The resolution of electron beam lithography

MicroelectronicEngineering 17 (1992) 17-20 Elsevier

17

THE RESOLUTION OF ELECTRON BEAM LITHOGRAPHY. M. I. Lutwyche. Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, United Kingdom. Abstract We present a model for the primary interaction of the electron beam with bound electrons in the resist. Time-dependent perturbation theory is used to calculate the probability that sites some distance from the electron beam will be exposed by the electromagnetic field around it. We show that, assuming our approximations to be correct, the range of this interaction is adequate to explain the resolution of P.M.M.A.

1. I N T R O D U C T I O N It is widely known that the resolution of electron beam lithography is determined by the range of the interaction between the electron beam and the resist and not by the resolution of the electron optical system.llll21[3] Complex theories have been used to simulate the paths of electrons and secondary electrons, and their interaction with the resist.141[5l They have not considered in depth the interaction with individual molecules in the resist. Experiments using a S.T.E.M. have also shown that electrons in the beam which pass close to, but do not enter, the surface of a material are capable of interacting with it. Theories have been developed to account for this. [6][7] The model used in this theory calculates the effect of the electromagnetic field generated by a passing electron on a bound electron in the resist. First order time-dependent perturbation theory is used to calculate the probability that this bound electron will be excited from its ground state tp to an excited state tp*, absorbing a quantum of energy. It is assumed that the transition from V to tp* causes exposure of the resist. Two initial approximations are made: that the incident electron is a classical point charge, and that it loses an insignificant amount of energy, thus travelling at constant velocity v as it passes. The closest approach of the electron is r. See figure 1. This model cannot take into account exchange potentials.

Electromagnetic field dp(t) andA(t). ~

tm~ ~ ~P' ~ . ~"~ ,.--.) Cldsest Electron travelling [ anrwoach r at velocity v. ",,,~ , ~ [ " - r ,-- ~-- --, -. V

Site in resist.

Path of electron. j

v

Figure 1. Model for the interaction of the electron and resist. 0167-9317/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved.

18

M.I. Lutwyche / The resolution of e-beam lithography

2. THE THEORY For a single electron charge e and mass m bound in a potential qbothe Hamiltonian is

Ho

112 V2 - e~o

-

{1)

8rr2m --

This Hamiltonian has as two of its eigenstates k0 and tp*. Suppose this site is exposed to a perturbing Hamiltonian due to a weak time-dependent electromagnetic field. The perturbing Hamiltonian can be shown to be ihe A(t).V + e~(t) 2rim-- --

^

H'(t) -

(2)

where h is Plank's constant and A(t) and ~ t ) are the vector and scalar potentials.[ 8] Thus using first order time-dependent perturbation theory, the probability of a transition from ~ to ~* is 2 (3) where AE is the energy difference between the eigenstates tp and tp*. For exposure of the resist by light, ~ t ) is zero since there is no free charge, but A(t) is nonzero. For the field generated by a free charge, ~ t ) is not zero. The potentials around a charge e, including relativistic effects, are:[9] ~(t) -

e 1 4neo q v 2 t 2 + (l_[~2)r 2

A(t)- ~

with 13 - v c

(4)

c~t)

(5)

where r is the closest approach, v the velocity of the charge, c the speed of light and eo the permitfivity of free space. Substituting these into eqn.(2) gives

i~i,(t)_

e2

1

(

ih ~

4neo q v 2 t 2+(1_152)r 2 - 2rim

1

c2 v . V _ + 1

(6)

The dimensions of a site in the resist will be similar to that of an atom, less than 4A. If r is much greater than this, then variations in r across the site will cause only a small variation in ^ H'(t). Thus the terms outside the brackets in eqn.(6) can be removed from the overlap integral. The overlap integral now contains the sum of two terms, one being , which is zero since tp* and tp are orthogonal. The other term is proportional to . For materials

M.I. Lutwyche / The resolution of e-beam lithography

19

exposed by light, such as optical resists, this term is large. One would expect all optical resists to be sensitive electron beam resists. Making the above approximations gives

p o~

e2 e h 471Eo ~]vEtE+(l_~E)r 2 ~ l p *

ih ~ t ' ~ v . v 271"m c 2 - - -

u)>

dt

(7)

We wish to know how P varies with r. The overlap integral is now independent of r and t so

P(r) o~

if::

e

_v2t2+ (

dt

I

(8)

The limits of this integral are :too since we want the probability for the passage of one electron. Unfortunately this integral is problematic so we make a binomial approximation for small t: 1

r

~a

4 v 2 t 2 + (1-132)r 2

(9)

v2t 2 ÷ 2(1-132)r 2

This approximation is best when t is small and r large, but we already require that r is greater than the size of the site. Figure 2 shows the approximate function. The important similarity is the half width of the function since eqn.(8) is a Fourier transform. Using complex contour integration, the result is IS~] p oc e x p

v h where ~ - ~-~1_132 ) xAE

-

cX/T2+ 2mc2T h 4%/'~mc2 xAE

(10)

T is the kinetic energy of the electron and m its mass. The probability of exposure falls to 0.37 of its value at the origin at a distance ~,. Thus ~, gives the 'resolution' of the process.

I

I

I

@

>

!

-2

i

!

0 Time (s)

2

xlO -15

Figure 2. Comparison of the original function (solid line) with the approximation (dashed line).

20

M.I. Lutwyche I The resolution of e-beam lithography

3. COMPARISON WITH E X P E R I M E N T A L DATA P.M.M.A. is exposed by light with a wavelength of 250 nm or less, so the energy required to expose a site in the resist, AE, is at least 4.9 eV. For incident electrons of 40 kV, Z is about 5.7 nm. A circle of diameter 11.5 nm is thus exposed by an electron beam of negligible diameter. The function obtained in eqn.(10) was used to predict the fractional exposure of P.M.M.A. at 50 kV and compared with data taken from reference [1]. See figure 3.

1.0

. . . .

I

. . . .

1(~

I

'0"

I

. . . .

l

re)

O

0.5 sults

I . . . . . . . . .

0

I

1

The°ry

. . . . . . . . .

I . . . . . . . . .

2

I

3

. . . . . . . . .

I1 /

xlO -8

Line width (m)

Figure 3. Comparison of fractional exposure values against line width for P.M.M.A.

4. C O N C L U S I O N S Molecules in the resist may be excited from tp to tp* by the electromagnetic field created when an electron passes within a range ?~. This range is consistent with the resolution of P.M.M.A. when other effects, such as electron scattering, are known to be negligible.

5. REFERENCES [1]. A.N. Broers. J. Electrochem. Soc. Volume 138, number 1, January 1981. [2]. M. Isaacson and A. Murray. J. Vac. Sci. Technol. 19(4), Nov/Dec 1981. Page 1117. [3]. I.G. Salisbury, R.S. Timsit, S.D. Berger, and C.J. Humphreys. Appl. Phys. Letters 45(12), 15 Dec 1984. Page 1289. [4]. D.F. Kyser J. Vac. Sci. Technol. B1(4), October to December 1983, page 1391. [5]. D.C. Joy. Journal of Microscopy, 136 (1984) page 241. [6]. J.M. Cowley. Surface Science, 114 (1982), pages 587 to 606. [7]. A. Howie and R.H. Milne, Journal of Microscopy, 136, page 279. [8]. B.H. Bransden and C.J. Joachain, Physics of Atoms and Molecules. (Wiley, New York, USA or Longman, UK, 1983) [9]. H. Arzeli~s, Relativistic Point Dynamics. (Pergamon Press, 1972).

Acknowledgement The author thanks A.N.Broers for discussions which motivated work on this theory.