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Electron inelastic mean free paths for plasmon excitations and interband transitions
Surface Science 293 (1993) 202-210 North-Holland
&i-face science
Electron inelastic mean free paths for plasmon excitations and interband transitions C.M. Kwei, Y.F. Chen Department of Electronics Engineering, National Chiao Tung University, Hsinchu, Taiwan, ROC
C.J. Tung’~* and J.P. Wang Institute of Nuclear Science, National Tsing Hua University, Hsinchu, Taiwan, RUC
Received 1 April 1993; accepted for publication 3 May 1993
Both plasmon excitations and interband transitions are important in the response of valence bands to fast electrons. An extended Drude dielectric function was established to describe such a response in solids of complex band structures. Parameters in this function were determined by a fit of the imaginary part of the function to experimental optical data. The real part of the dielectric function and the energy-loss function were also checked with experimental data to confirm critical-point energies in the interband transitions and plasmon energies in the collective excitations. In addition, sum-rules about the imaginary part of the dielectric function and the energy-loss function, respectively, were applied to assure the accuracy of these functions. Electron inelastic mean free paths in several solids were caicutated and compared to available e~e~mentai and theoretical data.
1. Introduction Information on electron inelastic mean free paths (MFPs) in solids is impo~ant in surface and interface analysis such as X-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy @ES). This information can be extracted from measured spectra in reflection electron-energy-loss spectroscopy (REELS) [l], determined by elastic peak electron spectroscopy 121or measured using the thin-film overlayer technique [3]. Corrections should be made for multiple elastic scatterings which modify the pathlengths and directions of electrons in solids [4,51. Alternatively, one may calculate electron MFPs using the dielectric response theory. The electron-gas model (EGM) which describes the response of valence-band (or conduction-band in * To whom correspondence should be addressed. i Present address: Department of Nuclear Engineering, Texas A & M University, College Station, TX 77843, USA. 0039-6028/93/$06.00
the case of a metal) electrons is widely used for these calculations [6-S]. The EGM is a good approximation for materials that exhibit simple energy-loss peaks with matched theoretical and experimental plasmon energies [9]. This model, however, is not good for solids that have complex band structures leading to strong interband transitions. In such cases, the ion-core lattice has a significant effect on electron motion to produce non-spherical Fermi surfaces and to support single-electron interband transitions [lo]. In addition, these transitions shift free-electron plasmon energies towards higher values and cause extra resonance peaks in the energy-loss spectrum 1113. For these solids, the dielectric function and the energy-loss function must be carefully examined with respect to their peak heights and positions. This can be done by applying sum-rules and identifying critical-point energies in the interband transitions. In this work, we calculate electron MFPs in several solids having complex band structures.
0 1993 - Elsevier Science Publishers B.V. All rights reserved
CM. Kwei et al. / Plasmon excitations and interband rransitiorw
Our emphasis is on the contribution from valence-band excitations. We employ an extended Drude dielectric function with parameters determined by a fit of its imaginary part to experimental optical data. The real part of the dielectric function and the energy-loss function are also checked with experimental data to confirm critical-point energies in the interband transitions and plasmon energies in the energy-loss spectra. In addition, sum-rules about the imaginary part of the dielectric function and the energy-loss function, respectively, are applied to verify the accuracy of these functions. Electron inelastic MFPs calculated in this work are comnared to avaitable experimental and theoretical data.
203
For a valence band of ni bound electrons per volume adding to the conduction band of no free electrons per volume, the imaginary part of the dielectric function is given by fll]
(4) where wP =I:(4+~7=n,)‘/~is the free-electron plasmon energy, Ri = n,/tts, wi is the gap-energy between conduction and valence bands or the critical-point energy for interband transitions, and yi is the damping constant associated with the valence band. The real part of the dielectric function is thus given by
2. Themy The Drude model works quite well for the conduction band of a free-electron-like metal such as aluminum [12]. In this model, conduction electrons are described by a free-electron gas constrained by the Fermi-Dirac statistics. Interaction with the ion-core lattice is assumed to be a minor effect leading the damping of plasma resonances to single-efectron excitations. The imaginary part of the dielectric function in the iongwavelength limit, i.e. 4 + 0, is given by
(1) where q is the momentum transfer, o is the energy transfer, oP is the free-electron plasmon energy, and y is the damping constant. Note that atomic units are used throughout this paper for all quantities and units unless otherwise specified. The real part of the dielectric function is given by the earners-Kronig anaiysis as q(0, w) = 1 -
is202 p to4+ ,=y2.
The energy-loss function, or the imaginary part of the negative inverse dielectric function, may then be calculated from -1 Im-------=-
( I 44
0)
E2 ET+&
(5) The effective plasmon energy of the system, defined by t-,(0, 01 = 0 in the limit y + 0 and yi -+ 0, becomes
For an insulator or a semiconductor with 120= 0 and ni = II (since the conduction band is psactitally empty), eq. (6) gives “); = 4nn + (0””
(“0
One sees that the plasmon energy of the system is shifted upward above the value for the free-electron gas. This situation is demonst~ted in Si and Ge [ll]. Also, interband tr~sitions cause a second resonance peak in the energy-loss function f12J. Since all electrons are in the valence band, the first term on the right-hand side of eq. (41 becomes zero. This leads to a continuous distribution of oscillator strengths above the band gap, For a solid having a complex band structure, the valence band may be composed of several subbands. Each ith subband is characterized by its own oscillator strength, Aj, damping constant, ‘yj, and criticai-point energy, wi. Interband transi-
204
C.M. Kwei et al. / Plasmon excitations and interband transitions
tions can be incorporated into the Drude model by adding these subband electrons to the freeelectron system. Based on the superposition of damped linear oscillators, one can write
Here Ai =A,wc with Ai established in eq. (4). The real part of the dielectric function is thus given by
E1(O,O) = EB
(9)
-
Note that we include an lB term in eq. (9) to account for the background dielectric function in the solid. The possible departure of lB from one is due to the fact that valence electrons are embedded in the background of polarizable ion cores [13]. By taking ~~(0, w) = 0 in the limit ri + 0, we can solve the plasmon energies of the system. We find that the L$, are the roots of the equation
(10) If all ni values are of comparable magnitude and the wi values are well distinct from one another, we should find as many interband transition peaks in the energy-loss function as there are subbands. In this work, we fit eq. (8) to experimental optical data to determine the parameters A:, yi and wi. To make sure that these results are accurate, we require also that ~~(0, w> and Im[ - l/40, WI in eqs. (9) and (3) are in agreement with experimental data. In addition, we check the validity of the sum-rule /0
?JE~(O, w’) dw’ = 2r*NZ,(
o),
(11)
where N is the number of atoms (or molecules) per volume and Z,(o) is the effective number of electrons per atom (or molecule) contributed by valence excitations up to the energy transfer o. Note that ZJm) approaches the number of valence electrons per atom (or molecule) and that
4rZVZJm) = L4:. We also check the validity of the sum-rule for the energy-loss function J',wu' Im[ .,o:t.,]
do’=Za’N~,
(12)
where Z$w> is an alternative effective number of electrons per atom (or molecule) contributed by valence excitations up to the energy transfer w. Again, Z:(m) approaches the number of valence electrons per atom (or molecule) and 4rNZ,Xm) = CA:. The sum-rules for the contribution from valence excitations, i.e. eqs. (11) and (12), are valid if one keeps the upper limits of the integration in these equations smaller than the binding energy of the most loosely bound inner shell. To extend the dielectric function into the q + 0 region of the q-o plane, we replace oi in eq. (8) by wi + q2/2 [14,151. This extension leads to a correct behavior of the dispersion relation at two extremes, i.e. the optical end, q + 0, and the Bethe-ridge region, q + m. The deviation of this extension from the true dispersion relation makes only minor difference in the determination of electron MFPs. The differential inverse mean free path (DIMFP), p(E, o), for an electron of energy E to lose energy w is given by [16]
(13) where q + = d%? f \/2E-20 is due to the conservation of energy and momentum. The inverse mean free path (IMFP), p(E), is obtained by integrating eq. (13) over all allowed energy transfers, i.e. p(E)
=
lEp(E, w) dw.
