Penn Algorithm Including Damping for Calculating the Electron

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Penn Algorithm Including Damping for Calculating the Electron Inelastic Mean Free Path Hieu Thanh Nguyen-Truong J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 12 Mar 2015 Downloaded from http://pubs.acs.org on March 12, 2015

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The Journal of Physical Chemistry

Penn Algorithm Including Damping for Calculating the Electron Inelastic Mean Free Path Hieu T. Nguyen-Truong∗,†,‡ Faculty of Materials Science, Ho Chi Minh City University of Science, 227 Nguyen Van Cu Street, Ho Chi Minh City, Vietnam E-mail: [email protected]

∗To

whom correspondence should be addressed of Materials Science, Ho Chi Minh City University of Science, 227 Nguyen Van Cu Street, Ho Chi Minh City, Vietnam ‡Faculty of Electronics and Computer Science, Volgograd State Technical University, 28 Lenin Avenue, Volgograd 400131, Russia †Faculty

1

ACS Paragon Plus Environment


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Abstract We present an approach for introducing damping into the Penn algorithm by using the Mermin dielectric function instead of the Lindhard dielectric function. We find that for a damping of 1.5 eV, the electron inelastic mean free path calculated by the present algorithm for Al is in excellent agreement with experimental values in the energy range 5–9 eV. Meanwhile for a damping of 2.0 eV our result for Au is consistent with the GW+T ab initio calculation at several eV. In particular, at an energy of 1 eV, our result for Au is 297 ˚A and lies within the range 220–330 ˚A obtained from measurements by ballistic electron emission microscopy.

Keywords electron inelastic mean free path, energy-loss function, dielectric formalism, Mermin dielectric function, Lindhard dielectric function

Introduction Knowledge of the electron inelastic mean free path (IMFP) is of importance for understanding electron transport in solids and is essential for surface analysis methods such as X-ray photoelectron spectroscopy or Auger electron spectroscopy.1 Recent attention has focused on the large discrepancy between theoretical and experimental IMFPs of low-energy electrons.2–6 Most theoretical models proposed so far are unsuitable for electrons of energy less than a few hundred eV. The reason is partially due to the neglect of the broadening effect of plasmon damping. The IMFP can be calculated within the dielectric formalism. Such a calculation requires determination of the energy loss function (ELF) which represents the probability that an incident electron loses energy and transfers momentum per unit path length traveled in the solid. In the optical limit, the ELF can be obtained experimentally by optical reflection or 2


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transmission measurements, 7 or by reflection electron energy loss spectroscopy.8 At fi momentum transfer, the ELF is extrapolated from the optical ELF by the two most common approaches: (a) the Penn algorithm,9 and (b) the linear combination of Drude-type10–12 or Mermin-type13,14 ELF using fi

parameters for oscillators. The latter has attracted

much attention in conjunction with the generalized oscillator strengths model,14,15 a review of which can be found elsewhere (e.g. Ref. 16). One of the main advantages of the fi ting approach is to take plasmon damping into account; meanwhile the plasmon lifetime in the Penn algorithm is infi due to the use of the Lindhard dielectric function 17 without damping. Although the Penn algorithm takes both single-electron and plasmon excitations into account and is a good approximation at high energies (typically larger than 1 keV), neglecting damping makes it unsuitable for low-energy electrons (less than about 100 eV) because plasmon excitation is dominant in this energy range. The implementation of damping is necessary due to the fi

plasmon lifetime in real materials. Another problem of this algo-

rithm is that the Lindhard dielectric function does not conserve the local number of electrons. Such a problem can be solved by using the Mermin dielectric function, 18 which is described in term of the Lindhard dielectric function of the complex frequency. Here we present an approach for introducing damping into the Penn algorithm by using the Mermin dielectric function 18 instead of the Lindhard dielectric function. Our previous study19 shown that such an approach can significantly improve the performance of the Penn algorithm. Nonetheless, the Mermin dielectric function used in previous work was modifi so that the corresponding Mermin-ELF behaves as a Dirac δ-function in the optical limit, like the Lindhard-ELF in the Penn algorithm. In the present work, we use the original Mermin dielectric function to ensure that the plasmon lifetime is fi even in the optical limit. To demonstrate the improvement of the present algorithm, we compare the calculated IMFPs for Al and Au with experimental values 20–22 and the GW(+T) ab initio results23 at several eV above the Fermi energy. Some recent sources of IMFP calculations 24–26 and measurements27–29 are also

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included to give an overview.

