- Email: [email protected]
Energy-loss function including damping and prediction of plasmon lifetime
Accepted Manuscript Title: Energy-loss function including damping and prediction of plasmon lifetime Author: Hieu T. Nguyen-Truong PII: DOI: Reference:
S0368-2048(14)00079-6 http://dx.doi.org/doi:10.1016/j.elspec.2014.03.010 ELSPEC 46267
To appear in:
Journal of Electron Spectroscopy and Related Phenomena
Received date: Revised date: Accepted date:
10-2-2014 23-3-2014 24-3-2014
Please cite this article as: Hieu T. Nguyen-Truong, Energy-loss function including damping and prediction of plasmon lifetime, Journal of Electron Spectroscopy and Related Phenomena (2014), http://dx.doi.org/10.1016/j.elspec.2014.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Research Highlights
Highlight • We present an approach to take damping into account of the energy loss function.
• We predict the values of damping and lifetime at low energies.
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• Influence of damping is significant in the low-energy region.
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• We calculate the electron inelastic mean free path for Al, Si, Cu, and Au.
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• The damping value is not only energy dependent but also material dependent.
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*Manuscript
Hieu T. Nguyen-Truong∗
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Energy-loss function including damping and prediction of plasmon lifetime
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Faculty of Electronics and Computer Science, Volgograd State Technical University, 28 Lenin Avenue, Volgograd 400131, Russia
Abstract
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An approach to include plasmon damping in the energy-loss function is described within the dielectric theory. Use of the energy-loss function included damping for calculating the electron inelastic mean free path yields results in good agree-
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ment with the experimental data and other theoretical results at medium-high energies. At a few eV above the Fermi energy, the present results are entirely consistent with those obtained from other measurements for Au. Also, a simple
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way to predict the values of damping and lifetime at low energies is described.
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Two values of lifetime for an electron with energy (above the Fermi energy) of 5 eV in Al and 6 eV in Au are predicted to be 2.18 fs and 1.70 fs, respectively.
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These predicted values are in reasonable agreement with those estimated from other measurements at the corresponding energies: 2.16 ± 0.22 fs in Al, and
1.91 ± 0.32 fs in Au.
Keywords: energy-loss function, dielectric response function, inelastic mean free path, plasmon damping, plasmon lifetime
1
1. Introduction
2
The energy-loss function (ELF) is the key quantity for calculating the elec-
3
tron inelastic mean free path (IMFP) within the dielectric theory. Information ∗ Permanent address: Faculty of Materials Science, Ho Chi Minh City University of Science, 227 Nguyen Van Cu Street, 5 District, Ho Chi Minh City, Vietnam. Email address: [email protected] (Hieu T. Nguyen-Truong)
Preprint submitted to Journal of Electron Spectroscopy and Related PhenomenaMarch 23, 2014
Page 2 of 23
on IMFP is important for surface analytical methods such as X-ray photoelec-
5
tron spectroscopy or Auger electron spectroscopy, and is frequently determined
6
by elastic peak electron spectroscopy with taking into account the surface ex-
7
citation effect [1, 2]. Experimentally, the ELF can be determined by analyzing
8
the reflection electron energy loss spectra [3, 4]. Theoretically, the ELF can
9
be extrapolated from the optical energy-loss function (OELF) by various algo-
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rithms [5–10].
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10
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4
One of the most popular algorithms for determining the ELF has been pro-
12
posed by Penn [7]. This algorithm is based on the statistical approximation
13
developed by Lindhard et al. [11] and has been extensively employed within the
14
single-pole approximation (SPA) for calculating the IMFP [7, 12–17] over either
15
200 or 300 eV energy range and the electron stopping power (SP) [18, 19] over
16
100 eV energy range. However, taking into account only the plasmon excitation
17
contribution makes the SPA only suitable for electrons of medium-high energy.
18
Recently, the ELF calculation in the Penn algorithm has been fully performed
19
by an integration of the Lindhard dielectric function without damping [20–23].
20
By including both single-electron and plasmon excitations contributions, this
21
“full Penn algorithm”, which is referred as LPA (stands for Lindhard–Penn
22
algorithm) hereafter, has extend the low-energy range down to the plasmon
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23
excitation energy.
24
However, neglecting damping in the Lindhard dielectric function [24] makes
25
the Penn algorithm not suitable for very-low-energy electron. As known, at-
26
tempt to include damping in the Lindhard dielectric function by turning the
27
real frequency into complex does not conserve the local number of electrons.
28
Mermin [25] has solved this problem by a phenomenological modification of the
29
Lindhard dielectric function. The Mermin dielectric function including damping
30
is described in term of the Lindhard dielectric function of the complex frequency.
31
In the present work, we describe an approach to include damping in the ELF
32
by using the Mermin dielectric function in the Penn algorithm instead of the
33
Lindhard dielectric function. Our approach, which is referred as MPA (stands
34
for Mermin–Penn algorithm) hereafter, is much simpler than the LPA but still 2
Page 3 of 23
takes advantage of its power. Furthermore, the MPA allows to predict the
36
values of damping and lifetime at low energies, knowledge of which is necessary
37
for understanding the electron transport properties. In the past two decades,
38
this problem has attracted much attention of researchers both in theoretical and
39
experimental studies [26–31].
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35
Including damping in the Lindhard dielectric function has been studied by
41
Ashley and Ritchie [32] in their IMFP calculations for Al. In two later works [33,
42
34] also for Al, they have used directly the Mermin-ELF in the role of the ELF
43
to take damping into account. Use of Mermin- or Lindhard-ELF instead of ELF
44
should only be considered as an approximation for elements that have only one
45
main plasmon peak like Al.
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40
Alternatively, the ELF can be also obtained by fitting the OELF to the
47
theoretical expression (Drude-type [5, 8, 35, 36], or Mermin-type [9, 10, 37]) to
48
determine the fitting parameters (weight, position, and width) corresponding to
49
the plasmon peaks, and then extrapolating into the non-zero momentum trans-
50
fer region by a linear combination of these fitting functions. The comparison
51
between approaches can be found elsewhere (e.g. Refs. [38, 39]).
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The MPA presented here does not demand any fitting parameters as well
53
as plasmon dispersion relation because the MPA-ELF is defined as an integral
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52
54
of the Mermin-ELF and a function linking with the OELF. Our approach only
55
requires knowledge of the optical dielectric function to obtain the OELF. In
56
the present work, the OELF is obtained from the experimental optical data in
57
the compilation of Palik [40], which has been checked and used in our previous
58
study [23] for SP calculations. The MPA-ELF then is employed in the IMFP
59
calculations for Al, Si, Cu, and Au in the energy range from 1 eV to 30 keV
60
above the Fermi energy. Comparisons show that our IMFPs are in reasonable
61
agreement with experimental values [1, 41–60] and other theoretical results [2,
62
7, 14, 21, 34, 61–65] at medium-high energies. Finally, a simple way to predict
63
the values of damping and lifetime at low energies is described. Some predicted
64
values of damping and lifetime in the energy range 2–9 eV are also reported,
65
two of which are entirely consistent with those obtained from experiment [46]. 3
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Also, it should be noted that the Hartree atomic units (~ = me = e = 1) is used
67
throughout this work.
68
2. Theory
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In the dielectric theory [24, 66, 67], the probability that an electron loses
70
energy ω and transfers momentum k per unit path length traveled in the solid
71
is given in term of the ELF Im[−1/ε(k, ω)], the complex dielectric function
72
ε(k, ω) is primarily responsible for the medium response. The main purpose of
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the present work is to obtain the ELF that is assumed to be given by Z ∞ −1 −1 = G(ωp )Im dωp , Im ε(k, ω) εM (k, ω) 0
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(1)
where ωp is the plasmon energy, G(ωp ) is an unknown function that will be
75
defined explicitly later, εM (k, ω) is the Mermin dielectric function [25] (1 + iγ/ω)[εL (k, ω + iγ) − 1] , εL (k, ω + iγ) − 1 1 + (iγ/ω) εL (k, 0 + iγ) − 1
(2)
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εM (k, ω) = 1 +
M
74
εL (k, ω + iγ) is the Lindhard dielectric function [24], and γ is a phenomenolog-
77
ical parameter that represents the damping rate which is closely related to the
78
plasmon lifetime.
