Transition from statistical Coulomb interactions to averaged space–charge effect

Optik 122 (2011) 1146–1151

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Transition from statistical Coulomb interactions to averaged space–charge effect Yoshio Takahashi ∗ Research Center for Ultra-High Voltage Electron Microscopy, Osaka University, Mihogaoka 7-1, Ibaraki, Osaka 567-0047, Japan

a r t i c l e

i n f o

Article history: Received 9 March 2010 Accepted 21 July 2010

Keywords: Electron beam Coulomb interaction Trajectory displacement Space charge Monte Carlo method

a b s t r a c t The statistical Coulomb interaction effect and the averaged space–charge effect, which broaden electronbeam size, have been studied to clarify their applicable regions and their transition region as a function of beam current. The beam size of a single beam segment can be analytically estimated by two theories based on different approaches: the expression of the trajectory displacement effect based on stochastic electron–electron interactions, and the beam radius equation based on the Coulomb repulsion force by averaged space charge. Both theories were compared with Monte Carlo simulation results for a beamcurrent ranging from 0.1 nA to 10 mA. It was confirmed that the beam size at the end of the focusing segment could be predicted by the expression of the trajectory displacement effect in a low beam current and by the beam radius equation in a high beam current. In addition, the beam sizes predicted by these theories were found to result in almost the same for a medium beam current. The theories transited as the beam-current increased. © 2010 Elsevier GmbH. All rights reserved.

1. Introduction Electron-beam apparatuses used for wafer inspection, device testing, and materials analysis require very fine electron-beam probes in order to achieve high spatial resolution. A probe with a large beam current is also required for inspection of high throughput and analysis of high signal-to-noise ratios. However, it is difficult to maintain a fine probe size in a high beam-current region because the Coulomb repulsion force becomes dominant with increasing beam current. Therefore, the Coulomb interaction should be properly evaluated for design of high-resolution and high-throughput probe-forming apparatuses. The broadening of the probe beam size by the Coulomb repulsion can be analytically predicted by two theories based on different approaches. One theory considers the Coulomb force from an averaged space charge produced by a large number of electrons. A cylindrical electron beam is broadened by the averaged space charge. This effect can be described as a ray equation [1]. In Refs. [2,3], the beam radius equation including not only the Coulomb effect but also the effect of the initial electron velocity spread is shown. This equation including the averaged space–charge effect is assumed not to be valid when there are few electrons around a ray electron. The other theory considers statistical Coulomb interactions among discrete electrons in a beam. A single electron traveling in a

∗ Present address: Central Research Lab., Hitachi, Ltd., Kokubunji, Tokyo 1858601, Japan. Fax: +81 423277677. E-mail address: [email protected]. 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.07.015

beam is affected by all the other electrons randomly distributed in that beam. Thus, the displacement of an electron ray is not deterministic but stochastic. Jansen deduced analytical expressions for the trajectory displacement effect and the Boersch effect under his extended two-particle model [4,5]. In this theory, most Coulomb interactions are required to be weak, and only a single strong collision is covered. Therefore, this theory is valid for the low and medium beam-current regions. There are many other statistical theories with different approaches, which are summarized in Ref. [4]. The theories based on the averaged space–charge effect and the statistical Coulomb interaction effect are expected to have different validity regions regarding the beam parameters. The aim of this paper is to clarify the applicable regions and the transition region of the theories for predicting probe-size broadening. For this purpose, the beam sizes estimated by both the theories are compared with the values calculated by a Monte Carlo simulation, depending especially on the beam current. 2. Review of analytical theories for beam-size broadening Beam-size broadening by the averaged space–charge effect in a field-free drift space is described by a ray equation including an averaged charge-density distribution [1,4]. With assumptions of a laminar flow and uniform distribution of electrons in a beam, the beam radius R satisfies the equation for a cylindrical beam along the z-axis: d2 R 1 = 4ε0 dz 2



m I 1 , 2e V 3/2 R

(1)

