Abstract— The complex scattering process of the high-energy electron beams in resist is simulated by Monte Carlo method. The energy deposition distributions are presented under different exposure conditions. The three-dimensional (3-D) development profiles are obtained with the developing threshold model. It is found that, in the high energy range, higher electron beam energy, thinner resist, appropriate dose and lower substrate's atom number will cause lower proximity effect. Based on the simulations, we can explain the proximity effect and the dose control on proximity effect correction via the three-dimensional development profiles. The results will be useful to optimize the exposure conditions in electron beam lithography, and to provide more accurate data for proximity effect correction.

I. INTRODUCTION

The merits of high-energy electron beam lithography are widely recognized as providing better resolution and greater accuracy, compared with optical lithography. Because of these advantages, electron beam lithography has been used for many years in photo mask patterning, direct writing on semiconductor devices, and fabrication of various nanometer devices. Although electron beam lithography can provide a focused electron beam at dimensions of less than 10nm, several factors besides the size of the electron beam determine the extent of the exposed volume in the resist. One of the major factors is the proximity effects due to the electron scattering in the resist and substrate. Simulation is an effective method to understand the proximity effect, and thereby improves the electron beam lithography process [1].

In order to investigate the influence on the proximity effect of many factors, for example electron beam energy, resist thickness, dose and substrate type, Monte Carlo method is utilized to simulate the complex scattering process of high-energy electron beam with Gaussian distribution in the resist. In the modeling process, we tracked electron scattering trajectories in the resist and substrate. Electron scattering behavior, deposition energy distribution and the 3-D development profiles of e-beam lithography on substrates

*The project is partly supported by the National S&T Major Project under

contact 2011ZX02507-001-003 and the Program for New Century Excellent Talents of the Ministry of Education, China，under contact NCET-11-0099.

Jiang-Yong Pan is with the Department of Electronic Science and Technology, Southeast University, Nanjing 210096, China.

Zai-Fa Zhou is with the Key Laboratory of MEMS of the Ministry of Education, Southeast University, Southeast University, Nanjing 210096, China ( corresponding author, telephone: 86-25-83792632-8817; fax: 86-25-83792939; e-mail: zfzhou@seu.edu.cn).

Qi Gan is with the Department of Electronic Science and Technology, Southeast University, Nanjing 210096, China.

Wen-Qin Xu is with the Department of Electronic Science and Technology, Southeast University, Nanjing 210096, China

have been obtained combined with developing threshold model. Dose control method is introduced at the end of paper so as to minimize the proximity effect.

II. MODELS AND SIMULATION PROCESS

In this paper, the subsection scattering model has been used. The elastic scattering include the screened Rutherford cross section [2] above 30keV as below

)1(

)1(.4)2cos1(

sin4

)1(22

4

022

4

NNNel E

eZZdE

eZZββ

πβθ

θθπσπ

++=

+−+= ∫

, (1)

E

ZN

32

31043.5 −×=β, (2)

where Z , E and e are the atomic number ,electron energy and electron charge respectively. When the energy is below and at 30keV, the Browning’ s Mott cross-section [3] is adopted

25.0

025.0

07.1

0

7.118

)/0007.0005.0(100.3 cm

EZEZEZ

M ++×=

−

σ , (3)

0

67.03

2

/104.3)1/(21cos

EZRR

−×=

−+−=

αααθ (4)

where Z and 0E are the atomic number and electron energy. Scattering angle can be calculated from (4). For the consideration of inelastic scatterings, we include the inner-shell scattering and the valence-shell scattering. The Gryzinsky cross-section below and at 20keV is used

⎥⎦⎤

⎢⎣⎡ −+−+×⎟

⎠⎞

⎜⎝⎛

+−×=

−

))1(7.2ln()2/11(321

11..1051.6 2

123

2

14

UUUU

UN

Ex

Binelσ

,(5)

where BEEU /0= , 0E and BE are the electron energy and binding energy. xN is the number of electrons in a particular shell. When the energy is above 20keV, the Moller cross-section [4] can be adopted

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −++

−−

−+⎟⎠⎞

⎜⎝⎛

+⎟⎠⎞

⎜⎝⎛ −== ∫

i

i

iii

MMinel Emv

edd

di ε

εττ

εεττεπε

εσσ

ε

1ln

)1(12

111

1212

2

2

02

45.0 (6)

0/ EEΔ=ε , (7)

)1( 1 −= −γτ , (8)

Monte Carlo simulation of high-energy electron beam lithography process*

Jiang-Yong Pan, Zai-Fa Zhou*, Qi Gan and Wen-Qin Xu

Proceedings of the 13thIEEE International Conference on NanotechnologyBeijing, China, August 5-8, 2013

978-1-4799-0676-5/13/$31.00 ©2013 IEEE 622

2

1 ⎟⎠⎞

⎜⎝⎛−=

cvγ , (9)

where EΔ is the energy loss, 24 /2 mveB π= , m and v are the rest mass and relativistic speed of electron.

