Ultramicroscopy 149 (2015) 26–33
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Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Modified Bethe formula for low-energy electron stopping power without fitting parameters Hieu T. Nguyen-Truong 1 Faculty of Electronics and Computer Science, Volgograd State Technical University, 28 Lenin Avenue, Volgograd 400131, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 29 August 2014 Received in revised form 28 October 2014 Accepted 6 November 2014 Available online 11 November 2014
We propose a modified Bethe formula for low-energy electron stopping power without fitting parameters for a wide range of elements and compounds. This formula maintains the generality of the Bethe formula and gives reasonable agreement in comparing the predicted stopping powers for 15 elements and 6 compounds with the experimental data and those calculated within dielectric theory including the exchange effect. Use of the stopping power obtained from this formula for hydrogen silsesquioxane in Monte Carlo simulation gives the energy deposition distribution in consistent with the experimental data. & 2014 Elsevier B.V. All rights reserved.
Keywords: Modified Bethe formula Electron stopping power Hydrogen silsesquioxane Energy deposition distribution
1. Introduction Electron stopping power (SP) determines the electron energy loss in solids and plays an important role in Monte Carlo (MC) simulation in continuous slowing down approximation (CSDA). The MC simulation in CSDA is widely applied in microscopy, microanalysis, and lithography to model electron scattering in solids. Nonetheless, knowledge of SP is not always available for materials of interest. The SP can be calculated by the Bethe formula [1,2], however this formula is not valid at low energies because it gives negative results for energies less than the mean excitation energy. There have been several attempts to modify the Bethe formula to avoid this problem [3–9] in which only the formula of Joy and Luo [5] retains the generality of the Bethe formula and has been widely used in MC models in CSDA. Nevertheless, these modified formulas are valid only for some elements in a limited energy range since the predicted SP is strongly overestimated or even negative as electron energy decreases. Theoretical calculations [10–13] and experimental data [14] show that at low energies the SP reduces to zero after reaching the maximum in the range 50–600 eV. In general, the use of these formulas for elements or compounds without fitting parameters is still questionable. In the present work we propose a modified Bethe formula that overcomes the negative value problem and avoids overestimation at low energies while maintaining the generality of the Bethe E-mail address: [email protected] 1 Permanent address: Faculty of Materials Science, Ho Chi Minh City University of Science, 227 Nguyen Van Cu Street, Ho Chi Minh City, Vietnam. http://dx.doi.org/10.1016/j.ultramic.2014.11.003 0304-3991/& 2014 Elsevier B.V. All rights reserved.
formula and giving reasonable agreement in comparing the predicted stopping powers for 15 elements and 6 compounds with the experimental data [14] and those calculated within dielectric theory including exchange effect [10,11,13]. We use this modified Bethe formula to calculate the SP for hydrogen silsesquioxane (HSQ) and then apply in MC simulation to determine the energy deposition distribution (EDD) of 3 keV electrons in 15 nm HSQ resist layer on Si substrate. The comparison of simulation result with the experimental data [15] shows a good agreement.
2. Modified Bethe formula The Bethe formula [1,2] for SP can be written as
−
⎛ e E⎞ dE ρZ ln ⎜ = 2π e 4 N A ⎟, ⎝ 2 I⎠ ds AE
(1)
where e is the elementary charge, NA is the Avogadro number, ρ is the mass density, Z is the atomic number, A is the atomic weight, E is the electron energy, e is the base of natural logarithm, and I is the mean excitation energy [16]. It can be seen that Eq. (1) gives the negative value for E < I 2/e and hence is not valid at low energies. In the present work we modify Eq. (1) by introducing a function G(E) as
−
⎞ ⎛ e E ρZ dE = 2π e 4 N A + G (E) ⎟, ln ⎜ ⎠ ⎝ 2 I AE ds
(2)
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
27
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
Fig. 1. Stopping power for 15 elements: thick solid line – present work [Eq. (2)], dashed line – Bethe formula [Eq. (1)], dash-dotted line – Joy and Luo [5], thin solid line – Nguyen-Truong [13], thin dotted line – Jablonski et al. [8], thick dotted line – Shinotsuka et al. [12], symbols – experimental data [14].
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H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
Fig. 2. Stopping power for 15 elements: thick solid line – present work [Eq. (2)], dashed line – Bethe formula [Eq. (1)], dash-dotted line – Joy and Luo [5], thin dotted line – Jablonski et al. [8], thick dotted line – Shinotsuka et al. [12].
