- Email: [email protected]
Physical approximants to electron scattering
MICROELSEVIER
Microelectronic
Engineering
34 (1997)
147-154
Physical approximants to electron scattering G. Messinaa,b, A. Paoletti”‘“,
S. Santangeloa’b’*, A. Tucciaronea’c
“Istituto Nazionale per la Fisica della Materia, 89100 Reggio Calabria, Italy bFac. Ingegneria dell’ Universitti, v.E.Cuzzocrea 48, Universith Roma Tor Vergata, Italy ‘Dpt. Scienze e Tecnologie Fisiche ed Energetiche, Universit2 Roma Tor Vergata, v.O.Raimondo, 00173 Roma, Italy Abstract In this paper, Monte Carlo modelling of electron scattering in a general composite substrate in the elastic regime is considered. In spite of the great number of physical variables involved in the process, a simplified formulation of the problem in terms of a limited number of dimensionless parameters is demonstrated by a modified application of Buckingham’s theorem of Dimensional Analysis. A single generalised Buckingham argument is introduced and demonstrated to ultimately intervene in determining both forward and backscattering. This approach is exemplified in the case of electron beam lithography, by evaluating the backscattering coefficient n and the forward scattering wi’dth w. By interpolation of the numerical data, simple Buckingham approximants to the physical laws governing the process are finally derived, and they are applicable to all elemental and composite materials with e-beam voltages from 5 to 150 kV and substrate thickness from 1 to 5000 nm. The unique position of diamond as a substrate material in electron beam lithography is emphasised. Keywords: Diamond
Monte
Carlo simulation;
Electron
scattering;
e-beam lithography;
Numerical
approximation
and analysis;
1. Introduction Quantitative evaluation of electron scattering in solid targets for lithographic applications is generally performed using numerical methods. However, the large number of variables usually involved makes it generally difficult to have a simple and intuitive knowledge about possible trends. A novel general approach to the solution of this problem [1,2], based on Buckingham’s theorem of Dimensional Analysis [3], has been recently proposed and applied to the evaluation of backscattering from single-component substrates by means of Monte Carlo (MC) simulation. According to this method, the existence of variable combinations is assumed, constituting independent dimensionless arguments (Buckingham arguments), which are the only ones that are
*Corresponding
author.
0 0167-9317/96/$17.00 PII SO167-9317(97)00076-7
1997 Elsevier Science B.V. All rights reserved
148
G. Messina et al. / Microelectronic
Engineering
34 (1997) 147-154
effective in describing the process and determining the system behaviour. Once the problem is formulated in terms of a limited number of Buckingham arguments, simple analytical functions of these arguments are derived by interpolating numerical data. They represent practical approximants (Buckingham approximants) to the physical laws governing the system. In this paper, such an approach is extended to the modelling of electron scattering in a general multi-component substrate, in the elastic regime. MC simulation of electron trajectories, aimed at the calculation of a few parameters that are descriptive of the system behaviour, separately evaluates forward scattering (FS) and backscattering (BS) electron distributions, at any depth within both elemental and composite substrates. In particular, the impact of forward scattering is described by the FS width w, defined as the mean radial distance over which forward scattered electrons are distributed; the backscattering is, as usual, characterised by the BS coefficient 7, defined as the ratio of backscattered to incident electrons. A single generalised Buckingham argument Q is introduced (Section 3, in terms of which a simplified picture of the electron scattering is obtained. By interpolating numerical results, very simple Buckingham approximants are derived, which reproduce well all the variations of 7 (Section 4 and w (Section 5 in the range of Q considered. Finally (Section 6, the behaviour of investigated substrate materials is compared and the unique position of diamond is demonstrated.
2. Electron
scattering
picture
The BS coefficient 7 and the FS width w in a given substrate may be calculated by means of a MC simulation of the electron trajectories [4-71. Both r~ and w are functions of a large number of variables, namely: e-beam voltage V, thickness t, and density 8 of the substrate; the individtral atomic numbers Z,, atomic weights Ai and stoichiometric factors of the various component species. As a consequence, the representation of the numerical results should, in principle, take place in a N + l-dimensional space (where N = 5 for elemental materials rising up to 9,12... in case of binary, ternary... compounds, respectively). By spanning the configurational space of the variables, q- and w-points cover N-dimensional hypersurfaces. The dependence of 7 and w on a specific process variable vi can be illustrated by projecting the hypersurfaces on the (vi, 7) and (vi, w) planes. However, since any arbitrary change of the remaining variables causes a confusing spread of the projected points, it is not easy to readily understand in what direction the effect of a variableconfiguration change goes. An example is given in Fig. 1 where the projection on the (Z, T>I plane is shown in the case of elemental materials. As can be seen, for a given material, several v-points are obtained by varying substrate thickness and e-beam voltage. Unfortunately, because of the great number of variables, even the recourse to parametric curves [S], emphasising the dependence of 77 and w on vi for given values of one of the remaining variables, does not provide a better insight into the problem. As previously demonstrated [ 1,2], a simpler picture of electron scattering may be obtained by giving up the idea of following the dependence of 7 and w on each of the variables involved, and replacing it by finding a dependence on a limited number of independent variable-combinations, such as Buckingham arguments.
