A Monte Carlo modeling of electron interaction ... - Wiley Online Library

SCANNING Vol. 18, 92–113 (1996) © FAMS, Inc.

Received March 31, 1995 Accepted May 5, 1995

A Monte Carlo Modeling of Electron Interaction with Solids Including Cascade Secondary Electron Production Z.-J. DING*† AND R. SHIMIZU† *Fundamental Physics Center, University of Science and Technology of China, Anhui, People’s Republic of China; †Department of Applied Physics, Osaka University, Osaka, Japan

Summary: A new Monte Carlo simulation approach has been developed to describe electron scattering and secondary electron generation processes in solids. This approach is based on the uses of Mott’s elastic scattering cross section and Penn’s dielectric function. A very good agreement has been found on the energy distribution of backscattered electrons between theoretical calculations and accurate experimental measurement recently made by Goto et al. (1994). This fact confirms that the present Monte Carlo model is very useful for more comprehensive understanding of basic phenomena in electron spectroscopy and microscopy, particularly in the sub-keV energy region where cascade secondary electrons play a dominant role. In this paper the details of the Monte Carlo procedure are described and further application to the mechanism of secondary electron generation is presented. Key words: Monte Carlo electron-trajectory simulations, dielectric function, cascade secondary electrons

Introduction The Monte Carlo electron trajectory simulation method has been extensively applied for many years to electron probe microanalysis, electron spectroscopy, and electron microscopy. Various Monte Carlo models, depending on the approaches to the treatment of electron elastic and inelastic scattering, have been proposed and used for specific purposes. Our aim was to develop a more accurate and comprehensive Monte Carlo model which can be applied to electron probe microanalysis, Auger electron spectroscopy, scanning electron microscopy (SEM), x-ray photoelectron spectroscopy, and reflection electron energy loss spectroscopy. Presentation of this paper was made possible through the support of the Foundation for Advances in Medicine and Science, Inc. Invited paper presented at SCANNING 95, Monterey, California, USA, April, 1995. Address for reprints: Z.-J. Ding Fundamental Physics Center University of Science and Technology of China Hefei 230026 Anhui, People’s Republic of China

First, the present status of the Monte Carlo calculation, particularly related to SEM, is outlined. For the treatment of electron elastic scattering, both the screened Rutherford formula and the Mott differential cross section have been available. Since it was been found (Ichimura and Shimizu 1981, Reimer and Krefting 1976) that the use of the Mott cross section is more satisfactory than the Rutherford cross section, particularly for heavier elements and at lower energies, the employment of the Mott cross sections in the keV and sub-keV energy region is now popular. Regarding the approach to electron inelastic scattering, the Bethe stopping power equation in the continuous slowing-down approximation (CSDA) has been widely used with considerable success. To include fast secondary electron generation, some modifications have also been made to the CSDA: the hybrid of the CSDA with the individual energy loss processes due to innershell ionization (Ichimura and Shimizu 1981), the utilization of the discrete inelastic scattering cross section of Moller (1931) (Murata et al. 1981), and the use of generalized oscillator strength in a hydrogenic approximation for inner shells (Desalvo and Rosa 1987), each of which permits the simulation of the fast knock-on electrons. However, any characteristic energy loss process specific to a sample is omitted under the CSDA. Considering that Bethe’s equation is valid only at sufficiently high electron energies, Rao-Sahib and Wittry (1974) have empirically extrapolated Bethe stopping powers to the low energy region by assuming a parabolic function, -dE/ds ∝ E −1/2, which has been extensively used in the simulation of the slowing down process of slow electrons (Joy 1987, Kotera 1989, Luo et al. 1987, Newbury et al. 1990). But this formulation gives energy dependence opposite of that predicated by the Lindhard theory for free electron gas (Lindhard 1954, Ritchie et al. 1969) and overestimates significantly the energy loss of low-energy electrons. Concerning secondary electron generation, both models of the secondary electron excitation assumed from the stopping power formula (Joy 1987, Matsukawa and Shimizu 1974, Murata et al. 1987) or from the Streitwolf (1959) equation (Koshikawa and Shimizu 1973, Kotera 1989, Kotera et al. 1990, Luo et al. 1987) have required fitting parameters in order to get the correct secondary electron yield. Furthermore, the secondary emission process was simply described by an exponential decay law (Joy 1985) or was hybridized with a cascade model of secondary production (Luo and Joy 1990) and with emission processes (Koshikawa

Z.-J. Ding and R. Shimizu: MC modeling of electron interaction with solids

and Shimizu 1973). Another unsatisfactory situation is that one usually divides the energy region for fast secondary electrons and the low secondaries to adopt different approaches for each (Ding and Shimizu 1988a, 1989a; Kotera 1989) because the available models of electron scattering and secondary generation were limited in a certain energy range. Therefore, a unified treatment of electron inelastic scattering and secondary electron generation is quite necessary. Perhaps the best approach should be based on a dielectric function which characterizes the specific excitation processes of a sample (Pines 1964). A dielectric function ε (q,ω) can provide us with detailed knowledge of energy loss cross section and scattering angular distribution for electron inelastic scattering. This has been achieved (Cailler and Ganachaud 1990, Ganachaud and Cailler 1979) for free electron metal, Al, using the well-known Lindhard dielectric function describing the plasmon excitation and electron-hole pair production. Unfortunately, the ideal Lindhard dielectric function of free electron gas in the random phase approximation (Fetter and Walecka 1971) is valid only for limited materials, that is, socalled free electron metals, and is hardly applicable to other materials such as transition and noble metals, for which the optical dielectric data have shown complexities due to interband transitions (Rather 1980). Some modified analytical dielectric functions in the plasmon-pole approximation with damping (Brandt and Reinheimer 1970) were also limited to materials, for which the damping plasmon dominates the energy loss processes of electrons, such as carbon and silicon (Desalvo et al. 1984). Furthermore, the theoretical calculation of q-dependent dielectric function, ε (q,ω), is difficult and has been numerically evaluated (Nizzoli 1978, Singhal 1975, Sramek and Cohen 1972, Walter and Cohen 1972) only for selected qs and the first few reciprocal lattice vectors using realistic band structure data for some simple metals and semiconductors (Sturm 1982). In fact, the comprehensive firstprinciple theoretical calculations for metals have been limited to ε(ω) = ε (q= 0,ω) (Maksimov et al. 1988). It is thus impractical to use ε (q,ω) derived from a band structure calculation for a Monte Carlo simulation, hence we have to use the optical dielectric data which are available experimentally from optical method and electron energy loss spectroscopy (Egerton 1986). Systematic data of the dielectric constants have been provided and compiled for a number of materials for practical use (Hagemann et al. 1975; Palik 1985, 1991) with advance in use of synchrotron radiation facilities. Our first attempt (Ding and Shimizu 1988a) had used the approach given by Powell (1985). The q-integrated excitation function for electron energy loss and production of secondary electrons is related to ε (ω) and a parameter, which was determined by fitting the calculated electron mean free path with experimental data. Reasonable accuracy has been achieved in the calculation of the energy distribution of backscattered electrons (Ding et al. 1988b). However, the angular information in electron inelastic scattering was accumulated by the integration over q, and the parameter-involved approach is not favorable for general use. Particularly, this parameter does not allow to describe the Bethe stopping powers at high energies, so that we

