Electron effective attenuation lengths in electron spectroscopies

Journal of Alloys and Compounds 362 (2004) 26–32

Electron effective attenuation lengths in electron spectroscopies A. Jablonski a,∗ , C.J. Powell b b

a Institute of Physical Chemistry, Polish Academy of Sciences, ul. Kasprzaka 44/52, 01-224 Warsaw, Poland Surface and Microanalysis Science Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8370, USA

Received 26 September 2002; received in revised form 27 November 2002; accepted 18 December 2002

Abstract An important measure of the opacity of a solid with respect to monoenergetic electrons in a solid is the effective attenuation length (EAL). However, there is much controversy in the literature concerning the definition of this parameter. It has been shown recently that different quantitative applications of electron spectroscopies require EALs resulting from different expressions. In the present report, these expressions for typical applications of X-ray photoelectron spectroscopy and Auger electron spectroscopy are briefly reviewed. The EALs needed for determination of overlayer thicknesses and for measurement of surface composition are compared for the same experimental configuration. These comparisons are made for selected photoelectron lines for which we expect strong electron elastic-scattering effects (Cu2s and Cu2p3/2 in Cu and Au4s and Au4f7/2 in Au). Substantial differences between EALs for these lines were found for the two applications. Synchrotron radiation can be used for experimental determination of the EAL values. These values can facilitate evaluation of the reliability of theoretical models used in calculations of EALs. © 2003 Elsevier B.V. All rights reserved. Keywords: Effective attenuation length; X-ray photoelectron spectroscopy; Auger electron spectroscopy

1. Introduction The electron attenuation length (AL) is commonly understood to be a measure of the opacity of a solid for electrons in surface-sensitive spectroscopies such as Auger electron spectroscopy (AES) and X-ray photoelectron spectroscopy (XPS). This term has been used to describe the rate at which the AES or XPS signal intensities from a substrate material or an overlayer film diminished as a function of the film thickness. The signal intensities often decreased nearly exponentially with film thickness and thus it was reasonable to refer to the exponential parameter in the AES and XPS experiments as an attenuation length for a particular material and a particular electron energy. The changes in AES and XPS signal intensities with overlayer-film thickness were believed to be due solely to inelastic-electron scattering, and the AL was then considered to be identical to the corresponding electron inelastic mean free path (IMFP), the latter being defined as the mean distance between inelastic collisions. However, about 20 years ago, it was realized that, as a result of elastic-electron scattering, the projection of an ∗

Corresponding author. E-mail address: [email protected] (A. Jablonski).

0925-8388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0925-8388(03)00558-9

electron trajectory on a given direction could then have a significantly different length than the length actually traveled, and thus the AL could markedly differ from the IMFP. More recently, it was recommended that the term effective attenuation length (EAL) be used instead to avoid potential confusion with the accepted use of the AL to describe absorption of a parallel beam of radiation. Despite an obvious need for accurate values of the EAL, there is continued controversy concerning its definition. The current status of this issue is briefly reviewed in this work. In Section 2, we present the current recommended definition of the EAL. Expressions defining EAL for two typical analytical applications of XPS and AES, the measurement of overlayer thickness and the measurement of surface composition, are discussed in Sections 3 and 4, respectively. The numerical values of EALs are compared in Section 5. Finally, brief information is provided on a database that provides EAL values.

2. Definition of the EAL The definition proposed in the terminology standard of the American Society of Testing and Materials (ASTM) is

A. Jablonski, C.J. Powell / Journal of Alloys and Compounds 362 (2004) 26–32

rather complex [1]. It involves knowledge of two additional parameters: 1. Emission depth distribution function (DDF)—for particles or radiation emitted from a surface in a given direction, the probability that the particle or radiation leaving the surface in a specified state originated from a specified depth measured normally from the surface into the material. 2. Average emission function decay length (EFDL)—the negative reciprocal slope of the logarithm of a specified exponential approximation to the emission depth distribution function over a specified range of depths, as determined by a straightline fit to the emission depth distribution function plotted on a logarithmic scale versus depth on a linear scale. According to the ASTM definition [1], the EAL is the average emission function decay length when the emission depth distribution function is sufficiently close to exponential for a given application. A similar definition has been accepted in a recent ISO standard [2]. For a given depth z, the EAL can be thus expressed by:
 d ln φ(z, α) −1 EAL = − cos α (1) dz where φ(z, α) is the emission depth distribution function, and α is the emission angle measured with respect to the surface normal. If we consider the finite range of depths between z1 and z2 , Eq. (1) can be transformed to: EAL =

z2 − z 1 1 cos α [ln φ(z1 , α) − ln φ(z2 , α)]

