NIM B
Beam Interactions
with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 257 (2007) 438–441
www.elsevier.com/locate/nimb
Fermi edge singularities in ion-induced electron emission
from plane metal surfaces
A. Sindona
a,b,*
, S.A. Rudi
a,b
, S. Maletta a, R.A. Baragiola c, G. Falcone
a,b
, P. Riccardi
a,b,c
a
Dipartimento di Fisica, Università della Calabria, Via P. Bucci 31C, 87036 Rende (CS), Italy
Istituto Nazionale di Fisica Nucleare (INFN), Gruppo Collegato di Cosenza, Via P. Bucci 31C, 87036 Rende (CS), Italy
Laboratory for Atomic and Surface Physics, University of Virginia, Engineering Physics, Charlottesville, VA 22901, United States
b
c
Available online 10 January 2007
Abstract
Auger electron emission – following low energy monoatomic ion impact on metal surfaces – is studied with a many body theory that
includes both the band structure and the Fermi singular response of metal electrons (to the sudden neutralization of the projectiles).
Application is made to the experimental kinetic energy distributions of electrons ejected from polycrystalline Al by Ar+-projectiles at
varying incident energies and angles. Excellent agreement is found between the theory and the measurements.
Ó 2007 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 manyelectron systems
1. Introduction
The emission of electrons from solids, induced by low
energy ion impact, involves a variety of physical processes
such as electronic excitation and ionization in the target
material, the transport of energetic electrons through samples, and electron capture and loss of the projectile during
the passage. It is normally divided into two different mechanisms, in which either the translational kinetic energy or
the internal potential energy of the incident projectile is
transferred to a target electron [1–3]. Potential electron
emission (PEE) is basically understood as the result of
two-electron Auger type processes, i.e. Auger neutralization (AN) and resonant neutralization (RN) followed by
inter-atomic Auger de-excitation (AD). In simple metal tar-
*
Corresponding author. Address: Dipartimento di Fisica, Università
della Calabria, Via P. Bucci 31C, 87036 Rende (CS), Italy. Tel.: +39 0984
496059; fax: +39 0984 496061.
E-mail address: Sindona@fis.unical.it (A. Sindona).
0168-583X/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.nimb.2007.01.024
gets, such as aluminum, the energy released by projectile
neutralization can also produce collective excitations in
the conduction band, such as plasmon-assisted neutralization [4] or many body shake-up [5]. The latter is a finalstate effect that parallels the sudden creation of a core hole
by absorption of a soft X-ray photon [6], in which the sudden change of charge of the projectile leads to a rearrangement of the ground state of conduction electrons on a long
time scale. The signature of this behavior is exponential
tailing of the kinetic energy distributions of ejected electrons responding to a non-Lorentzian broadening mechanism. Indeed, the study of the broadening of the Auger
peaks with the projectile velocity has been developed as a
unique tool to identify the leading physical processes
occurring in AN [1–5].
In this paper, we present a theory of ion-induced shakeup in metals, including the target band structure and specifically examining the Ar+/Al system at varying projectile
kinetic energies and incident angles. We use a semi-empirical one-electron potential to describe either the plane metal
surface, with projected band gap [7], or the impinging ion,
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 257 (2007) 438–441
439
assuming the electron–electron interaction to be of the Yukawa form. We compare our results to previous calculations
– using a jellium surface [5] – and to Hagstrum’s self-convolution model [1]. We discuss the variation of the broadening
parameters with the projectile perpendicular velocity and
show excellent agreement between the theory and the
experiments.