93

had to use this simple dielectric model only for slow electrons. According to Penn (1987) the q-dependent electron energy loss function may be derived from optical dielectric constants. This algorithm enables us to calculate energy loss cross section and scattering angular distribution required for a Monte Carlo simulation of discrete electron inelastic scattering processes (Ashley 1991, Ding and Shimizu 1989b). It has been shown that the method yields the Bethe stopping powers at high energies (Ashley 1988, Ding and Shimizu 1989b), and the calculated electron mean free paths fit the experimental data in a wide energy region for many elements and compounds. This fact indicates that the dielectric function modeling is very useful for Monte Carlo simulation of electron inelastic scattering. We have used it in the calculation of x-ray depth profiles (Ding and Wu 1993) and background in Auger electron spectroscopy (Ding et al. 1994a), and Tökési et al. (1995) have calculated the reflected electron energy loss spectrum. Jensen and Walker (1993) have also employed it to study the backscattering yield of positrons and electrons of high energies, but they failed to get good agreement with experimental data for electrons, partly because of the neglect of the secondary production. We shall demonstrate, by including cascade secondary electron generation, that the backscattering yields as well as the angular energy distribution describe the precise experimental curve down to low energies very well. This modeling describes the inelastic scattering reasonably well, covering the wide energy range from several eV above the Fermi energy to several tens keV. Furthermore, the simulation of cascade production of secondary electrons included with discrete electron inelastic collisions can be directly made in a simple way. The present Monte Carlo simulation model, therefore, is probably most useful for application to SEM.

Model of Electron Scattering Electron Inelastic Scattering

According to the dielectric theory, a dielectric function provides the comprehensive description of the response of a medium, as an assembly of interacting electrons and atoms, to the disturbance from an external point charge. The differential cross section for electron inelastic scattering is given by Pines and Nozières (1966)  −1  1 d 2 λ−in1 1 = Im   , d (hω )dq πa0 E  ε (q, ω )  q

(1)

where a0 is the Bohr radius, hω and hq are the energy loss and the momentum transfer, respectively, from an electron of kinetic energy E penetrating into a solid of dielectric function, ε (q, ω). λin is the electron inelastic mean free path. The energy loss function, defined as Im{−1/ε (q, ω )}, then completely determines the probability of an inelastic scattering event, the energy loss distribution, and the scattering angular distribution.

94

Scanning Vol. 18, 2 (1996)

Deriving wave vector- and frequency-dependent energy loss function: The problem is now how to derive the wave vector- and frequency-dependent energy loss function. A simple method is to extrapolate Im{−1/ε (ω )}, obtainable from optical data, into finite (q,ω)-plane. First, we may decompose the optical energy loss function into several terms of Drude-type energy loss function,   −1  −1  Im   = ∑ Gi Im  ,  ε D (ω ;ω p,i , γ i )   ε (ω )  i

(2)

dielectric function is approximated as

ε L (q, ω ;ω p ) = 1 +

ω 2p , ω (ω + iγ )

(3)

  π ω 2p −1 lim+ Im  δ (ω − ω q ). = γ q →0  ε L (q, ω ;ω p )  2 ω q

(9)

1.0

(4)

0.8

Im{−1/ε}

Cu 0.6

0.4

(5)

M2,3 0.2

0.0

100

101

(a)

∆E (eV)

102

103

1.2 Ag

1.0 0.8 Im{−1/ε}

where the dispersion ωp(q) = ωp + hq2/2m is chosen so that the Bethe ridge at ω = hq2/2m (Inokuti 1971) is reproduced for q → ∞. γ (q) represents the experimentally observed q-dependency of damping constant (Gibbons et al. 1976). The sum rule is conserved in this scheme. Although some authors (Tougaard 1987, Yubero et al. 1993) have practically used it, it is tedious to perform a numerical parameter fitting procedure. Particularly, most of the transition and noble metals present a very complex structure in Im{−1/ε (ω)} (Powell 1984), as can be seen from Figure 1 for Cu and Ag, which is difficult to be fitted if one uses only several Drude terms of finite value of γ(q). Therefore, Penn (1987) has proposed an improved scheme. The summation in Eq. (5) over finite number of Drude terms is then replaced by the integration   ∞  −1  −1 Im   = ∫0 dω p G(ω p ) Im  ,  ε L (q, ω ;ω p )   ε (q, ω ) 

(8)

In the limit γq→ 0, it becomes a δ-function,

with a damping constant γ of a plasmon loss peak at energy hωp. For q ≠ 0, the energy loss function may be extrapolated as (Ritchie and Howie 1977)  −1  −1   Im   = ∑ Gi Im  ,  ε D (ω ;ω p, i (q ), γ i (q ))   ε (q, ω )  i

(7)

  ωγ qω 2p −1 Im  = .  2 2 2 2 2  ε L (q, ω ;ω p )  (ω − ω q ) + ω γ q

The Drude dielectric function is given by

ε D (ω ;ω p , γ ) = 1 −

2 p

where q-dependence of the dielectric function is introduced through the damping term γq and the dispersion relation ωq = ωq(q,ωp). At q = 0, ωq = ωp, so that εL(q,ω ;ωp) is just an extended Drude dielectric function, εD(ω ;ωp,γ). The energy loss function is then

where Gi is the weight factor of ith Drude term,

ω γ ω 2p −1   Im  . = 2 2 2 2 2  ε D (ω ;ω p , γ )  (ω − ω p ) + ω γ

ω 2p , ω − ω − ω (ω + iγ q ) 2 q

M4,5

0.6 0.4 ×30 0.2

(6)

with the known Lindhard dielectric function εL(q,ω;ωp). The weight factor Gi becomes the expansion coefficient or the spectrum density, G(ωp). In a single-pole approximation (Hedin and Lundqvist 1969, Overhauser 1971), the Lindhard

0.0 100 (b)

101

∆E (eV)

102

103

FIG. 1 Plots of the energy loss function from optical data: (a) Cu, (b) Ag.


Therefore, Eq. (6) is just Eq. (5), but by summing over an infinite number of undamped Drude terms. Since each term is a δ -function, the procedure of extrapolating Im{−1/ε(ω)} into q ≠ 0 region will be greatly simplified. In fact, it is not necessary to perform any parameter fitting, since at each (q,ω) point the spectrum Im{−1/ε(ω)} determines completely Im{−1/ε(q,ω)}. Use of Eq. (9) for Eq. (6) at q = 0 leads to G(ω ) =

2  −1  Im  , πω  ε (ω ) 

ω p  −1   −1  ∞ Im  Im   = ∫0 dω p δ (ω − ω p ) ω q  ε (ω p )   ε (q, ω )  =

ω 0  −1  Im  , ω  ε (ω 0 )  (11)

where ω0 is the positive solution of

ω q (q, ω 0 ) = ω .

