Electronic band structure of zirconia and hafnia polymorphs from the GW perspective

The electronic structure of crystalline ZrO2 and HfO2 in the cubic, tetragonal, and monoclinic phase has been investigated using many-body perturbation theory in the GW approach based on density-functional theory calculations in the local-density approximation LDA . ZrO2 and HfO2 are found to have very similar quasiparticle band structures. Small differences between them are already well described at the LDA level indicating that the filled f shell in HfO2 has no significant effect on the GW corrections. A comparison with direct and inverse photoemission data shows that the GW density of states agrees very well with experiment. A systematic investigation into the structural and morphological dependence of the electronic structure reveals that the internal displacement of the oxygen atoms in the tetragonal phase has a significant effect on the band gap.


I. INTRODUCTION
Silicon-based microelectronic technology has undergone a paradigm shift at the beginning of this century replacing silicon dioxide ͑with a dielectric constant of = 3.9͒ by a gate material with a higher . 1,2Until recently the gate material of choice has been silicon's native oxide ͑SiO 2 ͒ and its nitride derivatives.The continual scaling of silica-based field-effect transistors has reduced the gate to ultrathin SiO 2 layers of ϳ1 nm thickness that exhibit excessive gate leakage currents resulting in intolerably high power dissipation and device breakdown.One of the most prominent solutions to this problem is the use of high-materials to achieve physically thicker but capacitively equivalent gate dielectrics. 1 Hafnia ͑HfO 2 ͒ and zirconia ͑ZrO 2 ͒ are currently among the most intensively studied high-materials.Both compounds combine a high dielectric constant of Ϸ 6 ϫ SiO 2 , with a large band gap, and, more importantly, high potential barriers to silicon for both holes and electrons.In a wider context, hafnia and zirconia are also frequently used in heterogeneous catalysis, as oxygen sensors and in solid oxide fuel cells.Yttria-stabilized zirconia is currently a prominent thermal barrier coating material in gas turbines for power generators and jet engines. 3[6][7][8][9][10] Zirconium ͑Zr͒ and Hafnium ͑Hf͒ belong to the same group in the periodic table and their main difference in terms of the electronic structure is the f states.Hf has a closed 4f subshell, whereas Zr has no f electrons.As a result of the "poor shielding" provided by the 4f electrons the atomic and ionic radii of Hf, although much heavier than Zr, are almost identical to those of Zr.Because of this "lanthanide contraction," Zr and Hf are often regarded as the two chemically most similar homogenesis elements. 11They do, however, differ in some minor but possibly significant aspects of their chemical and physical properties.At the atomic level, Hf has a slightly smaller electronegativity than Zr ͑1.16 vs 1.32 in terms of the spectroscopic electronegativity͒, 12 indicating a slightly stronger tendency to form ionic bonds with other more electronegative elements such as O.This small difference seems to affect the thermodynamic stability of zirconia and hafnia in contact with Si.A combined experimental and theoretical study 13 found that HfO 2 is more stable than ZrO 2 by 42 kJ/mol in the heat of formation, conversely hafnium silicides are less stable than zirconium silicides by ϳ25 kJ/ mol.As a result, the HfO 2 / Si interface is stable with respect to silicide formation, but the ZrO 2 / Si interface is not, which makes HfO 2 more suitable as a gate dielectric material.
Zirconia and hafnia can exist in several different polymorphic phases, depending on the growth conditions. 14In thin films zirconia and hafnia are typically amorphous in their as-deposited state [15][16][17][18] but nanocrystallites form during hightemperature annealing ͑typically at ϳ1000 K͒.The films preferentially crystallize in the most stable monoclinic structure but the presence of the tetragonal phase and other polymorphs is frequently observed as well. 16,17,19Different growth techniques ͑mostly physical-vapor deposition, chemical-vapor deposition, and atomic-layer deposition͒ ͑Ref.2͒ and different growth conditions produce films of varying quality with different phase compositions, which in turn influences their chemical and physical properties, including, in particular, properties that derive from the electronic structure. 27][18][19][20][21][22][23][24][25][26][27][28][29] Recent density-functional theory ͑DFT͒ calculations employing screened exchange 30 or hybrid functionals, 9,10 which partially correct the band-gap problem of more conventional DFT calculations, give band gaps in this range, too, but their dependence on adjustable parameters does not permit a unanimous identification.]31 In this work we report a first-principles study of the electronic band structure of ZrO 2 and HfO 2 .4][35] Applications to high-materials are not as frequent but are also slowly emerging. 4,5,36,37From the conceptual point of view, zirconia and hafnia are different from conventional sp semiconductors in the sense that the conduction bands ͑CBs͒ of these compounds are mainly of d character, for which the suitability of the GW approximation has not been fully established yet.9][40] Zirconia and hafnia-being the simplest transition-metal oxides-fall between normal ͑weakly correlated͒ sp systems and highly complex ͑strongly correlated͒ d-and f-electrons systems.][43][44][45][46] The paper is organized as follows.In Sec.II we briefly describe the theoretical framework and computational approach employed in this work.In Sec.III we present the electronic band structure of ZrO 2 and HfO 2 in different polymorphic phases and compare our theoretical studies with recent experiments.Section IV summarizes the main findings of this work.

