US 12,651,036 B2 · Method and apparatus for reconstructing atomic spatial distribution and electron beam function

A method for reconstructing an atomic spatial distribution and an electron beam function, comprising:

S101, obtaining, by controlling to move an electron beam to scan a sample, a diffraction intensity of the sample at each scanning position;
S102, describing a sample transmission function using local functions for the atomic spatial distribution, describing an electron beam function using aberration functions, establishing, based on the diffraction intensity, the sample transmission function, and the electron beam function, a forward propagation model containing to-be-optimized parameters, and calculating a value of a loss function;
S103, solving a derivative of the loss function with respect to the to-be-optimized parameters, to obtain gradients of the to-be-optimized parameters in the sample transmission function and the electron beam function, optimizing the to-be-optimized parameters based on the gradients, and updating the value of the loss function;
and
S104, determining whether a current iteration satisfies an iteration termination condition, performing S103 in response to the current iteration failing to satisfy the iteration termination condition, and outputting, in response to the current iteration satisfying the iteration termination condition, an optimized sample transmission function and an optimized electron beam function,
wherein the loss function is:
ℒ
=
1
2
⁢
∑
j
∑
u
,
v
(
❘
"\[LeftBracketingBar]"
ℱ
⁢
{
ψ
exit
}
❘
"\[RightBracketingBar]"
2
-
I
j
)
2
where j represents an ordinal number of the scanning position, u, v represents coordinates in real space, |⋅| represents calculating a modulus of each element in a matrix, {⋅} represents calculating a two-dimensional Fourier transform of a matrix, Ii represents the diffraction intensity, and ψexit represents an exit wave function;
the exit wave function ψexit is:
ψ
exit
=
𝒫
dz
⁢
{
…
⁢
𝒫
dz
⁢
{
𝒫
dz
⁢
{
P
⁡
(
r
-
r
j
)
⁢
O
1
(
r
)
}
⁢
O
2
(
r
)
}
⁢
O
3
(
r
)
⁢
…
}
⁢
O
1
(
r
)
,
where P(r) represents the electron beam function, P(r−ri) represents the electron beam scanning to a i-th position, dz{⋅} represents a Fresnel diffraction propagator, and Ol(r) represents the sample transmission function of the l-th slice;
the sample transmission function Ol(r) is:
O
l
(
r
)
=
Amp
1
(
r
)
*
e
i
⁢
σ
⁢
V
l
(
r
)
,
where Vl(r) represents a projection potential of the transmission function, Ampl(r) represents an amplitude of the transmission function,
V
l
(
r
)
=
s
l
(
r
)
+
p
l
(
r
)
+
d
l
(
r
)
+
f
l
(
r
)
+
…
,
and
Amp
l
(
r
)
=
∑
i
A
⁢
2
l
,
i
*
exp
⁢
{
-
[
(
M
l
,
i
-
y
)
2
+
(
N
l
,
i
-
x
)
2
]
B
⁢
2
l
,
i
}
+
D
l
,
where s(r) represents an s-orbital spatial distribution of electrons in the l-th slice, p(r) represents a p-orbital spatial distribution of electrons in the l-th slice, and sl(r) is represented by a local function of a Gaussian function as:
s
l
(
r
)
=
∑
i
⁢
A
⁢
1
l
,
i
*
exp
⁢
{
-
[
(
M
l
,
i
-
y
)
2
+
(
N
l
,
i
-
x
)
2
]
B
⁢
1
l
,
i
}
;
and
the electron beam function P(r) is:
P
⁡
(
r
)
=
ℱ
⁢
{
A
⁡
(
k
)
⁢
e
i
⁢
χ
⁡
(
k
)
}
,
χ
⁡
(
k
)
=
R
⁢
{
1
2
⁢
ω
⁢
ω
¯
⁢
C
1
,
0
+
1
2
⁢
ω
¯
2
⁢
C
1
,
2
+
ω
2
⁢
ω
¯
2
⁢
C
2
,
1
+
1
4
⁢
ω
2
⁢
ω
¯
2
⁢
C
3
,
0
+
…
}
,
where A(k) represents a diaphragm function, Cn,m represents an aberration coefficient, χ(k) represents an aberration function, R{⋅} represents taking a real part of a matrix in {⋅}, aberration ω=kx+i*ky, kx and ky are coordinates in reciprocal space, respectively, and ω is the complex conjugate matrix of ω, wherein dl(r) is the d-orbital spatial distribution of electrons in the l-th slice, Ml,i and Nl,i are the center coordinates of the i-th local function, A1 is the peak value of an s-orbital spatial distribution, A2 is a the peak value of the transmission function amplitude, B1/B2 are a spatial distribution range of the atomic spatial distribution, and a is an constant that scales the effect of the potential Vl(r) on the phase of the electron wave.

