First-principles Study of Electronic and Dielectric Properties of ZrO 2 and HfO 2
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(a)
(b) oI-HfO2
4
Hf
4
d
2
Hf d
2
0
0 O1
4 s DOS (# of states/cell-eV)
oII-HfO2
2
O1
4 s
p
p
2
0
0 O2
4 s
2
2
0 40
O2
4
p
p
s
0 Total
Total
20
20
10
0 -20
-15
-10
-5
0
Energy (eV)
5
10
15
0 -20
-15
-10
-5
0
5
10
15
Energy (eV)
Figure 3: Same as Fig. 1, but now for the two orthorhombic phases of HfO2 . 3.77 eV for the cubic and tetragonal phases, respectively. Again, both theoretical works agree quite well. The most significant difference between these two phases and the monoclinic phase is that the Hf d-like states separate into two bands in the cubic and tetragonal phases, but form a single band in the monoclinic phase. The DOS functions for the two high-pressure orthorhombic phases are presented in Fig. 3, from which one readily sees that the oxygen atoms exhibit two quite different types of DOS, as in the monoclinic phase. In fact, there is a surprising degree of resemblance of the monoclinic DOS with that of the Ortho-I and Ortho-II phases, especially the former. The DOS functions of the ZrO2 polymorphs (not shown) are qualitatively very similar to the corresponding ones for HfO2 . The most important differences are in the band gaps. Because of LDA error, the absolute values of the band gaps are not reliable, but trends should be meaningful. As shown in Table II, we find that the band gaps are systematically larger (by ∼0.5 eV) for HfO2 than for ZrO2 , and that variations of crystal structure can lead to band gap changes of order 1 eV. Also tabulated in Table II are our calculated dielectric constants (orientational average ¯0 ) for ZrO2 and HfO2 polymorphs. For the low-pressure cubic, tetragonal, and monoclinic phases, the dielectric constants are quoted from [2] and [3], where the full dielectric tensors are given. For the Ortho-I and II phases of ZrO2 , the linear-response scheme [4] is utilized in the calculation of the static dielectric tensors, which includes the lattice contributions as well as the electronic screening. In orthorhombic ZrO2 , ¯0 becomes even smaller, approximately 20 and 18 for the Ortho-I and II phases, respectively. The dielectric tensors for the orthogrombic phases are diagonal, with elements (22.6, 18.1, 19.6) and (18.8, 18.9, and 17.8) for Ortho-I and Ortho-II respectively. The average principal values of the Born effective charge tensors are about 5.0 and 2.5 for Zr and O atoms respectively for both orthorhombic phases. CONCLUSIONS We studied the structural, electronic, vibrational and dielectric properties for nearly
N7.2.6
Table II: Band gaps (Eg ) and orientationally averaged dielectric constants (¯0 ) calculated for crystalline phases of ZrO2 and HfO2 . Eg values in parentheses for HfO2 are from the theory of Ref. [11]. Phase Cubic Tetragonal Monoclinic Ortho-I Ortho-II
ZrO2 Eg (eV) ¯0 2.63 37 3.31 38 2.98 20 3.06 20.1 2.29 18.5
HfO2 Eg (eV) ¯0 3.15 (3.40) 29 3.84 (3.77) 70 3.45 (3.48) 16–18 3.75 2.94 -
all the currently well-recognized phases of ZrO2 and HfO2 , using advanced first-principles techniques, with a sp
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