As for the energy loss calculation, the Bethe formula modified by Joy and Lou [5], Bethe formula [6] and Bethe formula modified by relativistic effect were adopted to calculate the total energy loss respectively when the electron energy is below and at 10keV, above 10keV but below and at 20keV, above 20keV.

⎥⎦⎤

⎢⎣⎡ +=−

JkJEnZ

Ee

dsdE )(166.1ln2 0

0

4π , (10)

J

EnZE

edsdE 166.1ln2

0

4π=− , (11)

⎥⎦

⎤⎢⎣

⎡−++−−=− )1(

812ln)2(

2ln2 222

22

2

2

4

γγγγγ

πJ

EmvnZmv

edsdE , (12)

where n and J are the atomic density and mean ionization energy respectively. The Gryzinsky cross-section was used to calculate the discrete energy loss below and at 20kev, and the Moller cross-section was used to calculate the discrete energy loss above 20keV. The formula to calculate the discrete energy loss is as below [7]

dE

dEdE

ANN

dSdE ME

E

jAj

j

σρ∫=

2/0)( (13)

where AN is the Avogadro number, ρ is density of the material, A is mass number, jN is the number of electrons in the particular shell j .The electron scattering is calculated at every step of the beam scanning. A random number 1R is used to determine whether a scattering event was elastic or inelastic. If the random number 1R is less than Te σσ / , then the electron will undergo an elastic scattering. where cveT σσσσ ++= . eσ , vσ and cσ are the total elastic cross section, inelastic scattering cross section for the valence shell, and inelastic scattering cross section for the core shells, respectively. However, if 1R is greater than Te σσ / , then it will indicate that an inelastic event has occurred. The type of inelastic event is then decided by using another random number 2R . If )/(2 cvvR σσσ +< , then a valence shell excitation will be produced; otherwise a core shell excitation will occur. The high-energy electron beam is considered as Gaussian distribution. Generally, the simulation process is divided in three parts. Firstly, the Monte Carlo Point-spread function (PSF) is calculated using these formulas above. Secondly, The PSF is then combined with actual number of electrons to get the energy deposition distribution. The number of electrons is calculated with dose and Gaussian radius of the electron beam. Finally, combined with developing threshold model, development profiles can be obtained.

0 0.5 1 1.5 2 2.5 3 3.5

x 104

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Radius(nm)

Dep

osite

d E

nerg

y(ke

V/n

m2 )

25keV50keV100keV

Figure 1. the lateral energy distribution for different energy

(a)

(b)

(c)

Figure 2. (a) the 3-D development profile under 25keV, (b) the 3-D development profile under 50keV, (c) the 3-D development profile under

100keV.

III. RESULTS AND DISCUSSION

A. Beam Energy Effect The effect of beam energy on the resist profile has been

examined by using lateral energy distribution and 3-D development profiles for beam energies ranging from 25keV to 100keV. We choose 102μC/cm2 dose when plotting the lateral energy distribution [8]. The Gaussian radius of e-beam

623

is 10nm. The Gaussian radius of e-beam is 30nm and the dose is 500μC/cm2 when plotting 3-D development profiles.

In Fig. 1, the lateral energy distribution for 25keV, 50keV and 100keV e-beam energy in the case of 500nm PMMA on bulk Si is presented. It can be found that the curve declines more slowly under the low e-beam energy condition. Consequently the proximity effect is more serious. We can also get the conclusion from the 3-D development profiles [9].

In Fig. 2, the 3-D development profiles under different e-beam energy are presented. The serious proximity effect under low energy is explained by the fact that the low energy beam is more easily deflected and spreads out along the lateral direction. In addition, low energy beam has shown much higher secondary electron density, indicating more energy deposition.

B. Substrate Effect The substrate affects the e-beam processes due to their

backscattering effect. This is especially obvious for high-Z materials. In order to study this effect, we plot the lateral energy distribution, the longitudinal energy distribution [10] and 3-D development profiles for different kinds of substrates. The PMMA thickness is 500nm in the simulation.

0 2000 4000 6000 8000 10000 12000 14000 1600010

-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Radius(nm)

Dep

osite

d E

nerg

y(ke

V/n

m2 )

SiFe

Figure 3. the lateral energy distribution under different substrates

100

101

102

103

104

105

106

0

0.002

0.004

0.006

0.008

0.01

0.012

Depth• nm)

Dep

osite

d E

nerg

y•ke

V/n

m)

SiFe

Figure 4. the longitudinal energy distribution under different substrates

(a)

(b)

Figure 5. (a)the 3-D development profile in Si substrate, (b) the 3-D development profile in Fe substrate

In Fig. 3 and Fig. 4 the lateral energy distribution and the longitudinal energy distribution for Si and Fe substrate in the case of 102μC/cm2 dose and 10nm Gaussian radius are presented. It can be concluded that the curve in the lateral energy distribution declines more slowly when the substrate is a high-Z material, which indicates that the proximity effect is more serious for high-Z materials. It can also be found that the deposited energy in higher atomic number material is higher in the longitudinal energy distribution. We can verify the conclusion from the 3-D development profiles.