G. Messina et al. f Microelectronic
Engineering
34 (1997) 147-154
60 atomic
number
149
90 Z
Fig. 1. Points of the g-hypersurface projected in the (2, 7) plane, for team-voltages from 5 to 150 kV and substrate thicknesses from I to 5000 nm. Elemental materials from carbon to bismuth were examined. Note that the diamond-phase of carbon has been considered.
3. The generalised
Buckingham
argument
for electron
scattering
The average number of elastic events per incident electron Q, can be considered reasonably important in determining electron scattering. In a general composite substrate, the total elastic cross-section relative to an atom of the species i is ai =
5W4Z’ 4E’P,(P,
+ 1)’
(1)
where e is the electron charge, E, the energy of the incident electron and pi = 3.4Z2’3 lE, the effective screening parameter [9]. The average number of elastic events per incident electron relative to the species i is (2)
where yli = &V,.q /A 1 is the number of scattering centres per unit volume of the species whose weight percent is x,; NA denotes Avogadro’s number. The average number of elastic events per incident electron Q, , given by the sum of the individual contributions Q,i of all the component species,
Q, = xiQ,i =
(3)
is here assumed as a first Buckingham argument. Of course, Q , cannot be expected to be the only variable combination of importance in determining scattering. The backscattering probability per scattering event can be reasonably includ.ed in the formulation of the problem. Having introduced a large-angle (LA) cross-section [ 1,2]
G. Messim
1.50
et al. I Microelectronic
Engineering
34 (1997) 147-154
TW4ZY (7LAi =I 4E2 ’ as obtained by integrating differential elastic cross-section for scattering angles ranging between rr/2 and IT, we define the LA probability for an atom of the species i, as the ratio of the LA cross-section, ff LAi, to the total elastic cross-section, a,, namely
PLAi= fYL& ” Pi? a, with p,e 1 for all materials investigated in the energy range considered. The probability of elastic scattering from the species i is given by the ratio of the average number of elastic events per incident electron relative to the species i, Q I,, to the average number of elastic events per incident electron relative to the whole substrate, Q,,
Hence, the probability of a LA scattering event for the composite LA probabilities relative to each component species, namely, Q2
= @,P,,,
r
substrate is simply
the sum of the
zi
We find it reasonable to assume Q2 as a second Buckingham argument. We expect that the width w of forward scattering, as well as the backscattering coefficient v, both depend on the above Buckingham arguments through empirical laws to be derived by comparison with the given data. A single Buckingham argument has been demonstrated to be able, alone, to describe electron scattering in elemental materials [ 1,2]. This result can be reasonably extended to composite substrates, by assuming the existence of a single independent variable-combination, effective in determining the BS properties of either multi- and single-component substrates. Thus, we hypothesise that, in composite materials, 7 depends on Q, and Q2 through a single generalised Buckingham argument Q according to the same relationship as was derived in the case of the elemental materials, namely,
(8)
Q = Q,Q;“. If such an hypothesis
holds, the expression
of Q consistently
simplifies
to
7w4Z2nt,
Q= 4E2p0.2
(9)
for single-component substrates [2]. Then, wishing to extend to forward scattering the simplified formulation adopted for backscattering, we boldly assume that the dependence of w on Q, and Q, is through the same argument Q as was proposed for 7,~ All these assumptions have to be tested and compared with the numerical data. Note that all the data considered in Sections 4 and 5 refer to the elastic regime, where the energy loss in a to-thick substrate, roughly evaluated as AE = t,(dElds) [2], is of the order of 15% of E or less.
