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Monte Carlo simulation study of quasi-elastic electron

scattering from an overlayer/substrate system

Y.G. Li

, Z.M. Zhang

, S.F. Mao

and Z.J. Ding

Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics

Department of Astronomy and Applied Physics University of Science and Technology of China,

Hefei(230026)

*E-mail: zjding@ustc.edu.cn.

Abstract

Recent experimental results have shown that the recoil energies of electrons elastically scattered by

light and heavy atoms can be resolved for an energetic electron beam and at large scattering angles.

Full understanding of the scattering processes involved is helpful to sample characterization, and, for

providing more knowledge about electron inelastic mean free path. In this work we use a Monte Carlo

simulation method to quantitatively study the energy shift, the Doppler broadening and especially the

peak intensity ratio for an overlayer/substrate sample. Recoil energy in the electron elastic scattering

events is calculated based on our previous Monte Carlo simulation model by taking account of the

kinetic energy of atoms. An anisotropic distribution of the velocity direction of the atoms, the

Maxwell-Boltzmann thermal energy distribution and also the multiple scattering of electrons are

considered in the simulation. By introducing a polarized momentum a good agreement has been

obtained on the position shift of quasi-elastic peak between the calculation and experiment. The

calculation also shows a quantitative agreement with the experimental results on the peak intensity ratio

between different elements for a Ge/C overlayer sample. It is illustrated that the multiple scattering

effect is remarkable for a high energy beam.

PACS: 79.20.Ap Theory of impact phenomena; numerical simulation; 29.30.Dn: Electron

spectroscopy; 34.80.Bm: Elastic scattering; 68.49.Jk: Electron scattering from surface

Keywords: Monte Carlo, Quasi-elastic, EPRS, Overlayer/Substrate

1. Introduction

Electron elastic scattering at surface provides an effective way for determination of the

physical parameters related with electron transport processes, such as electron inelastic mean free

path (IMFP), by elastic peak electron spectroscopy (EPES) [1]. Recoil effect in quasi-elastic

scattering of electrons has been firstly described by Boersch et al. as early as 1967 [2]. It was

shown that there is an energy loss involved in large-angle elastic scattering events, consistent with

momentum transfer to a single atom in the scattering event. The quasi-elastic peak is a marked

feature in EPES spectra. From then on, recoil effect of the elastic peak in electron spectroscopy

has been paid much attention. A lot of experimental works have been performed to study the

energy shift and broadening of the quasi-elastic peak spectra for the elemental solids [3-5], organic

polymer solids [6-8], compounds [9] and overlayer systems [10-13] by varying atomic numbers of

the sample. Recently it has been shown that, by using an energetic electron beam (20-40 keV) and

at large scattering angles the recoil energies for atoms with large mass difference can be resolved

[9-13]. The technique is termed as electron Rutherford backscattering (ERBS) [9].

On the other hand, Monte Carlo simulations of the energy shift and recoil broadening of the

elastic peak in electron spectra have been performed later for the similar systems by using

classical approaches [14-15]. Also, a quantum scattering theory including recoil effect has been

proposed to deal with elastically scattered electrons and photoelectrons involved in electron

spectra [16]. However, these works have not yet derived the intensity ratio between peaks that

attributed to different elements, and, only one simple analysis of intensity ratio was given in Ref.

[12] Accurate description of the peak intensity ratio as well as peak shape provides an important

information on electron scattering process involved and physical mechanism related, which will be

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certainly helpful for providing more knowledge about electron IMFP and to sample

characterization, such as, for unique determination of the thickness of an overlayer, in-depth

distribution of elements and composition.

In this paper, we use a Monte Carlo simulation method to quantitatively study the energy

shift, the Doppler broadening and especially the peaks intensity ratio for an overlayer/substrate

system. The simulation is chiefly based on our previous Monte Carlo model of electron scattering

[17-19], but also considering the Maxwell-Boltzmann thermal energy distribution of atoms with

an anisotropic distribution of the velocity direction by introducing a polarized momentum

component. For comparison with experimental observation, the simulation of quasi-elastic

scattering of electrons of energies about 15-30 keV from a Ge/C overlayer either in reflection

geometry or transmission geometry has been performed. The simulated peak intensity ratio agrees

very well with the experimental data. The anisotropic velocity distribution has successfully

described the position of energy shift of the elastic peak.

