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First-principles study of thermodynamic stability and the electronic properties of intrinsic vacancy defects in barium hafnate
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 435501 (http://iopscience.iop.org/0953-8984/26/43/435501) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 435501 (9pp)
doi:10.1088/0953-8984/26/43/435501
First-principles study of thermodynamic stability and the electronic properties of intrinsic vacancy defects in barium hafnate S M Alay-e-Abbas1,2 and A Shaukat1 1 2
Department of Physics, University of Sargodha, 40100 Sargodha, Pakistan Department of Physics, GC University Faisalabad, Allama Iqbal Road, 38000 Faisalabad, Pakistan
E-mail: [email protected] Received 13 July 2014, revised 24 August 2014 Accepted for publication 8 September 2014 Published 9 October 2014 Abstract
The formation of intrinsic vacancy defects in barium hafnate, BaHfO3 and their corresponding electronic structures have been investigated using first-principles calculations. The thermodynamics of pristine and vacancy defects containing barium hafnate have been analyzed. Formation energies for neutral and fully charged Ba, Hf and O vacancies have been evaluated for determining their stability with respect to different chemical environments. From the calculated electronic structure and density of states, it is found that cation deficient barium hafnate is hole-doped, while the incorporation of oxygen vacancy retains the insulating nature of this material. The defect reaction energies for partial and full Schottky reactions are also computed, which controls the properties of non-stoichiometric barium hafnate. Keywords: vacancies, theories and models of crystal defects, electronic density of states and band structure (Some figures may appear in colour only in the online journal)
for utilization in positron emission tomography (PET) and computer aided tomography (CAT) scanners [7]. In addition to its applications in electronic and optical devices, BHO has also shown a remarkable increase in the current densities for SmBa2 Cu3 Oy and Gd1 Ba2 Cu3 O7−δ superconductors, where BHO is used as a flux-pinning material, making these compounds suitable for superconducting magnetic energy storage (SMES) and magnetic resonance imaging (MRI) applications [8]. From an application point of view, physical properties of solid state materials crystallizing in perovskite structure can be tuned to desired functionalities by incorporating point defects and impurities. The earliest statistical thermodynamic models of crystal defects considered vacancies in solids to be randomly distributed. However, later progress showed that the lattice energy can be lowered by introducing long range ordering of vacancies [9]. In the case of ABO3 perovskites, the electronic, optical, magnetic and transport properties can be tuned to an extremely large range by introducing dopants and vacancy defects [7, 10]. This can be achieved by
1. Introduction
Barium hafnate, BaHfO3 (BHO), has recently emerged as a potential candidate for utilization in important electronic technologies such as microprocessor and dynamic random access memory (DRAM) [1, 2]. The environmental and health hazards of commercially dominant lead-based PbZr1−x Tix O3 (PZT) have been driving the quest for identifying alternative piezoelectric materials for the electronics industry. In this regard, BHO has found renewed research interest as it has been proposed as a constituent of environmentally friendly piezoelectric ceramics BaTiO3 –CaTiO3 –BaHfO3 [3]. The optically inactive nature, high melting point, large thermal expansion coefficient [4] and sufficiently high dielectric constant [5] make pure BHO an excellent choice for utilization as buffer layers [6]. On the other hand, Ce3+ doping in BHO yields efficient gamma ray scintillation materials with high light output and decay time constants comparable to bismuth germinate [7]. This places BHO among other fast speed transparent scintillation materials that are suitable 0953-8984/14/435501+09$33.00
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J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
varying the growth conditions which dictate the formation of vacancy defects. For instance, varying the tensile and epitaxial strain during growth can be used for controlling the socalled ‘chemical expansion’ and vacancy types, respectively [10, 11]. Similarly, the thermal radiation treatment of samples is a useful technique for introducing isolated and clustered oxygen vacancies in perovskite oxides. The resulting nonstoichiometric modification of these materials is particularly interesting as, in addition to the possibility of altering their electronic conductivity, high ionic conductivity can be achieved through mobile oxygen vacancies that make defective perovskite oxides functional materials for numerous applications [9]. Depending upon synthesis conditions [10, 11], the type of vacancy and its concentration can lead to charge carriers which are central to achieving optical and electronic properties distinct from those available in the formal stoichiometric material [2, 12, 13]. There is an ever-increasing demand for smarter solid state materials for fulfilling the requirements of sensors and, importantly, different components of solid oxide fuel cells to ensure efficient conversion of chemical energy into electrical energy. This has resulted in an enormous upsurge in research activities focused on defective perovskite oxides [14]. In recent decades, many theoretical and experimental studies exploring the electronic, optical and electrical properties of BHO have appeared in literature [15–18]. Density functional theory (DFT) based first-principles techniques have been effectively utilized by Zhao et al [16], Liu et al [19] and Yangthaisong [20] in an attempt to enrich the information about the electronic, optical, mechanical and bonding properties of bulk barium hafnate. On the other hand, the bare (0 0 1) surfaces of BHO have also been explored in terms of thermodynamic stability and electronic structure [21]. However, intrinsic vacancy defects and the manipulation of electronic structure through vacancy defect incorporation in wide band gap BHO [22] are yet to be explored. In this work, we will perform a series of DFT calculations using the full-potential linear augmented plane wave plus local orbital (FP-LAPW + lo) method for vacancy formation in barium hafnate and investigate its related effects on the electronic structure. The outcome of this study is expected to be important in view of the rapidly increasing use of BHO in electronic, magnetic and optical devices.