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from randomly distributed field electrons is statistically calculated in a field-free segment. Although the average of each displacement of the test electron should be zero due to the axially symmetric distribution of the field electrons, finite deviation of the displacement occurs. ∗ ) at an arbitrary plane The scaled trajectory displacement (FW50 indicated by a location parameter Si is given in Eq. (17) of Ref. [5] as ∗ ¯ v∗ , r ∗ , Sc , Si ) ¯ 2/3 . = 4.2917HCT (, FW50 0 c

(8)

¯ v∗ , r ∗ , Sc , and Si are scaled linear electron density, scaled Here, , 0 c transverse velocity, scaled crossover radius, crossover location parameter, and image plane location parameter, respectively. The scaled parameters are defined as Fig. 1. Focusing segment for beam-size estimation. Beam size produced by unperturbed beam at terminal plane of segment becomes zero. Plane for obtaining minimum beam size of perturbed beam slightly shifts around terminal plane.

where m, e, ε0 , I, and V are the electron mass, electron charge, vacuum permittivity, beam current, and beam voltage, respectively. The relativistic effect is ignored. If the initial velocity spread of a beam is considered, the beam radius equation is generalized by taking the beam emittance into account [2,3]. If an electrostatic field is present and a magnetic field is absent in a beam segment, the beam radius equation is described as d2 R 1 dV dR 1 1 d2 V + R− + 2 2V dz dz 4V dz 2 4ε0 dz −

2 meV

 ε∗ 2 1 

R3



m I 1 2e V 3/2 R

= 0.

(2)

ε* is normalized beam emittance, which may be set to ∗

ε =



2mkT · R0 ,

1 8ε0



(3)

m IL2 2e R0 V 3/2

(4)

under the condition of I V 3/2



 2ε0

2e R02 . m L2

(5)

The minimum beam radius Rm is obtained at a different plane because of space–charge defocusing. Rm is exactly evaluated as



Rm = R0 exp

1 − 2ε0



m R02 2e L2



.

(6)

The beam size of full width median value (FW50 ) becomes FW50 = 0.80795 × R

¯ = 

 2ε 1/3 V 1/3 0

e

L2/3

m1/2

I · L2

27/2 ε0 e1/2

R02 V 3/2

v∗0 = R0 · rc∗ = rc ·

0

L2/3

 2ε 1/3 V 1/3 0

e

(7)

under a condition of uniform electron distribution. Beam-size broadening by statistical Coulomb interactions is given as the expression of the trajectory displacement effect by Jansen [5]. In this theory, stochastic displacement of a test electron running along the center axis of a beam by the Coulomb repulsions

(9)

,

 2ε 1/3 V 1/3 e

,

L2/3

(10)

,

(11)

.

(12)

The function HCT is also defined in Ref. [5].1 The beam size is obtained by adding the unscaled trajectory displacement (FW50 ) to the beam size of an unperturbed trajectory. The beam size at the terminal plane shown in Fig. 1 is directly estimated from Eq. (32) in Ref. [5] as ¯ v∗ , r ∗ , Sc , Si ) FW50 = 2.2135 × 10−20 HCT (, 0 c

where k, T, and R0 are the Boltzmann constant, temperature, and radius of an emission area, respectively. The laminar flow condition is lost. The focusing segment shown in Fig. 1 was considered. An unperturbed beam with a radius R0 is focused by a thin lens at the terminal plane of the segment with a focal length of L. The beam radius at the terminal plane can be obtained by numerically integrating Eq. (2). In cases of no external field (dV/dz = 0) and no initial velocity spread (ε* = 0), the beam radius Rt at the terminal plane is approximated as Rt ∼

∗ = FW50 · FW50

I 2/3 L2 V 4/3 R0 4/3

(13)

using rc < 10−12 , Sc = 0.999, and Si = 1 because the beam size produced by the unperturbed trajectory is zero. A minimum beam size is also obtained at the terminal plane as there is no space–charge defocusing for a test electron. If a uniformly external electrostatic field is present, Eq. (13) should be modified. The expression deduced by a slice method [4] is described in Appendix A. The Boersch effect, which causes the energy broadening produced in a beam segment, can also broaden the beam size through a chromatic aberration of a lens. However, in the single segment shown in Fig. 1, the Boersch effect does not contribute to the beamsize broadening because no lens action occurs during transmitting in the segment. Beam-size broadening by the Boersch effect should be considered in the case of successive segments combined by a lens. 3. Comparison of theories by Monte Carlo method 3.1. Monte Carlo method The beam sizes predicted by the averaged space–charge effect (the beam radius equation) and by the statistical Coulomb interaction effect (the expression for the trajectory displacement effect) were compared with the beam size calculated by a Monte Carlo simulation method in order to obtain the applicable regions of the theories and the transition behavior. The Monte Carlo method is a computer simulation method for tracing rays of randomly distributed electrons with a short time step of t. Since the Coulomb interactions among electrons running in a vacuum are determined