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It can be seen that the Mermin dielectric function presented here slightly
differences from the original form by the static limit term εL (k, 0 + iγ) instead of εL (k, 0) as of Mermin [25]. The absence of damping γ in εL (k, 0) causes the Mermin function behaves different from the Lindhard function in the longwavelength limit, which must be as a Dirac δ-function. This discrepancy can
be interpreted by considering the Lindhard dielectric function, which is more conveniently expressed in terms of the dimensionless variables z = k/(2kF ) and µ = (ω + iγ)/(kvF ) as [24] εL (z, µ) = 1 +
z−µ+1 χ2 1 1 + 1 − (z − µ)2 ln 2 z 2 8z z−µ−1 z+µ+1 + 1 − (z + µ)2 ln , z+µ−1
(3)
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where χ2 = 1/(πkF ), kF and vF is the Fermi momentum and velocity, respec-
80
tively. In the long-wavelength limit,
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79
lim Im(εL ) = 0+ ,
81
and the real part of εL can be approximated by
√
Ω2p χ2 1 = 1 − , z 2 3u2 ω2
(5)
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Re(εL ) ≃ 1 −
cr
(4)
k→0
82
where u = ω/(kvF ), Ωp =
83
tron density. Thus, the Lindhard-ELF Im(−1/εL ) = 0 elsewhere, except for
84
ω = Ωp where Re(εL ) = 0 and hence Im(−1/εL ) = +∞. In the other words,
85
the Lindhard-ELF behaves as a Dirac δ-function in the long-wavelength limit.
86
Consequently, the Mermin-ELF Im(−1/εM ) must also behave as a δ-function
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in the limit of k → 0. This property, however, will not take place if the term
M
87
4πne is the plasmon energy, and ne is the elec-
εL (k, 0 + iγ) in Eq. (2) is replaced by εL (z, 0), which is a non-zero quantity.
89
2.1. Function G and Kramer–Kronig sum rule
by the Drude-type ELF, Im
ωγΩ2p −1 = 2 , εM (k, ω) (ω − ̟k2 )2 + ω 2 γ 2
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In the long-wavelength limit, the function Im(−1/εM) can be approximated
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92
93
(6)
where ̟k is the plasmon dispersion relation. It is straightforward to see that γ ≃ FWHM assuming FWHM ≪ 4Ωp , where FWHM is the full width at half
94
maximum of the Drude-type ELF. Ritchie and Howie [5] have used directly
95
the Drude-type ELF for function Im(−1/ε). The presence of factor Ω2p in the
96
97
right-hand side of Eq. (6) is to satisfy the sum rule Z ∞ π −1 dω = Ω2p , ωIm ε(k, ω) 2 0
(7)
because the integral Z
0
∞
(ω 2
π ω2γ dω = − ̟k2 )2 + ω 2 γ 2 2
(8)
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does not depend on the plasmon dispersion relation ̟k . Substituting Eq. (6)
99
into Eq. (1), and using the sum rule (7), we obtain Z
∞
Z
G(ωp )dωp
0
0
Use of integral (8) in Eq. (9) leads to Z ∞ G(ω)dω = 1.
(10)
102
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0
101
(9)
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100
ω 2 γΩ2p π dω = Ω2p . 2 2 2 2 2 (ω − ̟k ) + ω γ 2
∞
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On the other hand, from Eq. (6), considering the δ-behavior of the MerminELF in the limit of γ → 0 and the sum rule (7), we can write −1 π ωp2 lim Im = δ(ω − ̟k ). γ→0 εM (k, ω) 2 ̟k
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(11)
As known, the ELF is identical to the OELF at the optical limit (k = 0), where
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and hence
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̟k=0 = ωp . Therefore, use of Eq. (11) in Eq. (1) for k = 0 leads to −1 −1 Im = Im ε(ω) ε(0, ω) Z ∞ π = G(ωp ) ωp δ(ω − ωp )dωp , 2 0
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G(ω) =
2 −1 Im . πω ε(ω)
(12)
(13)
104
Finally, the Kramer–Kronig sum rule can be obtained by substituting Eq. (13)
105
into Eq. (10),
2 π
Z
∞
0
1 −1 Im dω = 1. ω ε(ω)
(14)
106
It can be seen that the function G(ω) in Eq. (13) completely coincides with that
107
obtained by Penn [7], and the ELF (1) is an expansion of Penn-ELF, because in
108
the limit of zero damping, the Mermin dielectric function recovers the Lindhard
109
dielectric function,
lim εM (ω, k) = εL (ω, k),
γ→0 110
(15)
which is used in the Penn algorithm without damping.
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2.2. Dielectric function approximation
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The Lindhard dielectric function (3) can be rewritten in the form χ2 1 1 − z 2 − µ2 (z + 1)2 − µ2 + ln εL (z, µ) = 1 + 2 z 2 8z (z − 1)2 − µ2 µ z 2 − (µ − 1)2 . + ln 2 4 z − (µ + 1)2
cr
(16)
In the limit of z ≪ 1, the imaginary part of the first logarithm term approaches
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zero much faster than that of the second logarithm term. We therefore only approximate this first logarithm term by expanding it in a power series of the
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appropriate small parameter and present the real and imaginary parts of εL , respectively, as
χ2 1 1 − z 2 − µ2 Re + 2 2 z 2 2(z + 1 − u2 + g 2 ) 4z 2 × 1+ 3(z 2 + 1 − u2 + g 2 )2 µ z 2 − (µ − 1)2 , + ln 2 4 z − (µ + 1)2
χ2 µ z 2 − (µ − 1)2 Im(εL ) = 2 Im ln , z 4 z 2 − (µ + 1)2
(18)
where g = γ/(kvF ). It should be noted that though z ≪ 1, the presence of z 2
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and
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(17)
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M
Re(εL ) = 1 +
114
115
116
in this approximation is still essential. Additionally, two following conditions: z 2 + 1 − u2 + g 2 < 0, and z 2 + 1 − u2 + g 2 ≫ z must be satisfied. In Fig. 1, the SPA- and MPA-ELF for Al, Si, Cu, and Au are shown with
117
γ = 0.01 eV [68]. As can be seen, the tail of the SPA-ELF corresponding to the
118
plasmon excitation region lengthens along the dispersion line, meanwhile in the
119
MPA-ELF it quickly decays when entering into the single-electron excitation
120
region. This is because the SPA employs the Dirac δ-function and a dispersion
121
relation to extrapolate the OELF into the non-zero momentum transfer region,
122
neglecting the single-electron excitation. In contrast, the both single-electron
123
and plasmon excitations are taken into account in the MPA that is based on
124
the LPA.
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It is difficult to evaluate the difference between the MPA- and LPA-ELFs
126
on figure, though the plasmon damping is included in the MPA. The influence
127
of damping will be pointed out later in the IMFP results. In the ELF calcula-
128
tions, the MPA is much simpler than the LPA used in our previous study [23].
129
The ELF-LPA is described by a sum of two contributions, one associated with
130
the plasmon excitation and the other with the single-electron excitation, and
131
requires the fairly complicated procedures to obtain the desired result. Also, an
132
advantage of MPA is to allow to predict the values of damping and lifetime of
133
the material of interest at low energies, as will be shown at the end of the next
134
section.
135
3. Calculations and results
136
3.1. Inelastic mean free path
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Once the ELF is determined, the IMFP λin corresponding to the electron kinetic energy E is calculated by Z ωmax Z k+ 1 1 −1 −1 λin (E) = dω Im dk, πE 0 ε (k, ω) k− k where k± =
(19)
p p E(2 + E/c2 ) ± (E − ω)[2 + (E − ω)/c2 ] are the maximum (k+ )
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125
140
and minimum (k− ) momentum transfers, respectively; ωmax is the maximum en-
141
ergy loss. In the previous study [23], we have proposed an expression for ωmax
142
that takes into account the exchange effect between the incident electron and
143
electron in solids. This expression was originally suggested by V. A. Smolar
144
(private communication) based on the work of Hippler [69] for single-electron
145
excitation. We then added the plasmon excitation contribution in an appropri-
146
ate way and employed in the SP calculations with LPA for 10 elements. Using
147
these SPs in the Monte Carlo simulation, we obtained the backscattering elec-
148
tron yields in good agreement with the experimental measurement and much
149
better than the other simulation result. Therefore this expression of ωmax is
150
also employed in the present work.