(2)

This equation was originally proposed by Gries and Werner to define the EAL [3]. It has recently been shown [4,5] that EAL values resulting from the above definition are appropriate for determination of the depth of a thin marker layer in a matrix of another material. Quantitative applications of AES and XPS for this type of measurement are rather uncommon. If the elastic-electron collisions are neglected, the depth of the marker layer, z, should be calculated from the equation:   Im z = −λin cos α ln 0 (3) Im where λin is the IMFP, Im is the signal intensity due to a 0 is the signal intensity marker layer buried at a depth z, and Im from a marker layer deposited at the surface. We expect to obtain a more accurate value of the marker-layer depth from Eq. (3) if λin is replaced by EAL from Eq. (2). An important quantitative application of electron spectroscopies, AES and XPS, is the measurement of the thicknesses of overlayer films deposited on a substrate from measurements of substrate or overlayer-film signal-electron

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intensities. Generally, the EAL defined by Eq. (1) or (2) cannot be used for this (and other) quantitative applications. In view of these results, a new generalized definition of the EAL was proposed [4,5]: “The EAL is a parameter which, when introduced in place of the IMFP into an expression derived from the common AES or XPS formalism (in which elastic-electron scattering is neglected) for a given quantitative application, will correct this expression for elastic-scattering effects.” Obviously, the EAL will be described by different expressions for different quantitative applications. Let us consider two typical applications of electron spectroscopies.

3. Overlayer thickness measurements 3.1. Local EALs As shown by Jablonski and Powell [4,5], the EAL to be correctly used in the AES and XPS formalism for determining the overlayer thickness, t, should be calculated from: −1  ∞
d φ(z, α) dz (4) EAL = − cos α ln dt t If we assume that the DDF is exponential over the finite (but small) range of thickesses from t1 to t2 , Eq. (4) transforms to: EAL =

t2 − t1 1 ∞  cos α ln t φ(z, α) dz − ln t∞ φ(z, α) dz 1

(5)

2

Note that Eqs. (4) and (5) are different from Eqs. (1) and (2). For simplicity, we have assumed here that the elastic and inelastic scattering properties of the substrate and overlayer film are similar. The term ‘local EAL’ has been proposed [4,5] to indicate that the EAL values are characteristic of a narrow range of thicknesses over which the DDF is expected to be close to exponential. 3.2. Practical EALs In experimental practice, we often cannot specify in advance the expected range of thicknesses of overlayer films to be measured. It is then generally necessary to use an EAL values for a thickness for which the DDF is not exponential. Furthermore, EALs may be needed for a wide range of thicknesses. The largest useful range should vary from zero to the maximum thickness that attenuates the substrate signal to the value which is detectable by typical electron energy analysers. For many practical applications, the largest overlayer thicknesses that can be reliably measured will correspond to some overlayer thicknesses, tmax , that attenuate the substrate signal to, for example, 10, 5, or 1% of its initial value (for an uncovered substrate). The EAL

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value valid for such a thickness range can be calculated from [4,5]: 1 t   0 cos α ln Is − ln Is t 1  ∞

= cos α ln 0 φ(z, α) dz − ln t∞ φ(z, α) dz

L=

(6)

where Is0 is the signal intensity from the uncovered substrate, and Is is the signal intensity from the substrate covered by an overlayer with thickness t = tmax . The term ‘practical EAL’ has been proposed for L values defined by Eq. (6). We see that the L value is determined by the values of two integrals, one relating to the uncovered substrate and the second to the thickness equal to tmax . If the substrate signal intensity as a function of thickness deviates noticeably from an exponential function over that thickness range, it is then advisable to average the L values over the intermediate thicknesses. The following procedure has been proposed [4,5]: n

Lave =

1 1 ti

n cos α ln  ∞ φ(z, α) dz − ln  ∞ φ(z, α) dz i=1

0

ti

(7) where the thicknesses ti are evenly distributed over the relevant maximum range and n denotes the total number of considered thicknesses.

