Electronic structures and optical properties of monoclinic ZrO2 studied by first-principles local density approximation + U approach

The electronic structures and optical properties of the monoclinic ZrO2 (m-ZrO2) are investigated by means of first-principles local density approximation (LDA) + U approach. Without on-site Coulomb interactions, the band gap of m-ZrO2 is 3.60 eV, much lower than the experimental value (5.8 eV). By introducing the Coulomb interactions of 4d orbitals on Zr atom (Ud) and of 2p orbitals on O atom (Up), we can reproduce the experimental value of the band gap. The calculated dielectric function of m-ZrO2 exhibits a small shoulder at the edge of the band gap in its imaginary part, while in the tetragonal ZrO2 and cubic ZrO2 it is absent, which is consistent with the experimental observations. The origin of the shoulder is attributed to the difference of electronic structures near the edge of the valence and conduction bands.


Introduction
ZrO 2 is widely studied both experimentally and theoretically because of its excellent dielectric properties, wide band gap, high melting point, etc. [1,2]. Moreover, ZrO 2 has been proven to have one of the most high dielectric constants [3] and be promising in optical and protective coatings [4,5].
ZrO 2 exists in three polymorphs at atmospheric pressure [6]: the monoclinic, the tetragonal, and the cubic fluorite, denoted as m-, t-, and c-ZrO 2 , respectively. It has been known that the electronic structures and optical properties of the thermally annealed samples depend on their preparation process and the resulting structural details. In the thin films of ZrO 2 after annealing at 900 ℃, a small shoulder at the threshold of the absorption spectra has been detected [7].
The same characteristic appears in the dielectric function of as-deposited ZrO 2 films of 9.9 and 5.3 nm after rapid thermal anneals (RTA) at 600 ℃ [8]. On the other hand, no shoulder structure has been observed in both tetragonal phase and cubic phase. The exact reason about the small shoulder is still unclear. Therefore, it is essential to investigate the electronic structure of ZrO 2 and clarify the relations between the optical properties and lattice structures from the first-principles calculations.
The band structures of m-ZrO 2 have been calculated by taking into account spin-orbit interactions within the  framework of local density approximation (LDA) and general gradient approximation (GGA) [9]. The resulting band gap is about 3.58 eV, smaller than the experimental value, 5.8 eV [3]. The modified band structures incorporating the effect of electron correlations have been obtained by the GW approximation, giving rise to the band gap value of 5.34 eV, much closer to the experimental one. However, the GW calculation costs a lot in terms of numerical demands [10]. Another technique to include correlation effect with less computational effort is the so-called LDA + U or GGA + U approach [11,12]. The LDA (GGA) + U approach can give a qualitative improvement compared with the LDA (GGA) [11][12][13].
Further LDA + U f + U p , LDA + U d + U p , and GGA + U d + U p have been used for the description of CeO 2 , monoclinic HfO 2 , and cubic HfO 2 , respectively, leading to improved descriptions of the electronic structures [14][15][16]. Therefore, it is reasonable to use the approach for the investigation of m-ZrO 2 .
In this paper, we use the LDA + U scheme formulated by Loschen et al. [17], to calculate the electric structures and optical properties of m-ZrO 2 . The on-site Coulomb interactions of 4d orbitals on Zr atom (U d ) and of 2p orbitals on O atom (U p ) are determined so as to reproduce the experimental value of band gap for m-ZrO 2 . The calculated imaginary part of the dielectric function exhibits a shoulder structure at the edge of the band gap. Such a shoulder is absent in t-ZrO 2 and c-ZrO 2 , well agreeing with experimental results. The origin of the shoulder is attributed to the difference of the electronic structures near the edge of the valence and conduction bands. In our former paper [15], we have predicted that similar to m-HfO 2 , a small shoulder can also be found in the optical absorption of m-ZrO 2 . Here explicitly numerical evidence is provided.

Computational methodology
Density functional theory (DFT) calculations are performed with plane-wave ultrasoft pseudopotential, by using the LDA with CA-PZ functional and the LDA + U approach as implemented in the CASTEP (Cambridge Sequential Total Energy Package) code [18]. The ionic cores are represented by ultrasoft pseudopotentials for Zr and O atoms. For Zr atom, the configuration is [Kr] 4d 2 5s 2 , where the 4s 2 , 4p 6 , 4d 2 , and 5s 2 electrons are explicitly treated as valence electrons. For O atom, the configuration is [He] 2s 2 2p 4 , where 2s 2 and 2p 4 electrons are explicitly treated as valence electrons. The plane-wave cut off energy is 380 eV and the Brillouin-zone integration is performed over the 6 × 6 × 6 grid sizes using the Monkorst-Pack method for monoclinic structure optimization. This set of parameters assures the total energy convergence of 5.0×10 6 eV/atom, the maximum force of 0.01 eV/Å, the maximum stress of 0.02 GPa, and the maximum displacement of 5.0×10 4 Å. We calculate the electronic structures and optical properties of m-ZrO 2 by means of LDA without U and LDA + U d + U p after having optimized the geometry structure. The details of the calculation will be shown elsewhere [16].

