PHYSICAL REVIEW A 71, 052903 共2005兲
Many-body shake-up in Auger neutralization of slow Ar+ ions at Al surfaces
1
A. Sindona,1 R. A. Baragiola,2 G. Falcone,1 A. Oliva,1 and P. Riccardi1,2,*
Dipartimento di Fisica, Università della Calabria and INFN, Gruppo Collegato di Cosenza, Via P. Bucci, Cubo 30C, 87036 Rende (CS),
Italy
2
Engineering Physics, University of Virginia, Thornton Hall, Charlottesville, Virginia 22901, USA
共Received 2 July 2004; published 31 May 2005兲
Electron emission by 130–430-eV Ar+ ion impact on polycrystalline Al surfaces is studied extending the
theory of Auger neutralization to include the singular response of the metal conduction band to the sudden
change of charge in the incoming hole state, following its neutralization. The effect is manifested in the
high-energy tail of the electron energy distributions, where the theory accounts very well with experiments.
DOI: 10.1103/PhysRevA.71.052903
PACS number共s兲: 79.20.Rf, 34.50.Dy, 34.70.⫹e, 71.10.⫺w
I. INTRODUCTION
Neutralization of ions at the surface of a solid converts the
potential energy carried by incoming projectiles into electron
excitation and emission. Since the theoretical foundations
laid by Hagstrum 关1兴, potential electron emission 共PEE兲 has
been long discussed in terms of two-electron, Auger-type
processes, such as Auger neutralization 共AN兲 and resonant
neutralization followed by interatomic Auger de-excitation
共RN+ AD兲 关1–4兴. In AN, the electrostatic repulsion between
two target electrons leads to one of the electrons tunneling to
neutralize the incoming ion, and the other being excited. Recent studies have shown that the energy released by ion neutralization at metal targets can also produce collective excitations in the conduction band, such as surface plasmons,
whose decay occurs predominantly by excitation of a single
conduction electron 共plasmon-assisted neutralization 关5兴兲.
From this perspective, understanding how the potential energy of an incoming ion is transferred to the solid relates to
important aspects of fundamental physics. Furthermore, slow
ions neutralized outside the target provide a unique probe for
electronic excitations confined just to the surface region,
making the spectroscopy of emitted electrons one of the most
surface sensitive tools to study solids.
In this paper, we reconsider the theory of AN for slow,
singly charged, positive ions at metal targets and discuss another collective excitation, known as Fermi edge singularity
关6兴: the sudden change of charge of the projectile leads to a
rearrangement of the ground state of conduction electrons on
a long time scale; this final-state effect parallels the sudden
creation of a core hole by absorption of a soft x-ray photon
关6,7兴 and reflects in the broadening of the distributions of
ejected electrons with kinetic energy E, for a given incident
ion velocity.
We present measurements 共Sec. II兲 and calculations 共Secs.
III and IV兲 showing that each distribution is broadened by
the following components:
共i兲 asymptotic behavior of the matrix elements of the Auger potential;
共ii兲 initial-state lifetime;
*Electronic address: [email protected]
1050-2947/2005/71共5兲/052903共8兲/$23.00
共iii兲 final-state lifetime;
共iv兲 shift in atomic energy of the projectile near the surface;
共v兲 shift of the Fermi surface in the reference frame of the
projectile;
共vi兲 electron-phonon interaction at room temperature;
共vii兲 nonorthogonality between initial and final states due
to the sudden switching of a localized potential in the neutralized ion.
Effects 共i兲–共vi兲 have already been recognized and extensively
studied 关1,7兴; specifically, components 共i兲 and 共iii兲 produce a
dominant Lorentzian broadening, yielding a common intersection point 共i.e., a “magic energy”兲 for spectra acquired at
different ion kinetic energies 关2兴. The Fermi edge singularity
共vii兲 can be treated by the theory of Mahan, Nozieres, and De
Dominicis 共MND兲 关6兴 developed for x-ray studies 关7兴.
We measured the kinetic energy distributions of electrons
during the impact on Al surfaces of 130–430-eV Ar+ ions
共Sec. II兲, whose neutralization is not mediated by plasmon
excitation 关5兴 and involves only the ground state of the projectiles. In fact, although far from the surface the energies of
lowest excited states of Ar are nearly degenerate with the
Fermi energy of Al, the interaction with the surface yields a
positive shift of about 1.5 eV, making negligible the probability for RN+ AD processes. We observed that inclusion of
the Fermi edge singularity is needed to adequately reproduce
the high-energy behavior of the spectra 共Secs. III–V兲.
