3 Local Dielectric Constant Density Analysis of High-k Dielectric Nanomaterial
81
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
20
1
Hf
La
O
(a)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
20
1
Hf
La
O
(b)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
20
1
Hf
La
O
(c)
−0.05
0
1/4
0.15
i
i =1
i =2
i =3
Average
0.0
1.0
Hf
O
|sin|
(d)
−0.05
0
1/4
0.15
i [−]
i =1
i =2
i =3
Average
0.0
1.0
La
O
|sin|
(e)
Fig. 3.19 The eigenvalues of the polarizability density tensor of HfLaO x . Panels (a)–(c) are the
results of the first, second, and third eigenvalues (α 1,2,3 ) on a plane with the O atom near the central
vacancy and the next Hf and La atoms, respectively. The values are presented in descending order
of the real parts of the eigenvalues. The color map shows the real parts of the eigenvalues, and
bold lines represent the contour of the argument, and their units are in degree. The solid black lines
show the directions of the eigenvectors. The filled circles show the pseudopotential of the La and
Hf atoms. Panels (d) and (e) show all eigenvalues on the Hf-O and La-O lines. The upper part
of this panel is the real part of three eigenvalues and their average. The lower part of this panel
is the value of sine of the argument. The horizontal dotted lines on the upper panels represent
1/4π . The vertical dotted lines between the O atom and the metal atom show the boundary of the
pseudopotential, and the left region from this line is the inside of the pseudopotential
81
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
20
1
Hf
La
O
(a)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
20
1
Hf
La
O
(b)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
20
1
Hf
La
O
(c)
−0.05
0
1/4
0.15
i
i =1
i =2
i =3
Average
0.0
1.0
Hf
O
|sin|
(d)
−0.05
0
1/4
0.15
i [−]
i =1
i =2
i =3
Average
0.0
1.0
La
O
|sin|
(e)
Fig. 3.19 The eigenvalues of the polarizability density tensor of HfLaO x . Panels (a)–(c) are the
results of the first, second, and third eigenvalues (α 1,2,3 ) on a plane with the O atom near the central
vacancy and the next Hf and La atoms, respectively. The values are presented in descending order
of the real parts of the eigenvalues. The color map shows the real parts of the eigenvalues, and
bold lines represent the contour of the argument, and their units are in degree. The solid black lines
show the directions of the eigenvectors. The filled circles show the pseudopotential of the La and
Hf atoms. Panels (d) and (e) show all eigenvalues on the Hf-O and La-O lines. The upper part
of this panel is the real part of three eigenvalues and their average. The lower part of this panel
is the value of sine of the argument. The horizontal dotted lines on the upper panels represent
1/4π . The vertical dotted lines between the O atom and the metal atom show the boundary of the
pseudopotential, and the left region from this line is the inside of the pseudopotential