First-principles calculations of structural and electronic properties of monoclinic hafnia surfaces

First-principles calculations of structural and electronic properties of monoclinic hafnia surfaces Atashi B. Mukhopadhyay, Javier F. Sanz,*and Charles B. Musgrave Department of Chemical Engineering, Stanford University, 380 Roth Way, Stanford, California 94305, USA 共Received 10 November 2005; revised manuscript received 31 January 2006; published 24 March 2006兲 We have carried out a systematic theoretical study of the surfaces of monoclinic hafnia 共HfO2兲using plane waves and density functional theory based on the generalized gradient approximation. The fully relaxed structures of the bulk phases of HfO2are found to be in excellent agreement with experimental data, the monoclinic phase being the most stable. Simulations of the monoclinic phase surfaces indicate a large relaxation which reduces the total surface energy of all nine faces considered by between 23% and 36%, with a strong correlation between the unrelaxed and relaxed surface energies. Our calculations predict that the 共1 ¯ 11兲 and 共111兲faces of the monoclinic phase have the lowest surface energies and are hence the most stable faces. An analysis of the total and partial electronic density of states of bulk monoclinic HfO2reveals that the outer valence band significantly mixes the O 2pand Hf 5datomic states indicating some covalency of the Hf-O bonds. The total density and partial density of states of the monoclinic surfaces exhibit a surface state corresponding to the surface O 2sstates in the inner valence band region. DOI: 10.1103/PhysRevB.73.115330 PACS number共s兲: 68.35.Md, 68.47.Gh, 73.20.At, 81.15.Gh I. INTRODUCTION An intense search for a new high dielectric constant material to replace SiO2gate dielectrics in metal-oxide- semiconductor field-effect transistors, such as HfO2, has been undertaken to obtain insulating films with higher capacitance than thin SiO2films and with greater thicknesses to resist tunneling leakage.1–4 Hafnium dioxide 共HfO2兲has emerged as an excellent candidate due to its relatively high dielectric constant, wide band gap and stability on Si. The thermal stability of high-kmaterials in direct contact with Si and SiO2has become a key criterion in selecting suitable high-kchoices. In addition to its potential use as a gate dielectric, HfO2is also used as an optical coating because of its high refractive index and high transmission coefficient.5,6 Additionally, HfO2exhibits a relatively high laser damage threshold due to its high melting point, thermal and chemical stability and wide transparent range from the infrared to ultraviolet.7,8 Furthermore, HfO2serves as an excellent protective coating due to its thermal stability and hardness.9,10 Determination of stable surface structures of materials is required for the ab initio prediction of material properties as the relative stability of the surfaces will affect the orientations and sizes of crystallites, which in turn affect materials properties. This is especially true for the prediction of surface properties such as surface reactivity. HfO2thin films can be deposited on a variety of substrates using various methods. Techniques being explored include atomic layer deposition 共ALD兲,11 ion beam assisted deposition12 and jet vapor deposition.13 The structure of as-deposited HfO2films is not yet fully resolved. For instance, various experimental studies found that HfO2films deposited by ALD at 300 °C using HfCl4and H2O as precursors are polycrystalline, and mainly consist of the monoclinic phase with the 共1 ¯ 11兲surface exhibited as the main growth face.14,15 Upon annealing the intensities of the monoclinic peaks increase in the x-ray diffraction 共XRD兲spectra with the 共1 ¯ 11兲and 共111兲monoclinic surfaces primarily populated.14 On the other hand, Aarik et al. reported that reflections from the 共002兲,共200兲, and 共111兲 monoclinic surfaces at 300 °C displayed the highest intensities rather than the 共1 ¯ 11兲surface and that 共1 ¯ 11兲reflections were weak or absent in the plane in which the XRD pattern was recorded, including at elevated temperatures.16,17 In yet another ALD deposited film, Kukli et al. report that the film grown at low temperature shows only traces of crystallinity