II. EXPERIMENTS
The experimental setup was described previously 关5兴: Ar+
ions were produced in an electron bombardment source operated at low electron energies 共30 eV兲 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 共and therefore an approximately constant
transmission over the measured electron energy range兲 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 di-
052903-1
©2005 The American Physical Society
PHYSICAL REVIEW A 71, 052903 共2005兲
SINDONA et al.
FIG. 1. Kinetic energy distribution of electrons ejected from Al
by 230-eV Ar+ ions, with the sample biased at −2.5 V.
rectly form the sample 共and accordingly shifted to higher
energies兲 from a spurious peak of low energy electrons,
mainly arising from the grounded entrance grid of the analyzer, that tails exponentially and can be easily subtracted
共Fig. 1兲. The adequacy of this procedure can be seen by
comparison with the spectra acquired without the bias voltage 共Fig. 2兲.
The corrected energy distributions of emitted electrons
共Fig. 3兲, show the characteristic features of AN spectra
关1–4兴: constant areas, i.e., total electron emission yields, and
a magic energy. At emission energies larger than the 5.3 eV
magic value, each spectrum follows an exponential trend.
This behavior cannot be ascribed to electrons ejected by kinetic energy transfer from the projectile, since previous
measurements—of 1 keV Ar+ impact on Al surfaces at varying incident angle 关5兴—have shown that their contribution
FIG. 3. 共Color online兲 Kinetic energy distributions of electrons
ejected from Al by 130–430-eV Ar+ ions.
does not affect significantly the high-energy broadening of
the electron spectra. Furthermore, the exponential tailing
cannot be explained in terms of the mechanism 共i兲–共vi兲, suggesting presence of another broadening effect that will be
explained in terms of the Fermi edge singularity 共vii兲.
III. THEORY
We reexamine the basic interactions in a many electron
system probed by the positively charged background of the
atomic cores of the ion-metal structure and, unless stated
otherwise, we use atomic units.
A. Hamiltonian
Let R denote the position vector of the projectile in a
reference frame at the surface of the metal. Each electron at
position r interacts independently with the external potential,
vE共r,R兲 = vS共z兲 + vA共r − R兲 + ⌬vA共r,R兲,
FIG. 2. Corrected distribution of Fig. 1, shifted backwards by
the bias voltage, and spectrum acquired without the bias voltage.
The spectra are normalized to the same height.
共3.1兲
where vS共z兲 is the surface barrier of the metal band 关8兴,
vA共r − R兲 the effective central field of the impinging ion 关9兴,
and ⌬vA共r , R兲 the change of the surface potential induced by
the positive ion charge 关10,11兴. Thus the single-particle part
of the Hamiltonian writes hE共r , R兲 = −ⵜ2 / 2 + vE共r , R兲 and the
electrostatic repulsion between two electrons, at positions r
and r⬘, respectively, is described by a central potential of the
form vSC共兩r − r⬘兩兲 that takes into account the screening of
other electrons in the medium.
In this spinless approach, the total Hamiltonian is written
in second quantization as
052903-2
MANY-BODY SHAKE-UP IN AUGER NEUTRALIZATION …
H共R兲 =
冕
=
1
2
Other interactions such as the one-body intraband scattering potential of metal electrons, of matrix elements Vkk⬘
= 具k兩vE共r , R兲兩k⬘典, and the two-body electron-electron
d3r ⌿†共r,R兲hE共r,R兲⌿共r,R兲
冕 冕
d 3r
d3r⬘⌿†共r,R兲⌿†共r⬘,R兲
⫻vSC共兩r − r⬘兩兲⌿共r⬘,R兲⌿共r,R兲.
共3.2兲
Here ⌿共r , R兲, the electron field operator in real space, can be
expanded over the truncated orthonormal set 兵具r 兩 k典其 of the
metal Hamiltonian hM共z兲 = −ⵜ2 / 2 + vS共z兲, with spectrum 兵␧k其,
and the eigenfunction 具r 兩 a0共R兲典 = 具r − R 兩 a0典 of the atomic
Hamiltonian hA共r − R兲 = −ⵜ2 / 2 + vA共r − R兲, with eigenenergy
␧⬁a . hM共z兲 characterizes the conduction band, of Fermi energy
␧F—wave vector kF—and width ␰, including the continuous
spectrum above the vacuum level and below ⬃10 eV, while
hA共r − R兲 corresponds to the single atomic level active for
neutralization.