(14)
Using eqs. (13) and (14) with the energy-loss function given above, one can calculate electron DIMFPs and IMFPs in solids. For the contribution from inner-shell ionizations we apply either the local plasma approximation [17] or the sum-rule-constrained binary-collision model [18] using atomic generalized oscillator strengths from the Hartree-Slater approach
205
CM. Kwei et al. / Plasmon excitations and interband transitions
[19-211. The first use of the local plasma approximation in electron MFP theory was made with the EGM [7]. The inner-shell contribution, however, is very small for electron energies below several hundred eV in all solids studied here.
3. Results and discussion We now fit eqs. (31, (8) and (9) to measured optical data for several solids. Table 1 lists the fitting parameters for MgO, Cu, SiO, glass, Fe, SiO, crystal, Au, Pd, Ag and Ni. In these fits, we check not only the accuracy of ~~(0, o>, ~~(0, w) and Im[ - l/e(O, w)] determined by these equations but also the extent to which they satisfy the sum-rules of eqs. (11) and (12). These sum-rules are applied by setting the upper limits of integration to both finite and infinite values. In the case of finite-range sum-rules, we examine the compliance of the calculated effective number of valence electrons, at any given energy transfer, with the corresponding optical data. In the case of infinite-range sum rules, we check the satisfaction of the total oscillator strength to the condition L4; = 457NZ,. Since a separate treatment for inner-shells is used, we require that Z, equals the number of valence electrons per atom (or molecule). In all cases, we find that these Z, values agree quite well with those determined from observed plasmon energies. For Au, our fits cover the 5s and 5p inner-shell due to the strong overlapping of oscillator strengths between the valence band and these subshells in the vicinity of their binding energies. To avoid the ambiguity in the separation of cross contributions, we extend our fits to higher energy transfers covering these inner-shell excitations. Fig. 1 shows a comparison of ~~(0, 01, ~~(0, w) and Im[- l/e(O, w)] for MgO fitted presently (solid curves) and determined experimentally (dashed curves) [22,23]. It is seen from the figure that good agreement is found between present results and experimental data for all functions plotted. It is also seen that strong interband transitions occur in the region of critical-point energies between 8 and 25 eV. These transitions also induce plasmon peaks in the energy-loss function.
0 0
10
20 0
30
0.0 40
(ev)
Fig. 1. A plot of ~~(0, 01, ~~(0, 01 and Im[- 1/4,~)1 for MgO.Solid and dashed curves are results of present calculations and experimental measurements [22,231. Verticaldownward and upward arrows indicate critical-point energies and plasmon energies determined in this work.
Interband transition energies (vertical downward arrows) fitted in this work and plasmon energies (vertical upward arrows) calculated using eq. (10) are labeled in the figure. A comparison of these energies with corresponding experimental [241 and theoretical [25] data is made in table 1. Fig. 2 shows a comparison of ~~(0, w), ~~(0, w) and Im[ - l/e(O, w)] for Cu determined experimentally (dashed curves) [26-281 and obtained presently (solid curves) using eqs. (31, (8) and (9) with parameters listed in table 1. Although a similar fit by Yubero and Tougaard [29] to Im(- l/e) data of Cu is available, their fitting parameters produce poor results on e1 and Ed compared to experimental data, especially at small energy losses where interband transitions are important. This is because it is the imaginary part of the dielectric function rather than the energy-loss function that contains detailed information on interband transitions. Further, there was a missing term in their fitting parameters corresponding to the conduction band at wi = 0 for metals. This term leads to e1 + --cx) and Ed + co at w + 0, a common feature for all metals. Since Yubero and
Table 1 Parameters in eqs. (8) and (10) for several solids; values in parentheses for MgO are critical-point energies calculated theoretically [25] and plasmon energies determined experimentally [24] Afi (eV ‘1
yi (eV)
Oi (eV)
hp (eV)
A: (eV2)
7.8 10.1 11.7 13.9 15.3 18.8 20.2 22.1 24.5 36.5 47.1
Au (~a = 1) 1281 79 0.1 9 1 1.9 36 17 2.3 4 60 9 100 120 10 6 155 145 7.2 280 20 360 28 26 183
MgO (~a = 1.08) [22-251 0.3 8 3.1 70.6 1.5 45.7 1 47.7 1.9 25 2.5 98 1.2 13.9 1.5 15 2.6 12 15 105 30 152.1
7.7 9.5 10.8 13.1 14.6 17.2 19.2 20.8 24 35 45
Cu k, 64 20 6.5 5.5 4 55 42 172 80 240 100 80 412
0 0.3 2.5 3.1 3.7 5.05 8.93 14.74 25.6 40 55 65 85
0.2 2.3 2.9 3.5 4 7.2 10.4 19.3 27.3 43 56 65.7 87.6
SiO, glass kg = 1.03) [30-321 0.3 11 1.9 45 2.9 49 3.6 60 4 22 13 90 46 207
10.4 11.6 13.8 16.8 19.6 22 37
10.4 12.2 14.6 17.9 20 24.5 40.2
Fe kB = 1.31) [50,51] 0.18 52 5 198 2 10 4.8 41.4 3.2 28 15 200 26 100 150 305
0 2.5 6.04 9.56 12.9 18.4 24.8 48
1.1 5.8 8.1 11.2 13.8 21.8 27.9 50.9
SiO, crystal (~a = 1.01) [321 0.3 18 1.6 53 2.3 58 3.3 70 4 26 11 90 46 182
10.32 11.6 14.1 16.9 19.6 22 37
10.5 12.3 15 18.2 20.1 24.9 40
= 1.05) 126-281 0.01 0.1 0.65 0.7 0.7 2.6 4.76 10.18 8 32 30 30 37
(7.77) (10.89) (13.3) (14.02) (17.01) (18.99) (21.2) (24.76)
(8.2) (11.6) (14.3) (15.4) (18) (22.4) (25.6)
yi (eV)
w, (eV)
Gp (eV)
0.1 3.1 4.1 5.3 8.17 12 14 21.3 29.5 38.5 63 100
2.5 3.3 4.8 6.1 9.3 12.8 17.2 24.4 31.8 43.3 66.6 101
Pd kg = 1) [52-541 85 0.1 75 4.45 4.45 45 5 45 2 0.7 13 1.8 3 45 50 4.6 110 5.6 80 6 70 7 130 17 19 186
0 1.3 4.3 10.7 11.8 13.65 15 16.8 20.5 24.2 30.6 38 49
0.9 3.7 8.1 11.6 12.1 13.9 15.7 17.9 22.4 26.6 32.1 40.3 51.9
Ag kg = 1.03) [281 80 0.07 4 0.45 1.2 10 2 20 11 240 70 5.38 160 15 36 300
0 4.8 5.3 6.4 15 22 31.3 40
4 4.9 5.6 7.6 19 23.8 33.4 45.3
Ni (eB = 1.02)[281 0.01 25 2 50 4 30 6 60 1.3 18 4.2 37 7.2 95 18 200 70 200 80 545
0 0.45 1.5 3.6 4.6 8.3 14.6 24 45 58
0.2 1.2 2.7 4.4 7.1 10.5 17 28.1 46.8 63.7
C.M. Kwei et al. / Plasmon excitations and interband transitions
-0.3""' 0
I” 10
”
I ""I 20
”
30
“I’ 40
50
w (ev)
Fig. 2. A plot of ~~(0,w), ~~(0,o), Im[- l/6(0, w)l and Im[(c - 1j2/c(c + l)] for Cu. Solid, dashed and chain curves are results of present calculations, experimental measurements [26-281 and other calculations [29]. Vertical downward and upward arrows indicate critical-point energies and plasmon energies determined in this work.