(12)

It can be readily verified that the energy loss function for finite q satisfies the sum rule as required for optical data, ∞

 −1 

∞

 −1 

π

∫ ω Im  ε (q,ω ) dω = ∫ ω Im  ε (ω ) dω = 2 ZΩ 0

0

inelastic mean free paths, 1 ω q2 = ω 2p + υ F2 (ω p )q 2 + (hq 2 / 2 m)2 . 3

2 p

,

(13) where Ωp = (4πNe2/m)1/2, N = NAρ/A is the density of atoms, NA the Avogadro’s number, ρ the mass density, A the atomic weight, and Z the atomic number. In short, the expansion involved in Eq. (6) in a single-pole approximation treats the energy loss spectrum as being composed of multi-modes of localized plasmon of an energy hωp related to the localized electron densities with the expansion coefficients determined from the experimental optical constants. The q-dependence is introduced by extrapolating along plasmon dispersion, leading to derive the wave vector- and frequency-dependent dielectric function according to Eq. (11). The optical constants, n(ω) and k(ω), being, respectively, the real and imaginary parts of refractive index, can be used to derive ε1(ω) and ε2(ω), the real and imaginary parts of dielectric constant ε (ω) through the relations, εl (ω) = n2(ω) − k2(ω) and ε2(ω) = 2n(ω)k(ω). This method of extrapolating energy loss function, however, has still remained an arbitrariness in choosing the dispersion relation, since Eq. (11) is independent of the detailed expression of ωq(q,ωp). A reasonable expression of the dispersion relation is then required. Certainly, ωq should approach the free electron line hq2/2m as q → ∞, corresponding to the Bethe ridge. Penn has empirically used the plasmon dispersion for free-electron gas in their calculation of electron

(14)

Some authors (Ashley 1991, Kwei and Tung 1986, Ritchie and Howie 1977) have adopted another simpler relation,

ω q = ω p + hq 2 / 2 m,

(10)

and the extrapolated energy loss function is thus

95

(15)

which holds also for q → 0 and q → ∞. The resultant differences in inelastic mean free paths and stopping powers derived by these two equations are not significant since the universal Bethe surface dominates the scattering cross section. However, the true secondary electron yield is somewhat sensitive to the choice of dispersion relation since the variation on inelastic mean free path controls strongly the probability for secondary electron production through cascade multiplication processes. In Figure 2 we plotted the extrapolated energy loss function against the energy loss and momentum transfer for Cu and Au. Since the experimental investigations on the dispersion were mainly made for plasmons, it is difficult to access how accurately Eq. (14) and Eq. (15) describe interband transitions in transition and noble metals. We have compared the energy loss function derived from this model with those from Gryzinski’s classical theory of binary collision for Si L-shell (Ding et al. 1994a). One may note that the limit γq → 0 in deriving Eq. (9) implies that the q-dependency of the plasmon linewidth is omitted in Eq. (11). The possibility of interband transition occurring at plasmon energy causes a plasmon damping even at long wavelength limit. Experiments (Zacharias 1975) show that γq ∝ q2 for q < qc (qc ∼ − ωp/ υF) (Pines 1964). For q > qc, the linewidth increases rapidly with q due to the damping by single electron excitation. It is difficult to include such details in a general formula since γq usually is unknown except for plasmon excitation in simple metals. Nevertheless, we may use Eq. (8) or a Lorentzian damping function (Ding et al. 1989b) directly to replace the δ-function for known γq. This is necessary in the calculation of low-energy electron stopping powers for free electron-like materials such as Al. Excitation function: The energy loss distribution for electron inelastic scattering and excitation function for secondary electron generation, defined as the probability that an electron of kinetic energy E = (hk)2/2m shall lose an energy ∆E = hω per unit path length traveled in solids, is given by the integration over q of Eq. (1), dλ-1in 1 ∞ dq  −1   h Im  (2 kq − q 2 ) − ω  = Θ d (hω ) πa0 E ∫0 q   ε (q, ω )   2 m =

 −1  ∞ dq 1 ∞ ω p Im  δ (ω − ω q ) dω p ∫0 ∫ πa0 E 0 ε ω qω q ( )  p   h Θ  (2 kq − q 2 ) − ω ,   2m

(16)

96


where the step function Θ(x) (Θ(x) = 1 if x > 0; Θ(x) = 0 if x< 0) comes from the energy and momentum conservation. The δ-function in above the equation requires q = q– where q– is the solution of the dispersion equation ω = ωq( q–,ωp). Further simplification of the above equation can be made using the relation

δ [ f (q )] = ∑ δ (q − qi ) i

df (qi ) , dq

hω hq  (hq )2  (hω ) − (hω p ) +    2m  2

2

dλ−in1 1 ∞ = d ( ∆E ) πa0 E ∫0

 −1  Im    ε (ω p )   ( hq ) 2  ( ∆ E ) 2 − ( hω p ) 2 +    2m  hω p d ( hω p )

2

 h2  Θ  (2 kq − q 2 ) − ∆E .  2m 

(17)

where f (qi) = 0. For dispersion relation given by Eq. (14),

δ (ω − ω q ) =

Eq. (16) becomes

(19) For dispersion relation given by Eq. (15),

δ (q − q ),

δ (ω − ω q ) =

2

(18)

m δ (q − q ), hq

(20)

the excitation function is then  −1  ∞ hω d ( hω ) dλ-1in 1 p p = Im   ∫ d(∆E ) 2πa0 E∆E 0 ∆E − hω p  ε (ω p ) 

Cu

 h2  Θ (2 kq − q 2 ) − ∆E .  2m 

(21)

 −1  Im   ε (q, w) 

0.4 0.3 0.2 0.1 0

50

100 150 hω ( eV)

200

250

8

7

6

5

4 q

3 −1

(Å

2

1

0

)

(a)

 −1  Im   ε (q, w) 

Au

0.9 0.6 0.3 0

(b)

50

100 150 hω ( eV)

200

250

8

7

6

5

4 q

3 −1

(Å

2

1

0

)

The integration in Eq. (19) and Eq. (21) is much simpler than the method involving parameter fitting (Tougaard 1987). Figure 3 shows the calculated the energy dependency of excitation functions for Cu and Au. For a specific value of ∆E, the energy dependence of excitation function is analog to that of the ionization cross section, showing a maximum at an overvoltage ratio E/∆E ≈ 2. The initial energy of the excited secondary electrons is provided by this probability distribution in a Monte Carlo simulation. The energy distribution of fast secondary electrons emitted from a sample is therefore strongly correlated with the spectrum, Im{−1/ε(ω)}, and hence with a sample in question. Ashley (1991) has included exchange effect in his calculation of excitation function. Considering exchange efffect the maximum loss energy ∆E is 3⁄4 E but not E. The excitation function is smaller in values, for kinetic energies < 100 eV, when compared with that without including exchange term. This will result in smaller stopping powers and larger inelastic mean free paths for slow electrons. Electron mean free paths and stopping powers: Electron inelastic mean free path determines the probability of occurrence of a discrete inelastic scattering event in a Monte Carlo simulation. It can easily be calculated by integrating the excitation function, Eq. (19) or Eq. (21), over all possible energy losses as

λ−in1 = ∫

E − EF

0

FIG. 2 Perspective views of the electron energy loss function for finite values of q as extrapolated by Eq. (11) and Eq. (15): (a) Cu, (b) Au.

dλ-1in d ( ∆E ). d( ∆E)

(22)


In Figure 4 we compared the calculated electron mean free paths with experimental values for Cu and Au. Good agreement has been found for electron energies from several eV above Fermi energy to several tens keV and for a variety of materials. Tanuma et al. (1988, 1991a, 1991b, 1993) have proposed some analytical expressions for the calculated electron inelastic mean free path. Electron stopping power can also be obtained in a similar way but is weighted by energy loss −

E − EF dE dλ-1in d ( ∆E ). =∫ ( ∆E ) 0 ds d( ∆E )

(23)

Although the present Monte Carlo simulation of the discrete collision event does not need to use stopping power values directly, the stopping power provides a very useful estimate of the effective electron energy losses averaged over the

97

unit path length of electron trajectories. Thus, it is important to compare the stopping powers calculated by Penn’s modeling energy loss function with the Bethe stopping power formula. With respect to the electron inelastic mean free path valence excitations or the low-loss peaks in Im{−1/ε(ω)} dominate; the inelastic scattering channels through large energy loss have a negligible contribution to electron inelastic mean free path. However, this is not the case for the stopping power, since ∆E is weighted. A good model of electron inelastic scattering should describe the Bethe stopping power at high energies as well as the experimental data on electron inelastic mean free path. In Figure 5 we plot the calculated stopping powers for Cu and Au. It shows that the dielectric stopping power approaches the Bethe formula at sufficiently high energies. It can be analytically shown in the following that, in the limit k → ∞, Eq. (23) reduces to the nonrelativistic Bethe stopping power equa102

Cu

6

Cu 101

4

λin (nm)

dλ−1/d(∆E) (keV Å)−1

8

2

100 0 0.1 E V) (ke

1 10 0

(a)

20

40 ∆E

60 (eV)

100

80

10−1 100

101

(a)

102 E−EF (eV)

103

104

103

104

102 Au Au 101

6 λin (nm)

dλ−1/d(∆E) (keV Å)−1

8

4

100

2 0 0.1

10−1

E V) (ke

(b)

1 10 0

20

60 40 V) ∆E (e

80

100

FIG. 3 Perspective views of the energy loss distributions calculated by Eq. (19) and Eq. (14) for (a) Cu, (b) Au.