in the XRD pattern.18 However, reflection high-energy electron diffraction of the same film revealed 共002兲and 共1 ¯ 04兲 orientations of monoclinic phases. However, for films grown at higher temperature 共750 °C兲, the 共011兲,共1 ¯ 11兲, and 共111兲 diffraction peaks are strongest. Hence, direct comparisons to measured surface properties are difficult due to varying experimental conditions and film preparation methods. These experimental data thus only provide indirect and incomplete information about the detailed surface atomic structure. Moreover, these difficulties in interpretating different experimental studies are complicated by finite-size effects because samples are often polycrystalline, polymorphic, and contain impurities and dopants. Zr and Hf are remarkably similar elements and consequently the surface properties of HfO2are expected to be comparable to those of ZrO2. The surface properties of ZrO2 have been thoroughly investigated theoretically.19 Comparisons between our predicted properties for HfO2do show many similarities to those predicted for ZrO2, although we also find significant quantitative differences in the calculated properties. To the best of our knowledge, this contribution is the first theoretical investigation of the structural and electronic properties of HfO2surfaces. In this paper, we have investigated the surface properties of low-index surfaces of the most stable 共under the condition of atmospheric pressure and room temperature兲form of HfO2, that is, the monoclinic surfaces. We present surface energies and near surface ionic relaxations. We have also performed a detailed examination of the electronic structures of the nine inequivalent monoclinic sur- PHYSICAL REVIEW B 73, 115330 共2006兲 1098-0121/2006/73共11兲/115330共7兲/$23.00 ©2006 The American Physical Society115330-1 faces to offer a more complete understanding of the stable HfO2phases. This paper is organized as follows. The next section describes the computational aspects of this study. In Sec. III we discuss results pertaining to the bulk properties, the surface structures and ionic relaxation and surface electronic properties. In the last section we draw some general conclusions. II. COMPUTATIONAL DETAILS In order to model the extended nature of the surfaces, density functional theory 共DFT兲calculations under periodic conditions were carried out using the Vienna ab initio simulation package 共VASP兲.20,21 In these calculations the energy was calculated using the PW91 generalized gradient approximation 共GGA兲implementation of DFT proposed by Perdew et al.22,23 with the electronic states expanded using plane waves as basis set. The calculations were performed utilizing the projected augmented wave approach24 implemented in the VASP code. For Hf atoms, the semicore 5pelectrons have also been included in addition to the 5d2and 6s2valence electrons because we found that explicitly including the 5p electrons is required in order to obtain correct lattice parameters for bulk HfO2. For O atoms 2sand 2pelectrons were included. The plane-wave cutoff energy is 450 eV. The k points were generated using the Monkhorst–Pack method and 4⫻4⫻4 and 4⫻4⫻1 grid sizes were used for structural optimization of the bulk and surfaces, respectively. For calculation of electronic properties the grid sizes were increased to 10⫻10⫻10 and 10⫻10⫻1, for the bulk and surfaces, respectively. Forces on the ions were calculated using the Hellmann– Feynman theorem as the partial derivatives of the free electronic energy with respect to the atomic positions, and adjusted using the Harris–Foulkes25 correction to the forces. This approach for calculating the forces allows a geometry optimization using the conjugate-gradient scheme. Iterative relaxation of atomic positions was stopped when the change in total energy between successive steps was less than 0.001 eV. With this criterion, forces on the atoms were generally less than 0.1 eV/Å. To model the monoclinic surfaces we employed the wellknown slab approach consisting of a supercell that includes a portion of vacuum. After being replicated in the three directions of space, an array of slabs separated by a vacuum is obtained. The vacuum width was wide enough to prevent layer-to-layer interactions and we found