Tougaard also divided the contribution for the energy-loss function into valence band and inner-shells, their valence oscillator strength was expected to satisfy the condition CA: = 47rNZ,. Adopting 18 to be the number of inner-shell electrons, Z, = 11 is required in order to add them up to the total oscillator strength of Cu. The latter value agrees with the prediction using 35.9 eV as the plasmon energy [49]. An examination of Yubero and Tougaard’s results, however, shows that they do not completely make up the valence oscillator strength. It is unclear whether this deficiency of the oscillator strength has been incorporated into their inner-shell treatment. Interband transitions are enhanced in the energyloss function for surface plasmon excitations. A comparison of this function, Im[(e - l>*/~(e + 111, determined presently (solid curves), using Yubero and Tougaard’s parameters (chain curves), and using experimental data (dashed curves) is also shown in fig. 2. It is clear that our results are much better than those of Yubero and Tougaard for both volume and surface plasmon excitations.
207
In this figure, we again label critical-point energies (downward arrows) and plasmon energies (upward arrows) determined in this work. Similarly, we compare in fig. 3 ~~(0, w), ~~(0, 0) and Im[- l/e(O, o)] for SiO, glass fitted presently and measured experimentally [30-321. Prominent interband transitions occur in the region of critical-point energies between 10 and 20 eV. However, only a minor influence of these transitions on the energy-loss function can be seen because of reduced transition intensities in l2 above 15 eV. To assure the accuracy of dielectric functions obtained using eqs. (31, (8) and (91, we also plot in this figure results of Z,(w) and Z{(w>/~i calculated using eqs. (11) and (12). The agreement between present results (solid curves) and experimental data (dashed curves) is quite good. It is noticed that Z,(w) and Z:(w) merge to a constant value equal to the number of valence electrons per molecule at large w but differ significantly at small w.
5T_ 4s ,* 3 2
2-
= lwE2 0 -------,
Fig. 3. A plot of ~~(0,w), Iml- l/40, tions defined in eqs. (11) and (12) for
o)] and their integraSiO, glass. Solid and dashed curves are results of present calculations and experimental measurements [30-321. Vertical downward and upward arrows indicate critical-point energies and plasmon energies determined in this work.
CM
208
o~...“““““‘l”“l”“’ 0 10
20 0
Kwei et al. / Plasmon excitations and interband transitions
30
40
so
WI
Fig. 4. A plot of DIMFP, including contributions from both surface and volume plasmon excitations, for a 300 eV efectron in Fe. Solid, dashed, long-chain and short-chain curves are, respectively, results of present calculations, experimental measurements [29], model-A [29] and model-B calculations [291.
Using eq. (13) and the parameters listed in table 1, we can estimate electron DIMFPs in the listed solids for the contribution from valenceband excitations. Fig. 4 shows a comparison of present results (solid curves) with those determined e~erimentally (dashed curves) in REELS for a 300 eV electron at 20” incidence angle with respect to the surface normal of an Fe sample [29]. The sample thickness was much larger than the saturation depth for a constant probability of surface plasmon excitations. Both the present results and experimental data include contributions from volume and surface plasmon excitations and interband transitions. A detailed discussion about the calculation of differential probabilities for surface excitations will be presented elsewhere 1331. Good agreement is found between the present model calculations and the experimental measurements. To explore the im~~ance of interband transitions at smal1 energy losses, we also plot in this figure corresponding results (chain curves) calculated using dielectric functions fitted by Yubero and Tougaard [29]. Since these fits are less accurate for interband transitions, we can see that the agreement between their results (see ref. [27] for the description of model A and model B) and experimental data is poor in the energy-loss region between 0 and 30 eV.
Finally, we compare electron IMFPs calculated presently and determined experimentally. Fig. 5 shows such a comparison for fast electrons in SiO, crystal. Individual contributions from the valence band using the present approach and inner-shells using the sum-rule-constrained binary-collision model [ 181 are plotted separately. Present results (solid curves) are in reasonable agreement with experimental data (circles) [34,35] and with calculations by Tanuma et al. (dashed curves) [36] above N 100 eV. Their calculations made use of a model dielectric function proposed by Penn 1371based on a single-pole local plasma appro~mation and on the EGM of Lindhard. The basic ideas in the dielectric unction of Penn include the assumption of zero plasmon damping associated with the single-pole approximation and the application of a free-electron gas without band gap in the EGM. While these ideas should work successfully for high-energy electrons in the free-electron-like solids, they probably do not work equally well for low-energy electrons, especially in semiconductors and insulators. Although identical results on the energy-loss function at 9 + 0 are ensured by the application of the same optical data, the model dielectric function of Penn is different from that of present work at 4 > 0. Because of the inclusion of plasmon dampings
loo
,,I
,,I
SiO, (crystal)
102
IO3
E (ev) Fig. 5. A plot of electron IMFP in SiO, crystal. Individual (valence band, Si2s subshell and Si2p subshell) and total contributions are shown separately. Solid curves, dashed curves and symbols are results of present calculations, other calculations [36] and experimental measurements 134,351.
CM. Kwei et al. / Plasmon excitations and interhd
transitions
209
of the experimental optical data. Figs. 6 and 7 show a similar comparison of electron fMFPs in several solids calculated presently
4. Conclusims
Fig. 6. A plot of electron IMFP in Au, Fe, Cu and Pd. Solid curves, dashed curves and symhois are results of present calculations, other calculations 136,491and experimental measurements: open circles [39}, solid circles [38], open triangles [ItO], solid triangles [42] and open squares [41].
and the consideration of binding energies in our model dielectric function, it is anticipated that present results provide a better estimate for electron lMFPs at low energies. The reIiabili~ of these results depends, of course, on the accuracy
Research supported by the National Science Council of the Republic of China under Contract No. NSC82-0208-M-009-013.
loo
5*
10
References
fk
3
Electron inelastic mean free paths in solids are im~rtant in surface analysis. In this paper, we have calculated such quantities in several solids having complex band structures. Our description of plasmon excitations and interband transitions was based on an extended Drude dielectric function. Detailed examinations of critical-point energies, plasmon energies and of sum-rules were made to assure the accuracy of our calculations. These calculations apply not only to volume but also to surface plasmon excitations.