Atomic and metal states are orthonormalized as the ion
approaches the vicinity of the surface 关3兴 by introducing the
wave function 具r 兩 a共R兲典 = 具r 兩 a0共R兲典 − 兺k具r 兩 k典具k 兩 a0共R兲典 such
that
兺k 具r兩k典ck + 具r兩a共R兲典ca共R兲
⌿共r,R兲 =
共3.4兲
in which
H0共R兲 = ␧a共R兲c†a共R兲ca共R兲 +
兺k ␧kc†k ck
共3.5兲
refers to the unperturbed electron gas in the truncated orthonormal set 兵具r 兩 k典其 艛 兵具r 兩 a共R兲典其 with spectrum 兵␧k其 艛 兵␧a共R兲
= 具a共R兲兩hE共r , R兲兩a共R兲典其;
VH共R兲 =
兺k 兵Vak共R兲c†a共R兲ck + H.c.其,
共3.6兲
is the one-body hopping potential of for resonant charge
transfer 关12兴, of matrix elements Vak共R兲 = 具a共R兲兩hE共r , R兲兩k典;
VAU共R兲 =
兺
k,k⬘,k⬙
兵Vkk⬘⬙共R兲c†a共R兲c†k ck⬘ck⬙ + H.c.其
ak
共3.7兲
denotes the usual Auger potential 关1–4兴 of matrix elements
ak⬙
Vkk
共R兲 = 具k兩具a共R兲兩vSC共兩r − r⬘兩兲兩k⬙典兩k⬘典; and
⬘
VFE共R兲 =
兺 Vkka ⬘共R兲c†k ck⬘c†a共R兲ca共R兲
potential between the band states, of matrix elements Vkkk⬘k⬙⬘⬙
= 具k兩具k⬘兩vSC共兩r − r⬘兩兲兩k⬙典兩k⵮典, are neglected. The latter needs to
be considered when the plasmon-assisted neutralization
channel—arising from the dynamic screening by the metal
electrons 关4兴—is active, which is not the case of Ar+ / Al
where the energy released by neutralization is insufficient to
excite even a q = 0 surface plasmon 关5兴.
The exactly solvable part of the reference Hamiltonian
共3.2兲 is H0⬘共R兲 = H0共R兲 + VFE共R兲 that introduces a partition of
the Fock space into two subspaces: on the one hand, many
electron states with the ion state empty are still constructed
by antisymmetrizing the unperturbed set 兵具r 兩 k典其; on the other
hand, many electron states with the ion state occupied need
to be calculated from the eigenfunctions 兵具r 兩 ka共R兲典其 of the
final-state Hamiltonian hM
⬘ 共z兲 = hM共z兲 + vFE共r , R兲, with the
same spectrum of the unperturbed operator hM共z兲. The oneelectron potential activated by neutralization reads
vFE共r,R兲 =
共3.3兲
and annihilations operators 兵ck其 艛 兵ca共R兲其 satisfy the ordinary
algebraic rules of fermion operators.
Substituting the expansion 共3.3兲 into 共3.2兲 yields
H共R兲 = H0共R兲 + VH共R兲 + VAU共R兲 + VFE共R兲
PHYSICAL REVIEW A 71, 052903 共2005兲
d3r⬘vSC共r − r⬘兲兩具r⬘兩a共R兲典兩2 ,
共3.9兲
and the final states of AN are nonorthogonal to the initial
state.
B. Fermi’s golden rule formulation
The trajectory followed by the projectile is handled classically, thus the dynamic of the system is parametrically time
dependent. In the simplest case, the ion can be assumed to
reflect elastically form a plane at distance Z0 from surface of
the target, moving along a straight line R = R共t兲 of incident
velocity v = 共v储 , v⬜兲, parallel component R储共t兲 = v储t and perpendicular component Z共t兲 = v⬜兩t兩 + Z0. Since complete
knowledge of the eigenfunctions of both hM共z兲 and hM
⬘ 共z兲 is
available, we work in the interaction picture spanned by
H0⬘关R共t兲兴 and treat the Auger potential VAU关R共t兲兴 of Eq. 共3.7兲
as a small perturbation 关14兴.