II. COMPUTATIONAL METHOD
8][49][50] The self-energy in manybody perturbation theory that links the noninteracting with the interacting system is given by the product of the Green's function G and the screened Coulomb interaction W in the random-phase approximation ͑RPA͒.In practice the GW method is usually applied as a correction to Kohn-Sham ͑KS͒ density-functional theory, henceforth denoted G 0 W 0 .Further improvement can sometimes be gained by including partial self-consistency in the calculation of G with fixed RPA screening ͑W 0 ͒ ͑see, e.g., Ref. 51͒, henceforth denoted GW 0 .In this work we use both G 0 W 0 and GW 0 based on the KS single-particle Hamiltonian in the local-density approximation ͑LDA͒ and demonstrate that the GW 0 results are in general in better agreement with available experimental data.
All DFT calculations are performed using the WIEN2K package 52 in which the Kohn-Sham equations are solved in the full-potential ͑linearized͒ augmented plane wave plus local-orbital ͓FP-͑L͒APW+ lo͔ approach. 53GW calculations were performed using a recently developed all-electron GW code. 46,54Further details of the implementation will be presented elsewhere. 55,56ecently we have proposed to use the LDA with a local Hubbard-type correction ͑LDA+ U͒ as the starting point for GW calculations to overcome some of the pathologies of GW @ LDA for d-or f-electron systems. 46In this work we still apply the GW @ LDA approach for the following reasons: First, the LDA description of the materials considered here is still qualitatively correct and we expect the LDA single-particle Hamiltonian to be a suitable reference for G 0 W 0 and GW 0 calculations; second, it has been shown that GW @ LDA+ U results depend only very weakly on U for systems with empty or full d / f shells, 46,57 and GW @ LDA + U will therefore give essentially the same results as GW @ LDA.
The dielectric permittivity tensor of zirconia and hafnia is dominated by the ionic or static contribution ⑀ 0 . 14,58,59The electronic contribution ͑⑀ ϱ ͒ is a factor 5-6 smaller than ⑀ 0 . 14,58,59This is an indication for strong electron-phonon coupling in these materials, which should, in principle, be included in the dielectric function used in the GW calculations.1][62] For this reason we stick to the state of the art in GW and compute the screened Coulomb interaction with only the electronic dielectric function.We expect the band-gap renormalization due to electron-phonon coupling to be on the order of 0.1 eV or less, 63,64 which is smaller than the current experimental resolution for zirconia and hafnia.
Densities of states ͑DOS͒ are calculated using ϳ1000 k points in the Brillouin zone.A Gaussian broadening of 0.6 eV was chosen to mimic the typical instrumental resolution in the photoemission spectroscopy/inverse photoemission spectroscopy ͑PES/IPS͒ data.For GW DOS data, GW quasiparticle energies calculated on a sparse k mesh 56 are interpolated to the fine k mesh ͑ϳ1000͒ using the Fourier interpolation technique developed in Ref. 65.
͑Color online͒ Illustrations of cubic ͑left͒, tetragonal ͑central͒, and monoclinic ͑right͒ lattice structures.Large spheres represent M͑=Zr, Hf͒ atoms and small ones oxygen sites.Red ͑black͒ and pink ͑gray͒ colors distinguish oxygen atoms that are upward and downward displaced in the tetragonal structure, and the two nonequivalent oxygen atoms in the monoclinic structure.