1 Ill S. Tougaard and J. Kramer, Phys. Rev. B 43 11991~X651. 0.1
0.01 10’
102
10%
E (ev) Fig. 7. A plot of electron IMFP in Ag, Ni, Mg;Q and SiOz. Solid curves, dashed curves and symbols are results of present calculations, other calculations 136,491and experimental measurements: open circles 1431,solid circles [44], open triangles [481, solid triangles f453, open squares 1471,solid squares /461, open diamonds [34f and solid diamonds 1351.
121 3. Lesiak, A. Jablonski, Z. Prussak and P. Mrozek, Surf, Sci. 223 119891213. 131 C.J. Powell, J. Electron Spectrosc. 47 09881 197. I41 L. Tofterup, surf. Sci. 167 f1986) 70. [5] Y.F. Chen, CM. Kwei and C.J. Tung, J. Phys. D 25 (1992) 262. [61 C.J. Tung and R.H. Ritchie, Phys. Rev. B 16 (1977) 4302. [7] C.J. Tung, J.G. Ashley and R.H. Ritchie, Surf. Sci. 81 (1979) 427. 181D.R. Penn, Phys. Rev. B 3.5 (1987) 482. 191 C. Colliex, in: Advances in Optical and Electron Microscopy, Vol. 9, Fds. V.E. Cosslett and R. Barer (Academic Press, London, 1984) p. 65.
210
CM Kwei et al. / Plasmon excitations and interband transitions
1101 R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope (Plenum, New York, 1986). 1111H. Raether, Excitations of Plasmons and Interband Transitions by Electrons, Vol. 88 of Springer Tracts in Modern Physics (Springer, New York, 1980). 1121J. Daniels, C.V. Festenberg, H. Raether and K. Zeppenfeld, Optical Constants of Solids by Electron Spectroscopy. Vol. 54 of Springer Tracts in Modern Physics (Springer, New York, 1970) p. 78. [13] Y. Smith and E. Shiles, Phys. Rev. B 17 (1978) 4689. [14] R.H. Ritchie and A. Howie, Philos. Mag. 36 (1977) 463. [15] R.H. Ritchie, R.N. Hamm, J.E. Turner, H.A. Wright and W.E. Belch, in: Physical and Chemical Mechanisms in Molecular Radiation Biology, Eds. W.A. Glass and M.N. Varma (Plenum, New York, 1991) p. 99. 1161 R.H. Ritchie, Phys. Rev. 114 (1959) 644. 1171C.M. Kwei and L.W. Chen, Surf. Interface Anal. 11 (1988) 60. 1181 C.M. Kwei, Y.F. Chen and C.J. Tung, Phys. Rev. A 45 (1992) 4421. [19] R.D. Leapman, P. Rez and D.F. Mayers, J. Chem. Phys. 72 (1980) 1232. 1201S.T. Manson, Phys. Rev. A 6 (1972) 1013. 1211E.T. McGuire, Phys. Rev. A 3 (1971) 267. [22] D.M. Roessler and W.C. Walker, Phys. Rev. 159 (1967) 733. [23] D.M. Roessler and D.R. Huffman, in: Handbook of Optical Constants of Solids II, Ed. E.D. Palik (Academic Press, New York, 1991) p. 919. 1241V.E. Henrich, G. Dresselhaus and H.J. Zeiger, Phys. Rev. B 22 (1980) 4764. [25] C.Y. Fang, W. Saslow and M.L. Cohen, Phys. Rev. 168 (19681 992. 1261 B. Dold and R. Mecke, Optik 22 (19651435. [27] H.J. Hagemann, W. Gudat and C. Kunz, J. Opt. Sot. Am. 65 (1975) 742. [28] D.W. Lynch and W.R. Hunter, in: Handbook of Optical Constants of Solids I, Ed. E.D. Palik (Academic Press, New York, 19851 p. 275. [29] F. Yubero and S. Tougaard, Phys. Rev. B 46 (199212486. [30] H.R. Philipp, J. Phys. Chem. Solids 32 (19711 1935. [31] J. Rife and J. Osantowski, J. Opt. Sot. Am. 70 (1980) 1513. [32] H.R. Philipp, in: Handbook of Optical Constants of
Solids I, Ed. E.D. Palik (Academic Press, New York, 1985) p. 719. 1331C.M. Kwei, Y.F. Chen and C.J. Tung, manuscript in preparation. [34] R. Flitsch and S.I. Raider, J. Vat. Sci. Technol. 12 (1975) 305. 1351 M.F. Ebel and W. Liebl, J. Electron Spectrosc. Relat. Phenom. 16 (1979) 463. 1361S. Tanuma, C.J. Powell and D.R. Penn, Surf. Interface Anal. 17 (1991) 929. 1371D.R. Penn, Phys. Rev. B 35 (1987) 482. 1381Y. Bear, P.F. Heden, J. Hedmann, M. Klasson and C. Nordling, Solid State Commun. 8 (1970) 1479. 1391M. Klasson, J. Hedmann, A. Berndtsson, R. Nilsson and C. Nordling, Phys. Ser. 5 (1972) 93. 1401 I. Lindau, P. Pianetta, K.Y. Yu and W.E. Spicer, J. Electron Spectrosc. Relat. Phenom. 8 (1976) 487. 1411J.A. Knapp, F.J. Himpsel and D.E. Eastmann, Phys. Rev. B 19 (1979) 4952. 1421 M.P. Seah, J. Phys. F 3 (1973) 1538. 1431P.W. Palmberg and T.N. Rhodin, J. Appl. Phys. 39 (1968) 2425. 1441J.C. Tracy, NATO Summer School on Induced Electron Emission, Ghent, 1972, unpublished. 1451J. Brunner and H. Zogg, J. Electron Spectrosc. Relat. Phenom. 5 (1974) 911. 1461P. Mrozek, A. Jablonske and A. Sulyok, Surf. Interface Anal. 11 (1988) 499. 1471M.A. Burke and J.J. Schreurs, Surf. Interface Anal. 4 (1982) 42. 1481F. Wooten and W.M. Breen, Phys. Rev. 165 (19681 703. 1491S. Tanuma, C.J. Powell and D.R. Penn, Surf. Interface Anal. 17 (1991) 911. 1501M.A. Ordal, R.J. Bell, R.W. Alexander, L.L. Long and M.R. Querry, Appl. Opt. 24 (19851 1985. 1511T.J. Moravec, J.C. Rife and D.L. Dexter, Phys. Rev. B 13 (1976) 3297. 1521P.B. Johnson and R.W. Christy, Phys. Rev. B 9 (1974) 5056. [53] R.C. Vehse, E.T. Arakawa and M.W. Williams, Phys. Rev. B 1 (1970) 517. [54] D.L. Windt, WC. Cash, Jr., M. Scott, P. Arendt, B. Newman, R.F. Fisher and A.B. Swarzlander, Appl. Opt. 27 (1988) 246.