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
29
50–600 eV, the mean deviation
where
⎡ ⎛ ⎞ ⎛ E ⎞2⎤ I e 1 Z ln ⎢1 + ⎜ ⎟ ⎥ + ln ⎜⎜ ⎟⎟ G (E) = 1 − ⎝ I ⎠ ⎥⎦ E ⎢⎣ 2 3 ⎝2⎠ ⎡ 2⎤ 3 ⎛ 2 E⎞ E × exp ⎢− + ln ⎟ ⎥ . ⎜1 − ⎢⎣ I ⎠ ⎥⎦ I Z⎝ Z
ΔS =
(3)
The function G(E) is chosen so that the predicted SPs by Eq. (2) are always positive and consistent with other results [14,13] while maintaining the generality of Eq. (1). This function is found empirically based on the experimental data [14] and our SPs [13] calculated by the Lindhard–Penn algorithm [17] within the dielectric theory. This algorithm uses the experimental optical data and gives accurate results with energies down to the plasmon excitation energy. Recently, we have shown that the Lindhard– Penn algorithm can be improved by combining with the Mermin dielectric function [18] to take plasmon damping into account [19]. Fig. 1 shows the SPs obtained from the modified Bethe formula (2) for 15 elements, the theoretical results [19] and the experimental data [14] of SP and those given by Joy and Luo formula [5] are also included for comparison. It can be seen that the present modified Bethe formula not only overcomes the negative value problem and thus ensures the physical meaning of the predicted SP, but also gives reasonable results for a wide range of elements. In the energy range
1 N
N
∑
1−
n= 1
Spred (En ) Scalc (En )
, (4)
where Spred and Scalc are the predicted SPs by Eq. (2) and the calculated SPs [13], respectively, for 10 corresponding elements is Al (15.76%), Si (17.14%), Ni (14.42%), Cu (20.60%), Mo (12.81%), Rh (4.36%), Os (17.70%), Ir (8.75%), Pt (15.08%), and Au (16.59%). When extending the energy range from 50 eV to 30 keV, the mean deviations are Al (7.40%), Si (9.11%), Ni (7.08%), Cu (9.96%), Mo (7.77%), Rh (10.85%), Os (22.67%), Ir (8.68%), Pt (13.66%), and Au (11.34%). Thus, excepting for Rh and Os, the mean deviations are reduced, even by about half for Al, Si, Ni, Cu, and Mo. From these evaluations, we can believe in the predictive possibility of Eq. (2) with the mean deviation less than about 20%. Such mean deviation is acceptable when considering the simplicity and generality of the modified Bethe formula (2). Further comparisons with the SPs obtained from Eq. (16) of Ref. [8] and with those reported in Table 1 of Ref. [12] can be seen from Figs. 1–3. The authors of Ref. [8] introduced a so-called S-lambda expression to calculate the SP depending on the inelastic mean free path (IMFP) in the electron energy range from 200 eV to 30 keV (here we use the IMFP reported in Table 4 of Ref. [20]). Unfortunately, this dependence limits the application range of their expression and causes it cannot be used without knowledge of the IMFP. While the use of the S-lambda expression for energy
a
b
c
d
e
f
g
h
i
Fig. 3. Same as Fig. 2 except for 9 elements.
30
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
less than 200 eV is still questionable, one can see that this expression gives negative SP, which does not have physical meaning, as E < 1/d3 ≃ 18.34 eV . Additionally, the S-lambda expression is not satisfactory for low-Z elements, this is shown in Figs. 1(a), 2(a),
and (b) for graphite, Li, and Be, respectively. Figs. 1–3 show that the modified Bethe formula (2) does not suffer from these limitations, and in general, it gives the SPs smaller than those calculated by the complex dielectric function theory without the
a
b
c
d
e
f
g
Fig. 4. Stopping power for 7 compounds: solid line – present work [Eq. (2)], dash-dotted line – Joy and Luo [5], symbols: (a, b) – experimental data [14], (c)–(f) – Akkerman and Akkerman [11] (circle) and Ashley [10] (square).