A. Structural parameters
In this work, we consider the three low-pressure phases of MO 2 ͑M = Zr and Hf͒: cubic, tetragonal, and monoclinic, as illustrated in Fig. 1.In the simplest structure, the cubic one ͑c-MO 2 , space group Fm3 ¯m͒, M atoms form a face-centered cubic lattice and oxygen atoms occupy all tetrahedral sites, with one MO 2 formula unit in the primitive unit cell.The tetragonal structure ͑t-MO 2 , space group P4 2 / nmc͒ can be obtained by deforming the cubic unit cell along one direction ͑c / a Ͼ 1͒ and displacing alternate oxygen atoms in the direction along the tetragonal axis upward and downward with a relative distortion of d z .The monoclinic structure ͑m-MO 2 , space group P2 1 / c, also called the baddeleyite structure͒ has four MO 2 units in the primitive cell with a total of 13 struc-tural parameters; all four M sites are equivalent and sevenfold coordinated, and two nonequivalent oxygen sites are threefold and fourfold coordinated, respectively.The crystalline cubic and tetragonal structures are stable only at high temperature but can be stabilized by the presence of other lower valence metal oxides such as Y 2 O 3 , CaO, or MgO. 66In addition, the tetragonal structure has been observed in nanoparticles or ultrathin films. 67Since the main focus of this work is on the electronic properties of zirconia and hafnia, we use structural parameters from experiment taken from Ref. 14.They are listed in Table I.

B. Density of states
We first consider the DOS of different polymorphs of ZrO 2 and HfO 2 obtained in LDA, G 0 W 0 and GW 0 , as shown in Fig. 2. The most striking feature is the similarity between ZrO 2 and HfO 2 both in LDA and GW.The notable exception is the presence of the semicore f states in HfO 2 between the O 2p and O 2s bands.This similarity in the DOSs indicates that the filled f shell has no appreciable affect on the upper valence band ͑VB͒ of HfO 2 .This is also evident from the nearly identical quasiparticle corrections in zirconia and hafnia shown in Fig. 3.
Nevertheless there are several small but significant differences: ͑1͒ the band gaps in HfO 2 are slightly larger than those in ZrO 2 ͑by about 0.3 eV͒; ͑2͒ the VB width of HfO 2 is TABLE I. Experimental structural parameters for the different polymorphs of ZrO 2 and HfO 2 used in this work ͑Ref.14͒.Lattice constants a, b, and c are given in Å.In the tetragonal phase, the displacement of the oxygen atom ͑in units of c͒ with respect to the position in the ideal cubic phase is denoted as d z .The coordinates of the M ͑Zr or Hf͒ and the two nonequivalent oxygen atoms in the monoclinic phase ͑r M and r O1 , r O2 ͒ are given as internal coordinates.The experimental value of d z for t-HfO 2 is not available and due to the similarity between ZrO 2 and HfO 2 is set to that of t-ZrO 2 ͑for comparison, the optimized value by LDA is d z = 0.061͒.approximately 0.6 eV larger than that of ZrO 2 ; and ͑3͒ the crystal-field splitting in the CB is slightly stronger in HfO 2 than in ZrO 2 .In addition, c-HfO 2 has a direct minimal gap at the X point, whereas c-ZrO 2 has an indirect gap with the VB maximum ͑VBM͒ at X and the CB minimum ͑CBM͒ at ⌫.These differences are consistent with the known chemistry of Hf and Zr: the electronegativity of Hf is slightly smaller than in Zr and therefore HfO 2 is more ionic and exhibits stronger crystal-field effects than ZrO 2 . 68,69All these differences, however, are present already at the LDA level; the GW corrections ͑cf.Fig. 3͒ are nearly identical in ZrO 2 and HfO 2 , as mentioned above.