1

The constant C included in Eq. (34) in Ref. [5] should be replaced with 1/C.

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Table 1 Beam parameters used in beam-size calculation of focusing segment shown in Fig. 1. Beam voltage: V0 Segment length: L Beam radius incident to segment: R0 Beam current: I Scaled transverse velocity: v∗0 Scaled crossover radius: rc∗ Crossover location parameter: Sc Image plane location parameter: Si

10 kV 0.1 m 1 mm 0.1 nA to 10 mA 70.3 7.03 × 10−8 0.999 1

with few assumptions, a more realistic beam size can be estimated by this method than by the theories. The Monte Carlo program used was developed by referring to Refs. [4,6]. Displacements of the position (r) and velocity (v) of the ith-electron in a beam during a time period t were respectively calculated as 1 ri = vi t + 2

 vi =



d2 r i +a dt 2

d2 r i +a dt 2





t 2 +

1 d3 r i t 3 , 6 dt 3

1 d3 r i t 2 , 2 dt 3

t +

(14)

(15)

where a is the acceleration by an external electrostatic field. The components of the acceleration are az =

e V0 m L

V

1

V0



−1 ,

ax = ay = 0,

(16)

using a beam voltage V0 at the incident plane of a segment and V1 at the terminal plane. The second and third time-derivatives of ri in Eqs. (14) and (15) are given as Eqs. (14) and (15) in Ref. [6]. A minimum time step t was selected from the following three periods by each calculation step: t1 =

L N

 t3 =



m , 2eV0 63

d3 r i /dt 3 

 t2 =

1/3

22 d2 r i /dt 2 

1/2 ,

,

(17)

where X denotes the average of the absolute value of X. N, 2 , and 3 were preset to 10,000, 0.05 nm, and 0.005 nm, respectively. If a decelerating field (V1 < V0 ) existed, N was set to 100,000 to obtain a more accurate result. The position of the source electrons incident to the segment were assigned to be uniformly and randomly distributed using the pseudo-random number generator MT19937 [7]. The simulation started when the first electron entered the segment and stopped when all the electrons exited the segment. The displacement of an electron was calculated only when it was in the segment. The repulsion forces from all the other electrons prior to passing through the terminal plane were considered. 3.2. Beam-size calculation of focusing segment in drift space The parameters used in the beam-size calculation are listed in Table 1. The numbers of electrons used for the Monte Carlo simulation were 1000 for below 100 nA, 10,000 for 1 ␮A, and 100,000 for 10 ␮A; these were selected because the beam length was almost equal to the segment length. For larger beam currents, the number of electrons was limited to 200,000 because of computational power limitations. In those cases, the beam size was extrapolated to the value when the beam length was almost equal to the segment length. Fig. 2 shows the beam sizes (FW50 ) at the terminal plane of Fig. 1 as a function of the beam current. The solid line, the dashed line, and the circles indicate the sizes calculated by Eq. (13) of the trajectory displacement effect, by the numerical integration of the

Fig. 2. Beam size at terminal plane of focusing segment shown in Fig. 1 as function of beam current. Solid line is drawn by Eq. (8). Dashed line is drawn by numerical integration of Eq. (1).