G. Messina et al. I Microelectronic
4. Physical
approximant
to the backscattering
Engineering
34 (1997) 147-154
151
coefficient
In order to test the above assumptions, we firstly plotted, as a function of Q, the v-data obtained from a lot of materials, both elemental and composite, at different beam voltages (Fig. 2). In particular, elements from C to Bi and compounds from PMMA to PbTe were considered and their BS coefficient was evaluated for beam voltages varying from 5 to 150 kV and substrate thickness ranging from 1 to 5000 nm. Note that the diamond-phase D was considered for carbon. In spite of the independent variation of so many variables over such wide ranges, the q-points line-up onedimensionally along a common 7(Q) curve, demonstrating the existence of a generalised argument Q, able to account for the backscattering properties of any substrate. This represents a remarkable result. The same value of the argument Q, in fact, finally producing a desired BS coefficient 7, can be obtained by starting from a lot of variable configurations (the allowed ones being chosen from among the infinite possibilities by imposing the elasticity requirement, mentioned in Section 3, which has to be satisfied). Eq. (8), which indicates how the process variables can be modified in order to give a constant Q via a mutual compensation, represents a scaling law for the backscattering process. Furthermore, because of the need to be able to predict the effect produced by any variable configuration change on the backscattering, knowledge of the explicit dependence of 7 on Q can be useful. By interpolating numerical results, a practical approximant is obtained, representing in a simple way the physical law governing the backscattering process. The Buckingham approximant to the given q-data is
T(Q) = 83 Qy
(10)
where 77 is measured in percent. Eq. (10) overestimates v at small and high Q-values, while underestimating it at intermediate ones. Nevertheless, it is fully satisfactory for all practical purposes. However, it is worthwhile noticing that
o tantalum (5-150 keV) a other material (30 keV) o I
1
I
0.0
0.2
0.4
0.6
(1 (adim) Fig. 2. The universal dependence of the backscattering coefficient on the single argument Q. V-Data relative to composite materials from PMMA to PbTe are shown together with the same q-data as in Fig. 1. D stands for diamond phase of carbon. The straight line is the simple Buckingham approximant, q=83 Q.
152
G. Messina et al. I Microelectronic
a more accurate Buckingham elemental materials [2]:
approximant
Engineering
34 (1997) 147-154
has been deduced by a model derivation
q(Q) = 100( 1 - exp(- 1.3Q”“)),
(11)
v being measured in the same units as in Eq. (10). Such an expression, giving generalised BS argument, can also be utilised in case of composite substrates, (lo), whenever it is necessary to account for details of the variation of q with curvature near the origin and the successive negative curvature and saturating
5. Physical
approximant
of Q in the case of
to the forward
scattering
7 as a function of the as an alternative to Eq. Q, such as the positive trend for increasing Q.
width
According to Buckingham’s theorem, w can only enter a physical law if it is normalised to a variable having the dimensions of length. In principle, this variable may be a combination of the process variables. However, the simplest possibility, which also has an obvious physical basis, is that FS width w =: w/t,. t, alone provides a proper normalisation. Thus, we define the “normalised” In order to test the w(Q) law that was assumed in Section 3, we plot, as a function of the generalised argument Q, all the normalised w-data corresponding to the same variable configurations as in Fig. 2 (Fig. 3). As in the previous case, in spite of the independent variation of a large number of variables, w-points line up one-dimensionally along a single w(Q) curve, demonstrating the capability of the generalised BS argument of also describing the forward scattering properties of substrates. This again constitutes a quite remarkable result. The many process-variables ultimately intervene in determining the system behaviour in terms of both forward and backward scattering, through a single Buckingham argument, which thus suffices to describe the process. Eq. (8) can be consequently regarded as a scaling law for electron scattering in solids. As for the explicit dependence of w on Q, by interpolating the given data, a simple Buckingham approximant is derived, namely
100 1 DPbTe
Bi
3 o tantalum (5-150 keV) w other material (30 keV) 0.0
0.4
0.2
016
Q (adim) Fig. 3. The universal dependence of the normalised forward scattering width on the single argument Q. The w--data shown refer to the same cases as in Fig. 2. The solid line represents the simple Buckingham approximant, w= 85 (I -exp(-7.5Q)).
G. Messina et al. I Microelectronic
w = 85 (1 - exp(-7SQ))
Engineering
34 (1997) 147-154
1.53
(12)
where w is measured in percent. It is to be noticed that the smooth variation of w reflects the fact that the dimensions of the scattering roughly scale with the. geometry of the system. In particular, as demonstrated by the envelope of the electron trajectories, the width of the FS increases with the depth into the substrate according to an almost quadratic law, in the case of a weak-scattering regime (corresponding to small Q,-values). In the presence of intense scattering, a linear increase of w with depth is, in principle, expected, because of the asymptotic behaviour of o. However, it should be noted that such a situation is rarely achieved, the elastic limit being generally reached earlier than the w-asymptote.