0 2000 4000 6000 8000 10000 12000 14000 1600010

-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Radius(nm)

Dep

osite

d E

nerg

y(ke

V/n

m2 )

500nm300nm1000nm

Figure 6. the lateral energy distribution for different PMMA thickness

In Fig. 5, the 3-D development profiles in the case of 500μC/cm2 dose and 30nm Gaussian radius are presented. The serious proximity effect in high-Z material is explained by the fact that high-Z materials give a higher probability of large angle scattering of electrons due to their relatively larger nucleus. The number of backscattered electrons increases when increasing the substrate atomic number Z. The backscattered electrons enhanced the exposure energy of the resist, possessing close-to-threshold deposited energy from

624

forward scattered electrons. Thus above resist becomes more developable.

100

101

102

103

104

105

106

0

0.5

1

1.5

2

2.5

3

3.5x 10

-3

Depth• nm)

Dep

osite

d E

nerg

y•ke

V/n

m)

300nm

500nm

1000nm

Figure 7. the longitudinal energy distribution for different PMMA thickness

C. Resist Thickness Effect The resist profile dependence on resist thickness has been

simulated for a 50keV beam in a 300-1000nm thick resist on a silicon substrate.

(a)

(b)

(c)

Figure 8. (a) the 3-D development profile in 300nm resist, (b) the 3-D development profile in 500nm resist, (c) the 3-D development

profile in 1000nm resist

In Fig. 6 and Fig. 7, the lateral energy distribution and the longitudinal energy distribution in the case of 102μC/cm2 dose

and 10nm Gaussian radius are presented. We can find the phenomenon from the figure that the curve in the lateral energy distribution declines more slowly and the curve in the longitudinal energy distribution increases more slowly near the interface between the resist and substrate in thicker resist. This phenomenon indicates that the proximity effect is more serious in the thick resist. Also, the 3-D development profiles are plotted to verify the conclusion.

In Fig. 8, the 3-D development profiles in the case of 500μC/cm2 and 30nm Gaussian radius for different resist thickness are presented. The serious proximity effect in thick resist can be attributed to the fact that electrons motion longer and cumulative scattering effect increases at the resist bottom.

(a)

(b)

(c)

Figure 9. (a)the 3-D development profile of 200μC/cm2 dose, (b)the 3-D development profile of 500μC/cm2 dose, (c) the 3-D development

profile of 1000μC/cm2 dose

D. Dose Effect In order to investigate the effect of dose, the obtained 3-D

development profiles in the case of 500nm PMMA and 30nm Gaussian radius under different doses are as Fig. 9.

From Fig. 9, It can be found that the figure is like normal trapezoid in bigger dose, causing more serious proximity effect. When the dose is small, the profile is like reverse trapezoid. This phenomenon can be explained by the fact that as the dose is increased, the energy deposited into the resist increases and a greater percentage of the resist achieves the

625

threshold energy to become developable. The optimal dose that gives us the best profile is the minimum dose to open the trench.

IV. CORRECTION OF PROXIMITY EFFECT BY DOSAGE COMPENSATION

The e-beam of 10nm Gaussian radius is moved by certain step length, different 3-D etching development profile can be produced. In this part, 500μC/cm2 dose and 20nm step length were chosen. The e-beam energy is 50keV and Gaussian radius is 10nm.

(a)

(b)

(c)

(d)

Figure 10. (a)space diagram, (b) vertical view, (c) bottom view, (d) bottom view after dosage compensation

Because of the proximity effect, the actual width at the bottom becomes larger when several of e-beam forming a figure "—". The phenomenon can be easily found from Fig. 10 (b) and (c). In order to improve the situation, we could use

lower dose to make the development profile like reverse trapezoid. Fig. 10 (d) shows the development profile after optimization of dose, revealing an effective method to correct proximity effect.

V. CONCLUSION

In this paper, the Monte Carlo subsection scattering model is adopted to simulate the complex scattering of high-energy e-beam lithography and get the deposited energy distribution. The effects of the e-beam energy, substrate material, resist thickness and dose on the proximity effect have been investigated. In the high energy range, the proximity effect is smaller under the condition of higher energy of e-beam, substrate with lower atomic number and thinner resist. In addition, the actual e-beam lithography process has been simulated for a complex figure. The effect of dosage compensation on the proximity effect correction has also been studied. The results will be useful to optimize the exposure conditions in electron beam lithography, and to provide more accurate data for the proximity effect correction.

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