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
exchange effect [12]. Neglecting the exchange effect may cause the SP to be overestimated because the maximum energy loss is so large. In the previous study [13], we discussed the important role of the maximum energy loss in the SP calculations and proposed an assumption to determine this quantity by taking into account the exchange effect. The results based on this assumption are in good agreement with the experimental data of SP and backscattering coefficient [14]. Using the calculated SPs [13] to find a modified Bethe formula in the present work, we aim to a simple and effective modification while maintaining the generality of the Bethe formula because it is meaningful for elements and compounds that lack knowledge of the SP. The SP for compounds can be also calculated by Eq. (2), using the mean atomic number Z¯ and the mean atomic weight A¯ instead of Z and A, respectively. The mean excitation energy should be determined by experiment, or could be given by [21]
I = 9.76Z¯ +
58.5 . Z¯ 0.19
(5)
Fig. 4 shows the SPs obtained from the modified Bethe formula (2) for 7 compounds. It can be seen that the predicted SPs are consistent with the experimental data and the theoretical results calculated within the dielectric theory including the exchange effect [10,11]. These agreements demonstrate the generality of the present modified Bethe formula since it is found empirically based on the SPs for elements but not for compounds. Also, they show the correctness of the exchange effect included in the SP calculations [10,11,13] within the dielectric theory. The above comparisons show that the modified Bethe formula (2) overcomes the problems encountered in previous studies [3–9] and is able to predict the electron SP at low energies for elements and compounds with a reasonable accuracy.
31
3. Application to Monte Carlo simulation 3.1. Backscattering coefficient In order to give a more clear illustration on the predictive power of the modified Bethe formula (2) comparing with other approaches in a practical sense, we show in Fig. 5 the backscattering coefficients for Au calculated by our previous MC model in CSDA [13], using three different SP sources referred as (a) SBetheExc – SP predicted by Eq. (2), (b) SFitNoExc – SP predicted by Eq. (16) of Jablonski et al. [8] (using IMFP of Tanuma et al. [20]), and (c) SNoExc – SP of Shinotsuka et al. [12]. It can be seen from Fig. 5 that the SBetheExc tends to coincide with our previous results [13], meanwhile the SFitNoExc and the SNoExc seem to agree with the results of Zommer et al. [22]. This is because the three latter ones used the SP without the exchange effect, which has been mentioned in the previous section. Also, the comparison show that the differences between our approach and alternatives have a certain practical significance in the application.
3.2. Energy deposition distribution We use the modified Bethe formula (2) to predict the SP for HSQ and then apply to MC simulation to calculate the EDD of 3 keV electrons in 15 nm HSQ resist layer on Si substrate, the simulation result is subsequently compared with experimental data [15]. The MC model used here is extended from our previous model in CSDA [13]. It is assumed that the incident electron penetrating a solid loses energy continuously along its trajectory, which consists of a consecutive series of straight segments. The electron trajectory is determined by repeating the following steps: (1) generate path length, (2) calculate the remaining energy at the end of path, and (3) determine scattering angle.
Fig. 5. Backscattering coefficient for Au: solid circle – SBetheExc [SP predicted by Eq. (2)], solid square – Nguyen-Truong [13], solid up-triangle – SFitNoExc (SP predicted by Eq. (16) of Jablonski et al. [8]), solid down-triangle – SNoExc (SP of Shinotsuka et al. [12]), solid left-triangle – Zommer et al. [22], other open symbols – experimental data [14].
32
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
Fig. 6. The energy-point of 10,000 electron trajectories with initial energy of 3 keV in 15 nm HSQ on Si substrate: (a) view in the x–y plane, (b) view in the x–z plane. The energy-meter (in unit of eV) is shown on the right.
Fig. 7. The energy deposition distribution (EDD) of 3 keV electrons in 15 nm HSQ on Si substrate: (a) depth distribution, (b) lateral distribution at the resist/substrate interface. The (normalized) experimental values [15] are multiplied by the simulation value of EDD at radius 3.8 nm.