6. Additional
remarks
Eq. (12) and Eq. (10) respectively represent very simple approximants to the physical laws governing forward and backward scattering: they allow the quantitative effect of any change of the process variables on the system behaviour to be readily evaluated, as a function of the single argument Q. In this framework, the generalised argument Q, or, better, its reciprocal Q*, can be regarded as a universal quality factor for the electron scattering process in solids. In view of lithographic application, it may be interesting to compare results relative to some typical substrate materials. At a given e-beam voltage, the value of Q, obtained for a to-thick diamondsubstrate is lower than for several other substrates having the same thickness. This means that a higher quality factor Q* pertains to diamond. The more convenient lithographic performance of diamond with respect to these substrates (e.g., Si, MO, Ta etc.) is actually demonstrated by Eqs. (10,12), which correspondingly give a lower 77 and a smaller w. On the other hand, it is to be noted that the same value of Q may correspond to quite different experimental conditions depending on the substrate material utilised. In particular, by using substrates of fixed thickness t,, different e-beam voltages will be necessary in order to have the same v. For example, BS coefficients obtained by means of diamond at 30 kV, which is a typical operating voltage of standard lithographic systems, will require the use of -40, 120 and 210 kV in the case o-f Si, MO and Ta substrates, respectively. On the contrary, by operating at a given e-beam voltage, different substrate thicknesses will be required in order to produce the same result. For example, the 7 obtained by using a 1 pm-thick diamond-substrate will be matched by a 1Spm-BN or 4pm-PMMA substrates, but with the straightforward consequence of forward scattered electrons distributed in BN and1 PMMA over an average distance from the incidence direction that is 1.5 or 4 times larger than the FS width typical of diamond, respectively.
7. Conclusion Buckingham’s theorem of Dimensional Analysis, modified in its application with respect to the standard procedure, has been applied to the evaluation of electron scattering in a general composite substrate, as calculated by means of Monte Carlo simulation. We have considered the dependence of the forward scattering width w and backscattering
154
G. Messinn et al. / Microelectronic
Engineering
34 (1997)
147-154
coefficient 7 on the physical constants of the material, e-beam voltage and substrate thickness, for all elements of the periodic table and several typical compounds, for all variations of e-beam voltage and substrate thickness in the 5 to 150 kV and 1 to 5000 nm ranges, respectively. We have demonstrated that a single generalised Buckingham argument, dimensionless combination of the process variables, effectively determined both forward and backward scattering features. FinalIy, in the range of variables considered, simple Buckingham approximants have been obtained for both backscattering coefficient 77and normalised forward scattering width w, namely, 7 = 83 Q and w = 85 (1 - exp(-7.5Q)), respectively. The generalised Buckingham argument Q provides the rules according to which the variable configuration can be changed without affecting the resulting forward scattering and backscattering properties. Its reciprocal Q* can be regarded as a quality factor for electron scattering. Finally, from a comparative discussion, diamond, which exhibits a very high Q*, is demonstrated to play a leading role among substrate materials.
References [I] G. Messina, A. Paoletti, S. Santangelo and A.. Tucciarone, A single quality factor for electron backscattering from thin films, Microelectron. Engrg. 27 (1995) 183-I 86. [2] G. Messina, S. Santangelo, A. Paoletti and A. Tucciarone, Monte Carlo modelling of electron beam Ilthography: a scaling law, Microsys. Technol. 1 (1994) 23-29. [3] For an exhaustive discussion of the H theorem see, for example, P.W. Bridgman, Dimensional Analysis, YaJe University Press New Haven, 193 1. [4] K. Murata, Electron Beam interactions with Solids, SEM, Inc., AMF O’Hare, Chicago, pp. 31 l-329. [5] R.J. Hawryluk, A.M. Hawryluk and H.I. Smith, Energy dissipation in a thin polymer film by electron beam scattering, J. Appl. Phys. 45 (1974) 2551-2566. [6] D.C. Joy, The spatial resolution limit of electron lithography, Microelectron. Engrg. 1 (1983) 103-I 19. [7] G. Messina, A. Paoletti, S. Santangelo and A. Tucciarone, Electron scattering in microstructure processes, La Rivista de1 Naovo Cimento 15 (1992) I-57. [8] H. Niedrig, Electron backscattering from thin films, J. Appl. Phys. 53 (1982) R15-R49. [9] S. Leisegang, Zur Mehrfachstreuung von Elektronen in duennen Schichten, Z. Phys. 132 (1952) 183-198.