Theory Within the dielectric formalism, the IMFP λin of an electron of energy E is given by λ

in

r

1

−1

(E) =

πE

E−EF

dω

r

0

I k+ k−

−1

l

1 Im dk, k ε(k, ω)

(1)

where EF is the Fermi energy, and k± are are the largest and smallest momentum transfers. The ELF Im[−1/ε(k, ω)] represents the probability that an incident electron loses energy ω and transfers momentum k per unit path length traveled in the solid. We note that Hartree atomic units (n = me = e = 1) are used here. We follow Penn 9 to defi

the ELF in the form

l l r d 3r I −1 −1 = Im , Im Ω εM(k, ω; rsp(r)) ε(k, ω) I

(2)

where rp(r) = [3/4πnp(r)]1/3 is the pseudo-one-electron-radius, np(r) is the pseudo-electrons

e

e

density at radius r, and εM(k, ω; rps(r)) is the Mermin dielectric function. The integral (2) is taken over the Wigner–Seitz sphere of volume Ω. Changing the integration variable from r to ωp(r) = [4πnp(r)]1/2, we obtain e

I Im

−1

l

ε(k, ω)

r

∞

= 0

4π

r2

Ω ∂ωp /∂r

I Im

−1

l

dωp.

(3)

εM (k, ω; ωp)

We can integrate Eq. (3) without actually fi ding ∂ω p/∂r. Let us consider the Mermin dielectric function, 18 εM(k, ω) = 1 +

(1 + iγ/ω)[εL(k, ω + iγ) − 1] 1 + (iγ/ω)[ε L (k, ω + iγ) − 1]/[εL (k, 0) − 1]

4


,

(4)

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where γ is the damping coefficient, and εL is the Lindhard dielectric function, 17 εL(z, µ) = 1 +

z−µ+1 1 − (z − µ) ln 2 8z z−µ−1 z +µ+1 1 1 1 , + 1 − (z + µ)2 ln z + µ − 1 8z

χ2 1 z2

+

1 1

2

1

(5)

where z = k/(2kF), µ = (ω + iγ)/(kvF), χ2 = 1/(πkF), and kF and vF are the Fermi momentum and velocity, respectively. In the optical limit, we derive lim ε (k, ω) = 1 − k→0

and

M

γωp2 ωp2 , ω2 + i ω 3

(6)

I

l ω 3γωp2 −1 ) Im εM (0, ω) = ω3 − ωω 2 2 + γ2ω 4 . p

(7)

p

Eq. (7) shows that Im[−1/εM(0, ω)] is a smoothed Dirac δ-function peaked at ωp = ω (Fig. 1). Therefore the only significant contributions to the integral (3) will be in the vicinity of ωp = ω, where we can expect that ∂ω p/∂r is almost constant. Substituting Eq. (7) into Eq. (3), we obtain I

l −1 4π r2 Im = ε(0, ω) Ω ∂ωp /∂r ωp(r)=ω πω 2γ 1 I × . 2 2ω(ω2 + γ 2) ω2 + γ 2 − ω

(8)

Use of l l = Im I −1 −1 ε(ω) Im ε(0, ω) I

(9)

where Im[−1/ε(ω)] is the experimental optical ELF,7 and Eq. (8) in Eq. (3) yields the result I

l= −1 Im ε(k, ω)

r

∞

G(ωp)Im

0

5

I

l dωp, −1 εM(k, ω)


(10)


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Figure 1: Mermin energy-loss function (ωp = 35.89 eV, γ = 0.01 eV): transparent surface – Im[−1/εM(k, ω)] from Eq. (4), solid line – Im[−1/εM(0, ω)] from Eq. (7).

6


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where

I

2

Im −1 G(ω) = ε(ω) πω 2γ

In particular, in the limit of infi

lI 2ω(ω2 + γ 2)

ω2 + γ2 − ω .

(11)

damping, lim εM(k, ω) = εL(k, ω),

(12)

I l lim G(ω) = 2 Im −1 , γ→0 πω ε(ω)

(13)

γ→0

Eq. (10) recovers the result of Penn 9 .