340 more elements

An apparatus for reconstructing an atomic spatial distribution and an electron beam function, comprising:

an obtaining module configured to obtain, by controlling to move an electron beam to scan a sample, a diffraction intensity of the sample at each scanning position;
a construction module configured to describe a sample transmission function using local functions for the atomic spatial distribution, describe an electron beam function using aberration functions, establish, based on the diffraction intensity, the sample transmission function, and the electron beam function, a forward propagation model containing to-be-optimized parameters, and calculate a value of a loss function;
and
an optimization module configured to solve a derivative of the loss function with respect to the to-be-optimized parameters, to obtain gradients of the to-be-optimized parameters in the sample transmission function and the electron beam function, optimize the to-be-optimized parameters based on the gradients, update the value of the loss function, repeat an iteration process until an iteration termination condition is satisfied, and output an optimized sample transmission function and an optimized electron beam function,
wherein the loss function is:
ℒ
=
1
2
⁢
∑
j
∑
u
,
v
(
❘
"\[LeftBracketingBar]"
ℱ
⁢
{
ψ
exit
}
❘
"\[RightBracketingBar]"
2
-
I
j
)
2
,
where j represents an ordinal number of the scanning position, u, v represents coordinates in real space, |⋅| represents calculating a modulus of each element in a matrix, {⋅} represents calculating a two-dimensional Fourier transform of a matrix, Ij represents the diffraction intensity, and ψexit represents an exit wave function;
the exit wave function ψexit is:
ψ
exit
=
𝒫
d
⁢
z
⁢
{
…
⁢
𝒫
d
⁢
z
⁢
{
𝒫
d
⁢
z
⁢
{
P
⁡
(
r
-
r
j
)
⁢
O
1
(
r
)
}
⁢
O
2
(
r
)
}
⁢
O
3
(
r
)
⁢
…
}
⁢
O
l
(
r
)
,
where P(r) represents the electron beam function, P(r−rj) represents the electron beam scanning to a j-th position, dz{⋅} represents a Fresnel diffraction propagator, and Ol(r) represents the sample transmission function of the l-th slice;
the sample transmission function Ol(r) is:
O
l
(
r
)
=
Amp
l
(
r
)
*
e
i
⁢
σ
⁢
V
l
(
r
)
,
where Vl(r) represents a projection potential of the transmission function, Ampl(r) represents an amplitude of the transmission function,
V
l
(
r
)
=
s
l
(
r
)
+
p
l
(
r
)
+
d
l
(
r
)
+
f
l
(
r
)
+
…
,
and
Amp
l
(
r
)
=
∑
i
A
⁢
2
l
,
i
*
exp
⁢
{
-
[
(
M
l
,
i
-
y
)
2
+
(
N
l
,
i
-
x
)
2
]
B
⁢
2
l
,
i
}
+
D
l
,
where s(r) represents an s-orbital spatial distribution of electrons in the l-th slice, p(r) represents a p-orbital spatial distribution of electrons in the l-th slice, and sl(r) is represented by a local function of a Gaussian function as:
s
l
(
r
)
=
∑
i
⁢
A
⁢
1
l
,
i
*
exp
⁢
{
-
[
(
M
l
,
i
-
y
)
2
+
(
N
l
,
i
-
x
)
2
]
B
⁢
1
l
,
i
}
;
and
the electron beam function P(r) is:
P
⁡
(
r
)
=
ℱ
⁢
{
A
⁡
(
k
)
⁢
e
i
⁢
χ
⁡
(
k
)
}
,
χ
⁡
(
k
)
=
R
⁢
{
1
2
⁢
ω
⁢
ω
¯
⁢
C
1
,
0
+
1
2
⁢
ω
¯
2
⁢
C
1
,
2
+
ω
2
⁢
ω
¯
2
⁢
C
2
,
1
+
1
4
⁢
ω
2
⁢
ω
¯
2
⁢
C
3
,
0
+
…
}
,
where A(k) represents a diaphragm function, Cn,m represents an aberration coefficient, χ(k) represents an aberration function, R{⋅} represents taking a real part of a matrix in {⋅}, aberration ω=kx+i*ky, kx and ky are coordinates in reciprocal space, respectively, and ω is the complex conjugate matrix of ω, wherein dl(r) is the d-orbital spatial distribution of electrons in the l-th slice, Ml,i and Nl,i are the center coordinates of the i-th local function, A1 is the peak value of an s-orbital spatial distribution, A2 is a the peak value of the transmission function amplitude, B1/B2 are a spatial distribution range of the atomic spatial distribution, and a is an constant that scales the effect of the potential Vl(r) on the phase of the electron wave.