Microelectronic
Engineering
34 (1997)
147-154
Physical approximants to electron scattering G. Messinaa,b, A. Paoletti”‘“,
S. Santangeloa’b’*, A. Tucciaronea’c
“Istituto Nazionale per la Fisica della Materia, 89100 Reggio Calabria, Italy bFac. Ingegneria dell’ Universitti, v.E.Cuzzocrea 48, Universith Roma Tor Vergata, Italy ‘Dpt. Scienze e Tecnologie Fisiche ed Energetiche, Universit2 Roma Tor Vergata, v.O.Raimondo, 00173 Roma, Italy Abstract In this paper, Monte Carlo modelling of electron scattering in a general composite substrate in the elastic regime is considered. In spite of the great number of physical variables involved in the process, a simplified formulation of the problem in terms of a limited number of dimensionless parameters is demonstrated by a modified application of Buckingham’s theorem of Dimensional Analysis. A single generalised Buckingham argument is introduced and demonstrated to ultimately intervene in determining both forward and backscattering. This approach is exemplified in the case of electron beam lithography, by evaluating the backscattering coefficient n and the forward scattering wi’dth w. By interpolation of the numerical data, simple Buckingham approximants to the physical laws governing the process are finally derived, and they are applicable to all elemental and composite materials with e-beam voltages from 5 to 150 kV and substrate thickness from 1 to 5000 nm. The unique position of diamond as a substrate material in electron beam lithography is emphasised. Keywords: Diamond
Monte
Carlo simulation;
Electron
scattering;
e-beam lithography;
Numerical
approximation
and analysis;
1. Introduction Quantitative evaluation of electron scattering in solid targets for lithographic applications is generally performed using numerical methods. However, the large number of variables usually involved makes it generally difficult to have a simple and intuitive knowledge about possible trends. A novel general approach to the solution of this problem [1,2], based on Buckingham’s theorem of Dimensional Analysis [3], has been recently proposed and applied to the evaluation of backscattering from single-component substrates by means of Monte Carlo (MC) simulation. According to this method, the existence of variable combinations is assumed, constituting independent dimensionless arguments (Buckingham arguments), which are the only ones that are
*Corresponding
author.
0 0167-9317/96/$17.00 PII SO167-9317(97)00076-7
1997 Elsevier Science B.V. All rights reserved
148
G. Messina et al. / Microelectronic
Engineering
34 (1997) 147-154
effective in describing the process and determining the system behaviour. Once the problem is formulated in terms of a limited number of Buckingham arguments, simple analytical functions of these arguments are derived by interpolating numerical data. They represent practical approximants (Buckingham approximants) to the physical laws governing the system. In this paper, such an approach is extended to the modelling of electron scattering in a general multi-component substrate, in the elastic regime. MC simulation of electron trajectories, aimed at the calculation of a few parameters that are descriptive of the system behaviour, separately evaluates forward scattering (FS) and backscattering (BS) electron distributions, at any depth within both elemental and composite substrates. In particular, the impact of forward scattering is described by the FS width w, defined as the mean radial distance over which forward scattered electrons are distributed; the backscattering is, as usual, characterised by the BS coefficient 7, defined as the ratio of backscattered to incident electrons. A single generalised Buckingham argument Q is introduced (Section 3, in terms of which a simplified picture of the electron scattering is obtained. By interpolating numerical results, very simple Buckingham approximants are derived, which reproduce well all the variations of 7 (Section 4 and w (Section 5 in the range of Q considered. Finally (Section 6, the behaviour of investigated substrate materials is compared and the unique position of diamond is demonstrated.
2. Electron
scattering
picture
The BS coefficient 7 and the FS width w in a given substrate may be calculated by means of a MC simulation of the electron trajectories [4-71. Both r~ and w are functions of a large number of variables, namely: e-beam voltage V, thickness t, and density 8 of the substrate; the individtral atomic numbers Z,, atomic weights Ai and stoichiometric factors of the various component species. As a consequence, the representation of the numerical results should, in principle, take place in a N + l-dimensional space (where N = 5 for elemental materials rising up to 9,12... in case of binary, ternary... compounds, respectively). By spanning the configurational space of the variables, q- and w-points cover N-dimensional hypersurfaces. The dependence of 7 and w on a specific process variable vi can be illustrated by projecting the hypersurfaces on the (vi, 7) and (vi, w) planes. However, since any arbitrary change of the remaining variables causes a confusing spread of the projected points, it is not easy to readily understand in what direction the effect of a variableconfiguration change goes. An example is given in Fig. 1 where the projection on the (Z, T>I plane is shown in the case of elemental materials. As can be seen, for a given material, several v-points are obtained by varying substrate thickness and e-beam voltage. Unfortunately, because of the great number of variables, even the recourse to parametric curves [S], emphasising the dependence of 77 and w on vi for given values of one of the remaining variables, does not provide a better insight into the problem. As previously demonstrated [ 1,2], a simpler picture of electron scattering may be obtained by giving up the idea of following the dependence of 7 and w on each of the variables involved, and replacing it by finding a dependence on a limited number of independent variable-combinations, such as Buckingham arguments.