100 (b)

101

102 E−EF (eV)

FIG. 4 Electron inelastic mean free paths versus electron energy above Fermi energy. The solid line is calculated with dispersion relation, Eq. (14), and the dashed line with Eq. (15). Symbols represent experimental data. (a) Cu, (b) Au. The dotted line represents experimental data.

98


tion without including the exchange effect. Equation (23) may be expressed as

2 k 2 − kF2 − 2 mω p / h hk 2 2E q1 = lim = = . 2 k →∞ q k →∞ mω p hω p k − k − 4 mω p / h 2 lim

−

∞ dq ∞ dE 1 ∞  −1  = (hω )δ (ω − ω q ω p Im  dω p ∫0 ∫ ∫ 0 0 ds πa0 E qω q  ε (ω p ) 

h d (hω ) × Θ  (2 kq − q 2 ) − ω Θ( E − EF − hω )  2m 

Then the stopping power equation becomes  −1   2 E  dE 1 ∞ d ( hω p ) lim  −  = (hω p )Im   ln  ds  πa0 E ∫0  ε (ω p )   hω p 

1 ∞  −1   q1  = (hω p ) Im   ln  d hω p , ∫ 0 πa0 E  ε (ω p )   q2 

(

(25)

)

k →∞

=

(24)

2πe 4 NZ  2 E  ln ,  I  E

where q1 and q2 are integration limits for q. Using Eq. (15) and taking k → ∞,

(26) where we have used the sum rule, Eq. (13), and defined a mean ionization energy (Fano 1956) as

103

ln I =

2 1 π Z (hΩ p )2

∞

∫ (hω 0

p

Stopping power (eV/nm)

Cu 102

In fact, the concept of mean ionization energy is valid only for electron energies higher than the binding energy of the deepest inner shell of an atom, because less atomic shell electrons become visible for the scattering electrons of lower energies. This is the reason why the common Bethe equation with Berger and Seltzer (1964) ionization energy is ineffective in the low-energy region. Similar to the definition of the effective number of electrons per atom contributing to energy losses from 0 to ∆E = hω,

101

Bethe 100

10−1 100

101

(a)

 −1  ) Im   ln(hω p )d (hω p ).  ε (ω p )  (27)

102 103 E−EF (eV)

104

105

Zeff ( ∆E ) =

103

2 π (hΩ p )2

∫

∆E

0

 −1  (hω p ) Im  d (hω p ),  ε (ω p ) 

(28) Stopping power (eV/nm)

Au

which should approach the atomic number when ∆E → ∞, we may introduce the effective mean ionization energy per atom contributing to energy losses from 0 to ∆E,

102

101

ln[ Ieff ( ∆E )] = Bethe 100

ln(hω p )d (hω p )

10−1 100 (b)

 −1  ∆E 2 1 (hω p ) Im   2 ∫ π (hΩ p ) Zeff ( ∆E ) 0  ε (ω p ) 

101

102 103 E−EF (eV)

104

105

FIG. 5 Electron stopping powers versus electron energy above Fermi energy. The solid line is calculated with dispersion relation, Eq. (14), and the dashed line with Eq. (15). The dotted line represents Bethe stopping power. (a) Cu, (b) Au.

=

∫

∆E

0

 −1  (hω p ) Im   ln(hω p )d (hω p )  ε (ω p )  .  −1  ∆E ∫0 (hω p ) Im  ε (ω p ) d (hω p )  

(29)


Ieff (∆E) has a similar tendency as Zeff(∆E) and should approach experimental value of I when ∆E → ∞. Figure 6 shows the comparison of the calculated values of mean ionization energy for some elements with the equation of Berger and Seltzer (1964), I = 9.76 Z + 58.5 Z −0.19

(eV),

leff(∆E) (keV)

1.00

0.75

Au

Au

0.50

Cu

Ag

0.25

Ag

Cu

0.00

10−1

100

101 ∆E (keV)

102

Electron Elastic Scattering Our Monte Carlo simulations have employed the Mott (1929) cross section to describe elastic scattering. The relativistic representation of the differential cross section is expressed as 2 2  dσ e  = f (θ ) + g(θ ) ,  dΩ  Mott

(30)

and with the recommended data by Berger and Seltzer (1982). In the worst case, the calculated I-value by Eq. (29) for Cu exceeds the one by Eq. (30) by about 40%. This is probably due to uncertainties involved in optical constants, particularly at high frequencies above several hundreds eV through the multiplication terms, hωp and ln(hωp). Shiles et al. (1980) have obtained a very good agreement for Al based on the detailed Kramers-Kronig analysis of optical data. However, the ambiguity of optical data is less manipulated in deriving at an inelastic cross section. It is important to note from Figure 5 that the stopping power at low energies decreases with energy other than the E −1/2-dependence assumed by Rao-Sahib and Wittry (1974). Since experimental data on the low-energy stopping power have been rarely available, the inelastic mean free paths may cover the insufficiencies of the knowledge on the behavior of stopping powers. Equation (22) and Eq. (23) clearly predicate that stopping powers should have the similar energy dependency as that of the inverse of the inelastic mean free path. Compiled experimental data on inelastic mean free paths have given the relation (Seah and Dench 1979): λin = a/E2 + b E . Hence, for slow electrons of energies < 100 eV, λ in−1 ∝ E 2. A similar energy dependence for stopping powers has been found in previous theoretical calculations based on a statistical model of local Lindhard-electron gas (Ashley et al. 1979, Sugiyama 1976, Tung et al. 1979) as summarized by Berger and Seltzer (1982).

103

FIG. 6 Effective mean ionization energy as the function of photon energy for Cu, Ag, and Au. Open square mark represents value calculated with Eq. (30) and the closed circle with error bar is the recommended data by Berger and Seltzer (1982).