18, respectively. The iterative convergence of self-consistent calculations is carried out until the variation in ground state total energy is less than 10−4 Ry. To ensure elimination of residual forces on atoms resulting from vacancy incorporation in BHO, the forces on each atom have been minimized to a value less than 2 mRy/a.u. The cubic perovskite structure of BHO belongs to ¯ crystallographic space group #221 (P m3m) having an experimental lattice parameter of 4.167 Å [25]. In order to compute the electronic structure of pristine and defective BHO, we first performed volume optimization of the perovskite unit cell using PBE GGA and a dense 12 × 12 × 12 k-mesh. Subsequently, a 40 atom 2 × 2 × 2 supercell (SC), having composition Ba8 Hf8 O24 , has been derived from the optimized BHO unit cell for calculations involving pristine as well as defective structures. Neutral and fully charged isolated Ba 2− 4− 0 0 /VBa ), Hf (VHf /VHf ) and O (VO0 /VO2+ ) vacancies in (VBa BHO are realized by removing one Ba, Hf or O atom from 2×2×2 SC. The vacancy concentrations of cation and oxygen vacancy in 2 × 2 × 2 SC correspond to 12.5% and 4.167%, respectively, for which energy and forces minimization are achieved using a 6×6×6 k-mesh. In the current study, we have neglected spin-polarization after performing test calculations for vacancy containing SC, which confirmed that the inclusion of spin-polarization has a negligible effect on the calculated thermodynamic and electronic properties. 3. Results and discussion 3.1. Thermodynamic stability of barium hafnate
In order to obtain the stability diagram shown in figure 1, we first compute chemical potential limits for all atomic species in BHO, which dictate the formation energies of cation and oxygen vacancies [26]. This is because the chemical potential of an isolated atom, µx , is less than the chemical potential of solid/gas that atom in its stable solid or gaseous state (i.e. µx ) [27]: µx = µsolid/gas + µx . x
(1)
In the above equation µx 0 denotes the variation in the chemical potential of species x. For computing valid limits of µx for barium, hafnium and oxygen, the chemical potentials should be varied in such a way that their sum equals the enthalpy of formation of barium hafnate [28]:
2. Method of calculations
The generalized gradient approximation (GGA) parameterization scheme of Perdew, Burke and Ernzerhof (PBE [23]) has been employed for exchange-correlation energy contribution by using the WIEN2k implementation of the FP-LAPW + lo method [24]. A muffin-tin model for wave function, charge density and potential has been assumed within which spherical harmonics and plane wave basis sets are utilized for muffin-tin spheres and interstitial regions, respectively. The muffin-tin radii for Ba, Hf and O atoms are chosen to be 2.5 a.u., 1.98 a.u. and 1.75 a.u. respectively. For all the calculations, plane wave cut off (Ro ×Kmax ) and maximum values of angular momentum (lmax ) and charge density expansion (Gmax ) are set to 7.0, 10 and
HfBaHfO3 = µBa + µHf + 3µo .
(2)
Furthermore, stable production of BHO in equilibrium with binary compounds BaO and HfO2 can be achieved when the (µBa , µHf , µO ) values in equation (2) satisfy: µBa + µO HfBaO
(3)
µHf + 2µO HfHfO2 .