beam radius equation of Eq. (1), and by the Monte Carlo simulation, respectively. The unfilled circles represent the values obtained by extrapolation. By comparison of the beam size estimated by the three methods, the followings were found. (i) The expression of the trajectory displacement effect coincides with the Monte Carlo calculation for a beam-current ranging from 1 nA to 1 ␮A. (ii) The beam size evaluated by the beam radius equation coincides with the Monte Carlo calculation for a beam-current ranging from 10 nA to 10 mA. (iii) The beam sizes predicted by the trajectory displacement effect and by the beam radius equation overlap in the 10 nA to 0.5 ␮A range. Therefore, it was confirmed that the statistical Coulomb interactions are dominant for a low beam-current region and the averaged space–charge effect is dominant for a high beam-current region. In addition, the theory for predicting beam size at the terminal plane was found to transit from the theory of statistical Coulomb interaction (the trajectory displacement effect of Eq. (13)) to the averaged space–charge effect (the beam radius equation of Eq. (1)) with changes in beam current. The scaled beam size (Eq. (8)) predicted by the trajectory displacement effect becomes the same value when the parameters of ¯ v∗ , r ∗ , Sc , and Si are fixed. The beam radius equation (Eq. (1)) can , 0 c be rewritten using the scaling definitions of Eqs. (9)–(12) as ¯  2v∗2 d2 R ∗ 0 = , ∗ 2 R dς

(18)

where  = z/L. The scaled beam size at the terminal plane of Fig. 1 is



∗ = 0.80795 FW50 0

Si

dR∗ dς. dς

(19)

¯ v∗ , r ∗ , Sc , and Si are fixed, the Therefore, when the parameters of , 0 c scaled beam size does not change. These facts indicate that Fig. 2 ¯ and can be extended for different V0 , L, and R0 by replacing I to  ∗ if v is constant. Some numerical examples are listed FW50 to FW50 0 ∗ is constant even when V , L, in Table 2. It is confirmed that FW50 0 and R0 are respectively different. If V0 , L, and R0 change without keeping v0 constant, the beamcurrent range in which the predicted lines by the trajectory displacement effect and the beam radius equation overlap changes

Y. Takahashi / Optik 122 (2011) 1146–1151

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Table 2 ∗ predicted by trajectory displacement (TD) effect and by beam radius equation FW50 (BRE) with different beam voltage (V0 ), segment length (L), and beam radius (R0 ) at incident plane of segment. I (␮A)

V0 (kV)

L (m)

R0 (mm)

¯ 

v∗0

∗ FW50

TD 0.01 0.07 0.32 1.0 4.7

1 10 10 10 10

0.1 0.1 0.01 0.1 0.01

2.2 1.0 0.22 1.0 0.22

−6

5.2 × 10 5.2 × 10−6 5.2 × 10−6 7.6 × 10−5 7.6 × 10−5

70 70 70 70 70

BRE −4

8.2 × 10 8.2 × 10−4 8.2 × 10−4 6.1 × 10−3 6.1 × 10−3

5.9 × 10−4 5.8 × 10−4 5.8 × 10−4 8.5 × 10−3 8.5 × 10−3

in Fig. 2. For example, for V0 = 1 kV, L = 0.01 m, and R0 = 0.1 mm, the overlap range was 30 nA to 1 ␮A. Generally, the transition region of the theories (the overlap range) shifts to a lower beam current for lower V0 and larger R0 . This is presumably due to the space–charge force, which is effective with lower V0 and larger R0 . L had a weak dependency on the transition-range shift. The averaged space–charge effect shifts the plane forming a minimum beam size from the terminal plane in Fig. 1 because of space–charge defocusing. In practice, the minimum beam size is often used by refocusing of a probe-forming electron-beam system. Fig. 3 shows the minimum beam sizes by the theories, which were obtained at the different planes, and by the Monte Carlo method, which were found by searching the plane. The minimum beam size predicted by the trajectory displacement effect of Eq. (13) coincided with the values calculated by the Monte Carlo method for a wider beam-current range. The minimum beam size predicted by the beam radius equation of Eq. (1) decreased abruptly with decreasing beam current at about 1 mA in this calculation condition. This is because there were not a sufficient number of electrons at the minimum beam-size plane for a low beam current. Thus, the beam radius equation of Eq. (1) was not valid for predicting the minimum beam size in the low and medium beam-current ranges. 3.3. Effect of decelerating electrostatic field Since the expression of the trajectory displacement in a uniformly decelerating field is not given in Ref. [4], the beam size at the terminal plane of the focusing segment by the trajectory displacement effect was deduced by a slice method in Appendix A.