In the first step, a path length s is randomly generated: and
s = − λ el ln(rs ),
(6)
where rs is an uniform random number in the range (0, 1), λel is the elastic mean free path of electron and is given by
λ el− 1 = ρNA
∑ i
Ci σiel Ai
(7)
where ρ is the mass density, NA is the Avogadro constant, σ el is the elastic cross section [13], Ci and Ai are the weight fraction and the atomic weight of the ith element of the compound, respectively. Traveling along the length of s, electron loses energy continuously, its remaining energy at the end of the path is determined by using SP:
s=
∫E
En n +1
dE , −dE/ds
(8)
where En and En + 1 are the electron energy after the nth and (n + 1) th collisions, respectively. The SP for HSQ resist layer is obtained from the modified Bethe formula (2), whereas for Si substrate our SP [13] calculated within dielectric theory is reused. In the last step, the scattering angles θ and the azimuthal angles φ are given, respectively, by
2π ∑i Ci σi
el
dσ ∫0 ∑ Ci dΩi θ
el
i
sin θ′ dθ′ = rθ ,
(9)
φ = 2πrφ,
(10)
where rθ and rφ are uniform random numbers in the range (0, 1). Fig. 6 shows the energy-point of 10,000 electron trajectories. In simulation, one million electron trajectories are generated to ensure the stability of the result. The incident electron with initial energy of 3 keV is tracked down to 25 eV. The modified Bethe formula (2) allows us to choose a low cutoff energy without any trouble on the overestimation or the negative value problem. The use of a low cutoff energy can improve the accuracy of the simulation result and reduce the fracture of EDD at the resist/substrate interface as shown in Fig. 7(a). It can be seen from Fig. 7(b) that the simulation result of EDD at the resist/substrate interface is in good agreement with the experimental data of Manfrinato et al. [15]. We note that these authors normalized their experimental data by setting the measurement value at radius 3.8 nm to unity. Hence, before comparing, we multiply these normalized values by the simulation value of EDD at the corresponding radius. The consistency in comparison shows that the SP predicted by the modified Bethe formula (2) for HSQ is reasonable.
4. Conclusions We have proposed a modified Bethe formula for a wide range of elements and compounds without fitting parameters while maintaining the form of the Bethe formula. Its generality has been demonstrated by comparisons of the predicted SPs with the
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
experimental data and those calculated within dielectric theory including the exchange effect. Its application to HSQ shows a good agreement between the MC simulation in CSDA result and the experimental data of EDD. The modified Bethe formula (2) is useful to calculate the SP for elements and compounds, and can be applied to MC models in CSDA.
[10]
[11]
[12]
Acknowledgments The author is grateful to Professor V.A. Smolar for his help in the work. The author thanks Dr. V.R. Manfrinato for providing experimental data in Ref. [15].
[13]
[14] [15]
References [1] H. Bethe, Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie, Ann. Phys. 397 (1930) 325–400. [2] H. Bethe, J. Ashkin, Experimental Nuclear Physics, vol. 1, Wiley, New York, 1953. [3] T.S. Rao-Sahib, D.B. Wittry, X-ray continuum from thick elemental targets for 1050-keV electrons, J. Appl. Phys. 45 (1974) 5060. [4] G. Love, M.G. Cox, V.D. Scott, A versatile atomic number correction for electron-probe microanalysis, J. Phys. D: Appl. Phys. 11 (1978) 7–21. [5] D.C. Joy, S. Luo, An empirical stopping power relationship for low-energy electrons, Scanning 11 (1989) 176–180. [6] J.M. Fernandez-Varea, R. Mayol, D. Liljequist, F. Salvat, Inelastic scattering of electrons in solids from a generalized oscillator strength model using optical and photoelectric data, J. Phys.: Condens. Matter 5 (1993) 3593–3610. [7] T. Zhenyu, H. Yancai, An empirical energy loss equation of electrons, Scanning 24 (2002) 46–51. [8] A. Jablonski, S. Tanuma, C.J. Powell, Modified predictive formula for the electron stopping power, J. Appl. Phys. 103 (2008) 063708. [9] I.I. Maglevanny, V.A. Smolar, H.T. Nguyen-Truong, Extrapolation of the Bethe equation for electron stopping powers to intermediate and low electron
[16]
[17] [18] [19] [20]
[21]
[22]