Results and discussion
The space group of m-ZrO 2 is P2 1 /c and the local symmetry is 5 2h C . The lattice constants a, b, c, and β is experimentally determined to be a = 0.5145 nm, b = 0.5208 nm, c = 0.5311 nm, and β = 99.2°. The LDA calculation of the perfect bulk m-ZrO 2 is performed to determine the optimized parameters in order to check the applicability and accuracy of the ultrasoft pseudopotential. The optimized parameters are a = 0.5087 nm, b = 0.5175 nm, c = 0.5249 nm, and β = 99.6°, in good agreement with experimental [19] and other theoretical values [6,9,20]. However, the value of the band gap E g is around 3.60 eV, much smaller than the experimental value of 5.8 eV [3]. This is due to the fact that the DFT results often undervalue the energy of 4d orbitals on Zr atom, lowering the bottom level of conduction bands.
In order to reproduce the band gap, we first introduce U d for 4d orbitals on Zr atom. Using the experimental lattice parameters, we optimize geometry structure and calculate the band structure and density of states (DOS) of m-ZrO 2 . The band gap E g obtained from the band structure as a function of U d is shown in Fig.  1(a). It can be seen that E g firstly increases and then drops with increasing U d , showing a maximum value (4.93 eV) at U d = 11.0 eV, where the lattice parameters of the optimized structure are a = 0.5335 nm, b = 0.5201 nm, c = 0.5418 nm, and β = 98.8°. The maximum value is smaller than the experimental one (5.8 eV). The saturation of E g with U d may be related to the approach of 4d states toward 5s and 5p states, though microscopic mechanism is not yet fully understood. Next, we introduce U p for 2p orbitals on O atom, while keeping U d = 8.0 eV unchanged. The results in Fig. 1(b) show that E g monotonically increases with U p . When U d = 8.0 eV and U p = 4.35 eV, the calculated band gap of m-ZrO 2 is 5.8 eV, well coinciding with experiment. The lattice parameters of the optimized structure here are a = 0.5237 nm, b = 0.5145 nm, c = 0.5355 nm, and β = 99.2°.
The reason for adopting U d = 8.0 eV is based on the following arguments. First, note that when U d value is 11.0 eV, where the band gap reaches its maximum, the lattice parameter β is less than 90°, which indicates the occurrence of a lattice distortion in the m-ZrO 2 cell. Secondly, based on our previous results for the monoclinic HfO 2 (m-HfO 2 ), where the optimal value of U d for Hf atoms is 8.0 eV [15], we expect that the U d value for Zr atoms might be close to 8.0 eV as well. In Fig. 2, we further depict the evolution of the imaginary part 2  with the changes of U d for Zr atoms and of U p for O atoms. From Fig. 2(a), we can see that the behavior of 2  undergoes a conceivable change around U d = 8.0 eV. On the other hand, at the fixed U d = 8.0 eV, the development of 2  with the change of U p , as shown in Fig. 2(b), is quite smooth.
Besides, we present the calculations about the volume modulus of m-ZrO 2 changed with U d , and with U p while fixing U d = 8.0 eV in Fig. 3. From Fig. 3(a), it can be seen that the volume modulus changes non-monotonically with the increase of U d . The two turning points are around 2 and 10 eV. We suggest that the reasonable values for U d could be 2 eV ≤ U d ≤ 10 eV. On the other hand, as shown in Fig. 3(b), with the increase of U p (U d = 8.0 eV), the volume modulus experiences abrupt change after above 4.4 eV; while before 4.4 eV, it is almost flat. At U d = 8.0 eV and U p = 4.35 eV, the theoretical value of the bulk modulus obtained by LDA + U is 188.43 GPa, well agreeing with the experimental value 189 GPa in Ref. [21].
It should be noted that there exist different combinations of U d and U p which may reproduce the band gap value as well. For example, when U d = 11.0 eV, U p = 3.2 eV, the resulting E g is 5.807 eV; U d = 8.25 eV and U p = 4.25 eV give E g = 5.802 eV. It has been checked that these values produce similar electronic structures and optical properties. Based on the above arguments, and the reference to other numerical results, By adopting these U values (U d = 8.0 eV and U p = 4.35 eV), we perform the LDA + U calculation. The band dispersion is shown in Fig. 4(a). The bottom of the conduction band is located at the G/B point. Since the bottom shifts to higher energy with U d , accompanied by the reconstruction of the conduction band, the separated DOS at 5.6 and 8.7 eV obtained by LDA without U (not shown) merge to one sharp structure at 6.9 eV in Fig. 4(b). The conduction band is predominantly constructed by Zr 4d states, while the valence band is by O 2p states, as shown in Fig. 4(c) and Fig. 4(d), respectively. Therefore, the excitation across the gap is from the O 2p states to the Zr 4d states.  [15]. The calculated static dielectric constant is 3.98 by LDA + U d + U p (5.34 by LDA without U), coinciding with the experimental value 4.10 [22]. The imaginary part 2  shows a maximum at 9.83 eV. The maximum value (~7.99) is very close to the other theoretical value (~8.0) [9], in contrast with the value by the LDA without U (~9.78, not shown). Other optical properties can be computed from the complex dielectric function [23]. We obtain the refractive coefficient n = 1.99 of m-ZrO 2 by LDA + U d + U p (2.31 by LDA without U), which is close to the experimental value (n = 2.07 [24], 2.04 [25], and 1.932 [26]).