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The IMFPs for Al, Si, Cu, and Au are shown in Fig. 2 for three differ-
152
ence damping values: 0.01 eV, 0.1 eV, and 0.5 eV. Other available theoretical
153
results [2, 7, 14, 21, 34, 61–65] and experimental values [1, 41–60] of IMFP
154
are also included. It can be seen that, in general, the theoretical calculations
155
are in reasonable agreement with experimental measurements at high energies,
156
and there is no significant difference between theoretical values in this energy
157
region. At medium energies, the fitting formulas [2, 64] either over- or under-
158
estimate the IMFP value, excluding the two-term formula of Ziaja et al. [65].
159
The differences between the theoretical results become more apparent at low-
160
medium energies. Particularly, in the case of Cu and Au, the IMFPs of Ashley et
161
al. [34, 61] are totally different from those obtained from models based on the
162
Penn algorithm (SPA [7], LPA [21], and MPA) in the low-energy region. As
163
known, the Lindhard dielectric function εL is for the degenerate Fermi–Dirac
164
electron gas at absolute zero temperature, it is expected to be a good approxima-
165
tion for describing the momentum dependence of valence-electron excitations in
166
free-electron-like metal such as Al. The Lindhard-ELF Im(−1/εL) is employed
167
directly in the role of ELF Im(−1/ε) in the electron gas statistical model of Ash-
168
ley et al. [61]. In later work [34], they used Mermin-ELF Im(−1/εM) instead of
169
ELF Im(−1/ε) in the IMFP calculations for Al. Nonetheless, the Lindhard- as
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151
170
well as the Mermin-ELF are quite different from the ELF extrapolated from the
171
OELF. Use of Im(−1/εL ) or Im(−1/εM) as the ELF should only be considered
172
as an approximation for elements that have only one main plasmon peak, such
173
as Al and Si. This explains why for Cu or Au, which has many peaks in the
174
ELF as can be seen from Fig. 1, the IMFPs of Ashley et al. [61] are totally
175
different from our results at low energies.
176
It is important to emphasize that our approach may be less reliable for
177
non-free-electron-like materials or for semiconductor with a wide band gap and
178
complex band structure. The dielectric function theory is less likely to be cor-
179
rect for low-energy electrons, particularly in the view of possible failure of the
180
Born approximation in the Penn algorithm [7]. Furthermore, the lack of band
181
structure calculations makes the present IMFP in the low-energy region is ques9
Page 10 of 23
tionable since the plasmon energy loss is sensitive to the change in the band
183
structure [70]. Recent ab initio studies of Campillo et al. [30, 31] on hot-electron
184
lifetimes in metals have shown that the IMFP at low energies is very sensitive
185
to details of the band structure, which is important for the electron-electron
186
decay mechanism. These first-principles calculations were carried out using the
187
plane wave pseudopotential method with the local-density approximation for
188
the exchange-correlation functional. Also, the significant difference between our
189
IMFPs and those obtained by Ziaja et al. formula [65] has shown the effect of
190
band structure on the IMFP calculations at low energies. Ziaja et al. [65] fit-
191
ted their two-term formula to experimental data of IMFP for impact ionization
192
over a wide energy range, including the results of band structure calculations.
193
Therefore, although our calculations for Au at a few eV above the Fermi energy
194
are in reasonable agreement with the measurements of Sze et al. [42] as shown
195
in Fig. 2(d), the present results should only be considered as an illustration of
196
the IMFP trend in the low-energy region.
197
3.2. Plasmon damping and lifetime
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198
The role of damping included in the MPA is only observed at very low
199
energies for metals, whereas for semiconductor there is no clear difference in the IMFPs. The influence of γ on λin can be seen more clearly by comparing the
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201
202
IMFPs at a fixed energy E − EF for various damping values, typically in the range of 0.01–1.5 eV, as illustrated in Fig. 3 for Al and Au.
203
As can be seen from Fig. 3, the λin reaches a minimum in this damping
204
range. We believe this property of IMFP is useful for determining the lifetime
205
τ . According to the uncertainty principle, τ ∆E =
~ , 2
(20)
206
where ∆E is the energy uncertainty. It should be noted that the CGS system
207
is used hereafter. It is assumed that ∆E = γ/2, where γ is the FWHM as
208
mentioned in the discussion following Eq. (6). This assumption leads to the
10
Page 11 of 23
well-known formula of plasmon lifetime τ=
~ . γ
(21)
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209
Thus, an accurate value of damping is required to obtain the lifetime from
211
Eq. (21). In fact, the hot-electron lifetimes in noble metals are usually deter-
212
mined experimentally by time-resolved two-photon photoemission spectroscopy
213
[26–29], or can be calculated from first principles [30, 31].
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210
In the present work, we report an important property of IMFP that is be-
215
lieved to be useful for determining the value of γ and τ . Deriving from at-
216
tempts to improve the accuracy of the calculated IMFP at low energies, we
217
an
214
recognize that at a fixed energy E − EF , the IMFP depends on the damping as a parabolic function. We assume that the γ corresponding to the minimum
219
of λin is the desired damping value, and illustrate this observation with two
220
examples shown in Fig. 3. It can be seen that in the case of Al, λin reaches min-
221
imum at γAl, 5 eV = 0.30 eV, whereas of Au is at γAu, 6 eV = 0.39 eV. Using these
222
Al, 5 eV Au, 6 eV damping values in Eq. (21), we obtain τpredict = 2.18 fs, and τpredict = 1.70 fs.
223
On the other hand, the value of τ can be estimated from the electron-electron
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IMFP λe-e by relation [58]
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224
M
218
τ=
λe-e , vgr
(22)
225
where vgr is the group velocity of an electron of energy E that is assumed to be
226
given by
E=
1 ∗ 2 m v , 2 e gr
(23)
227
where m∗e is the effective electron mass. The calculation of m∗e /me (me being the
228
free-electron mass) requires a detailed knowledge of the band structure. Such
229
calculation has been performed by Ferrari et al. [71] using the Drude–Lorentz
230
model of the optical spectrum for carbon film. Here, we use m∗e = 1.06 me
231
for Al [72], and m∗e = 1.09 me for Au [73]. Use of Eq. (23) for an electron of
232
Al, 5 eV energy E − EF = 5 eV in Al (EF = 11.2 eV [21]) yields vgr = 23.19 ˚ A/fs.
233
Substituting this value into Eq. (22), and using the experimental value [46]
234
Al, 5 eV 5 eV λAl, = 50 ± 5 ˚ A, we obtain τestimate = 2.16 ± 0.22 fs. Comparing with e-e
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Page 12 of 23
236
237
Al, 5 eV τpredict = 2.18 fs obtained above, we see that the predicted value is consistent
with the estimated value. Similarly, for an electron of energy E − EF = 6 eV
Au, 6 eV 6 eV in Au (EF = 9 eV [21]), vgr = 22.00 ˚ A/fs. With λAu, = 42 ± 7 ˚ A [46], e-e
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235
Au, 6 eV Au, 6 eV we have τestimate = 1.91 ± 0.32 fs. Comparing with τpredict = 1.70 fs, we also
239
obtain a good agreement.
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238
Further calculations for Al and Cu have been performed to obtain τ in
241
the energy range 2–4 eV, and then compared with experimental values [26–29]
242
and theoretical results [31] from the first-principle calculations of hot-electron
243
lifetimes, as shown in Figs. 4 and 5. An approximate expression for τ in the limit of small energy E − EF and small electron-density parameter rs was derived by
an
244
us
240
245
Quinn and Ferrell [74] within the random-phase approximation (RPA) on the
246
free-electron gas (FEG) model as [30]
263
M
τ=
5/2 rs (E
− EF )2
,
(24)
where rs = (3/4πne )1/3 , and ne being the free-electron gas density. Result ob-
248
tained from this expression is also shown in Figs. 4 and 5 for comparison. It can
249
be seen that the present results seem to be consistent with those obtained from
250
FEG model rather than from ab initio calculations. Also, there are relatively
251
large deviations of experimental data from theoretical values. The difference be-
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247
252
tween theoretical results is significant for energies very near the Fermi energy,
253
this is most likely due to the band structure effect in the low-energy region, as
254
mentioned in the previous section.