346 more elements

An electronic device, comprising

a memory;
a processor;
and
a computer program stored in the memory and executable on the processor,
wherein the processor, when executing the computer program, implements:
S101, obtaining, by controlling to move an electron beam to scan a sample, a diffraction intensity of the sample at each scanning position;
S102, describing a sample transmission function using local functions for the atomic spatial distribution, describing an electron beam function using aberration functions, based on the diffraction intensity, the sample transmission function, and the electron beam function, a forward propagation model containing a to-be-optimized parameter, and calculating a value of a loss function;
S103, solving a derivative of the loss function with respect to the to-be-optimized parameters, to obtain gradients of the to-be-optimized parameters in the sample transmission function and the electron beam function, optimizing the to-be-optimized parameters based on the gradients, and updating the value of the loss function;
and
S104, determining whether a current iteration satisfies an iteration termination condition, performing S103 in response to the current iteration failing to satisfy the iteration termination condition, and outputting, in response to the current iteration satisfying the iteration termination condition, an optimized sample transmission function and an optimized electron beam function,
wherein the loss function is:
ℒ
=
1
2
⁢
∑
j
∑
u
,
v
(
❘
"\[LeftBracketingBar]"
ℱ
⁢
{
ψ
exit
}
❘
"\[RightBracketingBar]"
2
-
I
j
)
2
,
where j represents an ordinal number of the scanning position, u, v represents coordinates in real space, |⋅| represents calculating a modulus of each element in a matrix, {⋅} represents calculating a two-dimensional Fourier transform of a matrix, Ij represents the diffraction intensity, and ψexit represents an exit wave function;
the exit wave function ψexit is:
ψ
exit
=
𝒫
d
⁢
z
⁢
{
…
⁢
𝒫
d
⁢
z
⁢
{
𝒫
d
⁢
z
⁢
{
P
⁡
(
r
-
r
j
)
⁢
O
1
(
r
)
}
⁢
O
2
(
r
)
}
⁢
O
3
(
r
)
⁢
…
}
⁢
O
l
(
r
)
,
where P(r) represents the electron beam function, P(r−rj) represents the electron beam scanning to a j-th position, dz{⋅} represents a Fresnel diffraction propagator, and Ol(r) represents the sample transmission function of the l-th slice;
the sample transmission function Ol(r) is:
O
l
(
r
)
=
Amp
l
(
r
)
*
e
i
⁢
σ
⁢
V
l
(
r
)
,
where Vl(r) represents a projection potential of the transmission function, Ampl(r) represents an amplitude of the transmission function,
V
l
(
r
)
=
s
l
(
r
)
+
p
l
(
r
)
+
d
l
(
r
)
+
f
l
(
r
)
+
…
,
and
Amp
l
(
r
)
=
∑
i
A
⁢
2
l
,
i
*
exp
⁢
{
-
[
(
M
l
,
i
-
y
)
2
+
(
N
l
,
i
-
x
)
2
]
B
⁢
2
l
,
i
}
+
D
l
,
where s(r) represents an s-orbital spatial distribution of electrons in the l-th slice, p(r) represents a p-orbital spatial distribution of electrons in the l-th slice, and sl(r) is represented by a local function of a Gaussian function as:
s
l
(
r
)
=
∑
i
⁢
A
⁢
1
l
,
i
*
exp
⁢
{
-
[
(
M
l
,
i
-
y
)
2
+
(
N
l
,
i
-
x
)
2
]
B
⁢
1
l
,
i
}
;
and
the electron beam function P(r) is:
P
⁡
(
r
)
=
ℱ
⁢
{
A
⁡
(
k
)
⁢
e
i
⁢
χ
⁡
(
k
)
}
,
χ
⁡
(
k
)
=
R
⁢
{
1
2
⁢
ω
⁢
ω
¯
⁢
C
1
,
0
+
1
2
⁢
ω
¯
2
⁢
C
1
,
2
+
ω
2
⁢
ω
¯
2
⁢
C
2
,
1
+
1
4
⁢
ω
2
⁢
ω
¯
2
⁢
C
3
,
0
+
…
}
,
where A(k) represents a diaphragm function, Cn,m represents an aberration coefficient, χ(k) represents an aberration function, R{⋅} represents taking a real part of a matrix in {⋅}, aberration ω=kx+i*ky, kx and ky are coordinates in reciprocal space, respectively, and ω is the complex conjugate matrix of ω, wherein dl(r) is the d-orbital spatial distribution of electrons in the l-th slice, Ml,i and Nl,i are the center coordinates of the i-th local function, A1 is the peak value of an s-orbital spatial distribution, A2 is a the peak value of the transmission function amplitude, B1/B2 are a spatial distribution range of the atomic spatial distribution, and a is an constant that scales the effect of the potential Vl(r) on the phase of the electron wave.

352 more elements

Method and apparatus for reconstructing atomic spatial distribution and electron beam function

Life of this patent

Method and apparatus for reconstructing atomic spatial distribution and electron beam function

Life of this patent

Abstract

Claims12