99

(31)

with the scattering amplitudes f (θ ) =

[

(

) (

)]

+ − 1 ∞ ∑ (l + 1) ei 2δ l − 1 + l ei 2δ l − 1 Pl (cosθ ); 2ik l = 0

(32)

[

]

+ − 1 ∞ g(θ ) = −ei 2δ l + ei 2δ l Pl1 (cosθ ), ∑ 2ik l =1

(33) where Pl (cos θ ) and P1l (cos θ ) are Legendre and the first order associated Legendre functions, respectively. δ l+ and δ l− are spin ‘up’ and spin ‘down’ phase shifts of the lth partial wave, respectively; they are numerically evaluated (Yamazaki 1977) by solving the Dirac equation for the radial part of the wave function of the scattering electron in an atomic potential field whose analytical form has been given (Bonham and Strand 1963) for the Thomas-Fermi-Dirac atom. The calculated phase shifts agree with those tabulated, using a somewhat different numerical method (Fink and Ingram 1972, Fink and Yates 1970, Gregory and Fink 1974). In Figure 7 we compare the calculated differential cross sections with the experimental angular distribution of elastic scattering performed for Au vapor (Reichert 1963). It has been shown (Ichimura 1980) that some structures begin to appear in the differential cross sections for a heavy element like Au from about 4 keV. With decreasing energy, the structure becomes more intensive. But for a light element such as Al it is unobservable even down to a lower energy of 500 eV. The differential cross section plotted in a polar diagram then shows a lobe-like shape as illustrated in Figure 8. The main lobe is in the forward scattering direction, and there are some side lobes for large angle scattering and for backscattering. These lobes are caused by the interference of partial waves and may be destroyed when large amounts of partial waves are involved. This explains its tendency on the energy, since, at low energies, only several lower-order phase shifts are significant, but many more phase shifts of higher order begin to contribute the cross section at high energies. The structures in the differential cross sections affect not only the angular distribution of elastic peak (Ding et al. 1990, Murata 1988), but also the angularly resolved reflection electron energy loss spectrum (Ding 1990, Werner and Hayek 1994). This fact indicates the importance to use the Mott cross section in a Monte Carlo simulation at low primary beam energies. For true secondary electrons, any structure in the differential cross sections will be

100


500 eV

Relative intensity

400 eV

300 eV

250 eV

200 eV

175 eV

smeared out by multiple scattering processes. It should be mentioned that the nonrelativistic expression of the differential cross section in partial wave expansion, by putting δl+ = δl− = δl in Eq. (32), 2

dσ e 1 ∞ = ∑ (2l + 1)(ei 2δ l − 1)Pl (cosθ ) , dΩ 2 k l = 0

(34)

works only for light elements and fast electrons, similar to the Rutherford formula. Calculation results show that δl+ approaches δl− with increasing energy and decreasing atomic number. A light element such as Al hardly presents the difference between phase shifts of spin ‘up’ and spin ‘down’ in the energy region beyond sub-keV. For Au, the relativistic cross section differs from the nonrelativistic cross section up to very high energies. This observation seems to go against the concept that the relativistic effect is only prominent for fast elec-

150 eV 400 eV 150°120° 90° 60° 30° 150° 120° 90° 60° 30° Scattering angle Al

0.25

0.5

(Å2/rad)

Cu

0.5

(Å2/rad)

(a)

1900 eV

Relative intensity

1700 eV

1500 eV

1300 eV 400 eV 1100 eV Au

Ag

900 eV 0.25 700 eV

(b)

150°120° 90° 60° 30° 150° 120° 90° 60° 30° Scattering angle

FIG. 7 The angular distributions for electron elastic scattering from Au atoms in the energy range between 150 eV and 1.9 keV. (a) For energies 150–500 eV, (b) for energies 700–1900 eV. The vertical scale is such that the experimental maximum around 62° at 150 eV coincides with the theoretical one. Solid dots are experimental data (Reichert 1963) and the solid line is the smoothed experimental curve. The dotted line is the calculated curve.

FIG. 8 Polar plots of differential cross sections calculated by partial wave expansion method for Al, Cu, Ag, and Au at 400 eV.


trons. The point of this problem is that the spin–orbit interaction is the major relativistic effect, which is inversely related to electron speed in the case of electron-atom scattering. A moving charged particle can produce a magnetic field, and the particle views this field as if it were an external field, leading to an interaction between the field and the magnetic momenta, or spin, of a moving electron. The spin–orbit energy is approximately given by (Schiff 1968), 1  1 dV  L ⋅ S, 2 m 2 c 2  r dr 

(35)

where L and S are the angular momentum and spin vector of a scattering electron, respectively. This interaction potential results in the different phase shifts for spin ‘up’ and spin ‘down.’ Because the moving direction of slow electrons may be more strongly deflected by atomic potential than that of fast electrons, the lower-energy electrons have larger angular momentum. Therefore, the spin–orbit coupling is more pronounced for low-energy electrons. Furthermore, heavier elements tend to give a stronger coupling through their larger range of atomic potential. The integration of the differential cross section over whole solid angles gives the total cross section

101

accessed from the relation

λ−e1 = ∑ Cia / λie ,

(38)

i

where Cia represents the atomic concentration for ith component. However, the electron inelastic scattering is strongly related to the chemical environment and the total inelastic mean free path for unlocalized excitation in a sample should, in principle, be determined from the dielectric function which characterizes the sample in question. In practice, this is usually difficult to be realized because of the deficiencies of the experimental optical data for most alloys. Hence, we made the program to be able to handle two cases: If the material is compound-like, whose energy loss function is greatly different from the simple combination of constituting elements and can thus only be described by one set of dielectric data, the total inelastic mean free path is given directly by Eq. (22). For an alloy-like material described by the sets of dielectric data for each component, the total inelastic mean free path is obtained by

λ−in1 = ∑ Cia / λiin .

(39)

i

Electron Scattering dσ dσ σ e = ∫  e  dΩ = 2π ∫ sin θ  e  dθ , 0  dΩ  Mott  dΩ  Mott π

which relates to elastic mean free path through

λ-1e =

NA ρ σ e. A

(37)

Figure 9 shows the calculated results of the elastic mean free path for several elements. In the present calculation, we have ignored exchange and polarization effects which may be more pronounced than the relativistic effect below 101 eV (Walker 1971). We have also assumed that the isolated atomic potential is valid in condensed matter. This may be a problem for very slow electrons. A modification has been proposed to obtain the effective potential in a Wigner-Seitz cell (Czyzewski et al. 1990, Ohya et al. 1990, Valkealahti and Nieminen 1984).

Monte Carlo Procedure and Programming Before summarizing some of the basic procedures of the present Monte Carlo method in the following sections, we shall first consider the case where a sample is multilayered and each layer is either monoatomic or polyatomic. For an alloy or compound, the total elastic mean free path is simply

102

Elastic mean free path (Å)

(36)

The present Monte Carlo simulation of electron trajectories penetrating a sample is based on a description of individual electron scattering processes, as schematically shown in Figure 10. The problem is, then, reduced to the determination of values of physical quantities such as step length, scattering angle, energy loss, and so forth, in a particular scattering event. The Monte Carlo technique basically choses these values by random numbers according to respective cross sections. Given a probability distribution function P(x) for a variable x, we can derive a normalized accumulation function A(x), Mg Si Ti Cr Ag Au 101

100 101

102

103 Energy (eV)

FIG. 9 The energy dependence of electron elastic mean free paths.

104

102


A( x ) = ∫

x

x min

P( x ′)dx ′

∫

x max

x min

P( x ′)dx ′,

(40)

and determine a specific value of x from A(x)=R for a given value of uniform random number R ∈ [0,1]. Suppose that the step length, s, of a scattering electron between two successive collision events obeys the Possion stochastic process with the probability distribution

to inelastic collision. For a compound-like sample, the dielectric function is given such as to treat the material as a whole, and it is therefore not necessary to specify a particular element. In the next step we determine the amount of energy loss ∆E by the equation

R6 = ∫

∆E

0

P( s) = λ−m1e − s / λm ,

(41)

where λm is the total mean free path related to the corresponding elastic mean free path and the inelastic mean free path through

λ−m1 = λ−e1 + λ−in1 .

(42)

An electron will then suffer a scattering event when it passes s selected by a random number R1 via s = − λ m ln R1 .