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energies by empirical simulation of target effective mean excitation energy and atomic number, Nucl. Instrum. Methods B 316 (2013) 123–129. J.C. Ashley, Interaction of low-energy electrons with condensed matter: stopping powers and inelastic mean free paths from optical data, J. Electron Spectrosc. 46 (1988) 199–214. A. Akkerman, E. Akkerman, Characteristics of electron inelastic interactions in organic compounds and water over the energy range 20 10 000 eV, J. Appl. Phys. 86 (1999) 5809. H. Shinotsuka, S. Tanuma, C.J. Powell, D.R. Penn, Calculations of electron stopping powers for 41 elemental solids over the 50 eV to 30 keV range with the full Penn algorithm, Nucl. Instrum. Methods B 270 (2012) 75–92. H.T. Nguyen-Truong, Determination of the maximum energy loss for electron stopping power calculations and its effect on backscattering electron yield in Monte-Carlo simulations applying continuous slowing-down approximation, J. Appl. Phys. 114 (2013) 163513. D.C. Joy, A Database of Electron–Solid Interactions. Available at 〈http://web. utk.edu/ srcutk/htm/interact.htm〉, 2008. V.R. Manfrinato, L.L. Cheong, H. Duan, D. Winston, H.I. Smith, K.K. Berggren, Sub-5 keV electron-beam lithography in hydrogen silsesquioxane resist, Microelectron. Eng. 88 (2011) 3070–3074. S.M. Seltzer, M.J. Berger, Evaluation of the collision stopping power of elements and compounds for electrons and positrons, Int. J. Appl. Radiat. Isot. 33 (1982) 1189–1218. D.R. Penn, Electron mean-free-path calculations using a model dielectric function, Phys. Rev. B 35 (1987) 482–486. N.D. Mermin, Lindhard dielectric function in the relaxation-time approximation, Phys. Rev. B 1 (1970) 2362–2363. H.T. Nguyen-Truong, Energy-loss function including damping and prediction of plasmon lifetime, J. Electron Spectrosc. 193 (2014) 79–85. S. Tanuma, C.J. Powell, D.R. Penn, Calculations of electron inelastic mean free paths. IX. Data for 41 elemental solids over the 50 eV to 30 keV range, Surf. Interface Anal. 43 (3) (2011) 689–713. M.J. Berger, S.M. Seltzer, Tables of Energy Losses and Ranges of Electrons and Positrons. Available at 〈http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/ 19650002905.pdf〉, 1964. L. Zommer, A. Jablonski, G. Gergely, S. Gurban, Monte Carlo backscattering yield (BY) calculations applying continuous slowing down approximation (CSDA) and experimental data, Vacuum 82 (2) (2007) 201–204.
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Modified Bethe formula for low-energy electron stopping power without fitting parameters Hieu T. Nguyen-Truong 1 Faculty of Electronics and Computer Science, Volgograd State Technical University, 28 Lenin Avenue, Volgograd 400131, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 29 August 2014 Received in revised form 28 October 2014 Accepted 6 November 2014 Available online 11 November 2014
We propose a modified Bethe formula for low-energy electron stopping power without fitting parameters for a wide range of elements and compounds. This formula maintains the generality of the Bethe formula and gives reasonable agreement in comparing the predicted stopping powers for 15 elements and 6 compounds with the experimental data and those calculated within dielectric theory including the exchange effect. Use of the stopping power obtained from this formula for hydrogen silsesquioxane in Monte Carlo simulation gives the energy deposition distribution in consistent with the experimental data. & 2014 Elsevier B.V. All rights reserved.
Keywords: Modified Bethe formula Electron stopping power Hydrogen silsesquioxane Energy deposition distribution
1. Introduction Electron stopping power (SP) determines the electron energy loss in solids and plays an important role in Monte Carlo (MC) simulation in continuous slowing down approximation (CSDA). The MC simulation in CSDA is widely applied in microscopy, microanalysis, and lithography to model electron scattering in solids. Nonetheless, knowledge of SP is not always available for materials of interest. The SP can be calculated by the Bethe formula [1,2], however this formula is not valid at low energies because it gives negative results for energies less than the mean excitation energy. There have been several attempts to modify the Bethe formula to avoid this problem [3–9] in which only the formula of Joy and Luo [5] retains the generality of the Bethe formula and has been widely used in MC models in CSDA. Nevertheless, these modified formulas are valid only for some elements in a limited energy range since the predicted SP is strongly overestimated or even negative as electron energy decreases. Theoretical calculations [10–13] and experimental data [14] show that at low energies the SP reduces to zero after reaching the maximum in the range 50–600 eV. In general, the use of these formulas for elements or compounds without fitting parameters is still questionable. In the present work we propose a modified Bethe formula that overcomes the negative value problem and avoids overestimation at low energies while maintaining the generality of the Bethe E-mail address: [email protected] 1 Permanent address: Faculty of Materials Science, Ho Chi Minh City University of Science, 227 Nguyen Van Cu Street, Ho Chi Minh City, Vietnam. http://dx.doi.org/10.1016/j.ultramic.2014.11.003 0304-3991/& 2014 Elsevier B.V. All rights reserved.