Additional information is provided in Fig. 5(b) where the dielectric functions of m-ZrO 2 and m-HfO 2 obtained by LDA + U d + U p approach are shown. Similar shoulder structures at the edge of the absorption spectra can be found in the monoclinic phases of the two materials.
To further build up the connection between the emergence of the shoulder structure in the optical spectrum and the underlying lattice structure, in Fig.  6(a), the imaginary parts of complex dielectric functions for m-, t-, and c-ZrO 2 by LDA + U d + U p are shown. The values of U d and U p for both phases are chosen so that the experimental gap values can be reproduced: U d = 8.25 eV (8.25 eV) and U p = 5.0 eV (4.6 eV) for t-ZrO 2 (c-ZrO 2 ). We find similar spectra between t-ZrO 2 and c-ZrO 2 with only a small shift (~1 eV) of the position; while not only the global spectral distribution but also the spectral weight at the edge of the band gap in 2  show a significant contrast between m-ZrO 2 and t-ZrO 2 . As shown in the inset of Fig. 6(a), different from t-or c-ZrO 2 , a clear shoulder structure can be found at the edge of the band gap (~6.0 eV) in m-ZrO 2 .
In order to understand the origin of the shoulder, we show in Fig. 6(b) the comparison of the total DOS between m-ZrO 2 , t-ZrO 2 , and c-ZrO 2 . We find that for both valence and conduction bands, the DOS in t-ZrO 2 and c-ZrO 2 are similar, while the DOS in m-ZrO 2 is quite different from them. In particular, at the edge of the valence band, the DOS in m-ZrO 2 smoothly increases but there is a step feature in t-ZrO 2 and c-ZrO 2 at the edge instead. Furthermore, the conduction band in m-ZrO 2 shows a broad feature of DOS around 6.5 eV. In contrast, in t-ZrO 2 and c-ZrO 2 , a peak appears near 6 eV. Since for band insulators, 2  is basically given by the cross-band excitations, the difference of DOS between m-ZrO 2 and t(c)-ZrO 2 is indicative of the exclusive presence of the shoulder in 2  for m-ZrO 2 . A more detailed microscopic origin of the shoulder, such as the assignment of the momentum and band index, remains to be solved in the future. We note that the shoulder structure also appears at the edge of the band gap in the standard LDA calculation without U, as shown in Fig. 5(a). This could mean that the reconstruction of the electronic states due to the monoclinic structure is crucial for the shoulder structure.
The experimental studies for HfO 2 have shown that HfO 2 films reveal a shoulder-like feature in 2  [27][28][29]. But the experimental data of ZrO 2 films does not show a similar shoulder [30,31]. In many aspects, ZrO 2 resembles its twin oxide, HfO 2 , though it is generally believed that the electron correlations in ZrO 2 are weaker. In our results, the shoulder-like feature in the dielectric function is also identified in the monoclinic phase of ZrO 2 , which on the other hand is absent in the other two phases (tetragonal and cubic). It is consistent with other theoretical results, e.g., the one obtained by full-relativistic calculation [9]. Here we would like to point out that in order to prepare pristine crystallized ZrO 2 with monoclinic structure, the temperature of thermal annealing should reach over 900 ℃ [7] or even higher to 1000 ℃ [32], or prepare as-deposited ZrO 2 films of 9.9 and 5.3 nm after RTA at 600 ℃ [8]. It seems to indicate that for the ZrO 2 samples used in the optical measurements, the monoclinic component may not be dominant. If the monoclinic ZrO 2 (m-ZrO 2 ) is prevalent in the mixture of the three phases [7,8], the shoulder-like structure should appear in the imaginary part of the dielectric function.

Conclusions
In summary, the on-site Coulomb interactions have been introduced for 4d orbitals on Zr atom (U d ) and 2p orbitals on O atom (U p ) of m-ZrO 2 in the first-principles LDA band structure calculation. The optimal values of U d and U p which can reproduce the experimental value of band gap are obtained. The electron structures and optical properties of m-ZrO 2 are calculated accordingly. We find a shoulder structure at the edge of the band gap in the imaginary part of the dielectric function, consistent with the experimental observation in the crystalline samples of m-ZrO 2 . We have also confirmed that such a shoulder is absent in t-ZrO 2 and c-ZrO 2 . The origin of the shoulder is attributed to the difference of electronic structures near the edge of the valence and conduction bands between m-ZrO 2 and t(c)-ZrO 2 .