2. Theory
The formalism to calculate the kinetic energy distributions of electrons, ejected from a metal target by interaction with a one-state atom, is based on the reference
Hamiltonian
X y
HðtÞ ¼ ea ðtÞcya ðtÞca ðtÞ þ
ek ck ck
ð1aÞ
þ
X
k
fV
y
ak ðtÞca ðtÞck
þ H:c:g
ð1bÞ
k
þ
X
00
y
y
0
00
fV ak
kk0 ðtÞca ðtÞck ck ck þ H:c:g
ð1cÞ
k;k0 ;k00
þ
X
V akk0 ðtÞcyk ck0 cya ðtÞca ðtÞ;
ð1dÞ
k;k0
where energies are measured relative to the vacuum level
[5]. HðtÞ describes a system of mutually interacting electrons in a finite basis {jki}, with continuous spectrum
{ek}, of width n and Fermi energy eF , and a discrete state
ja(t)i of instantaneous energy ea(t). Its non-interacting part
– Eq. (1a) – contains the sum of the energy operators related with the metal band and the ion level. One-body resonant charge transfers – Eq. (1b) – are modeled by the
hopping potential V ak ðtÞ ¼ haðtÞjvE ðr; tÞjki [8], where
vE ðr; tÞ denotes the interaction of each electron, at position
r, with the surface barrier of the metal band and the effective potential of the impinging ion, together with its image
charge (Fig. 1(a)). Auger transitions – Eq. (1c) – are des00
cribed by the two-particle term V ak
kk0 ðtÞ ¼ hkjhaðtÞjvSC ðjr
00
0
0
0
r jÞjk ijk i, in which vSC ðjr r jÞ denotes the electrostatic
repulsion between two electrons, at positions r and r 0 ,
screened by other metal electrons [1,3]. Finally, Eq. (1d)
acts as an intraband scattering potential, suddenly activated by ion neutralization (yielding cya ðtÞca ðtÞ ¼ 1). It produces significant changes in conducting surfaces, which
reflects in the exponential tailing of ejected electron spectra.
The matrix elements V akk0 ðtÞ ¼ hkjhaðtÞjvSC ðjr r0 jÞjaðtÞijk0 i
can be approximated to a contact potential with the same
structure of the Mahan–Nozierés–De Dominicis (MND)
potential, that causes a Fermi edge singularity in soft Xray absorption from simple metals [6]. Other interactions
yielding one-body intraband scattering, due to the impinging ion and plasmon-assisted neutralization, are absent
from the treatment. Explicit time-dependence of (1), is a
consequence of the classical motion of the projectile, assumed to reflect elastically from a plane at distance Z0 from
the surface, with incident velocity v of parallel component
vk and perpendicular component v?.
Fig. 1. (a) Real space, one-electron potential and charge transfer processes
in the Ar+/Al(1 1 1)-system; (b) energy distribution of electrons excited
from Al(1 1 1) by Ar+-ions at vk ¼ 0:0112 a.u., corresponding to a
projectile energy of 130 eV and an incident angle of 78°. Other parameters
are explained in the text. The result is compared with the calculations of [5]
and the pure spectrum of [1].
The key-quantity of the formalism is N I ðk; vÞ, the distribution of electrons excited by the incident ion in a state jki
of energy ek > eF ; it is proportional to the differential crosssection for a transition from the unperturbed ground state
of the conduction band, with the ion state empty, to all
possible excited states, with the ion state occupied. Firstorder perturbation theory, with the correction of the average lifetimes of the initial and final states, as well as the
many atom response, giving rise to electron–phonon coupling [9], yields the convolution product
N I ðk; vÞ ¼
Z
1
1
deN 0I ðek e; k; vÞBðe; v? Þ;
ð2Þ
in which
N 0I ðek ; k; vÞ / qek
XZ
0
00
k ;k
1
1
00
2
i½ek þea ð0Þek0 ek00 qvk t
dtjV ak
kk0 ðtÞj e
ð3Þ
440
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 257 (2007) 438–441
is the rate for producing excited electrons, in the Fermi
golden rule approximation, and
Z 1
2 2
dteðiek C0 ÞtrPH t =2 F a ðt; v? Þ
Bðek ; v? Þ ¼
1
Rt
i
ds½e ðsÞea ð0Þ
ð4Þ
P a ðt; v? Þe 0 a
denotes a broadening function accounting for many body
effects that are outside the Fermi golden rule and related
to all the interactions in Eq. (1). In Eq. (3), qek labels the
density of states available to excited electrons,
q = k k 0 k00 is the momentum exchanged in a single
excitation process, and eiqvk t accounts for the shift of the
Fermi surface in the rest frame moving with the parallel
velocityR of the projectile. In Eq. (4), P a ðt; v? Þ ¼
t
expf 1 ds½wa ðsÞ þ 2Da ðsÞg defines the ion-survival
probability due to both Auger and resonant neutralization,
including either the AN transition rate wa(t) [1,3] or the virtual width of the atomic level Da(t) via hopping processes
[8]. C0, the lifetime width of the band holes created by
AN, and rPH , the (Gaussian) broadening due to the electron–phonon interaction at room temperature, describe
velocity-independent effects that are outside the theory. Finally, F a ðv? ; tÞ ½1 þ ie0 ðv? Þtaðv? Þ is a measure of nonorthogonality between the initial and the final states of
electrons in the metal that do not participate to AN. Its
Fourier transform defines the distribution of shake-up electrons of width e0(v?), corresponding to the energy range
where the matrix elements V akk0 ðtÞ are non-vanishing, in
the contact potential approximation. A fundamental
parameter is the singularity index a(v?), related to the
screening of band electrons to the sudden perturbation
activated by the neutralizing ion.