255
Considering all the obtained results, we believe that the MPA is useful to
256
predict the values of damping and lifetime at low energies by using the procedure
257
described above. It should be noted that in fact the IMFP does not always
258
increase or decrease monotonically as illustrated in Fig. 3. In that case, it is
259
necessary to fit the calculated IMFPs to a parabolic function, and then locate
260
the minimum of λin and corresponding γ. Finally, in Table 1 we report the
261
predicted values of γ and τ as a function of energy for Al, Si, Cu, and Au in the
262
energy range of 2–9 eV above the Fermi energy. These values may be useful for
263
evaluating the accuracy of our approach. 12
Page 13 of 23
Table 1: Predicted plasmon damping γ (eV) and plasmon lifetime τ (fs) versus energy for Al, Si
Cu
Au
τ
γ
τ
γ
τ
γ
τ
2
0.049
13.37
0.28
2.39
0.26
2.52
0.28
2.35
3
0.104
6.35
0.42
1.55
0.38
1.73
0.38
1.72
4
0.185
3.57
0.57
1.15
0.41
1.62
0.31
2.09
5
0.30
2.18
0.71
0.93
0.39
1.67
0.32
2.03
6
0.46
1.44
0.84
0.78
0.43
1.52
0.39
1.70
7
0.65
1.01
0.99
0.66
0.51
1.28
0.38
1.74
8
0.89
0.74
1.16
0.57
0.57
1.16
0.36
1.80
9
1.16
0.57
1.35
0.49
0.58
1.13
0.35
1.88
us
γ
an
264
Al
(eV)
cr
E − EF
ip t
Si, Cu, and Au.
4. Conclusions
We have described an approach to include plasmon damping in the ELF
266
calculation using the Penn algorithm and the Mermin dielectric function. The
267
MPA presented here is much simpler than the LPA but still takes advantage
268
of its power for calculating ELF, and allows to predict the values of damping
269
and lifetime at low energies. The calculated results show that plasmon damping
270
significantly influences the IMFP for metals in the low-energy region but declines
271
rapidly as the electron energy increases. In our approach, the damping value is
272
not only energy dependent but also material dependent.
Ac ce p
te
d
M
265
273
Acknowledgments
274
The author is grateful to Professor V. A. Smolar for his help in the work.
275
The author thank Dr. S. Tanuma for providing experimental data on inelastic
276
electron mean free paths in Refs. [1, 2].
277
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17
Page 18 of 23
ip t cr us an M d te Ac ce p
Figure 1: The SPA- and MPA-ELF for: ((a) and (b)) Al, ((c) and (d)) Si, ((e) and (f)) Cu, and ((g) and (h)) Au.
18
Page 19 of 23
3
cr
Ashley et al. (1976) Tanuma et al. (1991) Werner et al. (2000, 2001) Tanuma et al. (2011) Ziaja et al. (2006)
2
10
1
1
10
10 MPA. γ = 0.01 (eV) MPA. γ = 0.1 (eV) MPA. γ = 0.5 (eV)
0
10
1
10
2
3
10
4
10
MPA. γ = 0.01 (eV) MPA. γ = 0.1 (eV) MPA. γ = 0.5 (eV)
(b) Si 0
10
10
3
Gobeli and Allen (1962) Kane (1967) Pierce and Spicer (1972) Klasson et al. (1974) Flitsch and Raider (1975) Tanuma et al. (2005)
us
2
10
10
Kanter (1970) Kanter (1970) Callcott and Arakawa (1975)
an
Inelastic Mean Free Path (˚ A)
Ashley et al. (1976) Ashley et al. (1979) Penn (1987) Ashley (1988) Ashley (1990) Tanuma et al. (1991) Werner et al. (2000, 2001) Tanuma et al. (2011) Ziaja et al. (2006)
(a) Al
1
10
2
10
3
4
10
10
3
Ashley et al. (1976) Penn (1987) Ashley (1988) Ashley (1990) Tanuma et al. (1991) Werner et al. (2000, 2001) Tanuma et al. (2008) Tanuma et al. (2011) Ziaja et al. (2006)
2
2
10
1
10
MPA. γ = 0.01 (eV) MPA. γ = 0.1 (eV) MPA. γ = 0.5 (eV)
(c) Cu 0
1
10
2
3
10 10 E − EF (eV)
Ac ce p
10
te
d
10
10
Seah (1972) Pierce and Siegmann (1974) Knapp et al. (1978, 1979) Himpsel and Eberhardt (1979) Burke and Schreurs (1982) Tanuma et al. (2005)
M
10
Inelastic Mean Free Path (˚ A)
ip t
3
10
4
Sze et al. (1964) Palmberg and Rhodin (1968) Baer et al. (1970) Kanter (1970) Henke (1972) Klasson et al. (1972) Lindau et al. (1976) Gergely et al. (2004) Tanuma et al. (2005)
1
10
MPA. γ = 0.01 (eV) MPA. γ = 0.1 (eV) MPA. γ = 0.5 (eV)
(d) Au 0
10
Ashley et al. (1976) Penn (1987) Ashley (1988) Ashley (1990) Tanuma et al. (1991) Werner et al. (2000, 2001) Tanuma et al. (2008) Tanuma et al. (2011)
10
1
10
2
3
10 10 E − EF (eV)
4
10
Figure 2: Inelastic mean free path versus energy above the Fermi energy for: (a) Al, (b) Si, (c) Cu, and (d) Au. The MPA calculations for three different damping values: 0.5 eV (thick dashed line), 0.1 eV (thick dash-dotted line), and 0.01 eV (thick solid line), respectively. The other theoretical models: Ashley et al. [61] (⊳), Ashley et al. [34] (⊲), Penn [7] (◦), Ashley [62] (+), Ashley [63] (×), Tanuma et al. [14] (▽), and Tanuma et al. [21] (dot symbol). The fitting formulas: Werner et al. [64] (thin solid line), Ziaja et al. [65] (thin dashed line), and Tanuma et al. [2] (⋄). Other symbols are experimental data (Refs. [1, 41–60]).
19
Page 20 of 23
ip t cr (a) Al, E − E = 5 eV
(b) Au, E − E = 6 eV
F
61
F
24.5
59
M
λin (˚ A)
60
58
24
57
d
56
0.3
0.6
0.9
γ (eV)
1.2
1.5
23.5
23
0
0.3
0.6
0.9
γ (eV)
1.2
1.5
Ac ce p
0
te
55 54
us
25
an
62
Figure 3: Inelastic mean free path versus plasmon damping for: (a) Al at energy E − EF = 5 eV, and (b) Au at energy E − EF = 6 eV.
20
Page 21 of 23
ip t cr
14
us
Campillo et al. (2000) − Ab initio Campillo et al. (2000) − FEG Campillo et al. (2000) − Band structure Bauer et al. (1998) Quinn and Ferrell (1958, 1962) − FEG Present work
12
an
8
M
τ (fs)
10
d
6
2.5
Ac ce p
2 2
te
4
3 E − EF (eV)
3.5
4
Figure 4: Plasmon lifetime in Al (rs = 2.07). The solid squares are the predicted values by MPA. The dashed line is the result obtained from Eq. (24) for a FEG. The solid down triangles
are experimental data extracted from Fig. 3 in Ref. [29]. The solid circles, solid line, and open up triangles are data extracted from Fig. 2 in Ref. [31]: the solid circles represent the full ab initio calculations, the solid line represents the GW-RPA calculations for a FEG, and the open up triangles is the result of using plane-waves and band structure in the calculations.
21
Page 22 of 23
ip t cr
7
Campillo et al. (2000) − Ab initio Campillo et al. (2000) − FEG Bauer et al. (1997) Ogawa et al. (1997) − Cu(110) Ogawa et al. (1997) − Band structure Schmuttenmaer et al. (1994) − Cu(100) Quinn and Ferrell (1958, 1962) − FEG Present work
us
6
an
4
M
τ (fs)
5
3
te 2.5
3 E − E (eV)
3.5
4
F
Ac ce p
1 2
d
2
Figure 5: Plasmon lifetime in Cu (rs = 2.67). The solid squares are the predicted values by MPA. The dash-dotted line is the result obtained from Eq. (24) for a FEG. The solid circles and
solid line are data extracted from Fig. 2 in Ref. [27]: the solid circles are experimental values, and the solid line is the result calculated by Fermi-liquid theory using a band-structure model. The solid up and down triangles are experimental data extracted from Fig. 1 in Ref. [28] and Fig. 2 in Ref. [26], respectively. The dotted and dashed lines are data extracted from Fig. 10 in Ref. [31]: the dotted line represents the full ab initio calculations, and the dashed line is the result of using plane-waves in the FEG calculations.