(43)

Another random number, R2, determines the type of individual scattering event followed after passing s: If R2 < λ−e1 / λ−m1 ,

i −1

a j

i

/ λej

j =1

1/λe

< R3
EB, we can attribute it to an inner-shell ionization. We shall discuss this problem later once again. The polar and azimuthal angles of the secondary electron are decided from momentum conservation as sin θ ′ = cosθ , φ ′ = π + φ .

(52)

In small-angle inelastic events, which dominate the inelastic process, secondary electrons are produced perpendicularly to the direction of the incident electron trajectory. Sharing the kinetic energy of the incident electron, two outgoing electrons then have lower energies. The inelastic process proceeds with the successive production of secondaries, generating a large number of secondary electrons of low energies. Once a secondary electron is generated by a primary electron in an excitation event, its energy, coordinates, and moving direction are stored in memories of a computer. After a trajectory of the primary electron in simulation is terminated, all the stored information on the secondary electrons is recalled and the trajectories of those secondary electrons are simulated in the same way as the primary electron. This cascade process is traced in simulation until all the secondary electrons either escape from the surface or come to rest within the sample. We can classify the diffused primary electrons and cascade secondary electrons as follows. Among the trajectories for each primary electron incident into the target, only one trajectory that successively loses energy from the primary energy Ep and has the energy E1 after inelastic scattering is considered as the diffused primary electron and all the rest of electrons are regarded as secondaries. Since an electron of energy E can lose an energy ∆E from 0 to E − EF, the energy spectrum for cascade secondary electrons thus extends up to Ep. With respect to electron emission from the surface, it is necessary to consider the refraction of electrons by the potential barrier at the surface because surface barrier influences the energy and angular distribution of slow electrons. Quantum mechanical representation of transmission function is (CohenTannoudji et al. 1977)

104


 4 1 − U0 / E cos2 β 2  2 , if E cos β > U0 ; T ( E, β ) =  1 + 1 − U0 E cos2 β  0, otherwise,

[

]

simulation. The tracing of electrons is first made step by step until the cut-off condition is satisfied, and then one by one up to a total number of primary electrons. The last part processes all the accumulated results derived from the simulation and outputs these data into files.

(53)

Verification of the Model where the inner potential is the sum of Fermi energy and work function, U0 = EF + Φ, and β is the ejection angle of electrons measured from surface normal. Then, another random number R8 decides the fate of the electron: it is emitted if R9 < T and absorbed otherwise. When the electron is emitted from the surface, its kinetic energy measured from the vacuum level will be E lowered by U0 and the ejection angle will be changed accordingly will be changed to β0, as E − U0 sin β 0 = E sin β

(54)

according to the momentum conservation for the component parallel to the surface.

Random Number A meaningful Monte Carlo calculated result requires good statistics by simulating a large number of incident electron trajectories. Furthermore, the present simulation also includes secondary electron trajectories which are multiplied in a cascade process of secondary production. Therefore, one must make sure that none of the same electron trajectories, which can be occur occasionally by the repetition of the period of random number, will be calculated. We then generate random numbers to arrays, each one having its own seed and being used to select different physical quantities, such as array-1 for step length, array-2 for elastic scattering angle, array-3 for loss energy, and so on. This enables us to avoid repeating the same electron trajectory, because it is almost impossible to satisfy the requirement for the appearance of the same electron trajectory that all of these arrays, having used different times, should come into their next periods together in the first step of an electron trajectory. In this way one can also avoid the pseudo-period of random numbers since each array is regarded as a one-dimensional variable. The program for Monte Carlo simulation is composed of three parts. The first part is for initial conditions, including the physical constants for the specification of the sample and of the primary beam, and also the parameters for controlling options are inputted. Tables of cross sections are then prepared before the simulation to save time, enabling the process of tracing an electron to simply pick up the data in the table without further tedious arithmetic calculations. However, the present model of electron inelastic scattering requires very large computer memory space for tabulating cross sections as the functions of material, energy, energy loss, and scattering angle. The second part is the main routine of a Monte Carlo

To verify the present Monte Carlo model, it is more preferable to compare the calculation with the electron backscattering phenomena rather than with the generation of characteristic x-rays in electron probe microanalysis. In fact, x-ray depth profiles suffer many uncertainties in physical quantities used, such as ionization cross section and absorption coefficient, which are not inherent in a Monte Carlo model of electron scattering. Therefore, we shall compare the calculated energy distribution of backscattered electrons with the accurate experimental spectrum. Since old measurements of energy distribution with spherical retarding-field energy analyzers (such as Darlington 1975, Kulenkampff and Spyra 1954, Matsukawa et al. 1974) were performed in a rather low vacuum with lowenergy resolution at high accelerating voltages, these results are not suitable for the present purpose. We therefore have attempted to compare EN(E)-spectra newly measured by Goto et al. (1994) with a cylindrical mirror analyzer (CMA). Because a CMA has a specific angular acceptance detecting only those electrons emitted into the solid angle which is defined by the entrance angular aperture, the energy spectrum measured with a CMA contains also the angular information of backscattered electrons. It has been shown (Ding et al. 1988b, Shimizu and Ichimura 1984) that it is often difficult to make quantitative comparison with the CMA-measured curve in the low- energy region, where not only the contribution of secondary electrons produced in a sample has not been well clarified theoretically as yet, but also those electrons scattered inside an energy analyzer contribute significantly to the experimental background measured with a CMA. To overcome this problem, Goto et al. (1993) have developed a novel CMA system which has greatly reduced the contribution of the electrons scattered in the CMA estimated < 1% of the energy distribution. A Faraday cup allows to prevent the energy spectra from the distortion introduced by the energy dependence of the efficiency of the channel electron multiplier used conventionally. With this system they have reported the standard EN(E)-spectra with high-energy resolution for polycrystalline copper, silver, and gold, down to the energy region of the true secondary peak. Their results are very similar in shape to those measured with a concentric hemispheric analyzer with a Faraday cup (Smith and Seah 1989, Seah and Smith 1990) for the same materials but under different experimental configurations. In order to get an energy distribution with high-energy resolution and with low statistical fluctuation, we have performed Monte Carlo simulations by tracing a huge number of electron trajectories, as many as 2–3 × 107 primary electrons and about ten times more secondary electrons inside the sample. Eq. (14)


was used to calculate differential inelastic cross sections. For the calculation of CMA-measured energy distribution, we count only those electrons that are backscattered into the acceptance solid angles, of corn-like shape, with the polar angular aperture 42.3° ± 6°. In addition, considering that a CMA operates at constant relative energy resolution, we have convoluted the energy distribution with a Gaussian function as a weight corresponding to relative energy resolution ∆E/E = 0.24%. The Monte Carlo calculations were performed for a primary beam of 5 kV at normal incidence. Figure 11a shows the comparison of the calculated EN(E) distribution for Cu with the experimental spectrum. Very good agreement is found in this case over the whole energy range below elastic peak. By magnifying the intensities in the lower energy region where

105

the LMM Auger peaks are presented, Figure 11b has confirmed that the background under Auger peaks for Cu can be theoretically described with reasonable accuracy. The comparison suggests that the present Monte Carlo model has indeed attained reasonable accuracy in describing electron– solid interaction. We can further separate the contribution to the energy distribution of backscattered electrons by the diffused primary electrons from that by the cascade secondary electrons, as demonstrated by Figure 11c. In Figure 11d we compare the energy distribution calculated for CMA solid angles for detection with that for hemispheric solid angles. The spectra are similar in shape for normal incidence. Figure 12 illustrates the simulation results for Ag. In the case of Cu, we have not tried to identify the inner shell peak in optical energy loss function; all the secondary electrons are