Structure
Comparing LDA and G 0 W 0 , we note that the main effects of the G 0 W 0 corrections are ͑1͒ a significant increase in the band gap by ϳ1.4 eV, ͑2͒ a slight increase in the valence bandwidth, and ͑3͒ a significant downward shift of the O 2s states by ϳ1.5 eV and for HfO 2 a downwards shift of the Hf 4f states by ϳ2.5 eV.The GW 0 results are in general quite similar to those of G 0 W 0 except that the 4f states in HfO 2 are further pushed toward lower energy by ϳ1.0 eV.The O 2s binding energy and the band gaps also increase slightly.The fact that partial self-consistency ͑GW 0 ͒ increases the 4f binding energy by ϳ1 eV is consistent with a previous study by Shishkin and Kresse, 51 who observed a similar amount of increase for 3d semicore binding energies of IIB-VI and III-V semiconductors.In the remaining part of the paper we will therefore focus mainly on the GW 0 results since the G 0 W 0 results can be easily inferred from those of GW 0 due to their similarity.
With respect to different polymorphs we again observe a remarkable spectral resemblance.The upper valence band looks almost identical in all three phases, but the O 2s and Hf 4f states lie lower in energy in the cubic phase, compared to the tetragonal and monoclinic one.Moreover, the two pronounced peaks in the conduction band of the cubic and the tetragonal phase are washed out in the monoclinic phase as a result of its lower symmetry.

C. Band structures and band gaps
More detailed information on the electronic structure can be obtained from the band structure.Considering the similarity between ZrO 2 and HfO 2 , we only show the band structure of the three different phases of HfO 2 in Fig. 4. The GW 0 corrections to the LDA band structure are essentially k independent but energy dependent.This is reflected in the near linear behavior of the quasiparticle corrections shown in Fig. 3, which is typical for sp-bonded semiconductors. 48,50,71In other words, the further a state from the VBM, the larger the GW 0 correction.
The cubic phase has a direct gap at X ͓k = ͑1,0,0͔͒ that is nearly degenerate in energy with the indirect X to ⌫ gap.The highest valence band is almost flat in the X-W ͓k = ͑0,1/ 2,0͔͒ direction, which makes the indirect gaps of the W point close in energy, too.
In the tetragonal phase the fundamental gap is indirect ͓from Z ͓k = ͑0,0,1/ 2͔͒ to ⌫͔.Again two other gaps ͑⌫-⌫ and A-⌫͒ are close in energy.The highest valence bands along the M-⌫-Z line are nearly dispersionless and would also give rise to indirect transitions in this energy range.
In the monoclinic phase, the valence band has maxima at the two points ͓⌫ and B ͓k = ͑1 / 2,0,0͔͔͒ where the conduc-  I.The VBM is taken as energy zero.The nomenclature of the high-symmetry points is taken from Fig. 8 of Ref. 70.For clarity, the LDA band structure is only shown for the cubic phase since the GW 0 corrections mostly shift the valence with respect to the conduction bands and do not change the essential features of the band structure.tion band exhibits minima.This gives four combinations of gaps that are close in energy.
The fundamental band gaps of ZrO 2 and HfO 2 in the three different phases are summarized in Table II.The band gaps from G 0 W 0 are systematically smaller by about 0.3 eV than those of GW 0 .In both ZrO 2 and HfO 2 , the magnitude of the band gap follows the order cubicϽ monoclinicϽ tetragonal.Although the band gap itself is quite different for the different phases the GW corrections vary by less than 0.1 eV.Table II also lists theoretical band gaps for ZrO 2 from Ref. 4, in which a pseudopotential ͑PP͒ plane-wave approach was used for LDA and GW 0 @ LDA.While the PP-LDA band gaps for the cubic and tetragonal phase are almost identical to our all-electron ͑AE͒ results, they differ for the monoclinic phase.The PP-GW 0 corrections to the band gaps also differ from our AE calculations ͑the PP results are ϳ0.5 eV larger͒, which is consistent with recent findings that the PP approach has a tendency to overestimate GW band gaps as a result of core-valence partitioning and pseudoization errors. 54,82,83In addition, a generalized plasmon pole ͑GPP͒ model was employed in Ref. 4 to treat the frequency dependence of the polarization function, whereas we applied the more accurate imaginary frequency plus analytical continuation approach in our calculations.As demonstrated in a systematic study by Flezar and Hanke for IIB-VI semiconductors, 40 the GPP approximation consistently overestimates band gaps by ϳ0.1 eV.