&i-face science
Electron inelastic mean free paths for plasmon excitations and interband transitions C.M. Kwei, Y.F. Chen Department of Electronics Engineering, National Chiao Tung University, Hsinchu, Taiwan, ROC
C.J. Tung’~* and J.P. Wang Institute of Nuclear Science, National Tsing Hua University, Hsinchu, Taiwan, RUC
Received 1 April 1993; accepted for publication 3 May 1993
Both plasmon excitations and interband transitions are important in the response of valence bands to fast electrons. An extended Drude dielectric function was established to describe such a response in solids of complex band structures. Parameters in this function were determined by a fit of the imaginary part of the function to experimental optical data. The real part of the dielectric function and the energy-loss function were also checked with experimental data to confirm critical-point energies in the interband transitions and plasmon energies in the collective excitations. In addition, sum-rules about the imaginary part of the dielectric function and the energy-loss function, respectively, were applied to assure the accuracy of these functions. Electron inelastic mean free paths in several solids were caicutated and compared to available e~e~mentai and theoretical data.
1. Introduction Information on electron inelastic mean free paths (MFPs) in solids is impo~ant in surface and interface analysis such as X-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy @ES). This information can be extracted from measured spectra in reflection electron-energy-loss spectroscopy (REELS) [l], determined by elastic peak electron spectroscopy 121or measured using the thin-film overlayer technique [3]. Corrections should be made for multiple elastic scatterings which modify the pathlengths and directions of electrons in solids [4,51. Alternatively, one may calculate electron MFPs using the dielectric response theory. The electron-gas model (EGM) which describes the response of valence-band (or conduction-band in * To whom correspondence should be addressed. i Present address: Department of Nuclear Engineering, Texas A & M University, College Station, TX 77843, USA. 0039-6028/93/$06.00
the case of a metal) electrons is widely used for these calculations [6-S]. The EGM is a good approximation for materials that exhibit simple energy-loss peaks with matched theoretical and experimental plasmon energies [9]. This model, however, is not good for solids that have complex band structures leading to strong interband transitions. In such cases, the ion-core lattice has a significant effect on electron motion to produce non-spherical Fermi surfaces and to support single-electron interband transitions [lo]. In addition, these transitions shift free-electron plasmon energies towards higher values and cause extra resonance peaks in the energy-loss spectrum 1113. For these solids, the dielectric function and the energy-loss function must be carefully examined with respect to their peak heights and positions. This can be done by applying sum-rules and identifying critical-point energies in the interband transitions. In this work, we calculate electron MFPs in several solids having complex band structures.
0 1993 - Elsevier Science Publishers B.V. All rights reserved
CM. Kwei et al. / Plasmon excitations and interband rransitiorw
Our emphasis is on the contribution from valence-band excitations. We employ an extended Drude dielectric function with parameters determined by a fit of its imaginary part to experimental optical data. The real part of the dielectric function and the energy-loss function are also checked with experimental data to confirm critical-point energies in the interband transitions and plasmon energies in the energy-loss spectra. In addition, sum-rules about the imaginary part of the dielectric function and the energy-loss function, respectively, are applied to verify the accuracy of these functions. Electron inelastic MFPs calculated in this work are comnared to avaitable experimental and theoretical data.
203
For a valence band of ni bound electrons per volume adding to the conduction band of no free electrons per volume, the imaginary part of the dielectric function is given by fll]
(4) where wP =I:(4+~7=n,)‘/~is the free-electron plasmon energy, Ri = n,/tts, wi is the gap-energy between conduction and valence bands or the critical-point energy for interband transitions, and yi is the damping constant associated with the valence band. The real part of the dielectric function is thus given by
2. Themy The Drude model works quite well for the conduction band of a free-electron-like metal such as aluminum [12]. In this model, conduction electrons are described by a free-electron gas constrained by the Fermi-Dirac statistics. Interaction with the ion-core lattice is assumed to be a minor effect leading the damping of plasma resonances to single-efectron excitations. The imaginary part of the dielectric function in the iongwavelength limit, i.e. 4 + 0, is given by
(1) where q is the momentum transfer, o is the energy transfer, oP is the free-electron plasmon energy, and y is the damping constant. Note that atomic units are used throughout this paper for all quantities and units unless otherwise specified. The real part of the dielectric function is given by the earners-Kronig anaiysis as q(0, w) = 1 -
is202 p to4+ ,=y2.
The energy-loss function, or the imaginary part of the negative inverse dielectric function, may then be calculated from -1 Im-------=-
( I 44
0)
E2 ET+&
(5) The effective plasmon energy of the system, defined by t-,(0, 01 = 0 in the limit y + 0 and yi -+ 0, becomes
For an insulator or a semiconductor with 120= 0 and ni = II (since the conduction band is psactitally empty), eq. (6) gives “); = 4nn + (0””
(“0
One sees that the plasmon energy of the system is shifted upward above the value for the free-electron gas. This situation is demonst~ted in Si and Ge [ll]. Also, interband tr~sitions cause a second resonance peak in the energy-loss function f12J. Since all electrons are in the valence band, the first term on the right-hand side of eq. (41 becomes zero. This leads to a continuous distribution of oscillator strengths above the band gap, For a solid having a complex band structure, the valence band may be composed of several subbands. Each ith subband is characterized by its own oscillator strength, Aj, damping constant, ‘yj, and criticai-point energy, wi. Interband transi-
204
C.M. Kwei et al. / Plasmon excitations and interband transitions
tions can be incorporated into the Drude model by adding these subband electrons to the freeelectron system. Based on the superposition of damped linear oscillators, one can write
Here Ai =A,wc with Ai established in eq. (4). The real part of the dielectric function is thus given by
E1(O,O) = EB
(9)
-
Note that we include an lB term in eq. (9) to account for the background dielectric function in the solid. The possible departure of lB from one is due to the fact that valence electrons are embedded in the background of polarizable ion cores [13]. By taking ~~(0, w) = 0 in the limit ri + 0, we can solve the plasmon energies of the system. We find that the L$, are the roots of the equation
(10) If all ni values are of comparable magnitude and the wi values are well distinct from one another, we should find as many interband transition peaks in the energy-loss function as there are subbands. In this work, we fit eq. (8) to experimental optical data to determine the parameters A:, yi and wi. To make sure that these results are accurate, we require also that ~~(0, w> and Im[ - l/40, WI in eqs. (9) and (3) are in agreement with experimental data. In addition, we check the validity of the sum-rule /0
?JE~(O, w’) dw’ = 2r*NZ,(
o),
(11)
where N is the number of atoms (or molecules) per volume and Z,(o) is the effective number of electrons per atom (or molecule) contributed by valence excitations up to the energy transfer o. Note that ZJm) approaches the number of valence electrons per atom (or molecule) and that
4rZVZJm) = L4:. We also check the validity of the sum-rule for the energy-loss function J',wu' Im[ .,o:t.,]
do’=Za’N~,
(12)
where Z$w> is an alternative effective number of electrons per atom (or molecule) contributed by valence excitations up to the energy transfer w. Again, Z:(m) approaches the number of valence electrons per atom (or molecule) and 4rNZ,Xm) = CA:. The sum-rules for the contribution from valence excitations, i.e. eqs. (11) and (12), are valid if one keeps the upper limits of the integration in these equations smaller than the binding energy of the most loosely bound inner shell. To extend the dielectric function into the q + 0 region of the q-o plane, we replace oi in eq. (8) by wi + q2/2 [14,151. This extension leads to a correct behavior of the dispersion relation at two extremes, i.e. the optical end, q + 0, and the Bethe-ridge region, q + m. The deviation of this extension from the true dispersion relation makes only minor difference in the determination of electron MFPs. The differential inverse mean free path (DIMFP), p(E, o), for an electron of energy E to lose energy w is given by [16]
(13) where q + = d%? f \/2E-20 is due to the conservation of energy and momentum. The inverse mean free path (IMFP), p(E), is obtained by integrating eq. (13) over all allowed energy transfers, i.e. p(E)
=
lEp(E, w) dw.