that a width of 6 Å was enough to ensure that the energy was converged to within 0.001 eV/atom. The influence of slab thickness on the suitability of the surface models is crucial and will be analyzed in the discussion section. III. RESULTS AND DISCUSSION A. Bulk properties To check the reliability of the computational approach we apply it to calculate measured, well-known properties of the bulk phases of HfO2. At room temperature the equilibrium phase of HfO2is monoclinic which has the lowest free energy of formation and the largest volume.26–28 At approximately 1300 K monoclinic HfO2transforms into the tetragonal structure and transforms to the CaF2cubic structure near 2700 K.28 Here we fully relax both the cell lattice parameters as well as the ion positions. Table I reports our calculated structural parameters for the three phases of HfO2 we consider along with the energy of the system per HfO2 formula unit. We also include the data calculated in previous DFT studies by Zhao and Vanderbilt29 using ultrasoft pseudopotentials. We find that our values are in excellent agreement with the experimental results. In some cases and, especially for densities, our GGA results match experiment better than that of previous GGA results. We also calculate that the heat of formation of the monoclinic crystal, which is the most stable phase, is −1070.5 kJ/mol, in good agreement with the experimental value of −1144.7 kJ/mol 共7% error兲.30 Our energy per formula unit reproduces the correct energetic ordering of the phases 关EMONOCLINIC共HfO2兲 ⬍ETETRAGONAL共HfO2兲⬍ECUBIC共HfO2兲兴. B. Surface structure The ability of our approach to reproduce the experimental results for the bulk properties of HfO2indicates that it should be capable of accurately predicting HfO2surface properties. We emphasize that the purpose of this work is to determine surface properties of the perfect cut crystal. That is, we do not consider complex or extended reconstructions. The surface models we study include an integer number of HfO2 formula units and are thus stoichiometric. We follow the general rules provided by Christensen and Carter19 and design slabs to model surfaces with maximum surface compactness and minimized coordination loss. We have also avoided generating surface models that are significantly polar and thus artificially stable due to long-range electrostatic forces. Because monoclinic surfaces are more complex than other phases, the choice of a given surface index does not univocally define the surface structure in an obvious way. For our purpose, a slab corresponding to a given Miller face has an integer number of planes such that it is parallel to the surface and has a center of symmetry at the slab center. Consequently, it can be mapped onto all parallel planes below it by applying symmetry operators such as translations, screw axes or glide planes and thus the resulting slab has no dipole moment. The unrelaxed slabs have been cut from the bulk crystal, where bulk structures have been fully relaxed with respect to intracell and unit cell degrees of freedom. We have performed surface relaxations of all nine inequivalent lowindex faces of m-HfO2. All atoms in the slab have been allowed to relax, although both sides of the slab remain equivalent as they are related by an inversion, or a mirror/ glide plane, located at the center of the slab. In these calculations we do not allow the slab unit cell to relax. In Figs. 1 and 2 we only report the atomic displacements of the 共001兲 and 共1 ¯ 11兲surfaces with respect to slab thickness as these faces were most prominent during the growth of HfO2.14–17 In Figs. 1共a兲and 1共b兲we show the unrelaxed and relaxed 共001兲surfaces of a four layered slab. As can be seen in Fig. 1共a兲, there are two inequivalent surface hafnium atoms and MUKHOPADHYAY, SANZ, AND MUSGRAVE PHYSICAL REVIEW B 73, 115330 共2006兲 115330-2 two surface oxygen atoms. These inequivalent atoms are placed at different zplanes in the unrelaxed surface. However, upon relaxation these inequivalent atoms displace to almost align themselves in same zplane. Figure 1共c兲shows the quantitative displacements of inequivalent surface atoms for each direction. Both surface hafnium atoms move out of plane, however the second Hf atom 关Hf2 in Fig. 1共a兲兴 