(4)
The conditions in equations (1) through (4) can be met by computing the enthalpies of formations of BaO, HfO2 and 2
J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
Table 1. Comparisons of the calculated lattice parameters (Å) and enthalpy of formations (eV/formula unit) with experimental data.
Experiment ¯ BaO (space group # 225, F m3m) ao 5.539a Hf −5.680b HfO2 (space group # 14, P 21 c) ao , bo /ao , co /ao 5.117, 1.011, 1.034c Hf −11.864d ¯ BaHfO3 (space group # 221, P m3m) ao 4.167e Hf −18.933f a
BaHfO3 from DFT total energy calculations by using the following equations:
Hf hcp
HfHfO2 = EtHfO2 − Et
1 − 2 EtO2 2
5.577 −5.554 5.137, 1.014, 1.039 −11.268 4.208 −19.129
[30], b [31], c [32], d [33], e [25], f [34].
agreement with the stability region that can be obtained from the experimental Hf . The chemical potential values at point A and point B represent the extreme O-rich (oxidation) condition. Point A also represents the extreme Hf-poor condition, whereas the extreme Ba-poor condition is represented by point B. The O-poor/Hf-rich and Ba-rich (reduction) conditions are represented by point C and point D, respectively. The comparison of the oxygen chemical potentials of all the points shown in figure 1 depicts a large variation (−5.635 µO 0). Moreover, a series of lines parallel to A–B in the direction of the arrow pointing to the O-rich condition (µO = 0) also indicates the possibility of achieving a large variation in the chemical potentials of barium and hafnium. Therefore, the chemical potentials of Ba and Hf can also be tuned over a large range. In figure 1 the (µBa , µHf , µO ) coordinates of point X are also provided which is located in the center of the stability region (in between O-rich and O-poor conditions). The stability points shown in figure 1 provide us with the range of chemical potentials which dictate the formation of vacancy defects in barium hafnate.
Figure 1. Thermodynamic stability diagram of BaHfO3 showing chemical potential values at points A, B, C, D and X as an inset. The stability of barium hafnate is confined to the gray region enclosed in quadrangle ABCD. The O-rich condition is satisfied at both points A and B and for all points on lines drawn parallel to A–B away from the origin, while point C represents the extreme O-poor condition where the oxygen chemical potential is −5.635 eV.
1 HfBaO = EtBaO − EtBa fcc − EtO2 2
Calculated
(5) (6)
1 − 3 EtO2 (7) 2 where the minimum total energies of face centered cubic (fcc) Ba, hexagonal close packed (hcp) Hf, BaO, HfO2 and BHO for fcc hcp optimized unit cell volumes are represented by EtBa , EtHf , BaHfO HfO 3 EtBaO , Et 2 , and Et , respectively. On the other hand, O2 Et , is the minimum total energy of a relaxed O2 dimer. The calculated lattice parameters and enthalpies of formation for barium oxide, hafnium oxide and barium hafnate are listed in table 1. One can see that these values are in good agreement with experimental data and confirm the reliability of the formation energetics of vacancy defects in barium hafnate, as will be discussed later. A comparison of calculated and experimental enthalpy of formation values clearly indicates that HfBaHfO3 < HfBaO + HfHfO2 , is in accordance with the requirement for stable production of BHO from its binary constituents BaO and HfO2 [29]. Using the enthalpy of formation values presented in table 1, we have drawn the thermodynamic stability diagram shown in figure 1 where the equilibrium of BHO is maintained without any secondary phases for the (µBa , µHf , µO ) coordinates inside the quadrangle ABCD. From figure 1, it is evident that above the lines AD and BC, the chemical potential region for stable production of BHO is limited by BaO and HfO2 , respectively. The stability region of BHO is thus confined to the gray area which is in good Hf hcp
HfBaHfO3 = EtBaHfO3 − EtBa fcc − Et
3.2. Isolated neutral vacancies in BaHfO3
The structurally optimized pristine and defective SCs shown in figure 2 provide a visual description of the relaxation of atomic positions caused by the incorporation of anion and cation vacancies. After the minimization of Hellman–Feynman forces on the atoms surrounding vacancy sites, both bond lengths and bond angles of nearest neighbor (NN) and next-nearest neighbor (NNN) atoms deviate from their ideal values. The calculated NN and NNN distances in pristine as well as neutral cation and oxygen vacancies in a SC of BHO are compared in table 2. 0 0 In the case of VBa /VHf , the first NN O atoms move away from the vacancy site, whereas the second NN Hf/Ba atoms move inward. The outward relaxation of the NN O atom in the 0 0 case of VBa is smaller (2.967%) compared to VHf (4.363%). 0 0 In the case of NNN Hf/Ba atoms around VBa /VHf , vacancy, the inward relaxation towards the vacancy site amounts to 1.250%/5.135%. These values are in accordance with the stronger covalent bonding between linearly coordinated O and Hf atoms. In addition to the above effects, the Ba–O and 0 Hf–O bond lengths near the VBa site increase to 2.977 Å and 2.131 Å, whereas these bond lengths decrease to 2.837 Å and
