into solids,
exciting a core electron and switching on a core
hole. When a slow singly charged ion approaches
a metal surface, AN may occur as a two electron
process in which a band electron makes a transition to the unoccupied ion ground state, emitting
299
a longitudinal photon [2]. The photon is screened
rapidly by the solid, so it propagates towards to
the surface and excites another metal electron. In
quantum electro-dynamics, transverse photons
are described by a potential vector field solving
the vacuum Maxwell equations – under appropriate gauging conditions – while longitudinal
photons are described by the scalar field
u(r1) = òd3r2vsc(r12) W (r2)W(r2), where W is the
electron field operator in real space, W W the electron density operator and vsc a screened Coulomb
potential of two electrons in the many electron system. Transverse photons couple to the electron
current density operator, whereas longitudinal
photons coupleR to the electron density operator
via HI ¼ 1=2 d3 r1 Wy ðr1 Þuðr1 ÞWðr1 Þ. A convenient expansion for W is
X
WðrÞ ¼
hrjkick þ hrjaðtÞica ðtÞ;
ð1Þ
k
where: {hrjki} is the orthonormal set of the wavefunctions of the conduction band, of spectrum
{ek}, Fermi energy eF and width n and hrja(t)i is
the ground state of the ion, corresponding to the
ionization energy ea(t). Explicit time-dependence
is a consequence of the classical motion of the
projectile that follows a straight-line trajectory of
incident velocity v, perpendicular component
Z(t) = v?jtj + Z0 and turning point Z0, relative to
the surface. Atomic and metal states are orthonormalized as the ion approaches the vicinity of the
surface [4], so that electron annihilations operators
{ck, ca(t)} satisfy the ordinary algebraic rules of
fermion operators.
The representation of HI into the orthonormalized set yields two types of interaction: HAU ,
denoting the usual Auger potential [1,3,5] of ma00
00
0
trix elements V ak
kk 0 ðtÞ ¼ hkjhaðtÞjvsc ðr12 Þjk ijk i and
X
V akk0 ðtÞ cyk ck0 cya ðtÞca ðtÞ;
ð2Þ
HE ðtÞ ¼
k;k 0
in which V akk0 ðtÞ ¼ hkjhaðtÞjvsc ðr12 ÞjaðtÞijk 0 i, that describes the abrupt change of charge due to neutralization of the incident ion and injection of a band
hole. It should be noted that HI includes also the
two body potential between metal electrons, that
for certain projectile–target combinations may
give rise to plasmon assisted neutralization [5,7].
300
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 298–304
However, we shall consider applications in
which these effects have not been experimentally
observed.
Eq. (2) has the same structure of the MND potential in the X-ray problem, where the core-hole
potential operates as a scattering perturbation
upon conduction electrons after the core state
has been switched on. The Fock space is, therefore, partitioned into two subspaces: on the one
hand, many electron states with the ion state
empty are still constructed by anti-symmetrizing
the unperturbed set {hrjki}; on the other hand,
many electron states with the ion state occupied
need to be calculated from the set {hrjj(t)i}, in
which each hrjj(t)i diagonalizes the unperturbed
metal Hamiltonian plus the edge potential
òd3r 0 vsc(jr r 0 j)jha(t)jr 0 ij2. It turns out that the final
states of AN are not orthogonal the initial state.
The key quantity in our study is the distribution
NI (ek, v) of electrons excited above the Fermi level
by the incident ion. NI is proportional to the differential cross-section for a transition from the
unperturbed ground state jii of the conduction
band, with the ion state empty, to all possible excited states jfi, with the ion state occupied, that involve the excited electron, of energy ek > eF and
two band holes. We can use the Fermi goldenÕs
rule approximation, treating HAU as a small perturbation, to write
Z
XZ 1
00
2
2
N I ðek ;vÞ / qek d Xk
dtjV ak
kk 0 ðtÞj
k 0 ;k 00
1
F a ðt;v? ÞP a ðt;v? Þ exp C0 t r2PH t2
2
Z t
0
00
exp i ds ek þea ðsÞek ek q vk ;
0
ð3Þ
where qek is the density of states available to excited
electrons, q = k k 0 k00 is the momentum exchanged in a single excitation process and vk labels
the component of the ion velocity parallel to the
surface. Indeed, Eq. (3) comes from the Fermi
goldenÕs rule with the correction of three ingredients: the lifetime of the initial and final states, the
effect of the electron–phonon interaction and the
change in the final state of the metal due to the sudden switching of the edge potential (2), caused by
Rt
ion neutralization. P a ðt; v? Þ ¼ exp 1 ds ½wa ðsÞ þ 2Da ðsÞÞ is the probability that the ion
ground-state survives neutralization, due to both
Auger and resonant processes. It includes the AN
transition rate wa [1–4] and the virtual width of
the atomic state Da [10] due to hopping processes;
C0, the lifetime width of the band holes created
by AN and rPH, the (Gaussian) broadening due
to the electron–phonon interaction [8] at room
temperature, describe velocity-independent effects
that are outside the theory. Finally, Fa accounts
for the nonorthogonality between the initial and
the final state of electrons in the metal that do
not participate to AN. The distribution of electrons
ejected outside the solid is related to the distribution of excited electrons via the escape probability
or surface transmission function T(E) = E/(E + n)
[6], which applies to the case of normal emission.
