APL – Supplemental Material
Domain patterning by electron beam of MgO doped lithium
niobate covered by resist
V.Ya. Shur1,2,a), D.S. Chezganov1,2, A.R. Akhmatkhanov1,2, D.K. Kuznetsov1
1Institute
2Labfer
of Natural Sciences, Ural Federal University, Ekaterinburg, 620000, Russia
Ltd., 2-22 8 Marta str., 620014, Ekaterinburg, Russia
The computer simulation of the spatial distribution of electrons and electric field
The computer simulation of the space charge formation and spatial distribution of the
electric field was carried out for explanation of the experimental results. The simulation of
primary electron scattering in the target volume was performed by free software Casino
v.2.4.8.1 This method is based on the complete estimation of large number of single electron
interactions with a solid target. Monte-Carlo simulation of electron scattering was used for
estimation of the charge distribution in the bulk.
The simulated target represented a 1 μm thick resist layer on the irradiated polar surface
of 1 mm thick Z-cut LN plate. The studied electron energies were 7, 10, and 12 keV. The 105
electron trajectories were simulated in each case.
The obtained spatial distributions of electrons were sliced into a number of 10-nm-thick
layers. The radial electron distribution in each slice was fitted by:
𝑟−𝑟𝑐 𝑖 2
𝜌𝑖 (𝑟) = 𝐴𝑖 × 𝑟 × exp⁡[−2 (
𝜔𝑖
) ],
where Ai, rCi, and ωi are the fitting coefficients for ith slice.
We use the depth of the slice z = 10×i.
a)
Author to whom correspondence should be addressed. Electronic mail: vladimir.shur@urfu.ru.
(S1)
The depth dependences of the fitting coefficients A(z) for electron energies 7 and 10 keV
were fitted by:
𝑓(𝑧) =
𝑅
𝜔
𝑧−𝑧𝑐 + 1
2 )
1+exp⁡(−
𝜔2
× (1 −
1
𝜔
𝑧−𝑧𝑐 + 1
2 )
exp⁡(−
𝜔3
) + 𝑦0 ,
(S2)
where R, zc, y0, ω1, ω2, and ω3 are the fitting coefficients.
The depth dependences of the fitting coefficients rCi(z) and ωi(z) for electron energies
7 and 10 keV were fitted by polynomial and linear functions:
𝑓(𝑧) = 𝑐 + 𝑏𝑧 + 𝑎𝑧 2
(S3)
𝑓(𝑧) = 𝑞 + 𝑘𝑧
(S4)
The depth dependence of the electron distribution for electron energy of 12 keV is
essentially different due to change of the density and chemical composition at the resist to LN
interface. In this case A(z) dependence in each region was fitted by different functions:
𝑓(𝑧) = {
𝐵𝑒 𝑅0×𝑧 + 𝑦01 , 0 < 𝑧 ≤ 1000
𝑧−𝑧𝑐
𝐺 × exp⁡[−2 (
2
𝜔
) ] + 𝑦02 , 1000 < 𝑧 ≤ 1300
,
(S5)
The depth dependences of the fitting coefficients rCi(z) and ωi(z) for 12 keV were fitted
by polynomial function (Eq. S3).
Table S1. Fitting coefficients for electron distribution for 7 and 10 keV.
Electron
Energy,
keV
7
10
Depth,
nm
R,
107
zc ,
107
y0,
10-2
ω1,
107
ω2
ω3,
108
a,
10-5
b,
10-2
C,
10
k,
10-2
q,
10
560
1050
34
16
-7
-31
29
19
-13
-61
69
123
11
14
-77
-59
36
59
11
15
-46
38
35
64
Table S2. Fitting coefficients for electron distribution for 12 keV.
B,
10-4
55
R0,
10-4
52
y01,
10-2
11
G,
10-1
18
zc ,
102
11
y02,
10-3
51
a1,
10-5
30
b1,
10-2
20
c1,
10
34
a2,
10-5
30
b2,
10-3
72
c2,
10
78
The following function was used for spatial distribution of electrons:
𝑟−𝑟𝑐 (𝑧) 2
𝜌𝑑𝑖𝑠𝑡 (𝑟, 𝑧) = 𝐴(𝑧) × 𝑟 × exp⁡[−2 (
𝜔(𝑧)
) ],
(S6)
The spatial distribution of the electron density for electric field simulation was estimated
by:
𝜌(𝑟, 𝑧) = 𝑄 × (1 − 𝜂) × 𝑁 × 𝜌𝑑𝑖𝑠𝑡 (𝑟, 𝑧),
(S7)
where Q is injected charge, η = 0.04 is electron backscattered coefficient, N is the
normalization coefficient.
The normalization coefficient was calculated as:
2𝜋
𝑍
∞
𝑁 = 1/ ∫0 ∫0 𝑚𝑎𝑥 ∫0 𝜌𝑑𝑖𝑠𝑡 (𝑟, 𝑧) 𝑑𝜑𝑑𝑟𝑑𝑧,
(S8)
where Zmax is the electron maximum depth.
The simulated values of Zmax for 7 keV is 560 nm, for 10 keV is 1050 nm, and for
12 keV is 1300 nm.
The field distribution was simulated by COMSOL Multiphysics Modeling Software by
the finite element method for static charge. The spatial distribution of the field polar
component calculated for Q = 0.9 nC was presented in Fig. 6(a).
References
1
D. Drouin, A. R. Couture, D. Joly, X. Tastet, V. Aimez, R. Rauvin, Scanning 29, 92 (2007).