3.2.1. Structural and bonding properties.
3
J. Phys.: Condens. Matter 26 (2014) 435501
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Figure 2. (a) Pristine and defective 2 × 2 × 2 SC of BaHfO3 . The relaxation of atomic positions resulting from incorporation of charge neutral Ba, Hf and O vacancies are shown in (b), (c) and (d), respectively. Ba, Hf and O atoms are represented by black, gray and red spheres, respectively.
The variations in the bond lengths around the vacancy sites are also accompanied by changes in the cation–O–cation bond angles (Ba–O–Ba = 90◦ and Hf–O–Hf = 180◦ in bulk 0 BHO) which can also be seen in figure 2. For VBa both Ba–O–Ba and Hf–O–Hf bond angles around the vacancy site change to 91.761◦ and 172.932◦ . On the other hand, only the Ba–O–Ba bond angles decrease to 89.718◦ while Hf–O–Hf 0 . These changes in remains unchanged in the case of VHf the bond angles are in accordance with the relaxations and coordination environments of atoms surrounding the vacancy sites. The changes in the bond angles around the vacancy site, brought about by the incorporation of VO0 , are very small (Ba–O–Ba = 89.898◦ and Hf–O–Hf = 179.677◦ ). This is due to simple 2-fold linear coordination and 4-fold square planar coordination of O atoms with Hf and Ba atoms, respectively. In figure 3 the charge density contour plots are displayed, which reveal the nature of bonding between Ba/Hf and O atoms and the changes in bonding nature and charge distribution resulting from vacancy defects. It is evident that Hf–O bond is more covalent compared to the Ba–O bond. This can be attributed to the relatively larger electronegativity value of 0 Hf on the Pauling scale. The charge density plots for VBa 0 and VHf (figures 3(b) and (e)) compared to pristine BHO, demonstrate that the O-2p orbitals surrounding the vacancy sites become localized, thus giving rise to acceptor-like levels near the valence band maximum (VBM). A comparison of the charge density plots around the cation/anion sites in the
Table 2. Nearest neighbor distances (Å) for pristine (atom–atom) and defective (vacancy site–atom) BaHfO3 SC. In the case of defective 2 × 2 × 2 SC, distances larger/smaller than their corresponding pristine values depict outward/inward relaxation of the nearest neighbor atoms relative to the vacancy site. The neighboring atom, along with its coordination number, is shown in parenthesis.
Ba Pristine BaHfO3 0 VBa Hf Pristine BaHfO3 0 VHf O Pristine BaHfO3 VO0
NN
NNN
(O × 12) 2.976 3.067 (O × 6) 2.104 2.200 (Hf × 2) 2.104 2.108
(Hf × 8) 3.644 3.599 (Ba × 8) 3.644 3.466 (Ba × 4), (O × 8) 2.976, 2.976 2.980, 2.966
0 2.008 Å in the case of VHf . In accordance with the electrostatic interactions among cation and anion, the incorporation of VO0 results in an outward relaxation of NN and NNN cations, while the NNN O atoms undergo an inward relaxation towards the oxygen vacancy site. Consequently, the Ba–O/Hf– O bond length near the oxygen vacancy site decreases to 2.974 Å/2.010 Å. This may be further understood from the analysis, which is given below, of the bonding nature in Ba–O and Hf–O bonds.
4
J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
Figure 3. The charge density contour plots for pristine and barium, hafnium and oxygen vacancy containing BaHfO3 . The top and bottom panels display the charge densities in BaO and HfO planes, respectively. Ba, Hf and O atoms are represented by black, gray and red spheres, respectively.