2. Theory

2.1. Recoil Energy

The classical model of the electron quasi-elastic scattering from a free atom is used for its

simplicity and validity in most cases [14-15]. Went and Vos [20] have considered in detail that the

electron quasi-elastic scattering can certainly be described as scattering by a free atom, while the

fact that the atom is part of a crystal is assumed not to influence the result. Thus, it is convenient

and reasonable to use the binary encounter approximation instead of the electron-lattice interaction

for the electron quasi-elastic scattering for the impulse approximation is valid here. If an incoming

electron of mass

with momentum

elastically scatters over an angle

by an atom of

mass

, the momentum change is approximately about

(

)

2sin 2qP



. Assume that the

target atom is at rest initially before the collision the energy transferred from an electron to an

atom is,

(

)

()

2222

2sin sincos

mE M m M m

θθ

⎡

⎤

+−−

⎢

⎥

⎣

⎦

, (1)

where

Pm=

denotes the kinetic energy of the electron. As

mM

, the recoil energy

of the electron is approximated as,

()

2sin2

EqM E

= . (2)

More realistically, the target atoms have a kinetic energy due to thermal motion. The recoil

energy is then given by the following relation,

()

222

MMM

⋅

= −=+

(3)

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- 3 -

where

is the initial momentum of the target atoms. The momentum

here is usually

considered as isotropically distributed. The first term in above equation represents the energy shift

of the elastic peak and the second one is the Doppler broadening term due to the vibration of the

target atoms in different directions. The classical thermal vibration model has been used to

describe the motion of atoms in a solid by considering that the atoms will behave like a classical

gas and will have an average kinetic energy

32kT

in the present high temperature case. Only

in the low temperature limit the quantum physics should be employed to relate Doppler

broadening to the interaction of electrons with phonon modes of a lattice.

It has been shown that the calculated recoil energy by using the above classical approach could

not yield a good agreement with the experimental value, and a deviation of the peak position was

found in comparison [14-15, 21]. For the cause of such deviation Kwei et. al [15] considered that

the recoil energy should be centered at the most probable value of this loss for vibrating atoms

rather than for atoms at rest. However, even taking into account of the thermal motion of atoms

the simulation overestimates recoil energy when compared with experiments [14]. The reason is

quite obvious: the statistical averaging of the second term in Eq. (3) for isotropic vibrating atoms

gives vanishing contribution to the recoil energy and, thus, the simple treatment by using Eq. (3)

yields the same peak position as for rest atoms except that the peak is broadened. Vos and Went

[21] suggested that the classical thermal distribution cannot describe experiment, considering the

differences on the peak position and shapes between experimental and expected results. In order to

solve this problem, an improved scheme is introduced in this paper. The main idea is to consider

that the thermal vibration of target atoms may no longer be regarded as isotropically distributed,

for the Coulomb interaction between an energetic incident electron and the target atom can

influence the atomic motion through the polarization of atomic electron clouds.

Hence, a proper approach to the description of the anisotropic vibration of the target atoms

should be necessary. We assume in the classical approximation that the anisotropic vibration may

be described by an elliptical distribution and, accordingly, the momentum vector of atomic motion

is composed of an isotropic momentum

k and a preferential oriented momentum

k . In order

to estimate the value and the direction of the momentum component

k at least for qualitative

analysis, we now analyze the difference between the real momentum change

'q in an

experiment and the ideal momentum change

q for a binary collision in a classical theory. The

experimental fact indicates

'qq< for which we attribute to the effect of atomic polarization

during collision between charged particles. This is because the electron-atom scattering dynamics

accompanies with the change of electrostatic potential of an electron, and, with the induced

polarization of atomic electron clouds. These two factors then drive the motion of atomic nucleus.

As the polarization is a directional effect, it is quite reasonable to treat the thermal vibration of

target atoms as anisotropic distributed, but not as a simple isotropic distribution for a collision

between hard spheres. Then we can decompose the momentum of nucleus into two components,

an isotropic momentum

k and a preferential oriented momentum

k . This simple

phenomenological treatment averages the overall polarization effect for an amount of scattering

events into a single term within the framework of a classical binary collision theory.

The recoil energy is thus now rewritten as,

()()

222

MMMM

++ +

⋅

=−=++

qkk kk

, (4)

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