22
Page 23 of 23
S0368-2048(14)00079-6 http://dx.doi.org/doi:10.1016/j.elspec.2014.03.010 ELSPEC 46267
To appear in:
Journal of Electron Spectroscopy and Related Phenomena
Received date: Revised date: Accepted date:
10-2-2014 23-3-2014 24-3-2014
Please cite this article as: Hieu T. Nguyen-Truong, Energy-loss function including damping and prediction of plasmon lifetime, Journal of Electron Spectroscopy and Related Phenomena (2014), http://dx.doi.org/10.1016/j.elspec.2014.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Research Highlights
Highlight • We present an approach to take damping into account of the energy loss function.
• We predict the values of damping and lifetime at low energies.
cr
• Influence of damping is significant in the low-energy region.
ip t
• We calculate the electron inelastic mean free path for Al, Si, Cu, and Au.
Ac ce p
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M
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• The damping value is not only energy dependent but also material dependent.
Page 1 of 23
*Manuscript
Hieu T. Nguyen-Truong∗
ip t
Energy-loss function including damping and prediction of plasmon lifetime
us
cr
Faculty of Electronics and Computer Science, Volgograd State Technical University, 28 Lenin Avenue, Volgograd 400131, Russia
Abstract
an
An approach to include plasmon damping in the energy-loss function is described within the dielectric theory. Use of the energy-loss function included damping for calculating the electron inelastic mean free path yields results in good agree-
M
ment with the experimental data and other theoretical results at medium-high energies. At a few eV above the Fermi energy, the present results are entirely consistent with those obtained from other measurements for Au. Also, a simple
d
way to predict the values of damping and lifetime at low energies is described.
te
Two values of lifetime for an electron with energy (above the Fermi energy) of 5 eV in Al and 6 eV in Au are predicted to be 2.18 fs and 1.70 fs, respectively.
Ac ce p
These predicted values are in reasonable agreement with those estimated from other measurements at the corresponding energies: 2.16 ± 0.22 fs in Al, and
1.91 ± 0.32 fs in Au.
Keywords: energy-loss function, dielectric response function, inelastic mean free path, plasmon damping, plasmon lifetime
1
1. Introduction
2
The energy-loss function (ELF) is the key quantity for calculating the elec-
3
tron inelastic mean free path (IMFP) within the dielectric theory. Information ∗ Permanent address: Faculty of Materials Science, Ho Chi Minh City University of Science, 227 Nguyen Van Cu Street, 5 District, Ho Chi Minh City, Vietnam. Email address: [email protected] (Hieu T. Nguyen-Truong)
Preprint submitted to Journal of Electron Spectroscopy and Related PhenomenaMarch 23, 2014
Page 2 of 23
on IMFP is important for surface analytical methods such as X-ray photoelec-
5
tron spectroscopy or Auger electron spectroscopy, and is frequently determined
6
by elastic peak electron spectroscopy with taking into account the surface ex-
7
citation effect [1, 2]. Experimentally, the ELF can be determined by analyzing
8
the reflection electron energy loss spectra [3, 4]. Theoretically, the ELF can
9
be extrapolated from the optical energy-loss function (OELF) by various algo-
cr
rithms [5–10].
us
10
ip t
4
One of the most popular algorithms for determining the ELF has been pro-
12
posed by Penn [7]. This algorithm is based on the statistical approximation
13
developed by Lindhard et al. [11] and has been extensively employed within the
14
single-pole approximation (SPA) for calculating the IMFP [7, 12–17] over either
15
200 or 300 eV energy range and the electron stopping power (SP) [18, 19] over
16
100 eV energy range. However, taking into account only the plasmon excitation
17
contribution makes the SPA only suitable for electrons of medium-high energy.
18
Recently, the ELF calculation in the Penn algorithm has been fully performed
19
by an integration of the Lindhard dielectric function without damping [20–23].
20
By including both single-electron and plasmon excitations contributions, this
21
“full Penn algorithm”, which is referred as LPA (stands for Lindhard–Penn
22
algorithm) hereafter, has extend the low-energy range down to the plasmon
Ac ce p
te
d
M
an
11
23
excitation energy.
24
However, neglecting damping in the Lindhard dielectric function [24] makes
25
the Penn algorithm not suitable for very-low-energy electron. As known, at-
26
tempt to include damping in the Lindhard dielectric function by turning the
27
real frequency into complex does not conserve the local number of electrons.
28
Mermin [25] has solved this problem by a phenomenological modification of the
29
Lindhard dielectric function. The Mermin dielectric function including damping
30
is described in term of the Lindhard dielectric function of the complex frequency.
31
In the present work, we describe an approach to include damping in the ELF
32
by using the Mermin dielectric function in the Penn algorithm instead of the
33
Lindhard dielectric function. Our approach, which is referred as MPA (stands
34
for Mermin–Penn algorithm) hereafter, is much simpler than the LPA but still 2
Page 3 of 23
takes advantage of its power. Furthermore, the MPA allows to predict the
36
values of damping and lifetime at low energies, knowledge of which is necessary
37
for understanding the electron transport properties. In the past two decades,
38
this problem has attracted much attention of researchers both in theoretical and
39
experimental studies [26–31].
cr
ip t
35
Including damping in the Lindhard dielectric function has been studied by
41
Ashley and Ritchie [32] in their IMFP calculations for Al. In two later works [33,
42
34] also for Al, they have used directly the Mermin-ELF in the role of the ELF
43
to take damping into account. Use of Mermin- or Lindhard-ELF instead of ELF
44
should only be considered as an approximation for elements that have only one
45
main plasmon peak like Al.
an
us
40
Alternatively, the ELF can be also obtained by fitting the OELF to the
47
theoretical expression (Drude-type [5, 8, 35, 36], or Mermin-type [9, 10, 37]) to
48
determine the fitting parameters (weight, position, and width) corresponding to
49
the plasmon peaks, and then extrapolating into the non-zero momentum trans-
50
fer region by a linear combination of these fitting functions. The comparison
51
between approaches can be found elsewhere (e.g. Refs. [38, 39]).
te
d
M
46
The MPA presented here does not demand any fitting parameters as well
53
as plasmon dispersion relation because the MPA-ELF is defined as an integral
Ac ce p
52
54
of the Mermin-ELF and a function linking with the OELF. Our approach only
55
requires knowledge of the optical dielectric function to obtain the OELF. In
56
the present work, the OELF is obtained from the experimental optical data in
57
the compilation of Palik [40], which has been checked and used in our previous
58
study [23] for SP calculations. The MPA-ELF then is employed in the IMFP
59
calculations for Al, Si, Cu, and Au in the energy range from 1 eV to 30 keV
60
above the Fermi energy. Comparisons show that our IMFPs are in reasonable
61
agreement with experimental values [1, 41–60] and other theoretical results [2,
62
7, 14, 21, 34, 61–65] at medium-high energies. Finally, a simple way to predict
63
the values of damping and lifetime at low energies is described. Some predicted
64
values of damping and lifetime in the energy range 2–9 eV are also reported,
65
two of which are entirely consistent with those obtained from experiment [46]. 3
Page 4 of 23
Also, it should be noted that the Hartree atomic units (~ = me = e = 1) is used
67
throughout this work.
68
2. Theory
ip t
66
In the dielectric theory [24, 66, 67], the probability that an electron loses
70
energy ω and transfers momentum k per unit path length traveled in the solid
71
is given in term of the ELF Im[−1/ε(k, ω)], the complex dielectric function
72
ε(k, ω) is primarily responsible for the medium response. The main purpose of
us
the present work is to obtain the ELF that is assumed to be given by Z ∞ −1 −1 = G(ωp )Im dωp , Im ε(k, ω) εM (k, ω) 0
an
73
cr
69
(1)
where ωp is the plasmon energy, G(ωp ) is an unknown function that will be
75
defined explicitly later, εM (k, ω) is the Mermin dielectric function [25] (1 + iγ/ω)[εL (k, ω + iγ) − 1] , εL (k, ω + iγ) − 1 1 + (iγ/ω) εL (k, 0 + iγ) − 1
(2)
d
εM (k, ω) = 1 +
M
74
εL (k, ω + iγ) is the Lindhard dielectric function [24], and γ is a phenomenolog-
77
ical parameter that represents the damping rate which is closely related to the
78
plasmon lifetime.