Fig. 4. Beam size vs. beam current under condition with decelerating field. Although beam size is larger than that in Fig. 2, relation between trajectory displacement effect and beam radius equation is almost equivalent.

The slice method has also been applied for analyzing the energy broadening of the electron source [8,9]. As a result, the beam size was approximately obtained as the expression having an additional dependency on a function of ln(1/ ). The is defined as (V1 /V0 )1/2 . The beam size by the beam radius equation is directly evaluated by integrating Eq. (2) with ε* = 0 and d2 V/dz2 = 0. Fig. 4 shows the beam size as a function of beam current under the conditions of V0 = 10 kV, V1 = 1 kV, L = 0.1 m, and R0 = 1 mm. The decelerating field is 0.9 kV/cm. It was found that the beam size of a low beam current can be predicted by the trajectory displacement effect and that of a high beam current by the beam radius equation. In the medium beam-current range, the beam sizes predicted by the theories coincided with each other, regardless of the presence of the decelerating field. The theory for the beam-size estimation transits continuously from the trajectory displacement effect to the beam radius equation as the beam-current increases. 3.4. Effect of beam emittance Given an initial velocity spread of electrons entering the segment in Fig. 1, even if the Coulomb force is absent, the beam size becomes a finite value at the terminal plane. This unperturbed beam size is additively broadened by a Coulomb repulsion force. The initial velocity spread can be considered as a normalized beam emittance. The beam size at the terminal plane of the segment is obtained by numerical integration of Eq. (2) with finite ε*. In the statistical Coulomb interaction, the beam size at the terminal plane can be estimated as the square root sum of the square of a trajectory displacement broadening and the square of an unperturbed beam size. Fig. 5 indicates the ray in a focusing beam segment with an angular velocity spread of angle ˛1 . The beam voltage, segment length, and beam size at the incident plane of the segment were set to 10 kV, 0.1 m, and 1 mm, respectively. The normalized beam emittance ε* was ε∗ = a1

Fig. 3. Minimum beam size vs. beam current. Plane at which minimum size is obtained is different for trajectory displacement effect and beam radius equation.



2meV0 · R0 .

(20)

Fig. 6 shows the beam size at the terminal plane vs. beam current with ˛1 = 10 ␮rad. In a low beam current, the beam size is deter-

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Y. Takahashi / Optik 122 (2011) 1146–1151

Acknowledgements The author is grateful to Prof. K. Ura for valuable comments and suggestions on this research theme. He also thanks Prof. H. Mori, Prof. A. Takaoka, Associate Prof. R. Nishi, and Mr. A. Ikegami of Osaka University for fruitful discussions on the simulation results. This work was conducted in a collaborative research program between Osaka University, Hitachi High-Technologies, and Hitachi, Ltd. Appendix A. Fig. 5. Focusing segment of beam with initial velocity spread. Beam size at terminal plane of segment is not zero even without Coulomb interaction. Minimum size is determined by angular distribution of beam velocity at incident plane of segment.

The slice method calculates the total displacement at the arbitrary plane by summing up the displacement contributed by a thin slice of a segment. In the focusing segment in Fig. 1, the trajectory displacement at the terminal plane can be calculated using angular deflections (FW50˛ ) of a cylindrical beam segment. The two analytical expressions of the angular deflection are given as Eqs. (8.4.29) and (8.4.31) in Ref. [4] for different beam regimes: the Holtsmark regime and the pencil beam regime. The beam size (FW50 ) at the terminal plane becomes



L

FW50 =

(L − z) 0

FW50˛ dz, z

(A1)

where z and FW50˛ are the thickness of a slice and an angular deflection created in the slice, respectively. The angular deflection for each regime is FW50˛,H I 2/3 m1/3 , = 2.5386 z 28/3 ε0 r0 (z)4/3 V (z)4/3 for Holtsmark regime,

Fig. 6. Beam size at terminal plane of focusing segment vs. beam current in case of beam with initial velocity spread.