(14)
Using eqs. (13) and (14) with the energy-loss function given above, one can calculate electron DIMFPs and IMFPs in solids. For the contribution from inner-shell ionizations we apply either the local plasma approximation [17] or the sum-rule-constrained binary-collision model [18] using atomic generalized oscillator strengths from the Hartree-Slater approach
205
CM. Kwei et al. / Plasmon excitations and interband transitions
[19-211. The first use of the local plasma approximation in electron MFP theory was made with the EGM [7]. The inner-shell contribution, however, is very small for electron energies below several hundred eV in all solids studied here.
3. Results and discussion We now fit eqs. (31, (8) and (9) to measured optical data for several solids. Table 1 lists the fitting parameters for MgO, Cu, SiO, glass, Fe, SiO, crystal, Au, Pd, Ag and Ni. In these fits, we check not only the accuracy of ~~(0, o>, ~~(0, w) and Im[ - l/e(O, w)] determined by these equations but also the extent to which they satisfy the sum-rules of eqs. (11) and (12). These sum-rules are applied by setting the upper limits of integration to both finite and infinite values. In the case of finite-range sum-rules, we examine the compliance of the calculated effective number of valence electrons, at any given energy transfer, with the corresponding optical data. In the case of infinite-range sum rules, we check the satisfaction of the total oscillator strength to the condition L4; = 457NZ,. Since a separate treatment for inner-shells is used, we require that Z, equals the number of valence electrons per atom (or molecule). In all cases, we find that these Z, values agree quite well with those determined from observed plasmon energies. For Au, our fits cover the 5s and 5p inner-shell due to the strong overlapping of oscillator strengths between the valence band and these subshells in the vicinity of their binding energies. To avoid the ambiguity in the separation of cross contributions, we extend our fits to higher energy transfers covering these inner-shell excitations. Fig. 1 shows a comparison of ~~(0, 01, ~~(0, w) and Im[- l/e(O, w)] for MgO fitted presently (solid curves) and determined experimentally (dashed curves) [22,23]. It is seen from the figure that good agreement is found between present results and experimental data for all functions plotted. It is also seen that strong interband transitions occur in the region of critical-point energies between 8 and 25 eV. These transitions also induce plasmon peaks in the energy-loss function.
0 0
10
20 0
30
0.0 40
(ev)
Fig. 1. A plot of ~~(0, 01, ~~(0, 01 and Im[- 1/4,~)1 for MgO.Solid and dashed curves are results of present calculations and experimental measurements [22,231. Verticaldownward and upward arrows indicate critical-point energies and plasmon energies determined in this work.
Interband transition energies (vertical downward arrows) fitted in this work and plasmon energies (vertical upward arrows) calculated using eq. (10) are labeled in the figure. A comparison of these energies with corresponding experimental [241 and theoretical [25] data is made in table 1. Fig. 2 shows a comparison of ~~(0, w), ~~(0, w) and Im[ - l/e(O, w)] for Cu determined experimentally (dashed curves) [26-281 and obtained presently (solid curves) using eqs. (31, (8) and (9) with parameters listed in table 1. Although a similar fit by Yubero and Tougaard [29] to Im(- l/e) data of Cu is available, their fitting parameters produce poor results on e1 and Ed compared to experimental data, especially at small energy losses where interband transitions are important. This is because it is the imaginary part of the dielectric function rather than the energy-loss function that contains detailed information on interband transitions. Further, there was a missing term in their fitting parameters corresponding to the conduction band at wi = 0 for metals. This term leads to e1 + --cx) and Ed + co at w + 0, a common feature for all metals. Since Yubero and
Table 1 Parameters in eqs. (8) and (10) for several solids; values in parentheses for MgO are critical-point energies calculated theoretically [25] and plasmon energies determined experimentally [24] Afi (eV ‘1
yi (eV)
Oi (eV)
hp (eV)
A: (eV2)
7.8 10.1 11.7 13.9 15.3 18.8 20.2 22.1 24.5 36.5 47.1
Au (~a = 1) 1281 79 0.1 9 1 1.9 36 17 2.3 4 60 9 100 120 10 6 155 145 7.2 280 20 360 28 26 183
MgO (~a = 1.08) [22-251 0.3 8 3.1 70.6 1.5 45.7 1 47.7 1.9 25 2.5 98 1.2 13.9 1.5 15 2.6 12 15 105 30 152.1
7.7 9.5 10.8 13.1 14.6 17.2 19.2 20.8 24 35 45
Cu k, 64 20 6.5 5.5 4 55 42 172 80 240 100 80 412
0 0.3 2.5 3.1 3.7 5.05 8.93 14.74 25.6 40 55 65 85
0.2 2.3 2.9 3.5 4 7.2 10.4 19.3 27.3 43 56 65.7 87.6
SiO, glass kg = 1.03) [30-321 0.3 11 1.9 45 2.9 49 3.6 60 4 22 13 90 46 207
10.4 11.6 13.8 16.8 19.6 22 37
10.4 12.2 14.6 17.9 20 24.5 40.2
Fe kB = 1.31) [50,51] 0.18 52 5 198 2 10 4.8 41.4 3.2 28 15 200 26 100 150 305
0 2.5 6.04 9.56 12.9 18.4 24.8 48
1.1 5.8 8.1 11.2 13.8 21.8 27.9 50.9
SiO, crystal (~a = 1.01) [321 0.3 18 1.6 53 2.3 58 3.3 70 4 26 11 90 46 182
10.32 11.6 14.1 16.9 19.6 22 37
10.5 12.3 15 18.2 20.1 24.9 40
= 1.05) 126-281 0.01 0.1 0.65 0.7 0.7 2.6 4.76 10.18 8 32 30 30 37
(7.77) (10.89) (13.3) (14.02) (17.01) (18.99) (21.2) (24.76)
(8.2) (11.6) (14.3) (15.4) (18) (22.4) (25.6)
yi (eV)
w, (eV)
Gp (eV)
0.1 3.1 4.1 5.3 8.17 12 14 21.3 29.5 38.5 63 100
2.5 3.3 4.8 6.1 9.3 12.8 17.2 24.4 31.8 43.3 66.6 101
Pd kg = 1) [52-541 85 0.1 75 4.45 4.45 45 5 45 2 0.7 13 1.8 3 45 50 4.6 110 5.6 80 6 70 7 130 17 19 186
0 1.3 4.3 10.7 11.8 13.65 15 16.8 20.5 24.2 30.6 38 49
0.9 3.7 8.1 11.6 12.1 13.9 15.7 17.9 22.4 26.6 32.1 40.3 51.9
Ag kg = 1.03) [281 80 0.07 4 0.45 1.2 10 2 20 11 240 70 5.38 160 15 36 300
0 4.8 5.3 6.4 15 22 31.3 40
4 4.9 5.6 7.6 19 23.8 33.4 45.3
Ni (eB = 1.02)[281 0.01 25 2 50 4 30 6 60 1.3 18 4.2 37 7.2 95 18 200 70 200 80 545
0 0.45 1.5 3.6 4.6 8.3 14.6 24 45 58
0.2 1.2 2.7 4.4 7.1 10.5 17 28.1 46.8 63.7
C.M. Kwei et al. / Plasmon excitations and interband transitions
-0.3""' 0
I” 10
”
I ""I 20
”
30
“I’ 40
50
w (ev)
Fig. 2. A plot of ~~(0,w), ~~(0,o), Im[- l/6(0, w)l and Im[(c - 1j2/c(c + l)] for Cu. Solid, dashed and chain curves are results of present calculations, experimental measurements [26-281 and other calculations [29]. Vertical downward and upward arrows indicate critical-point energies and plasmon energies determined in this work.
