ANALYSIS OF ENERGY BROADENING IN CHARGED PARTICLE BEAMS

HAL Id: jpa-00224408

https://hal.archives-ouvertes.fr/jpa-00224408

Submitted on 1 Jan 1984

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ANALYSIS OF ENERGY BROADENING IN CHARGED PARTICLE BEAMS

M. Gesley, L. Swanson

To cite this version:

M. Gesley, L. Swanson. ANALYSIS OF ENERGY BROADENING IN CHARGED PARTICLE BEAMS. Journal de Physique Colloques, 1984, 45 (C9), pp.C9-167-C9-172.

�10.1051/jphyscol:1984928�. �jpa-00224408�

ANALYSIS OF ENERGY BROADENING IN CHARGED PARTICLE BEAMS

M.A. G e s l e y and L.W. Swanson

Department of Applied Physics and Electrical Engineering, Oregon Graduate Center, 19600 N.W. Walker Road, Beaverton, OR 97006, U.S.A.

Résumé

L'anomalie de distribution énergétique d'un faisceau d'ion monomères ou d'électrons est décrite par un modèle de file indienne à un paramètre, régi par l'émission de bruit de grenaille et par les interactions coulombiennes de paires. La dépendance de la largeur <àE> de la distribution et du décalage

<<)>> de l'énergie moyenne en fonction du courant, de la masse et de la tension est calculée* Une unification de plusieurs autres théories d'élargissement énergétique est faite à partir de la variation d'un paramètre libre qui repré- sente des effets de champ liés à la géométrie. Une caractéristique unique du modèle présent est le fait que <Afi> est inversement proportionnel à la charge.

Abstract

The anomalous energy distribution of a monomer ion or electron b e a m is described by a one p a r a m e t e r , single file model governed by shot noise emission and pairwise Coulomb interactions. The dependence of t h e width of t h e distribution <Af?>

and t h e shift in t h e m e a n energy <$> on current, mass, and voltage is calculated. A unification of a n u m b e r of other theories of energy broadening is made by variation of a free p a r a m e t e r t h a t r e p r e s e n t s field-geometrical effects. A feature unique to t h e p r e s e n t model is t h a t <hE> can be inversely related to charge.

1. Introduction

Since t h e work of Boersch [1] t h e anomalous energy distribution in charged parti- cle b e a m s h a s a t t r a c t e d t h e attention of a number of investigators. The distribution provides an insight to t h e c h a r a c t e r i s t i c s of t h e emission process t h a t govern its behavior as a function of c u r r e n t (1), mass (m), and charge (q=ne). Additional motiva- tion for studying this p r o p e r t y arises from the variance of t h e distribution being directly proportional to t h e chromatic image blur diameter [2]:

where Q is t h e chromatic aberration coefficient of t h e lens, o is t h e a p e r t u r e half angle s u b t e n d e d at t h e image, V0 t h e b e a m potential, and <hE> is t a k e n as t h e full width half maximum of t h e total energy distribution regardless of its n a t u r e . There- fore understanding t h e mechanisms t h a t effect t h e distribution m a y allow one to minimize t h e chromatic aberration limit on t h e b e a m size in microprobe applications.

This limit is particularly important in liquid metal ion source (LMIS) applications.

Here we consider an analysis of t h e energy distribution in charged particle beams with t h e hope of elucidating t h e fundamental physical mechanisms t h a t effect its behavior. A classical statistical model of charged particle emission is p r e s e n t e d in which t h e process is governed by shot noise. This is consistent with t h e m e a s u r e m e n t of the power s p e c t r a of Ga and Bi LMIS t h a t were found to be shot noise limited [3]. By further assuming t h a t pairwise Coulomb interactions dominate, expressions for <LE>

and t h e shift in t h e m e a n energy <$> a r e calculated. Results a r e compared t o LMIS m e a s u r e m e n t s and a n u m b e r of o t h e r theories concerned with t h e behavior of t h e dis- tribution.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984928

C9-168 JOURNAL

DE

PHYSIQUE

2. Model

Our analysis assumes a one dimensional (single file) current regime of charged particle emission. This regime can be defined a s those currents whose average inter- particle separation <6> is greater than the minimum beam radius A [4].