Thus, N(E,v) = T(E)NI(E, v) where E = ek eF /
is the electron kinetic energy and / the work function of the surface. In the following, we shall evaluate energies with respect to the vacuum level, so
that E = ek. The broadening behavior of N(E, v)
can be investigated by expressing Eq. (3) as a convolution product of the form
N I ðek ; vÞ ¼ N 0I ðek ; e; vÞ Bðe; v? Þe¼e
k
Z 1
deN 0I ðek ; ek e; vÞBðe; v? Þ:
ð4Þ
1
Here,
Z
N 0I ðek ; e; vÞ / d2 Xk
X 00
0
00
uak
kk 0 ðe þ ea ð0Þ ek ek q vk Þ
ð5Þ
k 0 ;k 00
is the spectrum of excited electrons broadened by
the AN potential
via the ‘‘internal functions’’
R1
00
00
2
iet
uak
dte
j V ak
0 ðeÞ ¼ qe
kk 0 ðtÞj =2p, and
1
kk
k
Z 1
1
Bðe; v? Þ ¼
dt exp ðie C0 Þt r2PH t2 F a ðt; v? Þ
2
1
Z t
P a ðt; v? Þ exp i dsðea ðsÞ ea ð0ÞÞ
0
ð6Þ
denotes a broadening function for the mechanisms
(ii)–(vi) listed in Section 1, plus the broadening due
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 298–304
the Fermi edge singularity. The broadening due to
00
uak
comes from the asymptotic behavior of the
kk 0
matrix elements of the AN potential. Using the
00
ak 00
approximate time dependence V ak
kk 0 ðtÞ ¼ V kk 0 ð0Þ
00
expðka v? jtj=2Þ, uak
becomes a Lorentzian funckk 0
tion of constant width kav?. Neglecting the effect
of the parallel velocity, the model simplifies to
the well-known Hagstrum convolution model [1].
This approach introduces a ‘‘universal’’ function
of the ion perpendicular velocity and the atomic level – convolution of B(e, v?) with a Lorentzian of
width kav? – taking into account the broadening
behavior of the electron spectra with the ion
motion.
(a)
301
(b)
3. Experiments and applications
We measured the kinetic energy distributions of
electrons ejected from polycrystalline Al surfaces,
during the impact of 130–430 eV Ar+ ions. Ar+
ions were produced in an electron bombardment
source operated at low electron energies (30 eV)
to prevent significant contamination of the ion
beam with doubly charged ions. The surface of
the samples were normal to the axis of the spectrometer and at 12° with respect to the ion beam
direction. The spectrometer was operated at a constant pass-energy of 50 eV (and, therefore, an
approximately constant transmission over the
measured electron energy range) and with a resolution of 0.2 eV. The high purity polycrystalline Al
surfaces were sputter cleaned by 4 keV Ar+ ions
at 12° glancing incidence. The spectra were acquired with the sample biased at 2.5 V to separate the contribution of electrons emitted directly
from the sample (which are accordingly shifted
to higher energies) from a spurious peak of low energy electrons, mainly arising from the grounded
entrance grid of the analyzer.
As shown in Fig. 1(a), the low energy peak
structure tails exponentially and can be easily subtracted. After such subtraction, the spectrum is
shifted backward by the bias voltage. The adequacy of this procedure can be seen by comparison
with the spectrum acquired without the bias voltage, shown in Fig. 1(b), normalized to the same
height. The corrected energy distributions of emit-
(c)
Fig. 1. (a) Kinetic energy distribution of electrons ejected from
Al by 230 eV Ar+-ions, with the sample biased at 2.5 V. (b)
Corrected distribution, without the spurious peak of low energy
electrons and spectrum acquired without the bias voltage. The
spectra are normalized to the same height for comparison. (c)
Corrected energy distributions of electrons ejected from Al by
130–430 eV Ar+-ions, obtained as in Fig. 1(a) and (b).
ted electrons are displayed in Fig. 1(c) and show
characteristic features of the AN process [1,3,5]:
constant areas, i.e. total electron emission yields
and a magic energy at 5.3 eV.