Results and Discussions The Penn algorithm is expected to be a good approximation for free-electron-like materials, which have a single dominant peak in the energy-loss spectrum, such as Al. However, there is a discrepancy between Penn’s result9 and the experimental value of the IMFP21,22 for Al in the energy range 5–9 eV above the Fermi energy (see inset of Fig. 2). In the present algorithm, by taking into account the plasmon damping, we are able to reduce this discrepancy. First, we start with two constant values of the damping coefficient: 0.1 and 1.5 eV. The latter is expected to give an IMFP lower than the former, because the increase of damping leads to a broadening of the plasmon linewidth, and thereby to a decrease of the IMFP. As a result, we obtain an excellent agreement with the experimental value of the IMFP21,22 for Al in the case of γ = 1.5 eV (solid line), whereas if γ = 0.1 eV (dotted line) our results seem to coincide with those of Penn 9 (open circles). From Fig. 2 we see that the diff

between these two lines occurs mainly for energies less than about 30 eV. This

is because the plasmon damping is effective only for electrons of energy comparable to the plasmon energy (15.79 eV for Al). The agreement between the calculated IMFP with γ = 1.5 eV and the experimental values 21,22 for Al should only be considered as a preliminary assessment of the role of damping 7



10 Inelastic Mean Free Path (˚ A)

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3

60 50 40 30

10

2

20 10 5

10

6

7

8

9 Ref. 5 Ref. 9 Ref. 24 Ref. 29 Ref. 21 Ref. 22 Ref. 102 in Ref. 25 Ref. 112 in Ref. 25 Ref. 115 in Ref. 25

1

10

0

10

1

2

10 E − E (eV)

10

3

10

4

F

Figure 2: Electron inelastic mean free path for Al (rs = 2.07). Present work: red solid line – γ = 1.5 eV, green dotted line – γ = 0.1 eV, blue dashed line – γk from Eq. (14). in the present algorithm. The use of constant damping is a rough approximation because, in fact, the plasmon linewidth broadens with increasing momentum transfer. Such a rough approximation not only underestimates the plasmon linewidth, but also fails to reproduce the asymmetry of the plasmon resonance at fi

momentum transfer.30

Let us consider the case of momentum-dependent plasmon damping. In general, it is diffi to precisely determine the damping parameter as a function of momentum transfer due to the infl

of the electronic structure of the material. In the limit of small momentum,

the function γk is found to be quadratic in k,31,32 γk =

πΩ03 30

(5 ln 2 + 1)

k2 kF2

+

πΩ0 100 8

30 ln 2 +

37 4


−

13

16 Ω 20

k4 , k F4

(14)

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√ for k < kc = α 3/rs, where kc is the critical momentum, Ω0 = α2 3rs, α = [4/(9π)]1/3, rs = [3/(4πne)]1/3 is the one-electron-radius, and kF = 1/(αrs) is the Fermi momentum. The expression (14) was obtained by Ninham et al.32 using the method developed by DuBois.31 This expression is exact in the high-density limit (rs → 0), and should only be considered as an approximation for metal (rs > 1). Here (rsAl = 2.07, rsAu = 3.01), we adopt the expression (14) for testing purposes of the present algorithm. The IMFP with γk is shown by the dashed line in Fig. 2. We see that the dashed line agrees well with the solid line (γ = 1.5 eV) rather than the dotted line (γ = 0.1 eV), and is also consistent with the experimental data21,22 in the energy range 5–9 eV. Our results (dashed and solid line) are quantitatively consistent with those obtained by Bourke and Chantler.5 These authors introduced a selfconsistent coupled-plasmon model to determine the plasmon damping depending on both transfer momentum and energy loss. In such a model, the Lindhard dielectric function is used to calculate the initial damping value, which is then updated by using the Mermin dielectric function in an iterative process until the ELF is converged. These comparisons show that the rough approximation of constant damping is likely to be appropriate for calculating the IMFP, as long as the damping value is suitable. Perhaps for this reason, the constant damping is widely adopted rather than the momentum-dependence plasmon damping, regardless of the limitations of this rough approximation, as mentioned above. To illustrate more clearly the role of damping in the present algorithm, we show in Fig. 3 the calculated IMFP for Au in two cases: γ = 2.0 eV (solid line), and γk from Eq. (14) (dashed line). We see that at several eV the dashed line tends to agree with the experimental data20 and the GW ab initio results,23 while the solid line is well consistent with the GW+T ab initio calculations. 23 The latter is expected to be more accurate than the former because measurements by ballistic electron emission microscopy (BEEM) shown that at about 1 eV the IMFP for Au lies in the range 220–330 ˚A, 33,34 here our result is 297 ˚A. The present result is closer to the BEEM measurements than other theoretical values: 400 ˚A from calculations based on the Boltzmann equation, 35 and 470 ˚A from GW+T ab initio calculations. 23 Zhukov

9



10 Inelastic Mean Free Path (˚ A)

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3

Penn (1987) Tanuma et al. (1991) Tanuma et al. (2008) Zhukov et al. (2010) − GW Zhukov et al. (2010) − GW+T Tanuma et al. (2011)