G. Messina et al. f Microelectronic
Engineering
34 (1997) 147-154
60 atomic
number
149
90 Z
Fig. 1. Points of the g-hypersurface projected in the (2, 7) plane, for team-voltages from 5 to 150 kV and substrate thicknesses from I to 5000 nm. Elemental materials from carbon to bismuth were examined. Note that the diamond-phase of carbon has been considered.
3. The generalised
Buckingham
argument
for electron
scattering
The average number of elastic events per incident electron Q, can be considered reasonably important in determining electron scattering. In a general composite substrate, the total elastic cross-section relative to an atom of the species i is ai =
5W4Z’ 4E’P,(P,
+ 1)’
(1)
where e is the electron charge, E, the energy of the incident electron and pi = 3.4Z2’3 lE, the effective screening parameter [9]. The average number of elastic events per incident electron relative to the species i is (2)
where yli = &V,.q /A 1 is the number of scattering centres per unit volume of the species whose weight percent is x,; NA denotes Avogadro’s number. The average number of elastic events per incident electron Q, , given by the sum of the individual contributions Q,i of all the component species,
Q, = xiQ,i =
(3)
is here assumed as a first Buckingham argument. Of course, Q , cannot be expected to be the only variable combination of importance in determining scattering. The backscattering probability per scattering event can be reasonably includ.ed in the formulation of the problem. Having introduced a large-angle (LA) cross-section [ 1,2]
G. Messim
1.50
et al. I Microelectronic
Engineering
34 (1997) 147-154
TW4ZY (7LAi =I 4E2 ’ as obtained by integrating differential elastic cross-section for scattering angles ranging between rr/2 and IT, we define the LA probability for an atom of the species i, as the ratio of the LA cross-section, ff LAi, to the total elastic cross-section, a,, namely
PLAi= fYL& ” Pi? a, with p,e 1 for all materials investigated in the energy range considered. The probability of elastic scattering from the species i is given by the ratio of the average number of elastic events per incident electron relative to the species i, Q I,, to the average number of elastic events per incident electron relative to the whole substrate, Q,,
Hence, the probability of a LA scattering event for the composite LA probabilities relative to each component species, namely, Q2
= @,P,,,
r
substrate is simply
the sum of the
zi
We find it reasonable to assume Q2 as a second Buckingham argument. We expect that the width w of forward scattering, as well as the backscattering coefficient v, both depend on the above Buckingham arguments through empirical laws to be derived by comparison with the given data. A single Buckingham argument has been demonstrated to be able, alone, to describe electron scattering in elemental materials [ 1,2]. This result can be reasonably extended to composite substrates, by assuming the existence of a single independent variable-combination, effective in determining the BS properties of either multi- and single-component substrates. Thus, we hypothesise that, in composite materials, 7 depends on Q, and Q2 through a single generalised Buckingham argument Q according to the same relationship as was derived in the case of the elemental materials, namely,
(8)
Q = Q,Q;“. If such an hypothesis
holds, the expression
of Q consistently
simplifies
to
7w4Z2nt,
Q= 4E2p0.2
(9)
for single-component substrates [2]. Then, wishing to extend to forward scattering the simplified formulation adopted for backscattering, we boldly assume that the dependence of w on Q, and Q, is through the same argument Q as was proposed for 7,~ All these assumptions have to be tested and compared with the numerical data. Note that all the data considered in Sections 4 and 5 refer to the elastic regime, where the energy loss in a to-thick substrate, roughly evaluated as AE = t,(dElds) [2], is of the order of 15% of E or less.