The key quantity in our study is the transition rate,
1 / ␶a共␧k , v兲, from the unperturbed ground state of the conduction band 兩0典N, composed of N band electrons with the ion
state empty, to all possible excited states 兩f a关␧k , R共t兲兴典N, with
N − 2 electrons below the Fermi energy, the ion state occupied, and an excited electron with energy ␧k ⬎ ␧F. The initial
state is the ground state of H0关R共t兲兴—with eigenenergy
E0—and each final state diagonalizes H⬘0关R共t兲兴—with
eigenenergy Eaf 关R共t兲兴. By Fermi’s golden rule,
共3.8兲
1
共␧ ,v兲 = 2␲
␶a k
k,k⬘
is a new interaction, of matrix elements Vakk⬘共R兲
= 具k兩具a共R兲兩vSC共兩r − r⬘兩兲兩a共R兲典兩k⬘典, that describes the sudden
change of charge due to neutralization of the incident ion and
injection of a band hole. It has the same structure of the
MND potential, where c†aca is the number operator of the
core hole 关6兴.
冕
冕
⬁
−⬁
dt
兺f ei兰 dt⬘兵E 关R共t⬘兲兴−E 其
t
0
a
f
0
⫻ 兩N具f a关␧k,R共t兲兴兩VAU关R共t兲兴兩0典N兩2 . 共3.10兲
The MND potential 共3.8兲 modifies significantly the many
electron states of the metal, when the atomic level is occupied, while it acts as a weak perturbation on single-particle
states. Figure 3 shows that a new broadening mechanism is
052903-3
PHYSICAL REVIEW A 71, 052903 共2005兲
SINDONA et al.
needed to explain the behavior of the energy tails of experimental electron energy distributions, above the magic energy.
Thus we intend to evaluate the effect of the Fermi edge singularity at the edge of AN, i.e., when the electrons of the
initial state, that participate to the process, lie close to the
Fermi energy. In this case, both the atomic and the excited
electron, in the final state, are negligibly perturbed by the
one electron potential 共3.9兲, since their energies, relative to
the Fermi energy, are large on the eV scale. For these reasons, we can approximate
兩f a关␧k,R共t兲兴典N ⬇ c†k c†a关R共t兲兴兩f 关R共t兲兴典N−2 ,
共3.11兲
in which 兩f 关R共t兲兴典N−2 is an excited state of the metal, with
time-independent eigenenergy E f , that involves the N − 2
band electron that do not participate to AN.
Using Eq. 共3.11兲 into Eq. 共3.10兲, and approximating the
energy of the band holes created by AN to ␧F, we obtain 共see
the Appendix兲
1
共␧ ,v兲 = 2␲␳共␧k兲
␶a k
冕
⫻ Fa共v,t兲e
d ⍀k
2
兺
k⬘,k⬙
冕
⬁
−⬁
1
␶a⬘
共Z兲 = 2␲
k,k⬘,k⬙
兺
k⬘,k⬙
冕
−⬁
兺k 兩Vak共Z兲兩2␦关␧a共Z兲 − ␧k兴.
,
NI共␧k,v兲 = N0␳共␧k兲
冕
d 2⍀ k
兺
k⬘,k⬙
冕
⬁
−⬁
dt兩Vkk⬘⬙关Z共t兲兴兩2
ak
t
ak
dt兩Vkk⬘⬙关Z共t兲兴兩2
NI共␧k,v兲 =
⬁
dx NI0共␧k,␧k − x,v兲B共x, v⬜兲,
共3.19兲
−⬁
in which
NI0共␧k,␧,v兲 = N0
冕
d 2⍀ k
兺 uakkk⬘⬙关␧ + ␧a共Z0兲 − ␧k⬘
k⬘,k⬙
− ␧ k⬙ − q · v 储 兴
共3.20兲
is the spectrum of excited electrons broadened by the Auger
potential 共i兲, via the internal functions
ukk⬘⬙共␧兲 = ␳共␧k兲
ak
t
B共␧, v⬜兲 =
共3.14兲
t
冕
and
In this relationship, q = k − k⬘ − k⬙ labels the momentum exchanged in a single excitation process and the factor e−iq·v储
accounts for the shift of the Fermi surface in the moving
frame 共v兲.
Then, we need to consider effects that are outside Fermi
golden’s rule formulation, i.e., the lifetime of initial 共ii兲 and
final 共iii兲 states and the effect of the electron-phonon interaction at room temperature 共vi兲. We introduce the probability
that the ion ground state survives neutralization, due to both
Auger and resonant transitions
共3.18兲
where the new effect, i.e., the Fermi edge singularity 共vii兲, is
contained in the propagator 共3.13兲.