From N I ðk; vÞ, the distribution of electrons which escape
from the metal with a kinetic energy E = ek can be written
as
Z
N ðE; vÞ ¼ d2 Xk T ðE; Xk ÞN I ðE; Xk ; vÞ;
ð5Þ
where T(E, Xk) denotes the surface transmission function,
or the probability that an excited electron escapes the metal
barrier. The calculation of N I ðk; vÞ depends either on the
model used to describe the electron–electron interaction,
vSC ðrÞ, in real space or on the choice of the basis, {jki},
for the metal band. In the simplest approximation, the
time-dependence of the matrix elements of the Auger potential – Eq. (1c) – is modeled by an exponential function
00
independent on electron momenta, i.e. V ak
kk0 ðtÞ ¼
ak00
ka v? jtj=2
V kk0 ð0Þe
, and the effect of the projectile parallel
velocity is neglected. This gives rise to Hagstrum’s self-convolution model [1], where N 0I ðek ; k; vÞ results from the convolution of a pure spectrum and a Lorentzian of
broadening kav? scaling linearly with the ion perpendicular
velocity. In [5], a jellium model was assumed for the metal
band and the electron–electron interaction was taken to be
of the Yukawa form vSC ðrÞ ¼ elr =r. The electron escape
probability was approximated to the spherical expression
T(E) = E/(E + n) and the spectrum of ejected electrons,
to a state of energy E, was computed
from N ðE; vÞ R
I ðE; vÞ ¼ d2 Xk N I ðE; Xk ; vÞ is the
I ðE; vÞ, where N
T ðEÞN
distribution of electrons excited to the same state. In the
present derivation, we explicitly consider the band structure of the surface, using the semi-empirical Chulkov’s potential [7] for calculating the electron wave-functions of the
target material. Accordingly, the transmission of electrons
is non-isotropic and T(E, Xk) can be obtained numerically
with the method reported in [10].
3. Application
We consider experiments of electron emission from
polycrystalline Al surfaces by impact of Ar+ ions, at different incident projectile energies and incident angles, that
have been reported in [4,5]. The target material is specified
by the surface barrier of [7] with the parameters of Al(1 1 1),
as sketched in Fig. 1(a). The image plane takes the value
zIM ¼ 3:49 a.u. and the surface work function is /
= 4.24 eV. The band spectrum, of width n = 15.70 eV,
has a narrow projected band gap in the range e1 < ek < e2,
where e1 = 8.89 eV and e2 = 8.64 eV. The atomic electron wave-function is calculated from a state dependent
pseudopotential, vA ½r RðtÞ, specific for the valence 3p
orbital of Ar with unperturbed energy e1
a ¼ 15:76 eV.
The nuclear image potential, DvA ½r; RðtÞ, is constructed
by the prescriptions of [5]. Fig. 1(a) shows a one-electron
picture of the process, with the potential, vE ðr; RÞ, used
to determine the unperturbed energy – Eq. (1a) – and the
hopping interaction – Eq. (1b) – in the ion/metal system.