D. Comparison with experiment
Since zirconia and hafnia are traded as very promising high-materials, their electronic properties have been extensively studied by different experimental techniques in recent years, including visible-ultraviolet absorption or spectroscopic ellipsometry ͑SE͒, 16,19,22,23,81 electron-energy-loss spectroscopy ͑EELS͒, 27,28,74,77 x-ray photoemission spectroscopy ͑XPS͒ or ultraviolet plus IPS ͑Refs.26, 72, and 73͒ or x-ray absorption spectroscopy. 84Reported band gaps from these experimental studies are included in Table II.We have not attempted to group the experimental results into the three columns because samples are often thin amorphous or polycrystalline films or no structural information is provided.
The experimental band gaps for both ZrO 2 and HfO 2 are scattered between 5 and 6 eV.This is in agreement with the range we find for the different phases in our GW 0 calculations.Compared to the most recent PES+ IPS experiment of Ref. 73, the band gaps from G 0 W 0 are systematically underestimated by ϳ0.3-0.6 eV, but the GW 0 results are in much better agreement.To be more specific, the GW 0 band gap for m-ZrO 2 is still slightly underestimated by approximately 0.3 eV, but that of m-HfO 2 agrees very well.We note, however, that some caution has to be applied when comparing theoretical band gaps with experimental data.Even in techniques that probe the quasiparticle and not the optical gap, such as PES+ IPS, uncertainties arise due to finite experimental res- olution and problems in band-edge determination.Band tails from imperfections in the sample, surface adsorbants, and substrate emission can wash out the band edges 73 and the final value may depend on the chosen energy region for the fitting scheme employed to extract the band edges. 73Other techniques, e.g., optical or x-ray absorption do not probe the fundamental gap directly and additional contributions coming from, e.g., the electron-hole interaction have to be taken into account.Last but not least approximations on the theory side ͑e.g., the neglect of electron-phonon coupling [60][61][62][63][64] or vertex corrections 40,[85][86][87] ͒ would have to be taken into account, too.
It is therefore illuminating to compare our calculations directly to PES+ IPS spectra.Figure 5 shows the GW 0 density of states and the PES+ IPS data from two different groups. 26,72,73In both studies the samples are amorphous thin films and the electron kinetic energy ͑in IPS͒ is in the same range ͑18-22 eV in Refs.26 and 72 vs 20.3 eV in Ref. 73͒ but the photon energies ͑in PES͒ differ ͑soft x rays of 120-400 eV in Refs.26 and 72 vs 40.8 eV in Ref. 73͒.In general, the agreement between the GW 0 results and the experimental spectra is quite remarkable.For ZrO 2 , the two experimental data sets differ quite significantly in their position of the main IPS peak, although their edges appear at almost the same energy, while the GW 0 conduction band is slightly lower in energy.For HfO 2 , the two experimental spectra are nearly identical, and the agreement with the GW 0 DOS is even better than for ZrO 2 .The remaining difference in the peak shape between theory and experiment could be due to final state effects that are not taken into account in the theory or the fact that the experimental samples are amorphous.
Another important quantity is the binding energy of the Hf 4f states, which is often used in the characterization of hafnia samples. 26,28,73,88Since we have not included spinorbit coupling in our GW calculations we will consider only the j-averaged 4f binding energy, ⑀ ¯4f ϵ 4 7 ⑀ 4f 7/2 + 3 7 ⑀ 4f 5/2 ͑the spin-orbit splitting obtained from LDA calculations is 1.7 eV, in good agreement with experiment͒.Different experimental measurements give essentially the same 4f level energies.The measured j-averaged 4f binding energy is approximately 14.5-14.9eV with respect to the VBM.The slight variation in the experimental data sets arises mainly from uncertainties in the determination of the VB edge. 26,28,73Our GW 0 studies give 4f binding energies of 14.5, 13.9, and 13.6 eV for the cubic, tetragonal, and monoclinic phases of hafnia, respectively.For the monoclinic phase, which is the most prevalent in experiment, our GW 0 4f binding energy is underestimated by about 1 eV.This is a considerable improvement over the LDA underestimation of Ӎ4.5 eV ͑see also Fig. 2͒.