1. Diagram depicts the sphere emitter model with single file emission of parti- cles 1 and 2. The emission half angle Q determines the beam width A near the emitting surface.

According to Figure ( 1 ) this implies that a t the emitter surface:

where d o and r0 a r e t h e emission half angle and emitter radius respectively and 60 is the initial position of particle 1 when particle 2 is just emitted. This qualitatively describes the boundary between pairwise and multiparticle, multidimensional interac- tions. Assuming for the LMIS that T O = 3 n m and Q

=

15' then <60>

>

0 . 8 n m delineates the condition for single file behavior. The values for T O and Q. are believed to be correct within a factor of two 15.61.

Assuming shot noise appropriately describes the emission process the charged particles can be considered as being emitted randomly with the number per unit time following a Poisson distribution.

0 ;k >+l

, T ,

I I S

L

I I + EMISSION TIME

n CO

Figure 2. Diagram representing Poisson distributed emission times. The random vari- able TO is exponentially distributed and corresponds t o the interevent time of emission.

The probability density function (PDF) for the interevent time ^{T O} is then given by [7]:

f

(701

=

A exp (-A701

.

⁽²⁾

where A = I/ q is the average rate of emission.

The electric field outside a sphere is :

F(60)

=

V 0 ~ & ' ( 1

+

6 0 / ~ 0 ) - ' When 60

<<

T O this reduces t o a uniform field :

F.

=

V o / ~ o

.

(4)

It is shown below t h a t during the interval TO, the emitted particle interacts, on the average, with a nearly uniform field for the range of currents and masses used in the analysis of

LMIS

data. Therefore the following relation holds:

60 = ( ~ F O TO "/ 2m

.

⁽⁵⁾

Using Eqs.(Z) and (5) the PDF for do is given by:

f (60) = ( k / 2 ) 6 ; " 2 e x p ( - k 6,

.

⁽⁶⁾

where k = I ( 2 m / q s F ~ ) " ~ . The mean initial spacing is:

<So>

=

q S ~ 0 / mlZ

.

( 7 ) For example, if one assumes the evaporation field for A l f is Fo= 20 V / n m , then a

finds that for typical ranges of current the uniform field approximation is reasonable when considering field electron emission.

The current, mass, and charge dependence of the shift in the mean energy

<@>

can be obtained as follows. The shift in the average energy per particle due to the pair- wise Coulomb interaction is:

i.e., only half the initial potential energy can be converted to kinetic energy a t the anode. Equations (6) and (8) lead t o the PDF for @:

The mean value of the shift is delined as:

*c

< a > = J a f ( m ) d m ,

0

where @,

=

q 2 / 6, and 8, is taken as the minimum possible value of the initial separa- tion 60. Without such an upper bound on the integration Eq.(lO) would diverge.

Evaluated in the low current limit this integral leads to:

<iP>

=

I d m q / FOGc (11)

l ~ ' " ' ' l l ' ' '

i l l p \

X I = 3 p A

a W 0

0 W ' - 0 1

10 50 100 500 1000

ATOMIC W T

1 1 , 1 1 1 I l l l l l U

5 10 50 100

I lp A )

Figure 3. Plots of experimental shifts in the TED peak position as a function of current (left) and mass (right) for the indicated

LMIS

species.

According t o Eq.(ll) a shift to larger kinetic energy with increasing current is expected. This prediction is borne out experimentally for AIC, GaC, InC,andBi* as shown in Fig. (3). For a given current t h e predicted relation < @ > = m ' / 2 agrees with the data for low mass ions ( I m(Ga)). For high mass ions only a qualitative agreement is obtained, possibly due to an implicit mass dependence of 6, in Eq.(ll).