formula and giving reasonable agreement in comparing the predicted stopping powers for 15 elements and 6 compounds with the experimental data [14] and those calculated within dielectric theory including exchange effect [10,11,13]. We use this modified Bethe formula to calculate the SP for hydrogen silsesquioxane (HSQ) and then apply in MC simulation to determine the energy deposition distribution (EDD) of 3 keV electrons in 15 nm HSQ resist layer on Si substrate. The comparison of simulation result with the experimental data [15] shows a good agreement.
2. Modified Bethe formula The Bethe formula [1,2] for SP can be written as
−
⎛ e E⎞ dE ρZ ln ⎜ = 2π e 4 N A ⎟, ⎝ 2 I⎠ ds AE
(1)
where e is the elementary charge, NA is the Avogadro number, ρ is the mass density, Z is the atomic number, A is the atomic weight, E is the electron energy, e is the base of natural logarithm, and I is the mean excitation energy [16]. It can be seen that Eq. (1) gives the negative value for E < I 2/e and hence is not valid at low energies. In the present work we modify Eq. (1) by introducing a function G(E) as
−
⎞ ⎛ e E ρZ dE = 2π e 4 N A + G (E) ⎟, ln ⎜ ⎠ ⎝ 2 I AE ds
(2)
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
27
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
Fig. 1. Stopping power for 15 elements: thick solid line – present work [Eq. (2)], dashed line – Bethe formula [Eq. (1)], dash-dotted line – Joy and Luo [5], thin solid line – Nguyen-Truong [13], thin dotted line – Jablonski et al. [8], thick dotted line – Shinotsuka et al. [12], symbols – experimental data [14].
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H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
Fig. 2. Stopping power for 15 elements: thick solid line – present work [Eq. (2)], dashed line – Bethe formula [Eq. (1)], dash-dotted line – Joy and Luo [5], thin dotted line – Jablonski et al. [8], thick dotted line – Shinotsuka et al. [12].
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
29
50–600 eV, the mean deviation
where
⎡ ⎛ ⎞ ⎛ E ⎞2⎤ I e 1 Z ln ⎢1 + ⎜ ⎟ ⎥ + ln ⎜⎜ ⎟⎟ G (E) = 1 − ⎝ I ⎠ ⎥⎦ E ⎢⎣ 2 3 ⎝2⎠ ⎡ 2⎤ 3 ⎛ 2 E⎞ E × exp ⎢− + ln ⎟ ⎥ . ⎜1 − ⎢⎣ I ⎠ ⎥⎦ I Z⎝ Z
ΔS =
(3)
The function G(E) is chosen so that the predicted SPs by Eq. (2) are always positive and consistent with other results [14,13] while maintaining the generality of Eq. (1). This function is found empirically based on the experimental data [14] and our SPs [13] calculated by the Lindhard–Penn algorithm [17] within the dielectric theory. This algorithm uses the experimental optical data and gives accurate results with energies down to the plasmon excitation energy. Recently, we have shown that the Lindhard– Penn algorithm can be improved by combining with the Mermin dielectric function [18] to take plasmon damping into account [19]. Fig. 1 shows the SPs obtained from the modified Bethe formula (2) for 15 elements, the theoretical results [19] and the experimental data [14] of SP and those given by Joy and Luo formula [5] are also included for comparison. It can be seen that the present modified Bethe formula not only overcomes the negative value problem and thus ensures the physical meaning of the predicted SP, but also gives reasonable results for a wide range of elements. In the energy range
1 N
N
∑
1−
n= 1
Spred (En ) Scalc (En )
, (4)
where Spred and Scalc are the predicted SPs by Eq. (2) and the calculated SPs [13], respectively, for 10 corresponding elements is Al (15.76%), Si (17.14%), Ni (14.42%), Cu (20.60%), Mo (12.81%), Rh (4.36%), Os (17.70%), Ir (8.75%), Pt (15.08%), and Au (16.59%). When extending the energy range from 50 eV to 30 keV, the mean deviations are Al (7.40%), Si (9.11%), Ni (7.08%), Cu (9.96%), Mo (7.77%), Rh (10.85%), Os (22.67%), Ir (8.68%), Pt (13.66%), and Au (11.34%). Thus, excepting for Rh and Os, the mean deviations are reduced, even by about half for Al, Si, Ni, Cu, and Mo. From these evaluations, we can believe in the predictive possibility of Eq. (2) with the mean deviation less than about 20%. Such mean deviation is acceptable when considering the simplicity and generality of the modified Bethe formula (2). Further comparisons with the SPs obtained from Eq. (16) of Ref. [8] and with those reported in Table 1 of Ref. [12] can be seen from Figs. 1–3. The authors of Ref. [8] introduced a so-called S-lambda expression to calculate the SP depending on the inelastic mean free path (IMFP) in the electron energy range from 200 eV to 30 keV (here we use the IMFP reported in Table 4 of Ref. [20]). Unfortunately, this dependence limits the application range of their expression and causes it cannot be used without knowledge of the IMFP. While the use of the S-lambda expression for energy
a
b
c
d
e
f
g
h
i
Fig. 3. Same as Fig. 2 except for 9 elements.