moves about 0.2 Å more in the zdirection than the first Hf atom 关Hf1 in Fig. 1共a兲兴 in order to place itself in the same plane. The displacement of O1 along the zdirection is negligible; however, the second oxygen 共O2兲is displaced by ⬃0.5 Å out of plane. Hence, the maximum displacement during relaxation is not necessarily exhibited by the outermost atoms. Furthermore, the displacement of atoms is not restricted to vertical movement, but atoms can also relax laterally. For example, O1 shifts almost by 0.4 Å in the xdirection. Because of significant subsurface relaxation a slab with at least four layers is necessary to accurately describe the surfaces to predict their surface properties. Similarly, in Figs. 2共a兲and 2共b兲we show the unrelaxed and relaxed 共1 ¯ 11兲surfaces, respectively. The 共1 ¯ 11兲face is a more complicated surface compared to the 共001兲surface. The 共1 ¯ 11兲surface has four inequivalent surface Hf and surface O atoms. The surface features are not visibly different in Figs. 2共a兲and 2共b兲. However, in Fig. 2共c兲we quantify the displacements of the surface atoms. Like the 共001兲surface, most surface atoms show out of plane relaxations. There are also significant lateral displacements of O atoms in the relaxed 共001兲structure. C. Surface energy The surface energy, Esurf共n兲, for a system comprising n layers is defined as Esurf共n兲=Etot共n兲−Ebulk共n兲 2A,共1兲 where Etot共n兲and Aare the total energy and total surface area per molecular 共HfO2兲unit, respectively.31 Ebulk refers to the energy of the bulk monoclinic system containing the same number of molecular units as the slab. Since the slab has two surfaces, the energy difference is normalized by twice the area of each surface in Eq. 共1兲. In Figs. 1共d兲and 2共d兲we show surface energy as a function of the number of layers. In Table II we have also summarized our results for all of the nine inequivalent faces. A useful tool in the analysis of re- TABLE I. Calculated structural parameters for three HfO2phases. Lattice parameters a,b,care Å, ␤ in degrees and V共volume per formula兲in Å3. Internal coordinates x,y, and zare unitless. Present GGA Previous LDAaPrevious GGAaExpt.b Cubic Energy/molecule −30.45 ¯¯¯ V32.11 31.95 36.15 32.77 a5.045 5.248 5.04 5.08 Tetragonal Energy/molecule −30.51 ¯¯¯ V32.69 32.77 37.74 a3.565 5.056 5.299 c5.146 5.127 5.373 Monoclinic Energy/molecule −30.69 ¯¯¯ V34.10 34.35 38.01 34.58 a5.079 5.106 5.291 5.117 b5.177 5.165 5.405 5.175 c5.250 5.281 5.366 5.220 ␤ 99.24 99.35 97.92 99.22 xHf 0.277 0.280 0.276 0.276 yHf 0.042 0.043 0.039 0.040 zHf 0.207 0.209 0.209 0.208 xO1 0.075 0.076 0.089 0.074 yO1 0.343 0.346 0.367 0.332 zO1 0.336 0.337 0.317 0.347 xO2 0.446 0.447 0.447 0.449 yO2 0.759 0.759 0.762 0.758 zO20.481 0.483 0.488 0.480 aReference 29. bReference 10 for cubic and Ref. 32 for monoclinic. FIRST-PRINCIPLES CALCULATIONS OF STRUCTURAL¼PHYSICAL REVIEW B 73, 115330 共2006兲 115330-3 laxation is the root mean squared 共rms兲ionic displacement reported in Table II. The rms ionic displacement is the average displacement of all N atoms in each slab relative to the geometrical center of mass ⌬rc.m: ⌬rrms =冑兺i=1 N共⌬ri−⌬rc.m.兲2 N.共2兲 This quantity will tend to decrease with increasing slab thickness because bulk ions have small or vanishing displacement upon relaxation. However, comparison of atomic displacement is a meaningful measure of relaxation as we have chosen all slabs to have approximately the same thickness of nearly 8 Å. The 共1 ¯ 11兲and 共111兲surfaces are the most stable faces of the monoclinic phase. These faces are also the ones corresponding to the most intense peaks in the XRD of various ALD films grown by various groups.14,15,18 However, some groups observed the peak corresponding to the 共111兲surface as the most intense peak along with that of the 共001兲face.4,16 This can be explained if the 共1 ¯ 11兲and 共111兲surfaces are ascribed as the thermodynamically favored surfaces while the 共001兲face could correspond to the kinetically favored surface, We predict a surface stability order for the surfaces we consider very similar to that predicted for the ZrO2sur- face. with a few exceptions.19 The predicted order of stability for ZrO2is 共1 ¯ 11兲⬎共1 ¯ 01兲⬎共111兲⬎共110兲⬎共011兲⬎共001兲 ⬎共100兲⬎共101兲⬎共010兲, whereas in HfO2we predict the stability order as 