Ac ce p
te
76
It can be seen that the Mermin dielectric function presented here slightly
differences from the original form by the static limit term εL (k, 0 + iγ) instead of εL (k, 0) as of Mermin [25]. The absence of damping γ in εL (k, 0) causes the Mermin function behaves different from the Lindhard function in the longwavelength limit, which must be as a Dirac δ-function. This discrepancy can
be interpreted by considering the Lindhard dielectric function, which is more conveniently expressed in terms of the dimensionless variables z = k/(2kF ) and µ = (ω + iγ)/(kvF ) as [24] εL (z, µ) = 1 +
z−µ+1 χ2 1 1 + 1 − (z − µ)2 ln 2 z 2 8z z−µ−1 z+µ+1 + 1 − (z + µ)2 ln , z+µ−1
(3)
4
Page 5 of 23
where χ2 = 1/(πkF ), kF and vF is the Fermi momentum and velocity, respec-
80
tively. In the long-wavelength limit,
ip t
79
lim Im(εL ) = 0+ ,
81
and the real part of εL can be approximated by
√
Ω2p χ2 1 = 1 − , z 2 3u2 ω2
(5)
us
Re(εL ) ≃ 1 −
cr
(4)
k→0
82
where u = ω/(kvF ), Ωp =
83
tron density. Thus, the Lindhard-ELF Im(−1/εL ) = 0 elsewhere, except for
84
ω = Ωp where Re(εL ) = 0 and hence Im(−1/εL ) = +∞. In the other words,
85
the Lindhard-ELF behaves as a Dirac δ-function in the long-wavelength limit.
86
Consequently, the Mermin-ELF Im(−1/εM ) must also behave as a δ-function
an
in the limit of k → 0. This property, however, will not take place if the term
M
87
4πne is the plasmon energy, and ne is the elec-
εL (k, 0 + iγ) in Eq. (2) is replaced by εL (z, 0), which is a non-zero quantity.
89
2.1. Function G and Kramer–Kronig sum rule
by the Drude-type ELF, Im
ωγΩ2p −1 = 2 , εM (k, ω) (ω − ̟k2 )2 + ω 2 γ 2
Ac ce p
91
In the long-wavelength limit, the function Im(−1/εM) can be approximated
te
90
d
88
92
93
(6)
where ̟k is the plasmon dispersion relation. It is straightforward to see that γ ≃ FWHM assuming FWHM ≪ 4Ωp , where FWHM is the full width at half
94
maximum of the Drude-type ELF. Ritchie and Howie [5] have used directly
95
the Drude-type ELF for function Im(−1/ε). The presence of factor Ω2p in the
96
97
right-hand side of Eq. (6) is to satisfy the sum rule Z ∞ π −1 dω = Ω2p , ωIm ε(k, ω) 2 0
(7)
because the integral Z
0
∞
(ω 2
π ω2γ dω = − ̟k2 )2 + ω 2 γ 2 2
(8)
5
Page 6 of 23
does not depend on the plasmon dispersion relation ̟k . Substituting Eq. (6)
99
into Eq. (1), and using the sum rule (7), we obtain Z
∞
Z
G(ωp )dωp
0
0
Use of integral (8) in Eq. (9) leads to Z ∞ G(ω)dω = 1.
(10)
102
us
0
101
(9)
cr
100
ω 2 γΩ2p π dω = Ω2p . 2 2 2 2 2 (ω − ̟k ) + ω γ 2
∞
ip t
98
On the other hand, from Eq. (6), considering the δ-behavior of the MerminELF in the limit of γ → 0 and the sum rule (7), we can write −1 π ωp2 lim Im = δ(ω − ̟k ). γ→0 εM (k, ω) 2 ̟k
an
(11)
As known, the ELF is identical to the OELF at the optical limit (k = 0), where
103
and hence
te
d
M
̟k=0 = ωp . Therefore, use of Eq. (11) in Eq. (1) for k = 0 leads to −1 −1 Im = Im ε(ω) ε(0, ω) Z ∞ π = G(ωp ) ωp δ(ω − ωp )dωp , 2 0
Ac ce p
G(ω) =
2 −1 Im . πω ε(ω)
(12)
(13)
104
Finally, the Kramer–Kronig sum rule can be obtained by substituting Eq. (13)
105
into Eq. (10),
2 π
Z
∞
0
1 −1 Im dω = 1. ω ε(ω)
(14)
106
It can be seen that the function G(ω) in Eq. (13) completely coincides with that
107
obtained by Penn [7], and the ELF (1) is an expansion of Penn-ELF, because in
108
the limit of zero damping, the Mermin dielectric function recovers the Lindhard
109
dielectric function,
lim εM (ω, k) = εL (ω, k),
γ→0 110
(15)
which is used in the Penn algorithm without damping.
6
Page 7 of 23
111
2.2. Dielectric function approximation
ip t
The Lindhard dielectric function (3) can be rewritten in the form χ2 1 1 − z 2 − µ2 (z + 1)2 − µ2 + ln εL (z, µ) = 1 + 2 z 2 8z (z − 1)2 − µ2 µ z 2 − (µ − 1)2 . + ln 2 4 z − (µ + 1)2
cr
(16)
In the limit of z ≪ 1, the imaginary part of the first logarithm term approaches
us
zero much faster than that of the second logarithm term. We therefore only approximate this first logarithm term by expanding it in a power series of the
an
appropriate small parameter and present the real and imaginary parts of εL , respectively, as
χ2 1 1 − z 2 − µ2 Re + 2 2 z 2 2(z + 1 − u2 + g 2 ) 4z 2 × 1+ 3(z 2 + 1 − u2 + g 2 )2 µ z 2 − (µ − 1)2 , + ln 2 4 z − (µ + 1)2
χ2 µ z 2 − (µ − 1)2 Im(εL ) = 2 Im ln , z 4 z 2 − (µ + 1)2
(18)
where g = γ/(kvF ). It should be noted that though z ≪ 1, the presence of z 2
Ac ce p
113
and
te
112
(17)
d
M
Re(εL ) = 1 +
114
115
116
in this approximation is still essential. Additionally, two following conditions: z 2 + 1 − u2 + g 2 < 0, and z 2 + 1 − u2 + g 2 ≫ z must be satisfied. In Fig. 1, the SPA- and MPA-ELF for Al, Si, Cu, and Au are shown with
117
γ = 0.01 eV [68]. As can be seen, the tail of the SPA-ELF corresponding to the
118
plasmon excitation region lengthens along the dispersion line, meanwhile in the
119
MPA-ELF it quickly decays when entering into the single-electron excitation
120
region. This is because the SPA employs the Dirac δ-function and a dispersion
121
relation to extrapolate the OELF into the non-zero momentum transfer region,
122
neglecting the single-electron excitation. In contrast, the both single-electron
123
and plasmon excitations are taken into account in the MPA that is based on
124
the LPA.
7
Page 8 of 23
It is difficult to evaluate the difference between the MPA- and LPA-ELFs
126
on figure, though the plasmon damping is included in the MPA. The influence
127
of damping will be pointed out later in the IMFP results. In the ELF calcula-
128
tions, the MPA is much simpler than the LPA used in our previous study [23].
129
The ELF-LPA is described by a sum of two contributions, one associated with
130
the plasmon excitation and the other with the single-electron excitation, and
131
requires the fairly complicated procedures to obtain the desired result. Also, an
132
advantage of MPA is to allow to predict the values of damping and lifetime of
133
the material of interest at low energies, as will be shown at the end of the next
134
section.
135
3. Calculations and results
136
3.1. Inelastic mean free path
cr
us
an
Once the ELF is determined, the IMFP λin corresponding to the electron kinetic energy E is calculated by Z ωmax Z k+ 1 1 −1 −1 λin (E) = dω Im dk, πE 0 ε (k, ω) k− k where k± =
(19)
p p E(2 + E/c2 ) ± (E − ω)[2 + (E − ω)/c2 ] are the maximum (k+ )
Ac ce p
139
te
d
138
M
137
ip t
125
140
and minimum (k− ) momentum transfers, respectively; ωmax is the maximum en-
141
ergy loss. In the previous study [23], we have proposed an expression for ωmax
142
that takes into account the exchange effect between the incident electron and
143
electron in solids. This expression was originally suggested by V. A. Smolar
144
(private communication) based on the work of Hippler [69] for single-electron
145
excitation. We then added the plasmon excitation contribution in an appropri-
146
ate way and employed in the SP calculations with LPA for 10 elements. Using
147
these SPs in the Monte Carlo simulation, we obtained the backscattering elec-
148
tron yields in good agreement with the experimental measurement and much
149
better than the other simulation result. Therefore this expression of ωmax is
150
also employed in the present work.