30
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
less than 200 eV is still questionable, one can see that this expression gives negative SP, which does not have physical meaning, as E < 1/d3 ≃ 18.34 eV . Additionally, the S-lambda expression is not satisfactory for low-Z elements, this is shown in Figs. 1(a), 2(a),
and (b) for graphite, Li, and Be, respectively. Figs. 1–3 show that the modified Bethe formula (2) does not suffer from these limitations, and in general, it gives the SPs smaller than those calculated by the complex dielectric function theory without the
a
b
c
d
e
f
g
Fig. 4. Stopping power for 7 compounds: solid line – present work [Eq. (2)], dash-dotted line – Joy and Luo [5], symbols: (a, b) – experimental data [14], (c)–(f) – Akkerman and Akkerman [11] (circle) and Ashley [10] (square).
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
exchange effect [12]. Neglecting the exchange effect may cause the SP to be overestimated because the maximum energy loss is so large. In the previous study [13], we discussed the important role of the maximum energy loss in the SP calculations and proposed an assumption to determine this quantity by taking into account the exchange effect. The results based on this assumption are in good agreement with the experimental data of SP and backscattering coefficient [14]. Using the calculated SPs [13] to find a modified Bethe formula in the present work, we aim to a simple and effective modification while maintaining the generality of the Bethe formula because it is meaningful for elements and compounds that lack knowledge of the SP. The SP for compounds can be also calculated by Eq. (2), using the mean atomic number Z¯ and the mean atomic weight A¯ instead of Z and A, respectively. The mean excitation energy should be determined by experiment, or could be given by [21]
I = 9.76Z¯ +
58.5 . Z¯ 0.19
(5)
Fig. 4 shows the SPs obtained from the modified Bethe formula (2) for 7 compounds. It can be seen that the predicted SPs are consistent with the experimental data and the theoretical results calculated within the dielectric theory including the exchange effect [10,11]. These agreements demonstrate the generality of the present modified Bethe formula since it is found empirically based on the SPs for elements but not for compounds. Also, they show the correctness of the exchange effect included in the SP calculations [10,11,13] within the dielectric theory. The above comparisons show that the modified Bethe formula (2) overcomes the problems encountered in previous studies [3–9] and is able to predict the electron SP at low energies for elements and compounds with a reasonable accuracy.
31
3. Application to Monte Carlo simulation 3.1. Backscattering coefficient In order to give a more clear illustration on the predictive power of the modified Bethe formula (2) comparing with other approaches in a practical sense, we show in Fig. 5 the backscattering coefficients for Au calculated by our previous MC model in CSDA [13], using three different SP sources referred as (a) SBetheExc – SP predicted by Eq. (2), (b) SFitNoExc – SP predicted by Eq. (16) of Jablonski et al. [8] (using IMFP of Tanuma et al. [20]), and (c) SNoExc – SP of Shinotsuka et al. [12]. It can be seen from Fig. 5 that the SBetheExc tends to coincide with our previous results [13], meanwhile the SFitNoExc and the SNoExc seem to agree with the results of Zommer et al. [22]. This is because the three latter ones used the SP without the exchange effect, which has been mentioned in the previous section. Also, the comparison show that the differences between our approach and alternatives have a certain practical significance in the application.