Exact evaluation of < m > requires detailed knowledge of the individual trajec- tories of the particles. This appears t o be possible only by numerical integration of the equations of motion. In order t o retain a closed expression for <AE> with the hope of understanding how the observable quantities I, m, q, and V. effect this variable we note that AE is a monotonic decreasing function of 60 [E]. We approximate this relationship by:

where a is a real positive parameter. Dimensional analysis indicates that, b ,

=

q V o ~ t . where t h e tip radius has been chosen as the length measure. A choice of any other length would only introduce a numerical constant that will not effect the functional dependence of the distribution. The PDF of <AE> is then calculated using Eqs.(9) and (12). The result is:

C9-170 JOURNAL

DE

PHYSIQUE

and d a = k b:lza. The expected value of <BE> is then found t o be:

<AE>= da2a{s-2ae-s&

.

(14)

For t h e case

K >

a > O , Eq.(14) becomes:

<AE>

= d,2a r(1-2a) ,

where r(1-2a) is the gamma function

191.

We may allow

a25

if a limit is imposed on the maximum possible value of t h e energy spread, <AE>-- E,. Again this is equivalent t o imposing a limit 6, on the initial separation. This type of approximation is typical of all classical theories dealing with charged particle interactions [4.8.10-161.

Assuming )$<a then <AE> is expressed as:

where r(l-Za,z) is an incomplete gamma function and z = d a ~ c V Z a . In the limit of low currents Eq.(16) is given by:

<AE> = 1 ~ 1 / 2 ( ~ ~ / q)(l-a)/2ar0 6$1-2a)/Za

.

(17) Variation of the parameter a results in a rather interesting unification of a number of energy broadening theories. To see this we first defme a rescaled length, lp = g / Vo, which will not effect t h e current or mass dependence of <AE>. One then finds that t h e theoretical expressions for the energy broadening, Eqs.(l5) and (17), have the general form:

<AE>

=

( p m ) a V O ~ ( ~ ) G ( U ) , (18) where G(a) is a product of the geometrical factors only and h ( a ) is some function of a .

Table I lists seven theories that predict an energy broadening of the form:

<AE>

=

I l / Z m 1 / 4 S/4 V1/4

P o G . (19)

Here the parameter is a = 1 / 4 and Gi (i=a,b,c,d) indicates the various forms the geometrical factor assumes in the different studies.

Table I. List of various theories t h a t predict an energy broadening of t h e form Eq.(19).

Various lengths are: do=truncated emitter diameter, ro=emitter radius. D=beam length, Ro=beam radius. 1, = q / Vo.

REFERENCE PRESENT STUDY (a=114) ROSE and SPEHR (1980) VAN LEEVWEN andJANSEN(lQ83) GESLEY,LARSON.SWANSON and HlNRlCHS ( l Q&) MASSEY.JONES,PLUMMER (1081)

KNAUER (1079)

LOEFFLER (lQS9)Od

1-0

Setting a = 3/ l 6 leads to:

<AE> = 131 8 m3/ 16 V. 5/ 4

G

GEOMETRICAL FACTOR G,=(r./lq)'12

%=

G,=(~'/RO')va

=( &do/Rb)'fl

DE CHAMBOST-HENNION (1079

MODEL SINGLE FILE CLOSED FORM

CROSSOVER CROSSOVER SINGLE FILE W1

NUMER. CALC.

PARALLEL BEAM CROSSOMR CROSSOVER

Table I1 compares t h e results of t h e present study with the two cases considered by de Chambost and Hennion [12].

Table 11. List of theories corresponding to Eq.(20).

MODEL SINGLE FlLE REFERENCE

PRESENT STUDY (a=3116)

EOMETRICAL FACTOR

)*l*e

This reproduces Knauer's result concerning a diverging point source [4], Table 111.

Table 111. Theories corresponding t o Eq.(21).

PRESENT S ~ Y 1 3

KNAUER (1081)

When a

= g

the energy broadening is given by:

<AE> = 1n2"~

G

(22)

This is also the result obtained in the low current limit of a circular crossover [l41 and in a model of single Coulomb deflections 1171, Table IV.