G. Messina et al. I Microelectronic
4. Physical
approximant
to the backscattering
Engineering
34 (1997) 147-154
151
coefficient
In order to test the above assumptions, we firstly plotted, as a function of Q, the v-data obtained from a lot of materials, both elemental and composite, at different beam voltages (Fig. 2). In particular, elements from C to Bi and compounds from PMMA to PbTe were considered and their BS coefficient was evaluated for beam voltages varying from 5 to 150 kV and substrate thickness ranging from 1 to 5000 nm. Note that the diamond-phase D was considered for carbon. In spite of the independent variation of so many variables over such wide ranges, the q-points line-up onedimensionally along a common 7(Q) curve, demonstrating the existence of a generalised argument Q, able to account for the backscattering properties of any substrate. This represents a remarkable result. The same value of the argument Q, in fact, finally producing a desired BS coefficient 7, can be obtained by starting from a lot of variable configurations (the allowed ones being chosen from among the infinite possibilities by imposing the elasticity requirement, mentioned in Section 3, which has to be satisfied). Eq. (8), which indicates how the process variables can be modified in order to give a constant Q via a mutual compensation, represents a scaling law for the backscattering process. Furthermore, because of the need to be able to predict the effect produced by any variable configuration change on the backscattering, knowledge of the explicit dependence of 7 on Q can be useful. By interpolating numerical results, a practical approximant is obtained, representing in a simple way the physical law governing the backscattering process. The Buckingham approximant to the given q-data is
T(Q) = 83 Qy
(10)
where 77 is measured in percent. Eq. (10) overestimates v at small and high Q-values, while underestimating it at intermediate ones. Nevertheless, it is fully satisfactory for all practical purposes. However, it is worthwhile noticing that
o tantalum (5-150 keV) a other material (30 keV) o I
1
I
0.0
0.2
0.4
0.6
(1 (adim) Fig. 2. The universal dependence of the backscattering coefficient on the single argument Q. V-Data relative to composite materials from PMMA to PbTe are shown together with the same q-data as in Fig. 1. D stands for diamond phase of carbon. The straight line is the simple Buckingham approximant, q=83 Q.
152
G. Messina et al. I Microelectronic
a more accurate Buckingham elemental materials [2]:
approximant
Engineering
34 (1997) 147-154
has been deduced by a model derivation
q(Q) = 100( 1 - exp(- 1.3Q”“)),
(11)
v being measured in the same units as in Eq. (10). Such an expression, giving generalised BS argument, can also be utilised in case of composite substrates, (lo), whenever it is necessary to account for details of the variation of q with curvature near the origin and the successive negative curvature and saturating
5. Physical
approximant
of Q in the case of
to the forward
scattering
7 as a function of the as an alternative to Eq. Q, such as the positive trend for increasing Q.
width
According to Buckingham’s theorem, w can only enter a physical law if it is normalised to a variable having the dimensions of length. In principle, this variable may be a combination of the process variables. However, the simplest possibility, which also has an obvious physical basis, is that FS width w =: w/t,. t, alone provides a proper normalisation. Thus, we define the “normalised” In order to test the w(Q) law that was assumed in Section 3, we plot, as a function of the generalised argument Q, all the normalised w-data corresponding to the same variable configurations as in Fig. 2 (Fig. 3). As in the previous case, in spite of the independent variation of a large number of variables, w-points line up one-dimensionally along a single w(Q) curve, demonstrating the capability of the generalised BS argument of also describing the forward scattering properties of substrates. This again constitutes a quite remarkable result. The many process-variables ultimately intervene in determining the system behaviour in terms of both forward and backward scattering, through a single Buckingham argument, which thus suffices to describe the process. Eq. (8) can be consequently regarded as a scaling law for electron scattering in solids. As for the explicit dependence of w on Q, by interpolating the given data, a simple Buckingham approximant is derived, namely
100 1 DPbTe
Bi
3 o tantalum (5-150 keV) w other material (30 keV) 0.0
0.4
0.2
016
Q (adim) Fig. 3. The universal dependence of the normalised forward scattering width on the single argument Q. The w--data shown refer to the same cases as in Fig. 2. The solid line represents the simple Buckingham approximant, w= 85 (I -exp(-7.5Q)).
G. Messina et al. I Microelectronic
w = 85 (1 - exp(-7SQ))
Engineering
34 (1997) 147-154
1.53
(12)
where w is measured in percent. It is to be noticed that the smooth variation of w reflects the fact that the dimensions of the scattering roughly scale with the. geometry of the system. In particular, as demonstrated by the envelope of the electron trajectories, the width of the FS increases with the depth into the substrate according to an almost quadratic law, in the case of a weak-scattering regime (corresponding to small Q,-values). In the presence of intense scattering, a linear increase of w with depth is, in principle, expected, because of the asymptotic behaviour of o. However, it should be noted that such a situation is rarely achieved, the elastic limit being generally reached earlier than the w-asymptote.