10

10

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2

Sze et al. (1964) Kanter (1970) Gergely et al. (2004) Tanuma et al. (2005) Ref. 102 in Ref. 25 Ref. 113 in Ref. 25 Ref. 115 in Ref. 25 Ref. 117 in Ref. 25

1

10

0

10

1

2

10 E − E (eV)

10

3

10

4

F

Figure 3: Electron inelastic mean free path for Au (rs = 3.01). Present work: red solid line – γ = 2.0 eV, blue dashed line – γk from Eq. (14). et al. 23 showed that by including the T-matrix terms with electron-hole scattering in the GW ab initio calculation, the IMFP at low energies is significantly improved and is in better agreement with the BEEM measurements. 34 These results again demonstrate that the plasmon damping greatly contributes to the IMFP of low-energy electron. We see that although the constant damping is a rough approximation, it is simpler and more efficient than the momentum-dependent plasmon-damping model in use. Figs. 2 and 3 show that the plasmon damping is effective only for energies less than a few tens of eV; for higher energies the Penn algorithm is recovered. As is known, the Penn algorithm is less reliable for non-free-electron-like materials36 or materials with a complex band structure. This may be the reason for the discrepancy between the theoretical and

10


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experimental IMFPs for Au at intermediate energies. To solve such a problem, Da et al. 6 used a mathematical function [1 − exp(−E/B)] to correct the calculated IMFP. Unfortunately, these authors did not discuss the physical meaning of this function. For more details, the reader is referred to, e.g., Ref. 5. A dielectric-function calculation that takes into account (inter/intra)band transitions can be found elsewhere (e.g., Ref. 37 and references therein). We note that neglect of the exchange effect may also be a reason for this discrepancy. In a previous study38 we proposed an approach to take the exchange effect into account through determining the maximum energy loss. Such an approach derives from a phenomenological point of view rather than from intrinsic physical properties. The exchange effect could be studied through a local-fi correction (LFC) within the random phase approximation. Once the LFC-ELF is determined, we can obtain the corresponding function G in a similar manner to that described above. For example, if we approximate the Mermin-ELF in the optical limit by the Drude-type ELF, l ωγωp2 −1 , )2 Im = εM (0, ω) ω2 − ωp2 + ω 2γ 2 I

then

2 G(ω) =

However, fi

πωγ

I Im

−1

(15)

lI 2ω

ε(ω)

ω2 + γ 2 − ω .

(16)

ing an appropriate LFC-ELF lies outside the main purpose of the present work.

On the technical side, there is no significant diff

in computation time between the

constant damping and the momentum-dependent plasmon-damping models. For constant damping model, we choose the damping parameter so that the calculated IMFPs agree with the experimental values. We expect that an appropriate form of momentum-dependent plasmon damping will further improve the calculation accuracy and make the present algorithm more effective for low- and intermediate-energy electrons.

11



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Conclusions We have presented an approach to introduce damping into the Penn algorithm by using the Mermin dielectric function instead of the Lindhard dielectric function. Both calculations of IMFP for Al and Au with the present algorithm have shown that: (1) the plasmon damping is effective only for energies less than a few tens of eV; and (2) the constant damping is simpler and more efficient than the momentum-dependent plasmon-damping model in use. We found that a damping of 1.5 eV for Al and of 2.0 eV for Au are appropriate for calculating IMFPs with the present algorithm.

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(35) Vlutters, R.; van ’t Erve, O. M. J.; Jansen, R.; Kim, S. D.; Lodder, J. C.; Vedyayev, A.; Dieny, B. Modeling of Spin-Dependent Hot-Electron Transport in the Spin-Valve Transistor. Phys. Rev. B 2001, 65, 024416. (36) Tanuma, S.; Powell, C. J.; Penn, D. R. Calculations of Stopping Powers of 100 eV–30 keV Electrons in 31 Elemental Solids. J. Appl. Phys. 2008, 103, 063707. (37) Chantler, C. T.; Bourke, J. D. Electron Inelastic Mean Free Path Theory and Density Functional Theory Resolving Discrepancies for Low-Energy Electrons in Copper. J. Phys. Chem. A 2014, 118, 909–14. (38) Nguyen-Truong, H. T. Determination of the Maximum Energy Loss for Electron Stopping Power Calculations and Its Effect on Backscattering Electron Yield in Monte-Carlo Simulations Applying Continuous Slowing-down Approximation. J. Appl. Phys. 2013, 114, 163513.

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Penn Algorithm Including Damping for Calculating the Electron

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