302
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 298–304
The conduction band of Aluminum is specified
by the parameters n = 15.95 eV and eF = /
= 4.25 eV, in the Jellium approximation, while
the ionization potential of Argon, at an infinite
distance from the target is e1
a ¼ 15:76 eV. We
use the eigenfunctions of the Jennings potential
as the starting basis set for the metal conduction
band and for continuous sates above the vacuum
level [11]. Resonant transfer processes to the excited states of Argon can be neglected although,
far from the surface, the energies of these levels
are close to the Fermi energy of Aluminum. In
fact, the interaction with the metal shifts ion energies positively by 1–2 eV. Thus, the most efficient
neutralization mechanism of ejection involves only
the ground state of the projectile, calculated by the
pseudopotential of [12].
We approximate the electron–electron potential
to the Thomas–Fermi expression vsc ðr12 Þ ¼ exp ðlr12 =r12 Þ, where l is the inverse screening length
due to other metal electrons. Both the unscreened
Coulomb potential (l = 0) and more advanced
dielectric screening models have been used for vsc
00
a
[3]. The matrix elements V ak
kk 0 and V kk 0 are worked
out by Fourier transforming in the coordinates
parallel to the surface [3] and numerically integrating over the coordinates perpendicular to the surface. Finally the internal distribution N 0I is
calculated using Monte-Carlo techniques for multiple wave-vector integrals in Eq. (5).
As a first check, we test the basic theory of AN,
corresponding to the replacement Fa(t, v?) ! 1 in
Eq. (6). The resulting external distributions
N(E,v) depend on four free parameters, i.e.
Z0, l, C0 and rPH. It should be pointed out that
no fitting procedure is required for their estimation. In fact, Z0 is fixed by the condition that the
pure spectrum coincides with the experimental
electron energy distributions at the magic energy.
Its value depends on the model used to calculate
the atomic level shift ea ðtÞ e1
a , at the time t. With
the approximations introduced in Section 1,
ea ðtÞ e1
a results from the expectation value, in
the state ja(t)i, of the surface potential [11] and
of the image potential of the scattering ion
[12,13]. This calculation allowed us to estimate
Z0 4 Å which corresponds to an energy shift of
1.45 eV, at t = 0. More sophisticated calcula-
tions, including the chemical interaction between
the ion and the metal at short ion-metal distances
[14], may produce larger values for Z0. l has a critical influence on the internal distribution N 0I at
excitation energies ek close to eF. On the other
hand, the line-shapes of both the internal and
external distributions, at emission energies above
2 eV, are unaffected by any choice of l in the
range 0.05kF 6 l 6 1.00kF, kF denoting the Fermi
wave-vector. Finally, Previous studies have attested that C0 is of the order of 0.01 eV [1] and
rPH 0.1 eV [8].
Fig. 2 shows the theoretical spectra – for
l = 0.1kF and C0 = 0.05 eV – broadened by the
experimental energy resolution, in which each
00
internal function uak
has been obtained numerikk 0
00
2
cally from the Fourier transform of jV ak
kk 0 ðtÞj .
For completeness we also report the solution for
a constant width ka = 1.83 a.u., extrapolated from
the asymptotic behavior of the numerical matrix
00
elements V ak
kk 0 . We observe that the approximated
distribution offers a qualitative good agreement
with data since, in this context, the ion velocity is
always small compared to the target Fermi veloc(a)
(b)
Fig. 2. The kinetic energy distribution of electrons ejected from
Al by 130 eV Ar+-ions is compared with theoretical distributions calculated from the basic theory of AN (Fa(t, v?) ! 1 in
Eq. (6)) and from the approach of this paper (Eqs. (3)–(6)).
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 298–304
ity. Consequently, the effective occupation of the
target states in the rest frame of the projectile is
very well described by the Fermi–Dirac distribution and shifted Fermi surface effects are small.
Fig. 2(b) shows the same distributions of Fig.
2(a), in the logarithmic scale. It is observed that
theoretical calculations are in excellent agreement
with the main part of experimental distributions
for electron kinetic energies less than 6 eV. However, the exponential behavior of spectra, at emission energies above the magic energy, is
overestimated.