3.2. Energy deposition distribution We use the modified Bethe formula (2) to predict the SP for HSQ and then apply to MC simulation to calculate the EDD of 3 keV electrons in 15 nm HSQ resist layer on Si substrate, the simulation result is subsequently compared with experimental data [15]. The MC model used here is extended from our previous model in CSDA [13]. It is assumed that the incident electron penetrating a solid loses energy continuously along its trajectory, which consists of a consecutive series of straight segments. The electron trajectory is determined by repeating the following steps: (1) generate path length, (2) calculate the remaining energy at the end of path, and (3) determine scattering angle.
Fig. 5. Backscattering coefficient for Au: solid circle – SBetheExc [SP predicted by Eq. (2)], solid square – Nguyen-Truong [13], solid up-triangle – SFitNoExc (SP predicted by Eq. (16) of Jablonski et al. [8]), solid down-triangle – SNoExc (SP of Shinotsuka et al. [12]), solid left-triangle – Zommer et al. [22], other open symbols – experimental data [14].
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H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
Fig. 6. The energy-point of 10,000 electron trajectories with initial energy of 3 keV in 15 nm HSQ on Si substrate: (a) view in the x–y plane, (b) view in the x–z plane. The energy-meter (in unit of eV) is shown on the right.
Fig. 7. The energy deposition distribution (EDD) of 3 keV electrons in 15 nm HSQ on Si substrate: (a) depth distribution, (b) lateral distribution at the resist/substrate interface. The (normalized) experimental values [15] are multiplied by the simulation value of EDD at radius 3.8 nm.
In the first step, a path length s is randomly generated: and
s = − λ el ln(rs ),
(6)
where rs is an uniform random number in the range (0, 1), λel is the elastic mean free path of electron and is given by
λ el− 1 = ρNA
∑ i
Ci σiel Ai
(7)
where ρ is the mass density, NA is the Avogadro constant, σ el is the elastic cross section [13], Ci and Ai are the weight fraction and the atomic weight of the ith element of the compound, respectively. Traveling along the length of s, electron loses energy continuously, its remaining energy at the end of the path is determined by using SP:
s=
∫E
En n +1
dE , −dE/ds
(8)
where En and En + 1 are the electron energy after the nth and (n + 1) th collisions, respectively. The SP for HSQ resist layer is obtained from the modified Bethe formula (2), whereas for Si substrate our SP [13] calculated within dielectric theory is reused. In the last step, the scattering angles θ and the azimuthal angles φ are given, respectively, by
2π ∑i Ci σi
el
dσ ∫0 ∑ Ci dΩi θ
el
i
sin θ′ dθ′ = rθ ,
(9)
φ = 2πrφ,
(10)
where rθ and rφ are uniform random numbers in the range (0, 1). Fig. 6 shows the energy-point of 10,000 electron trajectories. In simulation, one million electron trajectories are generated to ensure the stability of the result. The incident electron with initial energy of 3 keV is tracked down to 25 eV. The modified Bethe formula (2) allows us to choose a low cutoff energy without any trouble on the overestimation or the negative value problem. The use of a low cutoff energy can improve the accuracy of the simulation result and reduce the fracture of EDD at the resist/substrate interface as shown in Fig. 7(a). It can be seen from Fig. 7(b) that the simulation result of EDD at the resist/substrate interface is in good agreement with the experimental data of Manfrinato et al. [15]. We note that these authors normalized their experimental data by setting the measurement value at radius 3.8 nm to unity. Hence, before comparing, we multiply these normalized values by the simulation value of EDD at the corresponding radius. The consistency in comparison shows that the SP predicted by the modified Bethe formula (2) for HSQ is reasonable.
4. Conclusions We have proposed a modified Bethe formula for a wide range of elements and compounds without fitting parameters while maintaining the form of the Bethe formula. Its generality has been demonstrated by comparisons of the predicted SPs with the
H.T. Nguyen-Truong / Ultramicroscopy 149 (2015) 26–33
experimental data and those calculated within dielectric theory including the exchange effect. Its application to HSQ shows a good agreement between the MC simulation in CSDA result and the experimental data of EDD. The modified Bethe formula (2) is useful to calculate the SP for elements and compounds, and can be applied to MC models in CSDA.
[10]
[11]
[12]
Acknowledgments The author is grateful to Professor V.A. Smolar for his help in the work. The author thanks Dr. V.R. Manfrinato for providing experimental data in Ref. [15].
[13]
[14] [15]
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