E. Structural deformations
As mentioned in the previous section, experimental values for the band gaps of ZrO 2 and HfO 2 show a large variation.One of the many possible reasons is the dependence of the electronic on the atomic structure, i.e., morphology, crystal structure, and chemical composition of the sample.These factors, although often difficult to control experimentally, can be analyzed in theoretical calculations.Here we use the transformation of the cubic to the tetragonal phase as example to investigate the effects of structural changes.The two phases can be stabilized by bivalent or trivalent dopants, which may result in different equilibrium structures than those of the pure phases ͑that are only stable at high tem-peratures͒.The tetragonal phase of HfO 2 is of particular interest for the following reasons: ͑1͒ it can form a latticematched interface with silicon with a minimum number of dangling bonds and a lattice mismatch of less than 5%; 28 ͑2͒ it can be stabilized by bivalent or trivalent oxides with a lower doping concentration than the cubic structure, and therefore is more accessible at room temperature; and ͑3͒ it can exist in polycrystalline or amorphous thin films.
We first consider the cubic phase of ZrO 2 and HfO 2 and investigate how the volume deformation ͑hydrostatic strain͒ affects their electronic properties.The latter is usually quantified by the so-called volume deformation potential, a V ϵ dE g / d ln͑V / V 0 ͒, where V 0 is the equilibrium volume.Deformation potentials are one of the key parameters in practical materials engineering and device design. 89For different materials the magnitude of a V depends strongly on the chemical bonding 89 and larger ionicity and weaker covalency typically give smaller a V .
Figure 6 shows the LDA and GW 0 band gaps of cubic ZrO 2 and HfO 2 for different unit-cell volumes.Compared to sp semiconductors with pronounced covalent bonding, 89,90 the effect of the volume deformation on the band gap is relatively weak for ZrO 2 and HfO 2 .Reducing or increasing the volume by as much as 5% leads to band gap changes of less than 0.2 eV.A linear fit yields the hydrostatic volume deformation potentials listed in Table III.Three facts are noteworthy: ͑1͒ the magnitude of a V is noticeably larger for ZrO 2 than for HfO 2 ; ͑2͒ the difference between a V ͑X-X͒ and a V ͑X-⌫͒ is much larger in ZrO 2 than in HfO 2 ; and ͑3͒ The GW 0 corrections are more pronounced in ZrO 2 .These features are consistent with the notion that HfO 2 is more ionic than ZrO 2 .Wei and Zunger concluded from a systematic investigation of diamond and zinc-blende semiconductors that the deformation potential decreases as the ionicity increases. 89Stronger covalence implies that the VBM and CBM states have stronger bonding-antibonding character, which is sensitive to the change in the bond length and varies at different k in the Brillouin zone.
We will now address the question why the tetragonal phase has a much larger band gap than the other two phases by investigating the effects of the cubic-to-tetragonal transformation on the electronic band structure of HfO 2 .We first study the band-structure evolution for c / a deviations away from 1, i.e., that of the cubic phase.We found that even at c / a =1, d z is still finite ͑ϳ0.05͒ and increases linearly as c / a deviates from 1.A noticeable feature we have observed in t-HfO 2 is that while the CBM is always located at ⌫͓k = ͑0,0,0͔͒, k corresponding to VBM ͑k VBM ͒ is very sensitive to the variation in structural parameters c / a and d z .For example, fixing a and c to experimental values and varying d z , k VBM changes from ͑ 1 2 , 1 2 ,0͒ at d z = 0.01, to ͑ 1 2 , 1 2 , 1 2 ͒ at d z = 0.03, ͑0,0, 1 2 ͒ at d z = 0.05, and ͑ 1 4 , 1 4 ,0͒ at d z = 0.07.Despite the sensitivity of the VBM to variations in the structural parameters the direct gap at ⌫ is only slightly larger than the minimal indirect gap.We will therefore mainly focus on the direct gap at ⌫.
Figure 7͑a͒ shows the gap at ⌫ as a function of c / a with d z fixed at 0.05.The band gap changes only slightly for varying c / a, and interestingly shows a maximum at c / a Ӎ 1.03, the equilibrium c / a value.Since the equilibrium value of d z is strongly correlated with c / a, we further study the effect of varying c / a with relaxed d z ͓Fig.7͑b͔͒.In this case, the band gap changes much more strongly as c / a increases, in particular, when c / a is smaller than ϳ1.03.To further clarify the role of d z , we investigate the band gap of t-HfO 2 as a function of d z at fixed lattice constants ͑a and c͒ ͓Fig.7͑c͔͒.Increasing d z from 0.01 to 0.07, the direct gap at ⌫ increases almost linearly by more than 1 eV.We can conclude from these studies that the main factor that contributes to the large band gap of t-HfO 2 is the internal distortion of the oxygen atoms.