(A2)

FW50˛,P r0 (z)I 3 m2/3 = 494.36 , z 217/3 ε0 e7/2 V (z)5/2 for pencil beam regime,

mined by the initial velocity spread because of a small contribution by the Coulomb interaction. In a high beam current, the beam size is broadened by the averaged space–charge effect. Therefore, the beam size at the terminal plane with a finite beam emittance can be predicted by the beam radius equation of Eq. (2) in a wider beam-current range.

4. Conclusion The transition of the theories of the statistical Coulomb interaction effect and the averaged space–charge effect was investigated by comparing their results with the values calculated by the Monte Carlo simulation method. It was confirmed that the expression of the trajectory displacement effect predicts the beam size in a low beam current and that the beam radius equation predicts it in a high beam current at the terminal plane of the focusing beam segment. In addition, the beam sizes predicted by these theories coincided in the medium beam-current range. For a beam voltage of 10 kV, segment length of 0.1 m, and initial beam size of 1 mm, the transition region of the beam current was in the 10 nA to 1 ␮A range. The applicable regions and the transition region of the theories regarding on beam voltage, segment length and beam radius at an incident plane can be obtained by using the scaled beam size and the scaled linear electron density. Taking the initial velocity spread of a beam into account, the beam radius equation with the beam emittance can predict the beam size at the terminal plane of a segment in wider beam-current range.

(A3)

where beam voltage V and beam radius r0 at the slice are





z , L

V (z) = V0 1 − (1 − ␬2 )

r0 (z) = R0 1 −

1 1−k

(A4)

 1−

1 − (1 − 2 )

z L

 (A5)

using a decelerating parameter = (V1 /V0 )1/2 . The beam size at the terminal plane for each regime becomes FW50,H = 1.39329

I 2/3 L2 V0 4/3 R0 4/3

FW50,P = 2.59261 × 1031

H( ),

I 3 L2 R0 V0 5/2

for Holtsmark regime,

P( ),

(A6)

for pencil beam regime,(A7)

where H( ) and P( ) are the functions only on . Since simple expressions for H( ) and P( ) were not derived, the analytical forms were approximately obtained by fitting the numerical values of H( ) and P( ) with the function including the term ln(1/ ). For example, the approximate FW50 can be described as FW50,H = 3.48321

I 2/3 L2

1

V0 4/3 R0 4/3 3 L2 R 0 V0 5/2

31 I

FW50,P = 1.72840 × 10

ln



+



3 , 5

 3/2 1 ln

k

(A8)



1 + , 2

(A9)

Y. Takahashi / Optik 122 (2011) 1146–1151

for a range from 0.01 to 1. The interpolated expression of Eqs. (A8) and (A9) becomes

⎡

⎤1/ı

⎢

FW50 = ⎣ 

1 FW50,H

ı

1 +



1 FW50,P

⎥ ı ⎦

·

(A10)

ı may be set to 2. References [1] P. Kruitt, G.H. Jansen, in: J. Orloff (Ed.), Handbook of Charged Particle Optics, CRC Press, 1997, pp. 275–318. [2] G. Herrmann, Optical theory of thermal velocity effects in cylindrical electron beams, J. Appl. Phys. 29 (1958) 127–136.

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[3] K. Ura:, Nano-denshikougaku, Kyoritsu shuppan (2005) 91–130 (in Japanese). [4] G.H. Jansen, Coulomb interactions in particle beams, in: Advances in Electronics and Electron Physics, Suppl. 21, Academic Press, 1990. [5] G.H. Jansen:, Trajectory displacement effect in particle projection lithography system: modifications to the extended two-particle theory and Monte Carlo simulation technique, J. Appl. Phys. 84 (1998) 4549–4567. [6] E. Munro:, Computer programs for the design and optimization of electron and ion beam lithography systems, Nucl. Instrum. Methods A258 (1987) 443– 461. [7] M. Matsumoto, T. Nishimura:, Mersenne Twister: a 623-dimensionally equidistributed uniform pseudo-random number generator, ACM Trans. Model. Comput. Simul. 8 (1998) 3–30. [8] W. Knauer:, Energy broadening in field emitted electron and ion beams, Optik 59 (1981) 335–354. [9] T. Sasaki:, Upper bond of the beam energy broadening in acceleration region, J. Vac. Sci. Technol. B 4 (1986) 135–139.