By Eq. 共3.18兲, NI共␧k , v兲 can be expressed as the convolution product
⫻ ei兰0dt⬘兵␧k+␧a关Z共t⬘兲兴−␧k⬙−␧k⬘−q·v储其Fa共v⬜,t兲.
Pa共v⬜,t兲 = e−兰−⬁dt⬘兵共1/␶a⬘兲关Z共t⬘兲兴+2⌬a关Z共t⬘兲兴其 ,
2/2
⫻ ei兰0d␶兵␧k+␧a关Z共␶兲兴−␧k⬘−␧k⬙−q·v储其 ,
共3.12兲
共3.17兲
Finally, we take a simple exponential law, with average lifetime ⌫0, for the probability that the band holes created by AN
survive recombination 关1兴 and we model the electron-phonon
interaction by a Gaussian function of width ␴PH 关7兴.
With these prescriptions, we can write
2
is the propagator for the atomic electron, in the ground state
of the N − 2 band electrons that do not participate to AN, and
U共v ; 0 , t兲 is the time-development operator for the singular
potential 共3.8兲, in the interaction picture spanned by the freeelectron-gas Hamiltonian 共3.5兲.
The next step is the determination of NI共␧k , v兲, the distribution of electrons excited above the Fermi level by the incident ion, being proportional to Eq. 共3.12兲 that we reexpress
in the reference frame moving with the constant parallel velocity of the projectile,
d ⍀k
−⬁
⌬a共Z兲 =
共3.13兲
冕
ak
and the virtual width of the atomic state 关11,12兴 via hopping
processes
Fa共v,t兲 = N−2具0兩ca关R共0兲兴U共v;0,t兲c†a关R共t兲兴兩0典N−2
2
dt兩Vkk⬘⬙共Z兲兩2␦关␧a共Z兲 + ␧k − ␧k⬙ − ␧k⬘兴
⫻ Fa共v⬜,t兲Pa共v⬜,t兲e−⌫0t−␴PHt
where ␳共␧k兲 is the density of final states available to excited
electrons,
1
共␧ ,v兲 = 2␲␳共␧k兲
␶a k
冕
共3.16兲
ak
dt兩Vkk⬘⬙关R共t兲兴兩2
i兰t0dt⬘兵␧k+␧a关R共t⬘兲兴−␧k⬙−␧k⬘其
⬁
兺
⬁
冕
⬁
冕
⬁
−⬁
dt i␧t ak⬙
e 兩Vkk⬘ 关Z共t兲兴兩2 ,
2␲
2
共3.21兲
2
dte共i␧−⌫0兲t−␴PHt /2Fa共v⬜,t兲
−⬁
t
⫻ Pa共v⬜,t兲ei兰0d␶兵␧a关Z共␶兲兴−␧a共Z0兲其
共3.22兲
denotes the broadening function for mechanisms 共ii兲–共vii兲.
For normal emission, the distribution of electrons ejected
from the metal writes
N共E,v兲 = T共E兲NI共E,v兲.
共3.23兲
It depends on the electron kinetic energy outside the solid
E = ␧k − ␰ and is related to the distribution of excited electrons
via the escape probability or surface transmission function
关13兴,
共3.15兲
which includes the total AN transition rate 关1,2兴
052903-4
T共E兲 = ⌰共E兲
E
.
E+␰
共3.24兲
MANY-BODY SHAKE-UP IN AUGER NEUTRALIZATION …
PHYSICAL REVIEW A 71, 052903 共2005兲
FIG. 4. 共Color online兲 Schematics of AN from the Ar+ / Al
system.
IV. APPLICATION
We apply Eqs. 共3.20兲 and 共3.22兲 to Ar+ / Al, using ␧F =
−␾ = −4.25 eV, kF = 0.93 au, ␰ = 15.95 eV, and ␧⬁a =
−15.76 eV, where energies are measured relative to the
vacuum level and therefore E = ␧k 共Fig. 4兲. We approximate
the electron-electron potential to the Yukawa form,
vSC共兩r − r⬘兩兲 =
e−␮兩r−r⬘兩
,
兩r − r⬘兩
共4.1兲
in which ␮ is the inverse screening length of the electron
ak⬙
gas. Thus the matrix elements Vkk
and Vakk⬘ can be worked
⬘
out by Fourier transforming in the coordinates parallel to the
surface 关2兴 and numerically integrating over the coordinates
perpendicular to the surface. NI0共E , v兲 can be obtained using
Monte Carlo techniques for multiple wave-vector integrals
over ⍀k, k⬘ and k⬙ in Eq. 共3.20兲, where the internal functions
ak⬙
共␧兲 are calculated from the numerical Fourier transform
ukk
⬘
⬙
2
of 兩Vak
kk⬘ 关Z共t兲兴兩 .
