3 Local Dielectric Constant Density Analysis of High-k Dielectric Nanomaterial
85
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
20
1
Hf
La
O
(a)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
20
1
Hf
La
O
(b)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
20
1
Hf
La
O
(c)
−1
0
1
0 /
i [−]
i =1
i =2
i =3
0
1
Hf
O
|sin|
(d)
−1
0
1
0 /
i [−]
i =1
i =2
i =3
0
1
La
O
|sin|
(e)
Fig. 3.23 The inverse of the eigenvalues of the dielectric constant density tensor of HfLaO x .
Panels (a)–(c) present the results of the first, second, and third eigenvalues (
−1
1,2,3 ) on a plane
with the O atom near the vacancy and the next Hf and La atoms, respectively. The values are
presented in the 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 atom. Panels (d) and (e) show all eigenvalues on the Hf-O
and La-O atoms. 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 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
85
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
20
1
Hf
La
O
(a)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
20
1
Hf
La
O
(b)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
20
1
Hf
La
O
(c)
−1
0
1
0 /
i [−]
i =1
i =2
i =3
0
1
Hf
O
|sin|
(d)
−1
0
1
0 /
i [−]
i =1
i =2
i =3
0
1
La
O
|sin|
(e)
Fig. 3.23 The inverse of the eigenvalues of the dielectric constant density tensor of HfLaO x .
Panels (a)–(c) present the results of the first, second, and third eigenvalues (
−1
1,2,3 ) on a plane
with the O atom near the vacancy and the next Hf and La atoms, respectively. The values are
presented in the 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 atom. Panels (d) and (e) show all eigenvalues on the Hf-O
and La-O atoms. 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 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