3 Local Dielectric Constant Density Analysis of High-k Dielectric Nanomaterial
77
Table 3.3 The averaged value of the polarizability density tensor and dielectric constant density
tensor between O and metal atoms in HfLaO x obtained according to Eq. (3.14–3.15). Definition of
the cylindrical region is in the text
Bond
Polarizability
Dielectric constant
La−O
0.060
5.88
Hf−O
0.033
2.60
bohr. Results are shown in Table 3.3. The average polarizability density on a La-O
bond region was 0.06 and larger than that on the Hf-O bond region (0.03). The same
tendency was also observed in the average of the dielectric constant density. At the
La-O bond, the average was 5.88 and larger than that on the Hf-O bond (2.60). For
both polarizability and dielectric constants, an averaged value between La and O
atoms was large.
We focused on the distribution of the polarizability density tensor on a plane with
O and metal atoms. Figures 3.16, 3.17, 3.18, and 3.19 show results of La 2 O 3 , mHfO 2 , c-HfO 2 , and HfLaO x , respectively. The color map shows the real parts of the
eigenvalues. The black solid line segments show the directions of the eigenvectors.
The filled circle shows the pseudopotential of the metal atom. The green line
represents the argument of the eigenvalues. Panels (a)–(c) show the first, second, and
third eigenvalues, respectively. The values are presented in the descending order of
the real parts of the eigenvalues. Panel (d) shows all eigenvalues and their average
on the line between the metal and O atoms. In the case of the HfLaO x , panel (d)
shows the results between Hf and O atoms, and panel (e) shows results between
La and O atoms. Comparing Figs. 3.16 and 3.17, the large polarizability density
was distributed almost uniformly in the region in the first eigenvalue. Moreover,
the negative value was observed on the bond region in the third eigenvalues of mHfO 2 (Fig. 3.17c). Eigenvectors of these negative values were parallel to the bond.
As shown in panel (d), the large value around 1/4π was distributed widely in the
results of La 2 O 3 , while it was very restricted around the O atom in the results of
m-HfO 2 . The dielectric constant density was divergent and changes the sign of
the polarizability density at 1/4π . Hence, the regions where the eigenvalues were
almost 1/4π produced divergently large dielectric constant density, and internal
polarization was cancelled out with an external electric field. In the case of cHfO 2 (Fig. 3.18), more complex response was observed compared with m-HfO 2 .
These results were indicative of the basic response of metal oxides. In panel (d)
of Fig. 3.19, the region where first and second eigenvalues were about 1/4π spread
widely. However, in the region between Hf and O atoms, the first eigenvalues were
over 1/4π . Moreover, the second eigenvalues between Hf and O atoms were smaller
than those between La and O atoms. These make large difference in the cylindrical
average presented in Table 3.3. Next, we turned our attention to the distribution of
the eigenvalues, as shown in panels (a)–(c). In panel (a), large values in the first
eigenvalues were widely observed. The large values of the second eigenvalues can
be seen around O, Hf, and La atoms and the bond region between the O atom and
77
Table 3.3 The averaged value of the polarizability density tensor and dielectric constant density
tensor between O and metal atoms in HfLaO x obtained according to Eq. (3.14–3.15). Definition of
the cylindrical region is in the text
Bond
Polarizability
Dielectric constant
La−O
0.060
5.88
Hf−O
0.033
2.60
bohr. Results are shown in Table 3.3. The average polarizability density on a La-O
bond region was 0.06 and larger than that on the Hf-O bond region (0.03). The same
tendency was also observed in the average of the dielectric constant density. At the
La-O bond, the average was 5.88 and larger than that on the Hf-O bond (2.60). For
both polarizability and dielectric constants, an averaged value between La and O
atoms was large.
We focused on the distribution of the polarizability density tensor on a plane with
O and metal atoms. Figures 3.16, 3.17, 3.18, and 3.19 show results of La 2 O 3 , mHfO 2 , c-HfO 2 , and HfLaO x , respectively. The color map shows the real parts of the
eigenvalues. The black solid line segments show the directions of the eigenvectors.
The filled circle shows the pseudopotential of the metal atom. The green line
represents the argument of the eigenvalues. Panels (a)–(c) show the first, second, and
third eigenvalues, respectively. The values are presented in the descending order of
the real parts of the eigenvalues. Panel (d) shows all eigenvalues and their average
on the line between the metal and O atoms. In the case of the HfLaO x , panel (d)
shows the results between Hf and O atoms, and panel (e) shows results between
La and O atoms. Comparing Figs. 3.16 and 3.17, the large polarizability density
was distributed almost uniformly in the region in the first eigenvalue. Moreover,
the negative value was observed on the bond region in the third eigenvalues of mHfO 2 (Fig. 3.17c). Eigenvectors of these negative values were parallel to the bond.
As shown in panel (d), the large value around 1/4π was distributed widely in the
results of La 2 O 3 , while it was very restricted around the O atom in the results of
m-HfO 2 . The dielectric constant density was divergent and changes the sign of
the polarizability density at 1/4π . Hence, the regions where the eigenvalues were
almost 1/4π produced divergently large dielectric constant density, and internal
polarization was cancelled out with an external electric field. In the case of cHfO 2 (Fig. 3.18), more complex response was observed compared with m-HfO 2 .
These results were indicative of the basic response of metal oxides. In panel (d)
of Fig. 3.19, the region where first and second eigenvalues were about 1/4π spread
widely. However, in the region between Hf and O atoms, the first eigenvalues were
over 1/4π . Moreover, the second eigenvalues between Hf and O atoms were smaller
than those between La and O atoms. These make large difference in the cylindrical
average presented in Table 3.3. Next, we turned our attention to the distribution of
the eigenvalues, as shown in panels (a)–(c). In panel (a), large values in the first
eigenvalues were widely observed. The large values of the second eigenvalues can
be seen around O, Hf, and La atoms and the bond region between the O atom and