Tougaard also divided the contribution for the energy-loss function into valence band and inner-shells, their valence oscillator strength was expected to satisfy the condition CA: = 47rNZ,. Adopting 18 to be the number of inner-shell electrons, Z, = 11 is required in order to add them up to the total oscillator strength of Cu. The latter value agrees with the prediction using 35.9 eV as the plasmon energy [49]. An examination of Yubero and Tougaard’s results, however, shows that they do not completely make up the valence oscillator strength. It is unclear whether this deficiency of the oscillator strength has been incorporated into their inner-shell treatment. Interband transitions are enhanced in the energyloss function for surface plasmon excitations. A comparison of this function, Im[(e - l>*/~(e + 111, determined presently (solid curves), using Yubero and Tougaard’s parameters (chain curves), and using experimental data (dashed curves) is also shown in fig. 2. It is clear that our results are much better than those of Yubero and Tougaard for both volume and surface plasmon excitations.
207
In this figure, we again label critical-point energies (downward arrows) and plasmon energies (upward arrows) determined in this work. Similarly, we compare in fig. 3 ~~(0, w), ~~(0, 0) and Im[- l/e(O, o)] for SiO, glass fitted presently and measured experimentally [30-321. Prominent interband transitions occur in the region of critical-point energies between 10 and 20 eV. However, only a minor influence of these transitions on the energy-loss function can be seen because of reduced transition intensities in l2 above 15 eV. To assure the accuracy of dielectric functions obtained using eqs. (31, (8) and (91, we also plot in this figure results of Z,(w) and Z{(w>/~i calculated using eqs. (11) and (12). The agreement between present results (solid curves) and experimental data (dashed curves) is quite good. It is noticed that Z,(w) and Z:(w) merge to a constant value equal to the number of valence electrons per molecule at large w but differ significantly at small w.
5T_ 4s ,* 3 2
2-
= lwE2 0 -------,
Fig. 3. A plot of ~~(0,w), Iml- l/40, tions defined in eqs. (11) and (12) for
o)] and their integraSiO, glass. Solid and dashed curves are results of present calculations and experimental measurements [30-321. Vertical downward and upward arrows indicate critical-point energies and plasmon energies determined in this work.
CM
208
o~...“““““‘l”“l”“’ 0 10
20 0
Kwei et al. / Plasmon excitations and interband transitions
30
40
so
WI
Fig. 4. A plot of DIMFP, including contributions from both surface and volume plasmon excitations, for a 300 eV efectron in Fe. Solid, dashed, long-chain and short-chain curves are, respectively, results of present calculations, experimental measurements [29], model-A [29] and model-B calculations [291.
Using eq. (13) and the parameters listed in table 1, we can estimate electron DIMFPs in the listed solids for the contribution from valenceband excitations. Fig. 4 shows a comparison of present results (solid curves) with those determined e~erimentally (dashed curves) in REELS for a 300 eV electron at 20” incidence angle with respect to the surface normal of an Fe sample [29]. The sample thickness was much larger than the saturation depth for a constant probability of surface plasmon excitations. Both the present results and experimental data include contributions from volume and surface plasmon excitations and interband transitions. A detailed discussion about the calculation of differential probabilities for surface excitations will be presented elsewhere 1331. Good agreement is found between the present model calculations and the experimental measurements. To explore the im~~ance of interband transitions at smal1 energy losses, we also plot in this figure corresponding results (chain curves) calculated using dielectric functions fitted by Yubero and Tougaard [29]. Since these fits are less accurate for interband transitions, we can see that the agreement between their results (see ref. [27] for the description of model A and model B) and experimental data is poor in the energy-loss region between 0 and 30 eV.
Finally, we compare electron IMFPs calculated presently and determined experimentally. Fig. 5 shows such a comparison for fast electrons in SiO, crystal. Individual contributions from the valence band using the present approach and inner-shells using the sum-rule-constrained binary-collision model [ 181 are plotted separately. Present results (solid curves) are in reasonable agreement with experimental data (circles) [34,35] and with calculations by Tanuma et al. (dashed curves) [36] above N 100 eV. Their calculations made use of a model dielectric function proposed by Penn 1371based on a single-pole local plasma appro~mation and on the EGM of Lindhard. The basic ideas in the dielectric unction of Penn include the assumption of zero plasmon damping associated with the single-pole approximation and the application of a free-electron gas without band gap in the EGM. While these ideas should work successfully for high-energy electrons in the free-electron-like solids, they probably do not work equally well for low-energy electrons, especially in semiconductors and insulators. Although identical results on the energy-loss function at 9 + 0 are ensured by the application of the same optical data, the model dielectric function of Penn is different from that of present work at 4 > 0. Because of the inclusion of plasmon dampings
loo
,,I
,,I
SiO, (crystal)
102
IO3
E (ev) Fig. 5. A plot of electron IMFP in SiO, crystal. Individual (valence band, Si2s subshell and Si2p subshell) and total contributions are shown separately. Solid curves, dashed curves and symbols are results of present calculations, other calculations [36] and experimental measurements 134,351.
CM. Kwei et al. / Plasmon excitations and interhd
transitions
209
of the experimental optical data. Figs. 6 and 7 show a similar comparison of electron fMFPs in several solids calculated presently
4. Conclusims
Fig. 6. A plot of electron IMFP in Au, Fe, Cu and Pd. Solid curves, dashed curves and symhois are results of present calculations, other calculations 136,491and experimental measurements: open circles [39}, solid circles [38], open triangles [ItO], solid triangles [42] and open squares [41].
and the consideration of binding energies in our model dielectric function, it is anticipated that present results provide a better estimate for electron lMFPs at low energies. The reIiabili~ of these results depends, of course, on the accuracy
Research supported by the National Science Council of the Republic of China under Contract No. NSC82-0208-M-009-013.
loo
5*
10
References
fk
3
Electron inelastic mean free paths in solids are im~rtant in surface analysis. In this paper, we have calculated such quantities in several solids having complex band structures. Our description of plasmon excitations and interband transitions was based on an extended Drude dielectric function. Detailed examinations of critical-point energies, plasmon energies and of sum-rules were made to assure the accuracy of our calculations. These calculations apply not only to volume but also to surface plasmon excitations.