共1 ¯ 11兲⬎共111兲⬎共1 ¯ 01兲⬎共110兲⬎共001兲⬎共011兲 ⬎共101兲⬎共100兲⬎共010兲. However, the magnitude of relaxation energy for the HfO2surface is larger than that of the ZrO2surface.19 The surface energies reported in Table II indicate that the surface energy is highly anisotropic with a strong correlation between unrelaxed and relaxed surface energies. Consequently, the ordering of the unrelaxed surface stabilities is similar to that of the relaxed surface energies. The relaxation energy of the surfaces ranges from 23% to 36% of the total surface energy. We obtain a slight correlation between surface energies and the rms ionic relaxations. D. Electronic structure In this section we discuss the electronic properties of the surfaces. In Fig. 3, we report the bulk total density of states 共DOS兲obtained for the monoclinic crystal along with the partial DOS contributed by the three inequivalent atoms of the unit cell 共one Hf atom and two O atoms兲. The contribution to the DOS from Hf comes from the 5pand 5delec- trons, whereas for O, the 2sand 2pelectrons contribute. The total DOS is composed of two valence bands, a lower narrow band lying at about −16.5 eV composed of mainly O 2s states and an upper band lying between −6.6 and 0.0 eV mainly arising from O 2pstates, with a significant contribution of the Hf 5dorbitals. Note that the two inequivalent atoms have noticeably different DOS features, as expected from their different environments. The upper valence band is FIG. 1. 共Color online兲共a兲Unrelaxed and 共b兲relaxed 共001兲struc- ture of the four-layer surface model of monoclinic HfO2. The surface atoms 共Hf and O atoms兲are indicated. 共c兲Calculated displacements of surface Hf and O atoms as a function of slab thickness. 共d兲 Variation of surface energy as a function of slab thickness. MUKHOPADHYAY, SANZ, AND MUSGRAVE PHYSICAL REVIEW B 73, 115330 共2006兲 115330-4 separated by an energy gap of 3.8 eV from a conduction band composed of Hf 5dcharacter. This is in agreement with previous theoretical studies33–35 as well as with the photoemission and inverse photoemission data obtained from HfO2deposited on SiOxNy/pSi, which showed that the valence band mainly consists of O 2pnonbonding orbitals of ␲ symmetry while the conduction band is primarily Hf 5d-like nonbonding orbitals.35 However, our calculations show that the upper valence band exhibits mixing of the O 2pand Hf 5dstates, indicating covalency of the Hf–O bonds. In Fig. 4 we report the total DOS of both the unrelaxed and relaxed 共1 ¯ 11兲surfaces as well as that of the monoclinic crystal. For the sake of comparison, the energy values of the TABLE II. Surface energies for the nine inequivalent low-index faces of monoclinic HfO2. Face Surface energy 共J/m2兲%Relaxation Energy rms ionic relaxation ÅRelaxed Unrelaxed 共1 ¯ 11兲0.993 1.460 32 0.026 共111兲1.199 1.562 23 0.023 共1 ¯ 01兲1.322 1.858 29 0.033 共110兲1.388 2.043 32 0.030 共001兲1.416 2.169 35 0.060 共011兲1.484 2.100 29 0.024 共101兲1.550 2.412 36 0.047 共100兲1.667 2.165 23 0.021 共010兲1.878 2.782 33 0.040 FIG. 2. 共Color online兲共a兲Unrelaxed and 共b兲relaxed 共1 ¯ 11兲structure of the three-layer surface model of monoclinic HfO2.共c兲Calculated displacements of surface hafnium and oxygen atoms as a function of slab thickness. 共d兲Variation of the surface energy as a function of slab thickness. FIRST-PRINCIPLES CALCULATIONS OF STRUCTURAL¼PHYSICAL REVIEW B 73, 115330 共2006兲 115330-5 DOS of the surfaces in these plots have been shifted in order to align their valence band maxima with that of the bulk. Henceforth all comparisons will be made based on energies relative to the valence band maximum of the monoclinic bulk crystal. We do not present the calculated DOS of the other eight surfaces because we found them to exhibit similar features. The effect of relaxation of the slab on the total DOS is significant. Although the predicted band gaps may not be reliable 共according to the well-known DFT underestimation兲, trends in the predicted gap should be meaningful. As can be observed, the upper valence band of the bulk and relaxed 共1 ¯ 11兲surface look quite similar to each other while for the unrelaxed surface some states are pushed upwards into the lower part of the gap. As the zero of energy is referenced to the highest of these states it appears as