A single quality factor for electron backscattering from thin films

MICROEV-r.~ONIC ENGINEERING

ELSEVIER

Microelectronic Engineering 27 (1995) 183-186

A SINGLE QUALITY FROM THIN FILMS G.Messina*, A.Paolettl

FACTOR

FOR ELECTRON

BACKSCATTERING

, S.Santangelo* and A.Tucciarone**

* Fac. Ingegneria dell'Universit/i, v.E.Cuzzocrea 48, 89100 Reggio Calabria, Italy ** Dpt.Ingegneria Meccanica, Universit~i Roma Tor Vergata, v.O.Raimondo, 00173 Roma, Italy

Tel: +39-6-2025535

Fax: +39-6-2025538

In this paper, Buckingham's theorem on physically similar systems is applied for the first time to the derivation of interpolation curves of numerical data. A simplified dependence of the curves on a limited number of effective dimensionless parameters is found by a novel approach. In particular, the method is applied to Monte Carlo modelling and the calculation is considered of the backscattering coefficient 11 from a general substrate in the elastic regime. A single dimensionless backscattering parameter is introduced and a simple scaling law is determined, indicating how the configuration of the many variables involved can eventually change without affecting the result. The validity of the law is demonstrated in the 5 to 100 keV energy range, with substrate thicknesses ranging from 10 to 21000 A and for all the substrates of the periodic table.

1. I N T R O D U C T I O N Due to the complexity of the electron scattering process in solids, quantitative information about specific experimental conditions is often obtained, by means of Monte Carlo simulation methods, by performing numerical experiments. The great number of variables usually involved in the calculation, such as material physical constants (atomic number, atomic weight, density), electron energy, substrate thickness, makes it generally difficult to have a direct feeling about possible trends in case of a different configuration of the variables. This constitutes a hindering feature, particularly when a problem is considered where part of a configuration has to be changed, and it is necessary to readily understand how the other variables can eventually compensate for the required variation, without affecting the final result. This is often the case in

microsystem designing. A significant aid to the solution of this problem may come from Buckingham's theorem on physically similar systems [ 1], stating that dimensionless products of the variables eventually enter the formulation of a physical law. In this paper, a novel application of Buckingham's theorem is presented. The application of this theorem, usually limited to the inference of physical laws on a purely dimensional basis, is extended, for the first time, to the derivation of interpolation curves of numerical data. Furthermore, a simplified dependence of the curves on a limited number of effective dimensionless parameters is found by arbitrarily assuming parameters, on a physicalbesides a dimensional- basis, and by refining the resulting laws by a fitting procedure with the given data. In particular, the variables entering the calculation and some of their important

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G. Messina et al. /Microeleetronie Engineering 27 (1995) 183-186

combinations are briefly reviewed in Sect.2. In Sect.3, a single dimensionless backscattering parameter is introduced constituting a good scaling law for electron backscattering. In Sect.s 4 and 5, a finer formulation is presented and the range of validity of the scaling law is extended. 2. E L E C T R O N B A C K S C A T T E R I N G The backscattering (BS) coefficient 11 from a given substrate may be calculated by means of a Monte Carlo (MC) simulation of the electron trajectories. The penetration of an electron into a solid target is, as usual, simulated by assuming a single scattering model and utilising the continuous slowing-down approximation for evaluating the energy loss caused by the inelastic collisions [2-4]. Being interested in electron backscattering, in order to get a qualitative knowledge of the process, we focus our attention, among all scattering events, on those occurring at large angles. Consequently, we introduce a representative parameter, namely, we define the so-called large-angle crosssection

ffLA ~ L A = x Z 2 e4 /

4E2-

This can be derived from the Rutheford differential elastic cross-section and can be taken as independent on the screening parameter I ~ < < 1 , which is actually important only for small-angle scattering events. Since a great number of variables, such as the material physical constants (atomic number Z, atomic weight A, density p), the electron energy E, the substrate thickness z o, are involved in the BS calculation, the representation of the numerical results should, in principle, take place in a multi-dimensional space. By spanning the configurational space of the variables, Tl-points correspondingly cover a hypersurface. Unfortunately, the projection of this hypersurface on planes,

showing the dependence of 11 on each of the variables involved, often results in a confused spreading out of the projected points over an area, from which no direct feeling of possible trends can be acquired. The assumption of a separate T1dependence on each process-variable does not allow to get a ready understanding of the effect of a variable-configuration change. A simpler picture may derive from eventually finding a dependence of 11 on c o m b i n a t i o n s of the variables, such as the dimensionless parameters introduced by Buckingham in the theorem on physically similar systems. 3. THE P H Y S I C A L LAW We introduce a dimensionless combination of dimensional variables, that can be reasonably assumed, on a physical basis, as important in determining r I. In particular, we define a parameter PBS, giving the average number of BS events per incident electron in a z othick substrate, namely, PBS = CLA n z o