1 Ill S. Tougaard and J. Kramer, Phys. Rev. B 43 11991~X651. 0.1
0.01 10’
102
10%
E (ev) Fig. 7. A plot of electron IMFP in Ag, Ni, Mg;Q and SiOz. Solid curves, dashed curves and symbols are results of present calculations, other calculations 136,491and experimental measurements: open circles 1431,solid circles [44], open triangles [481, solid triangles f453, open squares 1471,solid squares /461, open diamonds [34f and solid diamonds 1351.
121 3. Lesiak, A. Jablonski, Z. Prussak and P. Mrozek, Surf, Sci. 223 119891213. 131 C.J. Powell, J. Electron Spectrosc. 47 09881 197. I41 L. Tofterup, surf. Sci. 167 f1986) 70. [5] Y.F. Chen, CM. Kwei and C.J. Tung, J. Phys. D 25 (1992) 262. [61 C.J. Tung and R.H. Ritchie, Phys. Rev. B 16 (1977) 4302. [7] C.J. Tung, J.G. Ashley and R.H. Ritchie, Surf. Sci. 81 (1979) 427. 181D.R. Penn, Phys. Rev. B 3.5 (1987) 482. 191 C. Colliex, in: Advances in Optical and Electron Microscopy, Vol. 9, Fds. V.E. Cosslett and R. Barer (Academic Press, London, 1984) p. 65.
210
CM Kwei et al. / Plasmon excitations and interband transitions
1101 R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope (Plenum, New York, 1986). 1111H. Raether, Excitations of Plasmons and Interband Transitions by Electrons, Vol. 88 of Springer Tracts in Modern Physics (Springer, New York, 1980). 1121J. Daniels, C.V. Festenberg, H. Raether and K. Zeppenfeld, Optical Constants of Solids by Electron Spectroscopy. Vol. 54 of Springer Tracts in Modern Physics (Springer, New York, 1970) p. 78. [13] Y. Smith and E. Shiles, Phys. Rev. B 17 (1978) 4689. [14] R.H. Ritchie and A. Howie, Philos. Mag. 36 (1977) 463. [15] R.H. Ritchie, R.N. Hamm, J.E. Turner, H.A. Wright and W.E. Belch, in: Physical and Chemical Mechanisms in Molecular Radiation Biology, Eds. W.A. Glass and M.N. Varma (Plenum, New York, 1991) p. 99. 1161 R.H. Ritchie, Phys. Rev. 114 (1959) 644. 1171C.M. Kwei and L.W. Chen, Surf. Interface Anal. 11 (1988) 60. 1181 C.M. Kwei, Y.F. Chen and C.J. Tung, Phys. Rev. A 45 (1992) 4421. [19] R.D. Leapman, P. Rez and D.F. Mayers, J. Chem. Phys. 72 (1980) 1232. 1201S.T. Manson, Phys. Rev. A 6 (1972) 1013. 1211E.T. McGuire, Phys. Rev. A 3 (1971) 267. [22] D.M. Roessler and W.C. Walker, Phys. Rev. 159 (1967) 733. [23] D.M. Roessler and D.R. Huffman, in: Handbook of Optical Constants of Solids II, Ed. E.D. Palik (Academic Press, New York, 1991) p. 919. 1241V.E. Henrich, G. Dresselhaus and H.J. Zeiger, Phys. Rev. B 22 (1980) 4764. [25] C.Y. Fang, W. Saslow and M.L. Cohen, Phys. Rev. 168 (19681 992. 1261 B. Dold and R. Mecke, Optik 22 (19651435. [27] H.J. Hagemann, W. Gudat and C. Kunz, J. Opt. Sot. Am. 65 (1975) 742. [28] D.W. Lynch and W.R. Hunter, in: Handbook of Optical Constants of Solids I, Ed. E.D. Palik (Academic Press, New York, 19851 p. 275. [29] F. Yubero and S. Tougaard, Phys. Rev. B 46 (199212486. [30] H.R. Philipp, J. Phys. Chem. Solids 32 (19711 1935. [31] J. Rife and J. Osantowski, J. Opt. Sot. Am. 70 (1980) 1513. [32] H.R. Philipp, in: Handbook of Optical Constants of
Solids I, Ed. E.D. Palik (Academic Press, New York, 1985) p. 719. 1331C.M. Kwei, Y.F. Chen and C.J. Tung, manuscript in preparation. [34] R. Flitsch and S.I. Raider, J. Vat. Sci. Technol. 12 (1975) 305. 1351 M.F. Ebel and W. Liebl, J. Electron Spectrosc. Relat. Phenom. 16 (1979) 463. 1361S. Tanuma, C.J. Powell and D.R. Penn, Surf. Interface Anal. 17 (1991) 929. 1371D.R. Penn, Phys. Rev. B 35 (1987) 482. 1381Y. Bear, P.F. Heden, J. Hedmann, M. Klasson and C. Nordling, Solid State Commun. 8 (1970) 1479. 1391M. Klasson, J. Hedmann, A. Berndtsson, R. Nilsson and C. Nordling, Phys. Ser. 5 (1972) 93. 1401 I. Lindau, P. Pianetta, K.Y. Yu and W.E. Spicer, J. Electron Spectrosc. Relat. Phenom. 8 (1976) 487. 1411J.A. Knapp, F.J. Himpsel and D.E. Eastmann, Phys. Rev. B 19 (1979) 4952. 1421 M.P. Seah, J. Phys. F 3 (1973) 1538. 1431P.W. Palmberg and T.N. Rhodin, J. Appl. Phys. 39 (1968) 2425. 1441J.C. Tracy, NATO Summer School on Induced Electron Emission, Ghent, 1972, unpublished. 1451J. Brunner and H. Zogg, J. Electron Spectrosc. Relat. Phenom. 5 (1974) 911. 1461P. Mrozek, A. Jablonske and A. Sulyok, Surf. Interface Anal. 11 (1988) 499. 1471M.A. Burke and J.J. Schreurs, Surf. Interface Anal. 4 (1982) 42. 1481F. Wooten and W.M. Breen, Phys. Rev. 165 (19681 703. 1491S. Tanuma, C.J. Powell and D.R. Penn, Surf. Interface Anal. 17 (1991) 911. 1501M.A. Ordal, R.J. Bell, R.W. Alexander, L.L. Long and M.R. Querry, Appl. Opt. 24 (19851 1985. 1511T.J. Moravec, J.C. Rife and D.L. Dexter, Phys. Rev. B 13 (1976) 3297. 1521P.B. Johnson and R.W. Christy, Phys. Rev. B 9 (1974) 5056. [53] R.C. Vehse, E.T. Arakawa and M.W. Williams, Phys. Rev. B 1 (1970) 517. [54] D.L. Windt, WC. Cash, Jr., M. Scott, P. Arendt, B. Newman, R.F. Fisher and A.B. Swarzlander, Appl. Opt. 27 (1988) 246.