84
M. Senami and A. Fukushima
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
20
1
20
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
20
1
20
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
20
1
20
1
Hf
O
O
(c)
−1
0
1
2
0 /
i [−]
i =1
i =2
i =3
0
1
Hf
O
|sin|
(d)
Fig. 3.22 The inverse of the eigenvalues of the dielectric constant density tensor of c-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 Hf atom and a next O 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 unit is 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. 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
3.6 Perspective
This analysis approach of the usage of local dielectric constant and local polarizability has been just started, and many subjects remain to be investigated. For
example, ordinary dielectric response is known to have the dependence on frequency
of electric field, and hence local dielectric constant and local polarizability have also
this dependence. However, this dependence has not been studied in any published
works. As another important issue, nuclear contribution to dielectric constant is
known to be important in hafnium dioxide. The works introduced in this article
have not included this contribution, and this important effect remains to be studied.
At last, the formalism of local dielectric constant and local polarizability is
based on quantum field theory. However, computations reviewed in this article
were performed by electronic structure computations based on quantum mechanics.
M. Senami and A. Fukushima
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−1.0
0.0
1.0
20
1
20
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
20
1
20
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
20
1
20
1
Hf
O
O
(c)
−1
0
1
2
0 /
i [−]
i =1
i =2
i =3
0
1
Hf
O
|sin|
(d)
Fig. 3.22 The inverse of the eigenvalues of the dielectric constant density tensor of c-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 Hf atom and a next O 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 unit is 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. 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
3.6 Perspective
This analysis approach of the usage of local dielectric constant and local polarizability has been just started, and many subjects remain to be investigated. For
example, ordinary dielectric response is known to have the dependence on frequency
of electric field, and hence local dielectric constant and local polarizability have also
this dependence. However, this dependence has not been studied in any published
works. As another important issue, nuclear contribution to dielectric constant is
known to be important in hafnium dioxide. The works introduced in this article
have not included this contribution, and this important effect remains to be studied.
At last, the formalism of local dielectric constant and local polarizability is
based on quantum field theory. However, computations reviewed in this article
were performed by electronic structure computations based on quantum mechanics.