3 Local Dielectric Constant Density Analysis of High-k Dielectric Nanomaterial
83
−1
0
1
2
0 /
i [−]
i =1
i =2
i =3
0
1
Hf
O
|sin|
(d)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
4
0.1
4
0.1
Hf
O
O
(a)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
4
0.1
4
0.1
Hf
O
O
(b)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
4
0.1
4
0.1
Hf
O
O
(c)
Fig. 3.21 The inverse of the eigenvalues of the dielectric constant density tensor of m-HfO 2 .
Panels (a)–(c) present the results of the first, second, and third eigenvalues (
−1
1,2,3 ) on a plane with
the central O atom and the next Hf atom, 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 circle shows the pseudopotential of the Hf
atom. Panel (d) shows all eigenvalues on the Hf-O line. This O atom is the upper one in panels
(a)–(c). 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 line between the
O atom and the Hf atom shows the boundary of the pseudopotential, and the left region from this
line is the inside of the pseudopotential
have also explained strong correlations of local dielectric response to internuclear
length, covalent radii, and radii of X atoms. As an example of application of these
quantities to the study of nanomaterial, hafnium dioxide was taken in this article
in viewpoints of gate dielectric thin film. Using monoclinic, cubic, and La-doped
cubic structures, we have evaluated the dielectric properties around the atoms and
the bond regions and clarified the effect of the La doping on the distribution of the
dielectric properties. We have shown that large polarizability density that makes the
negative eigenvalues of dielectric constant density can be seen around the nucleus
of the O atom, and the bond region affects the average dielectric constant strongly.
Moreover, we have also clarified that doping of La atoms makes the local electric
response of HfO 2 more complex. By focusing the local properties, we may obtain
the basic insights to construct new high-k materials.
83
−1
0
1
2
0 /
i [−]
i =1
i =2
i =3
0
1
Hf
O
|sin|
(d)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
4
0.1
4
0.1
Hf
O
O
(a)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
4
0.1
4
0.1
Hf
O
O
(b)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
4
0.1
4
0.1
Hf
O
O
(c)
Fig. 3.21 The inverse of the eigenvalues of the dielectric constant density tensor of m-HfO 2 .
Panels (a)–(c) present the results of the first, second, and third eigenvalues (
−1
1,2,3 ) on a plane with
the central O atom and the next Hf atom, 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 circle shows the pseudopotential of the Hf
atom. Panel (d) shows all eigenvalues on the Hf-O line. This O atom is the upper one in panels
(a)–(c). 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 line between the
O atom and the Hf atom shows the boundary of the pseudopotential, and the left region from this
line is the inside of the pseudopotential
have also explained strong correlations of local dielectric response to internuclear
length, covalent radii, and radii of X atoms. As an example of application of these
quantities to the study of nanomaterial, hafnium dioxide was taken in this article
in viewpoints of gate dielectric thin film. Using monoclinic, cubic, and La-doped
cubic structures, we have evaluated the dielectric properties around the atoms and
the bond regions and clarified the effect of the La doping on the distribution of the
dielectric properties. We have shown that large polarizability density that makes the
negative eigenvalues of dielectric constant density can be seen around the nucleus
of the O atom, and the bond region affects the average dielectric constant strongly.
Moreover, we have also clarified that doping of La atoms makes the local electric
response of HfO 2 more complex. By focusing the local properties, we may obtain
the basic insights to construct new high-k materials.