8
Page 9 of 23
The IMFPs for Al, Si, Cu, and Au are shown in Fig. 2 for three differ-
152
ence damping values: 0.01 eV, 0.1 eV, and 0.5 eV. Other available theoretical
153
results [2, 7, 14, 21, 34, 61–65] and experimental values [1, 41–60] of IMFP
154
are also included. It can be seen that, in general, the theoretical calculations
155
are in reasonable agreement with experimental measurements at high energies,
156
and there is no significant difference between theoretical values in this energy
157
region. At medium energies, the fitting formulas [2, 64] either over- or under-
158
estimate the IMFP value, excluding the two-term formula of Ziaja et al. [65].
159
The differences between the theoretical results become more apparent at low-
160
medium energies. Particularly, in the case of Cu and Au, the IMFPs of Ashley et
161
al. [34, 61] are totally different from those obtained from models based on the
162
Penn algorithm (SPA [7], LPA [21], and MPA) in the low-energy region. As
163
known, the Lindhard dielectric function εL is for the degenerate Fermi–Dirac
164
electron gas at absolute zero temperature, it is expected to be a good approxima-
165
tion for describing the momentum dependence of valence-electron excitations in
166
free-electron-like metal such as Al. The Lindhard-ELF Im(−1/εL) is employed
167
directly in the role of ELF Im(−1/ε) in the electron gas statistical model of Ash-
168
ley et al. [61]. In later work [34], they used Mermin-ELF Im(−1/εM) instead of
169
ELF Im(−1/ε) in the IMFP calculations for Al. Nonetheless, the Lindhard- as
Ac ce p
te
d
M
an
us
cr
ip t
151
170
well as the Mermin-ELF are quite different from the ELF extrapolated from the
171
OELF. Use of Im(−1/εL ) or Im(−1/εM) as the ELF should only be considered
172
as an approximation for elements that have only one main plasmon peak, such
173
as Al and Si. This explains why for Cu or Au, which has many peaks in the
174
ELF as can be seen from Fig. 1, the IMFPs of Ashley et al. [61] are totally
175
different from our results at low energies.
176
It is important to emphasize that our approach may be less reliable for
177
non-free-electron-like materials or for semiconductor with a wide band gap and
178
complex band structure. The dielectric function theory is less likely to be cor-
179
rect for low-energy electrons, particularly in the view of possible failure of the
180
Born approximation in the Penn algorithm [7]. Furthermore, the lack of band
181
structure calculations makes the present IMFP in the low-energy region is ques9
Page 10 of 23
tionable since the plasmon energy loss is sensitive to the change in the band
183
structure [70]. Recent ab initio studies of Campillo et al. [30, 31] on hot-electron
184
lifetimes in metals have shown that the IMFP at low energies is very sensitive
185
to details of the band structure, which is important for the electron-electron
186
decay mechanism. These first-principles calculations were carried out using the
187
plane wave pseudopotential method with the local-density approximation for
188
the exchange-correlation functional. Also, the significant difference between our
189
IMFPs and those obtained by Ziaja et al. formula [65] has shown the effect of
190
band structure on the IMFP calculations at low energies. Ziaja et al. [65] fit-
191
ted their two-term formula to experimental data of IMFP for impact ionization
192
over a wide energy range, including the results of band structure calculations.
193
Therefore, although our calculations for Au at a few eV above the Fermi energy
194
are in reasonable agreement with the measurements of Sze et al. [42] as shown
195
in Fig. 2(d), the present results should only be considered as an illustration of
196
the IMFP trend in the low-energy region.
197
3.2. Plasmon damping and lifetime
te
d
M
an
us
cr
ip t
182
198
The role of damping included in the MPA is only observed at very low
199
energies for metals, whereas for semiconductor there is no clear difference in the IMFPs. The influence of γ on λin can be seen more clearly by comparing the
Ac ce p 200
201
202
IMFPs at a fixed energy E − EF for various damping values, typically in the range of 0.01–1.5 eV, as illustrated in Fig. 3 for Al and Au.
203
As can be seen from Fig. 3, the λin reaches a minimum in this damping
204
range. We believe this property of IMFP is useful for determining the lifetime
205
τ . According to the uncertainty principle, τ ∆E =
~ , 2
(20)
206
where ∆E is the energy uncertainty. It should be noted that the CGS system
207
is used hereafter. It is assumed that ∆E = γ/2, where γ is the FWHM as
208
mentioned in the discussion following Eq. (6). This assumption leads to the
10
Page 11 of 23
well-known formula of plasmon lifetime τ=
~ . γ
(21)
ip t
209
Thus, an accurate value of damping is required to obtain the lifetime from
211
Eq. (21). In fact, the hot-electron lifetimes in noble metals are usually deter-
212
mined experimentally by time-resolved two-photon photoemission spectroscopy
213
[26–29], or can be calculated from first principles [30, 31].
us
cr
210
In the present work, we report an important property of IMFP that is be-
215
lieved to be useful for determining the value of γ and τ . Deriving from at-
216
tempts to improve the accuracy of the calculated IMFP at low energies, we
217
an
214
recognize that at a fixed energy E − EF , the IMFP depends on the damping as a parabolic function. We assume that the γ corresponding to the minimum
219
of λin is the desired damping value, and illustrate this observation with two
220
examples shown in Fig. 3. It can be seen that in the case of Al, λin reaches min-
221
imum at γAl, 5 eV = 0.30 eV, whereas of Au is at γAu, 6 eV = 0.39 eV. Using these
222
Al, 5 eV Au, 6 eV damping values in Eq. (21), we obtain τpredict = 2.18 fs, and τpredict = 1.70 fs.
223
On the other hand, the value of τ can be estimated from the electron-electron
d
te
IMFP λe-e by relation [58]
Ac ce p
224
M
218
τ=
λe-e , vgr
(22)
225
where vgr is the group velocity of an electron of energy E that is assumed to be
226
given by
E=
1 ∗ 2 m v , 2 e gr
(23)
227
where m∗e is the effective electron mass. The calculation of m∗e /me (me being the
228
free-electron mass) requires a detailed knowledge of the band structure. Such
229
calculation has been performed by Ferrari et al. [71] using the Drude–Lorentz
230
model of the optical spectrum for carbon film. Here, we use m∗e = 1.06 me
231
for Al [72], and m∗e = 1.09 me for Au [73]. Use of Eq. (23) for an electron of
232
Al, 5 eV energy E − EF = 5 eV in Al (EF = 11.2 eV [21]) yields vgr = 23.19 ˚ A/fs.
233
Substituting this value into Eq. (22), and using the experimental value [46]
234
Al, 5 eV 5 eV λAl, = 50 ± 5 ˚ A, we obtain τestimate = 2.16 ± 0.22 fs. Comparing with e-e
11
Page 12 of 23
236
237
Al, 5 eV τpredict = 2.18 fs obtained above, we see that the predicted value is consistent
with the estimated value. Similarly, for an electron of energy E − EF = 6 eV
Au, 6 eV 6 eV in Au (EF = 9 eV [21]), vgr = 22.00 ˚ A/fs. With λAu, = 42 ± 7 ˚ A [46], e-e
ip t
235
Au, 6 eV Au, 6 eV we have τestimate = 1.91 ± 0.32 fs. Comparing with τpredict = 1.70 fs, we also
239
obtain a good agreement.
cr
238
Further calculations for Al and Cu have been performed to obtain τ in
241
the energy range 2–4 eV, and then compared with experimental values [26–29]
242
and theoretical results [31] from the first-principle calculations of hot-electron
243
lifetimes, as shown in Figs. 4 and 5. An approximate expression for τ in the limit of small energy E − EF and small electron-density parameter rs was derived by
an
244
us
240
245
Quinn and Ferrell [74] within the random-phase approximation (RPA) on the
246
free-electron gas (FEG) model as [30]
263
M
τ=
5/2 rs (E
− EF )2
,
(24)
where rs = (3/4πne )1/3 , and ne being the free-electron gas density. Result ob-
248
tained from this expression is also shown in Figs. 4 and 5 for comparison. It can
249
be seen that the present results seem to be consistent with those obtained from
250
FEG model rather than from ab initio calculations. Also, there are relatively
251
large deviations of experimental data from theoretical values. The difference be-
Ac ce p
te
d
247
252
tween theoretical results is significant for energies very near the Fermi energy,
253
this is most likely due to the band structure effect in the low-energy region, as
254
mentioned in the previous section.