where β is the asymmetry parameter and ψ is the angle between the direction of X-rays and the analyser axis. It has been shown [5,8] that the parameter that should replace the IMFP in Eq. (8) to account for elastic-scattering effects should have the form: W (βeff , ψ) (10) LXPS = λin Qx W(β, ψ) where Qx and βeff are two correction parameters. Extensive tabulations [9] with these parameters or simple analytical expressions [6] are available in the literature. 4.2. Auger electron spectroscopy The Auger-electron current is described by a similar expression to Eq. (8) [6]:  I= (11) TDe I0 PA σi (E0 ) sec θ0 rλin Mx 4π where I0 is the primary electron current, PA is the probability that an Auger transition follows the ionization, σi (E0 ) is the ionization cross section at energy E0 , θ0 is the incidence angle of the primary beam, and r is the backscattering factor. It has been indicated [6] that elastic-electron collisions are accounted for if the IMFP in Eq. (11) is replaced by: LAES = λin QA

(12)

where QA is the correction parameter. This parameter can be calculated from simple analytical expressions [6].

4. Determination of surface composition

5. Comparisons

4.1. Photoelectron spectroscopy

Table 1 represents an ‘at a glance’ summary of expressions defining the EAL for different quantitative applications of electron spectroscopies. On inspection of Table 1, we see that all the EALs can be derived from the DDF. In the literature, several expressions for the DDF are available [10–12]. A recent analysis has indicated that the expression derived from solution of the kinetic Boltzmann equation within the so-called transport approximation is the most reliable [14]. This equation will be used in calculations of EALs considered in this section. Average practical EAL values, Lave , are calculated here from Eq. (7) for photoelectrons for which strong elastic-scattering effects are expected, i.e. photoelectrons with low kinetic energies or solids with medium or high atomic numbers. Thus, for analysis, we selected Cu2s, Cu2p3/2 , Au4s, and Au4f7/2 photoelectrons in the respective elemental solids. We also selected a typical experimental configuration in which the angle ψ is constant and equal to 54◦ while the emission angle, α, is varied. This corresponds to rotation of the sample in a chamber in which the locations of the analyser and the X-ray source are fixed. We consider the ratios REAL = Lave /λin calculated for three maximum overlayer thicknesses such that they attenuate the substrate signal down to 10, 5, and 1% of the initial

The theoretical relation between the Auger-electron or photoelectron signal intensity and the concentration of emitting atoms can be used to determine the surface composition of the studied sample. For photoelectrons, the following relation results from a theoretical model in which elastic-electron collisions are neglected [6]: I = TDe A0 Fx σx W(β, ψ) λin Mx

(8)

where T is the analyzer transmission function, De is the detector efficiency, A0 is the analyzed area when the analyzer is located along the normal to the surface, Fx is the flux of incident X-rays,  is the acceptance solid angle of the analyzer, σ x is the total photoelectric cross section, M is the atomic density of a given element (number of atoms per unit volume), and x is the concentration expressed as an atom fraction. The parameter W(β, ψ) in Eq. (8) is the normalized differential photoelectric cross section. For unpolarized radiation and random orientation of atoms or molecules, this cross section is expressed by [7]:
 dσx /d 1 β W(β, ψ) = = 1− (9) 3 cos2 ψ − 1 σx 4π 4

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Table 1 List of quantitative applications of AES and XPS, equations derived for determination of each quantity from the inelastic mean free path (IMFP), λin , under the assumption that elastic-electron scattering is insignificant, and the equations for the effective attenuation length (EAL), EAL, L, or LXPS , that should be used to replace the IMFP in the corresponding equation for the application of interest Quantitative application Thickness measurement of an overlayer film in a narrow range of thicknessesa

Equation used for application

Expression defining the EAL Local EAL





Is Is0

t = −λin cos α ln

1 t2 − t1 ln  ∞ φ(z, α) dz − ln  ∞ φ(z, α) dz EAL = cos t1 t2 α

  l t = −λin cos α ln 1 − I∞ Il Depth measurement of a thin layer in a narrow range of depthsb

Thickness measurement of an overlayer film over a wide range of thicknesses

Depth measurement of a thin layer over a wide range of depthsb

 z = −λin cos α ln

Im 0 Im

t1 < t < t2 Local EAL z2 − z1 1 EAL = cos α ln φ(z1 , α) − lnφ(z2 , α)



z1 < z < z2 Practical EAL

 t = −λin cos α ln

 z = −λin cos α ln

Is Is0



Im 0 Im

1 tmax ln L = cos α 0 < t < tmax



∞

 φ(z, α)dz − ln

0

0≤α≤

60◦

∞ tmax

φ(z, α)dz

Practical EAL zmax 1 L = cos α ln φ(0, α) − ln φ(zmax , α)