The basic theory of AN corresponds to the replacement
Fa共v⬜ , t兲 → 1 in Eqs. 共3.18兲 and 共3.22兲, which leads the external distributions N共E , v兲 to depend on four parameters: Z0,
␴PH, ␮, and ⌫0. Z0 is determined by the constraint that the
external distributions coincide at the experimental magic energy, yielding Z0 = 4 Å, and for ␴PH we use the same value of
x-ray studies on Al at room temperature, i.e., ␴PH ⬃ 0.1 eV
关7兴. Indeed, adjustment with the experiments of all the four
parameters confirmed the estimates for Z0 and ␴PH, and
yielded ␮ = 0.1 kF and ⌫0 = 0.05 eV. Variation of ␮ has a
strong influence on the position of the peak of NI共␧k , v兲, that
for ␮ ⱗ 1.00kF lies at ␧k ⱗ −2 eV, below the vacuum level.
However, it has a weaker influence on the high-energy tail of
the internal distributions, at ␧k ⲏ 1 eV above the vacuum
level, so that the external distribution N共E , v兲 is not significantly affected at E ⲏ 1 eV. We verified by calculations that
this is the case for any choice of ␮ in the range 0.05kF 艋 ␮
艋 1.00kF. Therefore ⌫0 is the only parameter that needs to be
fitted. The value reported above lies in the range 0.01–0.1
FIG. 5. Electron spectra of Fig. 3, for the ion incident energies
130–230 eV, are compared with theoretical distributions calculated
from 共I兲 the basic theory of AN; 共II兲 Hagstrum’s model, and 共III兲
Eqs. 共3.20兲 and 共3.22兲.
eV, estimated by the theory of mean free paths of hot holes in
solids 关1兴.
This choice of the four parameters corresponds to the distributions of Figs. 5 and 6, broadened by the experimental
energy resolution and labeled I. In the same figures are
shown the experimental spectra and the distributions II obtained from Hagtrum’s convolution model 关1兴 that we derived in our formulation by neglecting the effect of v储 and
using the approximated factorization
FIG. 6. Electron spectra of Fig. 3, for the ion incident energies
330–430 eV, are compared with theoretical distributions calculated
from 共I兲 the basic theory of AN, 共II兲 Hagstrum’s model, and 共III兲
Eqs. 共3.20兲 and 共3.22兲.
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PHYSICAL REVIEW A 71, 052903 共2005兲
SINDONA et al.
Vkk⬘⬙关Z共t兲兴 ⬇ Vkk⬘⬙共Z0兲e−␭av⬜兩t兩/2 .
ak
ak
共4.2兲
␭a is determined from the average asymptotic slope of
⬙
2
over occupied band states, yielding ␭a
ln兩Vak
kk⬘ 共Z兲兩
= 1.83 a . u, which introduces a universal broadening function
of the ion perpendicular velocity and the atomic level convolution of B共␧k , v⬜兲 with a Lorentzian of constant width
␭av⬜. The approximated results II, for the same choice of Z0,
␮, ␴PH, and ⌫0, do not differ significantly from the numerical
curves I. In fact, the ion velocity is always less than
⬃0.02 a . u., while the target Fermi velocity is 0.93 a.u. Consequently, the effective occupation of the target states in the
rest frame at velocity v储 is very well described by the FermiDirac distribution and shifted Fermi surface effects are small.
ak⬙
共Z兲兩2 follows a strict exponential law
Furthermore, each 兩Vkk
⬘
at ion-surface distances typically larger than ⬃15 Å.
In either case, theoretical calculations 共I, II兲 are in excellent agreement with the main part of experimental distributions of Fig. 3 for electron kinetic energies less than ⬃6 eV.
However, the logarithmic scale evidences that the highenergy exponential behavior of spectra cannot be reproduced
with the known broadening mechanisms.
