80
M. Senami and A. Fukushima
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
10
0.1
10
0.1
Hf
O
O
(a)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
10
0.1
10
0.1
Hf
O
O
(b)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
10
0.1
10
0.1
Hf
O
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)
Fig. 3.18 The eigenvalues of the polarizability density tensor of c-HfO 2 . Panels (a)–(c) present
the results of the first, second, and third eigenvalues (α 1,2,3 ) on a plane with the central Hf atom
and the next O 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 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 horizontal
dotted line on the upper panel represents 1/4π . 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
density tensor over 1/4π were observed in the surrounding region of the nucleus of
the O atom. Due to its large polarizability density, the dielectric constant density
tensor has negative eigenvalues. The second one is that in the region where the
polarizability density tensor has the negative eigenvalues, one of the eigenvalues
of the dielectric constant density tensor shows the negative sign, and this negative
eigenvalue may affect the total dielectric constant of the material. At last, doping of
La atoms makes the local electric response of HfO 2 more complex compared with
pure materials. Focusing the local properties, we can clarify what decreases the
total dielectric constant and polarizability, like the negative values of polarizability
density and dielectric constant density and may obtain the knowledge to construct
new high-k materials.
M. Senami and A. Fukushima
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
10
0.1
10
0.1
Hf
O
O
(a)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
10
0.1
10
0.1
Hf
O
O
(b)
−6.0
0.0
6.0
[bohr]
−6.0
0.0
6.0
[bohr]
−0.1
0.0
0.1
10
0.1
10
0.1
Hf
O
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)
Fig. 3.18 The eigenvalues of the polarizability density tensor of c-HfO 2 . Panels (a)–(c) present
the results of the first, second, and third eigenvalues (α 1,2,3 ) on a plane with the central Hf atom
and the next O 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 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 horizontal
dotted line on the upper panel represents 1/4π . 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
density tensor over 1/4π were observed in the surrounding region of the nucleus of
the O atom. Due to its large polarizability density, the dielectric constant density
tensor has negative eigenvalues. The second one is that in the region where the
polarizability density tensor has the negative eigenvalues, one of the eigenvalues
of the dielectric constant density tensor shows the negative sign, and this negative
eigenvalue may affect the total dielectric constant of the material. At last, doping of
La atoms makes the local electric response of HfO 2 more complex compared with
pure materials. Focusing the local properties, we can clarify what decreases the
total dielectric constant and polarizability, like the negative values of polarizability
density and dielectric constant density and may obtain the knowledge to construct
new high-k materials.