255
Considering all the obtained results, we believe that the MPA is useful to
256
predict the values of damping and lifetime at low energies by using the procedure
257
described above. It should be noted that in fact the IMFP does not always
258
increase or decrease monotonically as illustrated in Fig. 3. In that case, it is
259
necessary to fit the calculated IMFPs to a parabolic function, and then locate
260
the minimum of λin and corresponding γ. Finally, in Table 1 we report the
261
predicted values of γ and τ as a function of energy for Al, Si, Cu, and Au in the
262
energy range of 2–9 eV above the Fermi energy. These values may be useful for
263
evaluating the accuracy of our approach. 12
Page 13 of 23
Table 1: Predicted plasmon damping γ (eV) and plasmon lifetime τ (fs) versus energy for Al, Si
Cu
Au
τ
γ
τ
γ
τ
γ
τ
2
0.049
13.37
0.28
2.39
0.26
2.52
0.28
2.35
3
0.104
6.35
0.42
1.55
0.38
1.73
0.38
1.72
4
0.185
3.57
0.57
1.15
0.41
1.62
0.31
2.09
5
0.30
2.18
0.71
0.93
0.39
1.67
0.32
2.03
6
0.46
1.44
0.84
0.78
0.43
1.52
0.39
1.70
7
0.65
1.01
0.99
0.66
0.51
1.28
0.38
1.74
8
0.89
0.74
1.16
0.57
0.57
1.16
0.36
1.80
9
1.16
0.57
1.35
0.49
0.58
1.13
0.35
1.88
us
γ
an
264
Al
(eV)
cr
E − EF
ip t
Si, Cu, and Au.
4. Conclusions
We have described an approach to include plasmon damping in the ELF
266
calculation using the Penn algorithm and the Mermin dielectric function. The
267
MPA presented here is much simpler than the LPA but still takes advantage
268
of its power for calculating ELF, and allows to predict the values of damping
269
and lifetime at low energies. The calculated results show that plasmon damping
270
significantly influences the IMFP for metals in the low-energy region but declines
271
rapidly as the electron energy increases. In our approach, the damping value is
272
not only energy dependent but also material dependent.
Ac ce p
te
d
M
265
273
Acknowledgments
274
The author is grateful to Professor V. A. Smolar for his help in the work.
275
The author thank Dr. S. Tanuma for providing experimental data on inelastic
276
electron mean free paths in Refs. [1, 2].
277
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17
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ip t cr us an M d te Ac ce p
Figure 1: The SPA- and MPA-ELF for: ((a) and (b)) Al, ((c) and (d)) Si, ((e) and (f)) Cu, and ((g) and (h)) Au.
18
Page 19 of 23
3
cr
Ashley et al. (1976) Tanuma et al. (1991) Werner et al. (2000, 2001) Tanuma et al. (2011) Ziaja et al. (2006)
2
10
1
1
10
10 MPA. γ = 0.01 (eV) MPA. γ = 0.1 (eV) MPA. γ = 0.5 (eV)
0
10
1
10
2
3
10
4
10
MPA. γ = 0.01 (eV) MPA. γ = 0.1 (eV) MPA. γ = 0.5 (eV)
(b) Si 0
10
10
3
Gobeli and Allen (1962) Kane (1967) Pierce and Spicer (1972) Klasson et al. (1974) Flitsch and Raider (1975) Tanuma et al. (2005)
us
2
10
10
Kanter (1970) Kanter (1970) Callcott and Arakawa (1975)
an
Inelastic Mean Free Path (˚ A)
Ashley et al. (1976) Ashley et al. (1979) Penn (1987) Ashley (1988) Ashley (1990) Tanuma et al. (1991) Werner et al. (2000, 2001) Tanuma et al. (2011) Ziaja et al. (2006)
(a) Al
1
10
2
10
3
4
10
10
3
Ashley et al. (1976) Penn (1987) Ashley (1988) Ashley (1990) Tanuma et al. (1991) Werner et al. (2000, 2001) Tanuma et al. (2008) Tanuma et al. (2011) Ziaja et al. (2006)
2
2
10
1
10
MPA. γ = 0.01 (eV) MPA. γ = 0.1 (eV) MPA. γ = 0.5 (eV)
(c) Cu 0
1
10
2
3
10 10 E − EF (eV)
Ac ce p
10
te
d
10
10
Seah (1972) Pierce and Siegmann (1974) Knapp et al. (1978, 1979) Himpsel and Eberhardt (1979) Burke and Schreurs (1982) Tanuma et al. (2005)
M
10
Inelastic Mean Free Path (˚ A)
ip t
3
10
4
Sze et al. (1964) Palmberg and Rhodin (1968) Baer et al. (1970) Kanter (1970) Henke (1972) Klasson et al. (1972) Lindau et al. (1976) Gergely et al. (2004) Tanuma et al. (2005)
1
10
MPA. γ = 0.01 (eV) MPA. γ = 0.1 (eV) MPA. γ = 0.5 (eV)
(d) Au 0
10
Ashley et al. (1976) Penn (1987) Ashley (1988) Ashley (1990) Tanuma et al. (1991) Werner et al. (2000, 2001) Tanuma et al. (2008) Tanuma et al. (2011)
10
1
10
2
3
10 10 E − EF (eV)
4
10
Figure 2: Inelastic mean free path versus energy above the Fermi energy for: (a) Al, (b) Si, (c) Cu, and (d) Au. The MPA calculations for three different damping values: 0.5 eV (thick dashed line), 0.1 eV (thick dash-dotted line), and 0.01 eV (thick solid line), respectively. The other theoretical models: Ashley et al. [61] (⊳), Ashley et al. [34] (⊲), Penn [7] (◦), Ashley [62] (+), Ashley [63] (×), Tanuma et al. [14] (▽), and Tanuma et al. [21] (dot symbol). The fitting formulas: Werner et al. [64] (thin solid line), Ziaja et al. [65] (thin dashed line), and Tanuma et al. [2] (⋄). Other symbols are experimental data (Refs. [1, 41–60]).
19
Page 20 of 23
ip t cr (a) Al, E − E = 5 eV
(b) Au, E − E = 6 eV
F
61
F
24.5
59
M
λin (˚ A)
60
58
24
57
d
56
0.3
0.6
0.9
γ (eV)
1.2
1.5
23.5
23
0
0.3
0.6
0.9
γ (eV)
1.2
1.5
Ac ce p
0
te
55 54
us
25
an
62
Figure 3: Inelastic mean free path versus plasmon damping for: (a) Al at energy E − EF = 5 eV, and (b) Au at energy E − EF = 6 eV.
20
Page 21 of 23
ip t cr
14
us
Campillo et al. (2000) − Ab initio Campillo et al. (2000) − FEG Campillo et al. (2000) − Band structure Bauer et al. (1998) Quinn and Ferrell (1958, 1962) − FEG Present work
12
an
8
M
τ (fs)
10
d
6
2.5
Ac ce p
2 2
te
4
3 E − EF (eV)
3.5
4
Figure 4: Plasmon lifetime in Al (rs = 2.07). The solid squares are the predicted values by MPA. The dashed line is the result obtained from Eq. (24) for a FEG. The solid down triangles
are experimental data extracted from Fig. 3 in Ref. [29]. The solid circles, solid line, and open up triangles are data extracted from Fig. 2 in Ref. [31]: the solid circles represent the full ab initio calculations, the solid line represents the GW-RPA calculations for a FEG, and the open up triangles is the result of using plane-waves and band structure in the calculations.
21
Page 22 of 23
ip t cr
7
Campillo et al. (2000) − Ab initio Campillo et al. (2000) − FEG Bauer et al. (1997) Ogawa et al. (1997) − Cu(110) Ogawa et al. (1997) − Band structure Schmuttenmaer et al. (1994) − Cu(100) Quinn and Ferrell (1958, 1962) − FEG Present work
us
6
an
4
M
τ (fs)
5
3
te 2.5
3 E − E (eV)
3.5
4
F
Ac ce p
1 2
d
2
Figure 5: Plasmon lifetime in Cu (rs = 2.67). The solid squares are the predicted values by MPA. The dash-dotted line is the result obtained from Eq. (24) for a FEG. The solid circles and
solid line are data extracted from Fig. 2 in Ref. [27]: the solid circles are experimental values, and the solid line is the result calculated by Fermi-liquid theory using a band-structure model. The solid up and down triangles are experimental data extracted from Fig. 1 in Ref. [28] and Fig. 2 in Ref. [26], respectively. The dotted and dashed lines are data extracted from Fig. 10 in Ref. [31]: the dotted line represents the full ab initio calculations, and the dashed line is the result of using plane-waves in the FEG calculations.
22
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