0.5

0.020 1 Cu 5 kV Cu 5 kV

0.015

0.3 0 4.7

0.2

4.8

4.9

E(dη/dE)

E(dη/dE)

0.4

5.0

0.005

simulation experiment

0.1

LMM 0.010

0.000

0.0 0

1

2 3 Energy (keV)

(a)

4

5

0

0.4

(b)

0.8

1.2

Energy (keV) 0.10

0.020 0.08

Cu 5 kV

Cu 5 kV Total

(dη/dE) (keV)−1

E(dη/dE)

0.015

Diffused primaries 0.010

0.06

0.04 CMA

0.005

Cascade secondaries

0.02 HMA 0.00

0.000 0 (c)

1

2 3 Energy (keV)

4

0

5 (d)

1

2 3 Energy (keV)

4

FIG. 11 (a) Comparison of the calculated EN(E) distribution for Cu (thin line) with the experimental spectrum (solid line) for a 5 kV primary beam at normal incidence. Inset figure shows the elastic peak with low loss peak. (b) Comparison of the shape of the background under the LMM Auger peaks. (c) Energy distribution of backscattered electrons contributed by diffused primary electrons and by cascade secondary electrons. (d) Comparison of the simulated energy distributions measured with a CMA and those measured with a hemispherical analyzer.

5

106


assumed to be excited from valence electrons, that is, their initial energy is ∆E + EF. For Ag, optical energy loss function shows an M-edge at about 360 eV (Fig. 1b). Thus, any inelastic scattering for hω0 > 360 eV must be related to an inner shell ionization, and the initial energy of knock-on secondary electron is taken as ∆E − EB, as we stated before, where the binding energy EB is simply assumed to be hω0. We empirically found that this identification of the inner shell edge in optical energy loss function is necessary to obtain quantitative agreement on the energy distributions in the low-energy region. If we assume that all of the secondary electrons are produced in Fermi electron excitation, then the energy loss due to inner shell ionization will result in a higher energy of the secondary electron and the overestimation of the secondary electrons is 0.5 1.5

E(dη/dE)

0.4

Ag 5 kV

0.3 0.0 4.7

0.2

4.9

5.0

simulation experiment

0.1

0.0

4.8

0

1

(a)

2 3 Energy (keV)

4

5

0.020

Ag 5 kV

E(dη/dE)

0.015

0.010 MNN

0.005

0.000 0 (b)

0.5 1.0 Energy (keV)

1.5

FIG. 12 (a) Comparison of the calculated EN(E) distribution for Ag (thin line) with the experimental spectrum (solid line) for a 5 kV primary beam at normal incidence. Inset figure shows the elastic peak with low loss peak. (b) Comparison of the shape of the background under the MNN Auger peaks.

unavoidable, particularly at the high-energy side of the corresponding Auger peaks.

Monte Carlo Calculation for Scanning Electron Microscopy Yield, Energy, and Depth Distributions

To see how the present Monte Carlo simulation model can describe the generation and emission processes of cascade secondary electrons, we should compare the calculated yield and energy distribution for true secondary electrons with experimental data. Figure 13 shows the variations of backscattering yield η (E > 50 eV), true secondary electron yield δ (E < 50 eV), and total yield σ = δ + η with the primary beam energy for Cu. The simulated results are compared with some selected experimental data which were measured in a UHV chamber. It can be seen that the agreement for η is very good. But the simulation gave a higher secondary electron yield and so the total yield. A recent compilation of electron yields made by Joy (1995), by including individual data collected from literature as early as 1898 but without judging the reliability of data, has shown that the scatter among the available experimental data is very large. Particularly, the discrepancy is about a factor of 2 for Cu at low energies. It is then difficult to judge a Monte Carlo model solely from electron yields without comparing energy distribution as done above for backscattered electrons. According to the present model, the energy of secondary electrons is strongly correlated with the optical electron energy loss function. Some features presented in Im{−1/ε (ω)} may be seen in the energy distribution of secondary electrons, such as by comparing Figure 1 with the experimental EN(E) distribution for true secondaries (Goto et al. 1994). However, these features are only weakly illustrated in energy distribution because only those secondary electrons elastically emitted from the surface after their birth can carry such information. Furthermore, inelastic collisions with large scattering angle or momentum transfer integrate, in effect, energy loss functions shifted by about (hq)2/2m; hence, any characteristic peak in Im{−1/ε (ω)} will be greatly smeared out. Figure 14 shows the comparison of normalized energy distribution of true secondary electrons for Cu (EF = 7.57 eV, Φ = 4.65 eV) between the calculation and the experiments. Simulation yields a distribution shifted to higher energies by 1–2 eV which may be understood if we consider the difference on the inner potential between target and detector materials. In Figure 15 we compare the experimental energy distribution measured with a CMA (Goto et al. 1994) with the present Monte Carlo-calculated results for CMA solid angles and for hemispheric solid angles. Except for a small peak shift due to experimental factors, the agreement on the normalized intensities has been obtained, as shown in Figure 15a. This suggested that the energy distribution of secondary electrons is weakly angular-dependent. Figure 16 demonstrates the variation of energy distribution with the change of work function.


σ

0.75

0.50

0.25

0.00 0

Cu

5

10

15 20 Energy (eV)

25

30

1.00 1.0

η

0.5

0.0 0

1

(a)

2 3 Primary energy (keV)

4

5

Intensity (arb. units)

Electron yields

Cu 1 kV

FIG. 14 Normalized energy distribution of true secondary electrons emitted from Cu at 1 keV primary energy. Histogram is the present Monte Carlo-calculated result (solid line). Experimental curves measured with hemispheric analyzers are: (dashed line) Koshikawa and Shimizu (1973), (thin solid line) Pillon quoted by Ganachaud (1977), (thin dashed line) Goto et al. (1975).

2.0

1.5

1.00


With the increase in work function, the maximum peak intensity decreases and peak width increases. Sharper secondary peak then means smaller work function. It is interesting to study the correlation between energy loss of incident electrons and the energy of secondary electrons. Similar to an (e,2e) experiment (Vos et al. 1995), we can simulate the coincidence distribution of the energy loss spectrum for primary electrons with energy distribution for secondary electrons. From Figure 17 we can see a small ridge at ∆Ep = Es, which is attributed to the secondary electrons directly emitted without suffering further inelastic collisions. Other parts of the distribution in the region Es < ∆Ep obviously represent cascade secondary electrons. The counts in the figure have not been enough to resolve the relation between optical energy

107

2.0

0.75

Cu 5 kV

0.50

0.25

0.00 0

20

(a)

Cu

40 60 Energy (eV)

80

100

1.00

δ 1.0 Intensity (arb. units)

Electron yields

1.5

0.5

0.0 0 (b)

1

2 3 Primary energy (keV)

4

0.75

Cu 5 kV

0.50

0.25

5

FIG. 13 (a) Primary energy dependence of backscattering yield η and total yield σ for Cu. Solid lines with open circles are the present Monte Carlo-calculated results. Closed circles (Koshikawa and Shimizu 1973) and dashed lines (Goto et al. 1975) are experimental data. (b) Primary energy dependence of secondary yield δ for Cu. Solid lines with open circles are the present Monte Carlo-calculated results. Closed circles (Bindi et al. 1980) are experimental data. Hatched region represents the variation of experimental curves for δ using experimental data for δ max and Epmax and an empirical equation δ (Ep) (Seiler 1983).