CONCLUSIONS
To summarize, in this work we have employed manybody perturbation theory in the GW approach to investigate the quasiparticle band structure of crystalline ZrO 2 and HfO 2 in the cubic, tetragonal, and monoclinic phase.We found that the GW 0 approximation, with partial self-consistency in constructing G, gives band gaps of ZrO 2 and HfO 2 that are in better agreement with experiment than G 0 W 0 .ZrO 2 and HfO 2 have very similar electronic band structures; small differences between them can be explained by the difference in ionicity, which is already captured by the LDA.The filled f states in HfO 2 have little effect on the GW corrections.On the other hand, GW significantly increase the binding energy of the semicore f states compared to LDA, bringing them much closer to the experimental value.We observe nearly identical GW corrections for the three different phases, despite band-gap differences between them of almost 1 eV.The GW 0 density of states agrees very well with available direct and inverse photoemission spectra within the uncertainty of the experimental resolution and sample quality.With respect to structural deformations we find that the magnitude of the volume deformation potential is significantly smaller in cubic ZrO 2 and HfO 2 than in conventional sp semiconductors.For tetragonal HfO 2 a pronounced sensitivity between the relative oxygen displacement and the electronic band structures was observed that explains the larger gap of the tetragonal phase.Our work therefore indicates that the GW approach can describe the quasiparticle band structure of widegap oxides with empty d states and/or fully filled semicore f  states well.We note however, that the electron-hole ͑exci-tonic͒ and the electron-phonon interaction, which have been neglected in this study, may play a considerable role in ZrO 2 and HfO 2 , as indicated by a recent study on TiO 2 . 64Further studies in this direction and more refined experiments are clearly needed for a more unambiguous comparison between experiment and theory.

FIG. 4 .
FIG.4.GW 0 band structures of HfO 2 in the cubic ͑upper͒, tetragonal ͑middle͒, and monoclinic ͑lower͒ phase calculated at the experimental lattice parameters given in TableI.The VBM is taken as energy zero.The nomenclature of the high-symmetry points is taken from Fig.8of Ref.70.For clarity, the LDA band structure is only shown for the cubic phase since the GW 0 corrections mostly shift the valence with respect to the conduction bands and do not change the essential features of the band structure.

FIG. 6 .FIG. 7 .
FIG.6.͑Color online͒ Band-gap variation in LDA and GW 0 as a function of volume for cubic ZrO 2 and HfO 2 .V 0 is the experimental equilibrium volume.

TABLE II .
Band gaps of ZrO 2 and HfO 2 ͑in units of eV͒ obtained in this work compared to previous calculations and experimental data.

TABLE III .
Hydrostatic band-gap deformation potential ͑eV͒ of cubic ZrO 2 and HfO 2 in LDA and GW 0 .