The MND theory 关6,7兴 predicts an asymmetric line shape
of the form
FMND共E⬘兲 = ⌰共E⬘兲
e−E⬘/␧0
,
⌫共␣兲 ␧0␣ E⬘␣−1
共4.3兲
for photoemitted electrons with energy E⬘, relative to the
core-hole energy, where ␧0 is a cutoff parameter of the order
of the conduction band width and ␣ a singularity index.
FMND共E⬘兲 results from the Fourier transform of Eq. 共3.13兲
for a contact core-hole potential perturbing band electrons in
a range of width ␧0 from the Fermi energy 关6,14兴. In the
present context, we considered a contact potential of the
form 共3.8兲 with the matrix elements
a
Vkk⬘共Z兲
=
再
Vka k 共Z兲, if 兩␧k − ␧k⬘兩 艋 ␧0
F F
0,
otherwise
冎
,
共4.4兲
obtaining for the correlation function Fa the same structure
of the Fourier transform of FMND, i.e.,
Fa共v⬜,t兲 ⬇
冉
1
1 + i␧0共v⬜兲t
冊
␣共v⬜兲
.
共4.5兲
Such a function seems to be the right candidate to cope with
the high-energy discrepancies of the Lorentzian broadening
model. The singularity index depends on the Fourier transform ␾l共v⬜ , ␧兲 of the instantaneous phase shifts 关15兴 of the
potential 共3.9兲 at ␧ = ␧F,
␣ 共 v ⬜兲 = 兺
l
共2l + 1兲
兩⌽l共v⬜,␧F兲兩2 .
␲2
共4.6兲
Calculation of ␣, in the S-wave approximation, and adjustment ␧0 show that both parameters increase with increasing
v⬜. ␣ ranges from 0.12 to 0.16, the same order of the estimated value for the singularity indices of the measured 2s
and 2p core lines of Al 关7兴, and 0.9艋 ␧0 艋 1.5 eV.
The modified electron distributions III, determined with
ak⬙
the numerical internal functions ukk
and reported in Figs. 5
⬘
and 6, offer the correct exponential trend at higher energies.
We find that ⌫0 is now reduced to 0.01 eV, being consistent
with the range of values reported in literature 关1兴. In Fig.
5共a兲, we also show the true unbroadened spectrum of the
ion-metal system 关1兴, i.e., the static response of the target to
a projectile at a fixed distance Z0 from the surface. The latter,
defined by
N0共E兲 = T共E兲NI0共E,v → 0兲,
共4.7兲
crosses other theoretical distributions at the magic energy, as
predicted by Monreal and Apell 关2兴.
V. CONCLUSIONS
In summary, we extended the theory of AN to include the
effect 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 effect is
manifested in the high-energy tail of the electron energy distributions, where the theory accounts very well with measurements reported in this work and elsewhere 关5兴. 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 Fermiedge singularity.
APPENDIX: APPROXIMATIONS INVOLVED IN THE
EVALUATION OF THE AN TRANSITION RATE
The approximation 共3.11兲 is reasonable when the energies
of both the excited and the atomic electrons in an Auger
transition are such that
兩␧k − ␧F兩 Ⰷ ␧0共v⬜兲,
兩␧a共Z兲 − ␧F兩 Ⰷ ␧0共v⬜兲,
共A1兲
where ␧0共v⬜兲 is the width of the broadening function 共4.3兲,
responsible for the effect 共vii兲. In relation with the Ar+ / Al
system, we have shown that the experimental distributions
electrons, in Figs. 3, 5, and 6, are reproduced with ␧0共v⬜兲
ⱗ 1.5 eV. In addition, the maximum shift to the binding energy of the ion level is about ⬃1.5 eV, corresponding to the
turning distance Z0 ⬃ 4 Å 共Fig. 4兲. Thus 兩␧a共Z兲 − ␧F兩 is generally larger than 10 eV. As for the energy of the excited electron, Eq. 共A1兲 is satisfied for ␧k ⲏ 5 eV. This means that
shake-up electrons participating to AN, in the initial state,
have energies close to ␧F. Therefore we can limit our estimation to their effect on the energy distributions above ⬃5 eV,
where the basic theory of AN is unable to reproduce the
studied experiments.