The MND theory [9,8] predicts an asymmetric
line-shape for photoemitted electrons of the form
ðE eb Þa1 expððE eb Þ=e0 Þ, in which a is a critical exponent depending on the phase-shifts of the
screened core-hole potential, eb the core-state binding energy and e0 a cut-off parameter. Such a
broadening function seems to be the right candidate to cope with the observed high energy behavior of the spectra. We have modeled the
correlation function Fa with the same structure
of the Fourier transform of the MND broadening
function, i.e.
F a ðt; v? Þ ð1 þ ie0 ðv? ÞtÞ
aðv? Þ
:
ð7Þ
Adjustment of the parameters in Eq. (7) shows
that e0 increases with increasing v?, being of the
order of the virtual width of the atomic state, i.e.
e0 0.9–1.5 eV. a is found also to increase in the
range 0.12–0.16, the same order of the estimated
value for the critical exponents of the measured
2s and 2p core lines of Al in X-ray photoelectron
spectroscopy [8]. The modified electron distributions, calculated with the numerical internal func00
tions uak
– using l = 0.1kF and C0 = 0.01
kk 0
eV – offer the correct exponential trend at higher
energies, as reported in Figs. 2(b) and 3. It should
be noted that C0 was slightly overestimated in the
previous analysis to compensate the absence the
MND mechanism.
In summary, we extended the theory of Auger
neutralization to include the effect of many-body
shake-up of the valence electrons of the metal
due to the abrupt change of the surface potential
caused by electron capture by the incident ion outside the surface. The effect is manifested in the
high-energy tail of the electron energy distribu-
(a)
303
(b)
Fig. 3. Kinetic energy distributions of electrons ejected from Al
by 230 and 430 eV Ar+-ions are compared with theoretical
distributions calculated from the basic theory of AN
(Fa(t, v?) ! 1 in Eq. (6)) and from the approach of this paper
(Eqs. (3)–(6)).
tions, where the theory accounts very well with
the measurements. As a corollary, we expect the
effect to be absent in materials with a band-gap.
In such cases, although some form of shake-up will
still occur, the effect on the electron energy distributions will be markedly different because of the
absence of a Fermi-edge singularity.
References
[1] H.D. Hagstrum, Phys. Rev. 96 (1954) 325, p. 336;
Inelastic Ion–Surface Collisions, Academic Press, NY,
1977, p. 1;
H.D. Hagstrum, Y. Takeishi, D.D. Pretzer, Phys. Rev. 139
(1965) A526.
[2] F.M. Propst, Phys. Rev. 129 (1963) 7;
V. Heine, Phys. Rev. 151 (1966) 561;
J.A. Appelbaum, D.R. Hamann, Phys. Rev. B 12 (1975)
5590;
L.A. Salmi, Phys. Rev. B 46 (1992) 4180.
[3] R. Monreal, S.P. Apell, Nucl. Instr. and Meth. B 83 (1993)
459;
R. Monreal, N. Lorente, Phys. Rev. B 52 (1995) 4760;
R. Monreal, Surf. Sci. 388 (1997) 231.
304
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 230 (2005) 298–304
[4] B.L. Burrows, A.T. Amos, Z.L. Miskovic, S.G. Davidson,
Phys. Rev. B 51 (1995) 1409.
[5] M.A. Vicente Alvarez, V.H. Ponce, E.C. Goldberg, Phys.
Rev. B 57 (1998) 14919.
[6] R.A. Baragiola, Low Energy Ion–Surface Interaction,
Wiley, New York, 1994, 187.
[7] P. Riccardi, P. Barone, A. Bonanno, A. Oliva, R.A.
Baragiola, Phys. Rev. Lett. 84 (2000) 378.
[8] P.H. Citrin, G.K. Wertheim, Y. Baer, Phys. Rev. Lett. 35
(1975) 885;
Phys. Rev. B 16 (1977) 4256.
[9] G.D. Mahan, Phys. Rev. 163 (1967) 612;
P. Nozieres, C.T. De Dominicis, Phys. Rev. 178 (1969)
1097.
[10] P. Kürpic, U. Thumm, Phys. Rev. A 54 (1996) 1487.
[11] P.J. Jennings, R.O. Jones, M. Weinert, Phys. Rev. B 37
(1998) 6113.
[12] A.E.S. Green, D.L. Sellin, A.S. Zachor, Phys. Rev. 184
(1969) 1.
[13] A. Sindona, G. Falcone, Surf. Sci. 529 (2003) 471.
[14] W. More, J. Merino, R. Monreal, P. Pou, F. Flores, Phys.
Rev. B 58 (1998) 7385.