0 ≤ α ≤ 60◦

0 < z < zmax

Determination of surface composition by XPSc I = TDe A0 Fx σx W(β, ψ) λin Mx

EAL for quantitative surface analysis  ∞ φ(z, α)dz 0 LXPS = λin  ∞ φnel (z, α)dz 0

LXPS = λin Qx Determination of surface composition by AESc I =  4π TDe I0 PA σi (E0 ) sec θ0 rλin Mx

W(βeff , ψ) W(β, ψ)

EAL for quantitative surface analysis  ∞ φ(z, α)dz 0 LAES = λin  ∞ φnel (z, α)dz 0

LAES = λin QA a b c

Il and Il0 are the signal intensities from an overlayer film of thickness t and from a homogeneous overlayer-film material, respectively. 0 are the signal intensities from a thin marker layer in a matrix and from the marker layer at the surface, respectively. Im and Im Index ‘nel’ in the DDF indicates that the function φnel (z, α) is derived using a model in which elastic-electron scattering is neglected.

signal for an uncovered substrate. Plots of REAL are shown in Figs. 1 and 2 as a function of emission angle for the Cu and Au photoelectron lines, respectively. We see that, in all cases, the EAL is a relatively weak function of the emission angle between zero and about 60◦ . The same conclusion was published in recent reports [4,5]. In these experimental configurations, the EAL may be smaller by up to 38% than the corresponding IMFP. Similar results have been obtained by Seah and Gilmore [13] who developed an empirical formula for the EAL. For emission angles larger than 60◦ , the EALs depend strongly on the emission angle and on the maximum film thickness. In such cases, the EAL should be evaluated for a particular emission angle. Let us calculate now the ratio RXPS EAL = LXPS /λin for the same experimental configuration using Eq. (10). The corrections Qx and βeff derived within the transport approximation [6] were used in these calculations. As shown in Fig. 3, the dependence of the EAL on emission angle is different than

that for overlayer-thickness determinations shown in Figs 1 and 2. The variation with emission angle is much less pronounced, and the ratio is closer to unity. These differences between the EAL results for each line in Figs. 1 and 2 and the corresponding EAL results in Fig. 3 are due to the different magnitudes and dependencies on emission angle of L from Eq. (6) and LXPS from Eq. (10). It can therefore be seen that the EALs for different analytical applications can have different numerical values, and that these should not be used interchangeably. The reliability of EAL values is of considerable importance for quantitative applications of electron spectroscopies. The issue of the accuracy of EALs for overlayer-thickness determinations has been addressed in recent reports [14,15]. It is expected that EALs derived from Monte Carlo simulations of electron transport in solids would be more accurate than EALs calculated from approximate solutions of the Boltzmann equation due to the

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Fig. 1. Dependence of the ratio REAL = Lave /λin calculated from Eq. (7) on emission angle, α, for photoelectron lines in copper. The electron energies correspond to photoelectrons emitted with the use of Mg K␣ radiation. The numbers for each curve indicate the percentage to which the substrate signal intensity is attenuated by the overlayer. (a) Cu2s photoelectrons; (b) Cu2p3/2 photoelectrons.

Fig. 2. Dependence of the ratio REAL = Lave /λin calculated from Eq. (7) on emission angle, α, for photoelectron lines in copper. The electron energies correspond to photoelectrons emitted with the use of Mg K␣ radiation. The numbers for each curve indicate the percentage to which the substrate signal intensity is attenuated by the overlayer. (a) Au4s photoelectrons; (b) Au4f7/2 photoelectrons.