CREWE SINGLE

= r Gb=(b/rOlh)

Table IV. Theories corresponding t o Eq.(22).

SINGLE FILE DIVERGING POINT

SOURCE

Figure (4) plots LMIS data of <AE> as a function of (Im'") for singly and doubly charged monomer ions using data from [17-191. This corresponds t o the behavior predicted by Eq.(22), a

=B.

The present work diverges with the other theories when the charge dependence of the energy broadening is considered. Figure (4) and [l81 indicate that monomer ions in the low current regime exhibit the following behavior:

<AE>,,=I-

.

(23)

By setting a = % we find Eqs.(l7) and (22) yield precisely this relationship. Interest- ingly, no other theory (including t h e others listed in Table

IV)

predict a n inverse rela- tionship between <AE> and q.

.

^{Go' 3 0 0 X}

,

^{I"' 3 7 5 K}

1"' 9331 r Bat 46311

Figure 4. Plots of experimental values of <AE> according to the predictions of Eq.(22) .for singly (left) and doubly (right) charged monomer ions.

3. Conclusion

Analysis of a model of one dimensional pairwise Coulomb interactions in a charged particle beam governed by shot noise has led to closed expressions for the anomalous energy broadening <BE> and shift in mean energy

<$>.

The theoretical current

C9-172 JOURNAL DE PHYSIQUE

dependence of

<a>

is in agreement with LMIS measurements of Al. Ga. In, and Bi mono- mer ions. For low masses the predicted mass dependence of

<@>

also agrees with the LMIS data. The use of a free parameter in the expression for <U> has unified a number of energy broadening theories. The unique feature of the present study is its ability to explain t h a t <BE> has an inverse relationship with charge a s found in mono- m e r ion emission from LMI sources.

We a r e greatful t o Dr. A.E. Bell for useful discussions.

4. References

[l] H. Boersch. 2. Phys.

139

(1954) 115.

[2] P. Grivet, Electron Gptics, part 1 chap.? (2nd English ed. Pergamon Press, 1972).

[3] L.W. Swanson, G.A. Schwind, A.E. Bell, and J.E. Brady, J. Vac. Sci. Technol.

16

(1979) 1864.

[4] W. Knauer, Optik 59 (1981) 335.

[5] R. Gomer, J. Appl. Phys.

B

(1979) 365.

[6] P. Sudraud, 30th Int. Field Emission Symp., Philadelphia

[7] W. Feller. A n Introduction t o Probability Theory a n d I t s Applications. Vol.11 (Wiley and Sons. 1966).

[E] M.A. Gesley, D.L. Larson, L.W. Swanson. and C.H. Hinrichs, SPIE Microlithography Conferences. 11-16 Mar 1984. Santa Clara. Calif.

[g] M. Abramowitz and 1.k Stegun. Handbook of Mathematical r n n c t w n s AMS-55 (National Bureau of Standards 1972).

[l01 W. Knauer, Optik

2

(1971) 211.

[l11 G.A. Massey, M.D. Jones, and B.P. Plummer, J. Appl. Phys.

2

(1981) 3780.

1121 E. de Chambost and C. Hennion, Optik

B

(1980) 357.

[l31 J.M.J. van Leeuwen and G.H. Jansen, O p t i k a (1983) 179.

[l41 K.H. Loeffler, 2. angew. Phys.

27

(1969) 145.

[l51 H. Rose and R. Spehr, Optik (1980) 339.

[l61 A.V. Crewe. Optik

M

(1978) 205.

[l71 L.W. Swanson, G.A. Schwind, andA.E. Bell, J. Appl. Phys. 51 (1980) 3453.

[l81 L.W. Swanson, Nucl. Inst. and Methods,

m

(1983) 347.

[l91 M. Komuro, Proc. Int. Ion Eng. Conf., Kyoto (1983)337.