Substituting Eqs. 共3.7兲 and 共3.11兲 in Eq. 共3.10兲, we obtain
N−2具f共R兲兩ca共R兲ckVAU共R兲兩0典N
=
兺 Vkkak⬘⬙共R兲 N−2具f共R兲兩ck⬘ck⬙兩0典N
k⬘,k⬙
since ␧k ⬎ ␧F. Then,
052903-6
共A2兲
MANY-BODY SHAKE-UP IN AUGER NEUTRALIZATION …
1
共␧ ,v兲 = 2␲␳共␧k兲
␶a k
冕 冕
⬁
d 2⍀ k
dt
−⬁
⫻ ei兰0dt⬘兵␧a关R共t⬘兲兴+␧k其
t
兺f
冏兺
k⬘,k⬙
k⬙¯k⬙
aFk⬘¯k⬙共v,t兲
ei共E f −E0兲t
Vkk⬘⬙关R共t兲兴
ak
冏
PHYSICAL REVIEW A 71, 052903 共2005兲
2
⫻ N−2具f关R共t兲兴兩ck⬘ck⬙兩0典N .
f
Ũ共v;t,0兲 = TeiK0tŨS共v;t,0兲
共A3兲
t
共A4兲
k⬙¯k⬙
aFk⬘¯k⬙共v,t兲
共A5兲
兺k ␧kc†k ck + 兺 Vkka ⬘关R共t兲兴c†k ck⬘ ,
共A6兲
k,k⬘
and T is the time ordering operator 关16兴. 关ŨS共v ; 0 , t兲兴N−2 operates on the subspace of the N − 2 metal electrons that do not
partecipate to AN and K0, the diagonal part of K0⬘关R共t兲兴, is
such that
e−iE0t兩0典N = e−iK0t兩0典N .
1
共␧ 兲 = 2␲␳共␧k兲
␶ k
⫻
冕 冕
⬁
d 2⍀ k
t
¯
¯
共A12兲
can be interpreted as atomic electron propagator in the
ground state of a metal band with N − 2 electrons. In fact, it
can be put in the form
Fa共v,t兲 = N−2具0兩ca关R共0兲兴U共v;0,t兲c†a关R共t兲兴兩0典N−2 ,
共A13兲
k⬙¯k⬙
k
k⬙a
k⬙k⬙
兺
aFk⬘¯k⬘共v,t兲V¯k⬘k 关R共t兲兴.
¯ ¯
Fa共v,t兲 = N−2具0兩Ũ共v;0,t兲兩0典N−2
In the same approximation, Lk ¯k 共v , t兲 tends to ␦k⬙¯k⬙␦k⬘¯k⬘,
⬘⬘
yielding Eq. 共3.12兲.
Using more accurate models for the matrix elements
兺 e−i共␧ ⬘+␧ ⬙兲t⬘Vkkak⬘⬙关R共t兲兴
k⬘,k⬙
⫻
共A11兲
⬘⬙
with U共v ; t , 0兲 the time development operator for H0⬘关R共t兲兴,
in the interaction picture spanned by H0关R共t兲兴. Equation
共A12兲 converges to Eq. 共4.5兲 for the contact potential 共4.4兲.
dtei兰0dt⬘兵␧k+␧a关R共t⬘兲兴其
−⬁
k
k⬙¯k⬙
= Lk ¯k 共v,t兲Fa共v,t兲,
共A7兲
Thus Eq. 共A3兲 can be rewritten in the form
共A10兲
where
is the time-development operator—in the Scrhödinger
representation—for the one-body Hamiltonian
K0⬘关R共t兲兴 =
共A9兲
the time-development operator 共A5兲 in the interaction picture
spanned by K0, and ck共t兲 = e−i␧ktck. By the linked cluster theorem 关6,16兴, the many-body diagram 共A9兲 can be written as
where
ŨS共v;t,0兲 = Te−i兰0dt⬘K0⬘关R共t⬘兲兴
†
is the propagator for the core holes created by AN,
By the completeness relation over the final states 兵兩f关R共t兲兴典其,
we can write
兺f 兩f关R共t兲兴典N−2eiE tN−2具f关R共t兲兴兩 = 关ŨS共v;0,t兲兴N−2 ,
†
= N具0兩ck⬙ck⬘Ũ共v;0,t兲c¯k⬘共t兲c¯k⬙共t兲兩0典N
共A8兲
k⬘,k⬙
Here,
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a
共R兲, in Eq. 共3.8兲, Lk ¯k 共v , t兲 produces another broadening
Vkk
⬘
⬘⬘
component to Eq. 共3.20兲 that seems to affect negligibly the
high-energy behavior of the electron energy distributions.
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