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Fig. 3. Dependence of the ratio RXPS EAL = LXPS /λin calculated from Eq. (10) on emission angle. (a) Copper; (b) gold.

fact that the exact shape of the differential elastic-scattering cross sections are accounted for in the Monte Carlo algorithms. Comparison of EALs from both sources showed that the expression of Tilinin et al. [11] is closest to the Monte Carlo results for common XPS measurement configurations [14]. The reliability of EAL values can also be evaluated by comparison with experimental data. The EALs for Si2p photoelectrons in SiO2 overlayers on Si, with energies between 140 and 1000 eV, were recently measured using synchrotron radiation [16]. This approach is valuable because EALs could be measured over a wide range of electron energies and for different photoelectron emission angles. These parameters were also calculated from transport theory for the same system and experimental configuration. Reasonably good agreement between experimental and theoretical values was found, particularly in the energy range from 400 to 1000 eV [15].

netic Boltzmann equation. One should be aware of the fact that the EALs are dependent on a number of parameters: the electron energy, the composition of the solid, and the experimental configuration. For this reason, it would be impractical to attempt a preparation of extensive tabulations with these parameters. To facilitate determination of EAL values, a database for use in surface-sensitive spectroscopies has been recently released by NIST [17]. This database, to be operated on a personal computer, provides values of EALs in solid elements and compounds at selected electron energies between 50 and 2000 eV. These values are determined for any experimental configuration specified by the user. The EALs are calculated from an algorithm based on electron transport theory [11] that is considered to be the most reliable [14]. All the parameters listed in Table 1 can be obtained from the database, although the database was designed primarily to provide EALs for measurements of overlayer-film thicknesses.

6. NIST effective attenuation length database 7. Conclusions The summary expressions in Table 1 are relatively simple but evaluation can be complex. Although the DDF, and consequently all the correction parameters, can be derived from Monte Carlo simulation of electron trajectories in solids, such an approach generally requires a considerable amount of computer time. Analytical expressions have been fitted to selected results of simulations to facilitate the use of calculated parameters [9,13]. As mentioned, the EAL values can also be derived from an approximate solution of the ki-

There is no universal expression for defining electron effective attenuation lengths in solids. Different analytical applications of AES and XPS require EALs calculated from different expressions. For a given experimental configuration, these values may differ considerably. The present report is intended as a practical guide for selecting the proper defining expression for a particular type of analysis.

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In principle, all the EAL values can be calculated from the depth distribution function. The analytical expression published by Tilinin et al. [11] is presently the most reliable such expression for common XPS configurations. EALs can be conveniently obtained from the recently released NIST EAL database. Acknowledgements One of the authors (AJ) wishes to acknowledge partial support by KBN Grant 2P03B 039 18, and the other author (CJP) wishes to acknowledge partial support by the NIST Standard Reference Data Program. References [1] ASTM Standard E 673-98, Annual Book of ASTM Standards vol. 3.06, American Society for Testing and Materials, West Conshohocken, PA, 2001, p. 735.

[2] International Organization for Standardization, ISO 18115: 2001. Surface chemical analysis: Vocabulary, ISO, Geneva, 2001. [3] W.H. Gries, W. Werner, Surf. Interface Anal. 16 (1990) 149. [4] C.J. Powell, A. Jablonski, Surf. Interface Anal. 33 (2002) 211. [5] A. Jablonski, C.J. Powell, Surf. Sci. Rep. 47 (2002) 33. [6] A. Jablonski, Surf. Sci. 364 (1996) 380. [7] R.F. Reilman, A. Msezane, S.T. Manson, J. Electron Spectrosc. Relat. Phenom. 8 (1976) 389. [8] A. Jablonski, C.J. Powell, Phys. Rev. B 50 (1994) 4739. [9] A. Jablonski, Surf. Interface Anal. 23 (1995) 29. [10] I.S. Tilinin, W.S.M. Werner, Phys. Rev. B 46 (1992) 13739. [11] I.S. Tilinin, A. Jablonski, J. Zemek, S. Hucek, J. Electron Spectrosc. Relat. Phenom. 87 (1997) 127. [12] V.I. Nefedov, J. Electron Spectrosc. Relat. Phenom. 100 (1999) 1. [13] M.P. Seah, I.S. Gilmore, Surf. Interface Anal. 31 (2001) 835. [14] A. Jablonski, C.J. Powell, Surf. Sci. 520 (2002) 78. [15] C.J. Powell, A. Jablonski, Surf. Sci. 488 (2001) L547. [16] H. Shimada, N. Matsubayashi, M. Imamura, M. Suzuki, Y. Higashi, H. Ando, H. Takenaka, S. Kurosawa, S. Tanuma, C.J. Powell, Surf. Interface Anal. 29 (2000) 336. [17] C.J. Powell, A. Jablonski, NIST Electron Effective-AttenuationLength Database (SRD 82), Version 1.0, National Institute of Standards and Technology, Gaithersburg, MD, 2001.