0.00 0 (b)

20

40 60 Energy (eV)

80

100

FIG. 15 Normalized energy distribution of true secondary electrons emitted from Cu at 5 keV primary energy. Solid line is derived from EN(E) spectrum experimentally measured with a CMA (Goto et al. 1994) and data points represent Monte Carlo-calculated result for: (a) CMA solid angles for detection, (b) hemispheric solid angles.

108


loss function and true secondary peak for energies above several tens eV. Figure 18a shows the intensity distribution of secondary production as the function of initial energy and depth. The cascade process produces many very low-energy electrons, and the number of secondary electrons increases exponentially with lowering initial energy. Most of them, however, can hardly be emitted from the surface to be true secondary electrons. This is because, first, the surface barrier prevents the emission of low-energy electrons and, second, a very large number of secondary electrons is produced at deep depths. Figure 18b shows that the depth distribution for secondary

electron production is very similar in shape to that for x-ray production in electron probe microanalysis.

Spatial Distribution and Multifractal Property A Monte Carlo simulation of electron scattering processes yields zigzag electron trajectories whose quantitative description was difficult. For example, the globe shape of electron trajectories in a heavy element is very different from that in a light element, particularly when cascade secondary electrons are included. But how can this difference be quantitatively described? A fractal analysis (Mandelbroat 1982) has provided an answer to this question (Gauvin and Drouin 1992, 1994, Li et al. 1996a). Furthermore, multifractal analysis can tell us how spongy the trajectories are. It is then interesting to know how dense the signals produced by incident electron tra-

Cu 1 kV

0.8 Cu 5 kV

0.6

2.0

0.4 0.2 0.0

1 ∆Φ

2 3 (e V)

4 5

0

2

4

14 12 0 1 ) 8 (eV gy 6 r e En

16

1.5

20

18

1.0

Intensity


1.0

0.5 0.0

FIG. 16 Variation of energy distribution of true secondary electrons with the increment of work function.

25

Ag 1 kV

80

20

(a)

15 10 Ene rgy (eV )

20

5 0

60 ) m 0 4 th (n p e D

0

0.6 0.4

80 12 E (e 0 s V)

160

200

FIG. 17 Simulated coincidence distribution of reflected electron energy loss spectra with secondary electron energy distribution. ∆Ep denotes the energy loss of primary electrons and Es the energy of secondary electrons.

15 Cu 5 kV

10

5

0

16 0

40

20 0

0

12 0 ∆E 80 p ( eV 40 )

0

0.2

20


0.8


1.0

0 (b)

50

100

150

Depth (nm)

FIG. 18 (a) Intensity of cascade secondary electrons versus their initial depth and energy. (b) Depth distribution of secondary electron production.


jectories distribute within the interaction volume and how they can be described. We have shown that the spatial distribution of inner shell ionization for x-ray production can also be described by a fractal dimension (Ding et al. 1994b). The multifractal parameters for surface distribution of secondary electron emission depend on accelerating voltage and relate to resolution in an SEM (Li et al. 1995). Applying the similar multifractal analysis to the spatial distribution of secondary electron production can give us a deeper understanding of electron interactions with specimen in an SEM. Figure 19 demonstrates the simulated electron trajectories including cascade secondary electrons and the spatial distribution of secondary electron production sites for a limited number of primary electrons. The pear-shaped spatial distribution is similar to that of ionization, except that some local intense regions can be observed because the cascade process of sec-

Cu 5 keV

4 Primary electrons

50

109

ondary electron production is limited in a small local volume. We have used a standard box counting method to calculate fractal dimension and generalized fractal dimension. The interaction volume is divided by n3 (n = 180) cubes of lattice constant δ0. Denoting Φi,j,k the number of secondary electrons in the (i,j,k)th cell, the occupation probability is introduced as n

∑Φ

pi , j ,k = Φ i , j ,k

i, j ,k

,

i , j , k =1

(55)

and a partition function is defined by

χ q (δ ) =

n

∑p

q i, j ,k

∝ δ −τ ( q ) ,

i , j , k =1

(56)

where δ is varied as being integer times of the unit cell size δ0, δ = mδ0, as long as n/m is an integer. q is the moment order and the generalized fractal dimension is given as (Hentschel and Procaccia 1983) Dq =

log χ q (δ ) τ (q) 1 lim . = 1 − q q − 1 δ → 0 log δ

30 20

depth z (nm)

40

10 0

40 30 20 10 0 Dist −10 ance −20 y (n −30 m) (a)

10

20

0

0 −3

ce 0 −1 stan 0 Di −2

x

30 ) m (n

(57)

For q = 0, D0 is the Hausdorff dimension that simply describes the geometrical property of Φi,j,k. The larger q, the more the intense regions of Φi,j,k are enhanced in Dq. The multifractal dimension Dq for smaller q emphasizes on the fractal property of the outer part of spatial distribution where the intensities are weak. The variables (q, τ (q)) can be changed to (α, ƒ(α)) by performing the Legendre transformation and the Hölder exponent α, and the multifractal spectrum f(α) can be calculated by −τ (q ) = αq − f (α ),

α=−

(58)

dτ d = (q − 1) Dq . dq dq

[

]

(59)

Cu, 5 kV, 106 Cascade electrons

6 4 2

Z- axis (nm)

10 8

0 8

6

(b)

4

2

2

4

6

0 0 ) −2 Y- a −2 (nm xis −4 4 s i − (nm −6 ax ) −8 8 −6 X−

FIG. 19 View of Monte Carlo simulated (a) electron trajectories including cascade secondary electrons and (b) spatial distribution of secondary electron birth location.

Figure 20 shows χq versus δ curves for several values of q. Note that for q = 0 the curve is a linear straight line, whose negative slope is just the Hausdorff dimension, in the double logarithmic plot. In the case of Cu bombarded by a 5 keV primary beam, the spatial distribution has a fractal dimension, D0 = 2.805, with correlation coefficients about 99.99% for line fitting. This value is smaller than 3 and can be easily understood if we consider that the interaction volume for secondary production has no definite boundary surface. From the linear portion of Figure 20 we have obtained the multifractal dimension Dq shown in Figure 21. It can be seen that with the increase of q from a minus value, Dq increases first from a small value to a maximum and then falls. This can be understood as follows: under the limits q→ −∞ and q → +∞, Dq represents, respectively, the fractal dimension for the distribution of very low and very high intensities, of which the spatial distribution is like the isolated islands; therefore, Dq would be very small at both limits. For moderate q, the inten-


sity distribution is like the linked island and the greater fractal dimension is then expected. However, the maximum Dq is still smaller than 3. Because of this behavior of Dq, τ (q), the counterpart quantity f(α ) shows an anomalous character as demonstrated in Figure 22. Normal f(α ) for a restricted multifractal should be shaped like a bell with a maximum at q = 0, as demonstrated by Halsey et al. (1986) with some examples of exact fractal. This kind of anomaly has been found in other Monte Carlo simulation distributions, such as the spatial distribution of ionization (Ding et al. 1994b) and surface distribution of secondary electron emission (Li et al. 1995). These distributions have a common character in χq(δ ) curves: there are two linear portions. f (α ) is anomalous in smaller scaling size regime but is normal in greater scaling size regime. If we take the portion in the regime, 3 × 101 ≤ δ ≤ 2 × 102 (nm), we can then obtain

q = −6

Log (χq(δ))

40

Cu 5 kV

10

q=0 3.0 q=1 2.0 Cu 5 kV q>0

1.0

0.0 0.0

0.5

1.0

1.5 α

2.0

2.5

3.0

FIG. 22 Multifractal spectrum f (α).

Summary −2

0

0 2 −10

q