We use a Yukawa potential, for the electron–electron interaction with the inverse screening length l ¼ 0:1kF , where
kF labels the Fermi wave-vector. Indeed, the external distribution N(E, v) changes negligibly with variations of l in the
00
range (0.05–0.5)kF. The matrix elements V ak
kk0 ðtÞ are worked
out by Fourier transformation to the coordinates parallel
to the surface and numerically integrating over the coordinates perpendicular to the surface [3]. The turning distance
of the projectile, Z0, is fixed to 4 Å, in agreement with
previous studies [5]. This allows to calculate N 0I ðek ; k; vÞ
using Monte–Carlo integration for multiple wave-vector
integrals over k 0 and k00 in Eq. (3).
We introduce the distribution of electrons excited to a
state jki (of
R energy ek) by an (unneutralized) ion, viz.,
0 ðek ; vÞ ¼ d2 Xk N 0 ðek ; k; vÞ. This quantity allows to visuN
I
I
alize the differences between the pseudopotential calculations of this paper (I) and the jellium description (II [5])
of Al. Fig. 1(b) shows the results of the two models, in comparison with the pure spectrum, analytically derived from
the self-convolution model (III) by neglecting the momentum dependence of the static matrix elements of the Auger
00
a
potential: V ak
kk0 ð0Þ ¼ V 0 . The distribution II is more broadened to higher energy than the distribution I, because of the
different decaying rates of metal wave-functions outside the
solid in the two descriptions. This reflects in the tails of the
distributions of ejected electrons, calculated form Eqs. (2)
A. Sindona et al. / Nucl. Instr. and Meth. in Phys. Res. B 257 (2007) 438–441
441
for a and n increase with increasing the projectile perpendicular velocity, which clearly emphasizes the non-negligible effect of shake-up electrons in AN. As a further
evidence, we report in Fig. 2(b) the theoretical distributions
calculated for an incidence energy of 1 keV, and incidence
angles of 70° and 0°, in comparison with the experiments of
[4]. The latter show a low energy peak due to kinetic electron emission that is fitted out with a Gaussian function.
Even if the results are more qualitative, owing to the
absence of a reliable model to reproduce the raising front
of the signals, we continue to observe an increase of the
parameters of the Fa(t,v?). More importantly, the tail of
the distributions clearly show a non-Lorentzian behavior.
Gaussian effects are more relevant due to the decreased resolution of the analyzer.
4. Conclusion
We have reported on electron energy distributions
ejected from plane metal surfaces, of well defined Miller
indices, by low energy ion impact. Comparison with experiments has confirmed the role of many body shake-up of
metal electrons due to the abrupt change of the surface
potential caused by electron capture by the incident ion
outside the surface. The efficiency of the singular response
of the metal band increases with increasing the particle perpendicular velocity, as a consequence of the increase in the
neutralization rates of AN. The effect is manifested in the
high-energy tail of the electron energy distributions, where
the theory accounts very well with measurements.
Fig. 2. Kinetic energy distributions of electrons ejected from Al by: (a)
130 and 430 eV Ar+-ions, with an incident angle of 78°, relative to the
surface normal, (b) 1 keV Ar+-ions, with an incident angle of 70° and 0°,
relative to the surface normal. Comparison is made with the theoretical
distributions obtained from Eqs. (2)–(5), with the calculations of [1,5].
and (5), using a fast Fourier transform algorithm. Fig. 2(a)
shows N(E, v), obtained from the models I–III, in comparison with the experimental distributions acquired from Al
by 130–430 eV Ar+ ions [5]. The only adjustable parameters used in the fitting are related to the broadening function of shake-up electrons – Eq. (4) – since for rPH we
have used the same value of X-ray studies on Al at room
temperature, i.e. rPH 0:1 eV [9], while C0 is fixed to
0.01 eV [1]. As for the model III, the average rate of electron decay outside the solid is assumed to be ka 1 a.u.
We observe that in each model both the optimized values
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