where n = p N A/A denotes the scatteringcentre density of the given material, N A being Avogadro's number. We tentatively assume a ~-dependence on PBS, tO be verified by comparison with the numerical data. In particular, MCresults for tungsten are utilised. In Fig. l, the existence of a simple T1 ( P B s ) law is demonstrated, in case of 20 and 40 keV electron energies. The Ti-points line-up one-dimensionally describing, in a g o o d approximation and within the l i m i t e d v a r i a b l e range c o n s i d e r e d , a common ( P B s ) curve. Thus, PBS represents a fairly good quality factor for backscattering. PBS, which can be explicitly expressed as PBS = x Z 2 e 4 p N A z o / 4 A E 2,

actually constitutes a scaling law for electron BS: for example, it indicates, if the electron energy is varied, how the

G. Messina et al. /Microelectronic Engineering 27 (1995) 183-186 50, o

o x o e

~

x

x

40 keV • x • " 20keV

25.

x

v

0'

o.oo

o.g

PBs (adim) Fig.l -Plots of 11 versus PBS for tungsten data at 20 and 40 kcV. thickness of a given substratc should correspondingly scale in order to compensate for the desired variation with no or very small changes in the resulting BS coefficient. Hence, PBS contains most of the dependence of vl on the process variables, so that it can bc sufficient for many applications. However, as one can see, there is a slight variation of slope in the ~ curves for different electron energics, suggesting that, for a finer formulation of the problem, the definition of a second dimensionless combination of dimensional variables may bc helpful. 4. F U R T H E R LAW R E F I N E M E N T It is natural trying including the screening parameter ix, whose variations we have previously neglected. The slope of the ~ ( P B s ) law smoothly varies with ~, thus, we assume that T1 is proportional to the PBS parameter, slightly "corrected" by it, namely, oc PBS / ~ , where y<
185

that ~ should not be influenced by simultaneous variations of PBS and p.T giving the same PBS / ttT product. In particular, we firstly make a 11 ~tv vs PBS plot relative to the 20 and 40 keV data and find y = 0.2; then, in order to demonstrate the wider applicability of the assumed law, we extend the plot to a much wider energy range than that utilised in the law derivation, and to different substrate materials. In Fig.2, the pv data as a function of the PBS parameter are shown for tungsten substrate at energies varying from 5 to 100 kcV and for substratc thicknesses ranging from 10 to 21000 A and, at 20 kcV, for Si, Ti, Zn, Mo, Tc and Bi substratcs. The TI try points line-up along a single straight line, thus effectively demonstrating the assumed law. 5. C O N C L U S I O N Thus, within a very wide range of validity, we have demonstrated that T1 depends on a single quality factor QBS defined as QBS = PBS / I~0'2, or explicitly, QBS = x Z 2e 4 p N A z o / 4 A E 21x 0.2, which is the only variable combination that ultimatcly counts in dctermining the final electron B S effect. In addition, all the variation of 11, in the Q B S range considered, is well accounted for by the simple empirical law vI = 83 QBS, corresponding to the straight line of Fig.2. In conclusion, by a novel application of the theorem on physically similar systems, based on a two-way and trial-and-error, rather than the conventional one-way and necessary, approach, we have discussed MC modelling of electron beam lithography, with regard to backscattering from a general substrate in the elastic regime and in the energy range 5 to 100 keV. We have considered the dependence of the BS coefficient T1 on the

186

G. Messina et al. /Microelectronic Engineering 27 (1995) 183-186

15 Bi 0

10

Zn

5 Ti

20 keV (D) : Si, Ti, Zn, Mo, Te, Bi

w: 5(+) 10(o)20(x) 40(0) 50(*) loo(-) keY O l~ 0.00

,

I

i

I

0.06

0.12

I

I 0.18

PBs(odim) Fig. 2- The TI !x't data plotted as a function of a single parameter PBS, dimensionless combination of variables. In particular: W substrate, for energies (in keV) 5 (+), l0 (4~), 20 (x), 40 (o), 50 (,) and 100 (I); furthermore, for E = 20 keV, the following different substrates (El), in increasing order: Si, Ti, Zn, Mo, Te and Bi. (1931), Dimensional Analysis. Haven, Yale University Press.

New

material constants, electron energy and substrate thickness, finally demonstrating that TI is determined by a single quality factor QBs" A practical equation is found representing the results in the range of variables considered. QBs effectively constitutes a scaling law.

2. K.Murata, T.Matsukawa, R.Shimizu, Jpn. J. Appl. Phys. 1__O,0678 (1971).

6. R E F E R E N C E S

4. G.Messina, A.Paoletti, S.Santangelo, A.Tucciarone, La Rivista del Nuovo Cimento ~ 1 (1992).

1. See,

for

example,

P.W.Bridgman

3. R.J.Hawryluk, A.M.Hawryluk, H.I. Smith, J.Appl. Phys. ~ 2551 (1974).