6. Additional
remarks
Eq. (12) and Eq. (10) respectively represent very simple approximants to the physical laws governing forward and backward scattering: they allow the quantitative effect of any change of the process variables on the system behaviour to be readily evaluated, as a function of the single argument Q. In this framework, the generalised argument Q, or, better, its reciprocal Q*, can be regarded as a universal quality factor for the electron scattering process in solids. In view of lithographic application, it may be interesting to compare results relative to some typical substrate materials. At a given e-beam voltage, the value of Q, obtained for a to-thick diamondsubstrate is lower than for several other substrates having the same thickness. This means that a higher quality factor Q* pertains to diamond. The more convenient lithographic performance of diamond with respect to these substrates (e.g., Si, MO, Ta etc.) is actually demonstrated by Eqs. (10,12), which correspondingly give a lower 77 and a smaller w. On the other hand, it is to be noted that the same value of Q may correspond to quite different experimental conditions depending on the substrate material utilised. In particular, by using substrates of fixed thickness t,, different e-beam voltages will be necessary in order to have the same v. For example, BS coefficients obtained by means of diamond at 30 kV, which is a typical operating voltage of standard lithographic systems, will require the use of -40, 120 and 210 kV in the case o-f Si, MO and Ta substrates, respectively. On the contrary, by operating at a given e-beam voltage, different substrate thicknesses will be required in order to produce the same result. For example, the 7 obtained by using a 1 pm-thick diamond-substrate will be matched by a 1Spm-BN or 4pm-PMMA substrates, but with the straightforward consequence of forward scattered electrons distributed in BN and1 PMMA over an average distance from the incidence direction that is 1.5 or 4 times larger than the FS width typical of diamond, respectively.
7. Conclusion Buckingham’s theorem of Dimensional Analysis, modified in its application with respect to the standard procedure, has been applied to the evaluation of electron scattering in a general composite substrate, as calculated by means of Monte Carlo simulation. We have considered the dependence of the forward scattering width w and backscattering
154
G. Messinn et al. / Microelectronic
Engineering
34 (1997)
147-154
coefficient 7 on the physical constants of the material, e-beam voltage and substrate thickness, for all elements of the periodic table and several typical compounds, for all variations of e-beam voltage and substrate thickness in the 5 to 150 kV and 1 to 5000 nm ranges, respectively. We have demonstrated that a single generalised Buckingham argument, dimensionless combination of the process variables, effectively determined both forward and backward scattering features. FinalIy, in the range of variables considered, simple Buckingham approximants have been obtained for both backscattering coefficient 77and normalised forward scattering width w, namely, 7 = 83 Q and w = 85 (1 - exp(-7.5Q)), respectively. The generalised Buckingham argument Q provides the rules according to which the variable configuration can be changed without affecting the resulting forward scattering and backscattering properties. Its reciprocal Q* can be regarded as a quality factor for electron scattering. Finally, from a comparative discussion, diamond, which exhibits a very high Q*, is demonstrated to play a leading role among substrate materials.
References [I] G. Messina, A. Paoletti, S. Santangelo and A.. Tucciarone, A single quality factor for electron backscattering from thin films, Microelectron. Engrg. 27 (1995) 183-I 86. [2] G. Messina, S. Santangelo, A. Paoletti and A. Tucciarone, Monte Carlo modelling of electron beam Ilthography: a scaling law, Microsys. Technol. 1 (1994) 23-29. [3] For an exhaustive discussion of the H theorem see, for example, P.W. Bridgman, Dimensional Analysis, YaJe University Press New Haven, 193 1. [4] K. Murata, Electron Beam interactions with Solids, SEM, Inc., AMF O’Hare, Chicago, pp. 31 l-329. [5] R.J. Hawryluk, A.M. Hawryluk and H.I. Smith, Energy dissipation in a thin polymer film by electron beam scattering, J. Appl. Phys. 45 (1974) 2551-2566. [6] D.C. Joy, The spatial resolution limit of electron lithography, Microelectron. Engrg. 1 (1983) 103-I 19. [7] G. Messina, A. Paoletti, S. Santangelo and A. Tucciarone, Electron scattering in microstructure processes, La Rivista de1 Naovo Cimento 15 (1992) I-57. [8] H. Niedrig, Electron backscattering from thin films, J. Appl. Phys. 53 (1982) R15-R49. [9] S. Leisegang, Zur Mehrfachstreuung von Elektronen in duennen Schichten, Z. Phys. 132 (1952) 183-198.