though the entire DOS of the unrelaxed surface is shifted towards lower energy. The unrelaxed surface has a smaller band gap of 2.1 eV compared to the bulk band gap of 3.8 eV. We predict two surface states near −15.0 and 0.1 eV for the unrelaxed surface corresponding, respectively, to the 2sand 2pstates of surface oxygen. Upon relaxation of the slab, the latter disappears, however the former state is retained. The DOS of the relaxed slab is almost identical to the DOS of the bulk except for one feature at −15.0 eV. A more extensive analysis of the partial DOS shows that these extra surface states arise from an oxygen 2sorbital corresponding to one of the surface O atoms. The band gap of the relaxed surface is approximately 0.5 eV lower than that of the bulk. These features are very similar to those predicted for each of the other eight monoclinic surfaces. IV. CONCLUSIONS In summary, we have predicted structural and electronic properties of different m-HfO2surfaces. Our GGA based DFT results are in excellent agreement with the existing experimental results of structural bulk properties. Calculated surface energies demonstrate that there is a strong correlation between unrelaxed and relaxed surface energies. We also found a weak correlation between surface energies and rms ionic relaxations. We predict that relaxations are not only limited to the outer surface atoms, but that subsurface atoms undergo significant relaxation. The calculated surface energies predict that the 共111兲and 共1 ¯ 11兲surfaces are the most stable surfaces for the m-HfO2system in agreement with the appearance of these surfaces in ALD grown HfO2films. However, the surface energy of the 共001兲face, which is observed in some of the ALD grown films, is estimated to be about 0.7 J/m2larger than the most stable surfaces suggesting that such a surface could be kinetically favored under given experimental conditions. Analysis of the partial DOS of the bulk phase indicates that there is a noticeable hybridization of O 2pand Hf 5dstates, indicating a significant covalent contribution to the Hf–O bond. The electronic DOS of the relaxed and unrelaxed surfaces highlights the importance of surface relaxations and describes the nature of surface dominant states. For instance, we predict a prominent surface state near −15.0 eV contributed by surface O 2sor- bitals. The present work is intended as a preliminary step towards the understanding of HfO2surfaces and their reactivity which we hope will ultimately lead to a better understanding of the principles that govern the surface chemistry responsible for the ALD of HfO2films. A subsequent publication will provide an in depth analysis of the surface reactions of ALD precursors on these relaxed monoclinic surfaces. ACKNOWLEDGMENTS We gratefully acknowledge the support of the Stanford Initiative for Nanoscale Materials and Processing and the Materials Structures and Devices SRC/DARPA MARCO Center. This work was partially supported by the Spanish Ministerio de Educación y Ciencia, and the European FEDER, Project No. MAT2005-01872. J.F.S. thanks the Ministerio de Educación y Ciencia, the Universidad de Sevilla and Stanford University for supporting his sabbatical stay at Stanford. FIG. 3. 共Color online兲Total and partial DOS of bulk monoclinic HfO2.EVBM represents the maximal energy of the valence band. FIG. 4. 共Color online兲Total DOS of bulk, unrelaxed and relaxed 共1 ¯ 11兲surfaces of monoclinic HfO2. MUKHOPADHYAY, SANZ, AND MUSGRAVE PHYSICAL REVIEW B 73, 115330 共2006兲 115330-6 *On leave from Departamento de Química Fisica, Facultad de Quimica, E-41012 Sevilla, Spain 1P. A. Packan, Science 285, 2079 共1999兲. 2C. T. Kuo, R. Kwor, and K. M. Jones, Thin Solid Films 213, 257 共1992兲. 3M. Ritala, M. Leskela, L. Niinisto, T. Prohaska, G. Friedbacher, and M. Grasserbauer, Thin Solid Films 250,72共1994兲. 4J. Aarik, A. Aidla, A. A. Kiisler, T. Uustare, and V. Sammelselg, Thin Solid Films 340,110共1999兲. 5H. O. Sankur and W. Gunning, Appl. Opt. 28, 2806 共1989兲. 6M. Reisse, B. Keiper, S. Weissmantel, H. Johansen, R. Scholz, and T. Martini, Thin Solid Films 241,119共1994兲. 7M. Alvisi, S. Scaglione, S. Martelli, A. Rizzo, and L. Vasanelli, Thin Solid Films 354,19共1999兲. 8M. 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