BaO and HfO planes for pristine and cation/anion deficient cases shows maximum charge localization/delocalization. For instance, it is clear that more charge localization occurs for hafnium vacancy compared to barium vacancy. This means that the presence of electrons in the unoccupied states above 4− would Fermi level (EF ) for a charged hafnium vacancy VHf 2− . result in higher positioning of these states compared to VBa A comparison of charge densities for pristine (figures 3(a) and (d)) and VO0 containing BHO (figures 3(c) and (f )) depicts the elimination of the covalent bonding between the metal atom and O vacancy site. In the case of oxygen vacancy, the charge density shows more delocalization in the HfO plane, which can be attributed to dangling Hf-5d orbitals on the creation of VO0 . Consequently, this yields an occupied defect level near the conduction band minimum (CBM) in the band structure of oxygen deficient BHO (see next section).
are located 1.160 eV above the occupied defect level at the symmetry point. The shifting of the EF near the CBM occurs to conserve the semiconductor nature of barium hafnate, whereas bulk-like states of valence band appearing around −3 eV in figure 4(d) are separated from this defect level by an energy difference of 2.478 eV. In order to elucidate the contribution of cation and anion orbitals for pristine and defective BHO in the vicinity of EF , figure 5 shows the calculated partial density of states (PDOS) corresponding to the band structure diagrams (figure 4). It can be seen that the valence band states near EF are predominantly made up of the O-2p orbitals with insignificant contributions from Ba and Hf, while the CBM is dominated by the Hf-5d orbitals. Consequently,the direct (–) band gap appears between the occupied O-2pand unoccupied Hf−5d orbitals. In the case of cation vacancies, the defect levels appearing just above the VBM at point assume an anion orbital-like character, resulting in the creation of triply degenerate O-2p hole states. These hole states appear 0.117 eV and 0.186 eV 0 0 above the EF for VBa and VHf , respectively. The higher position 0 of the VHf unoccupied hole-doped states is in accordance with its higher charge state and lower oxygen coordination. Until the synthesis of polycrystalline barium hafnate by Maekawa et al in 2006 [4], only a few studies were available in literature addressing its physical properties. For this reason, theoretical investigations exploring the electronic properties of BHO like the ones by Vali [35] and Yangthaison [20] assumed that the experimental band gap of BHO would be around 6.0 eV. However, in recent spectroscopy ellipsometry and refractometry studies of BaHfO3 thin films grown on TiN and Si revealed a fundamental band gap of 4.8 eV [15]. As
The band structures of pristine and defective 2×2×2 barium hafnate SCs calculated using the PBE GGA are displayed in figure 4. The VBM and CBM of pristine BHO are both located at the symmetry point and have a band gap (Eg ) amounting to 3.616 eV which is comparable with previous pseudopotential calculations (3.783 eV [35] 0 0 and VHf in and 3.610 eV [20]). The incorporation of VBa BHO gives rise to unoccupied energy levels which are located slightly above the EF as evident from figures 4(b) and (c). These unoccupied energy levels lead the cation deficient BHO into a hole-doped state. On the other hand, neutral oxygen vacancy yields a defect level, occupied by 2 electrons, which appears below the CBM. Consequently, EF is shifted near the CBM such that the unoccupied conduction band states 3.2.2. Electronic properties.
5
J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
0 0 Figure 4. Calculated electronic band structures of (a) pristine BaHfO3 SC and defective SCs containing (b) VBa , (c) VHf and (d) VO0
vacancies.
expected from GGA band structure calculations, our calculated band gap is underestimated compared to the experimental data. Calculations of formation energy for anion defects in perovskite require band gap correction for predicting correct trends of vacancy formation [28, 36]. Consequently, a rigid shift of the GGA CBM (∼1.184 eV) is required as band gap correction places the defect level for VO0 containing BHO at 3.662 eV above the bulk-like valence band states shown in figure 4(d). This demonstrates that BHO is inactive in the visible region in accordance with experimental observations [15] supporting the band gap correction for formation energy calculations of oxygen vacancies in the next section.
where EtBaHfO3 and Et [X q ] are the minimum total energies of pristine and defective SCs. The vacancy type and its charged state are represented by X and q, respectively. The quantity µx is the chemical potential of species X which can be varied according to the stability points shown in figure 1. EF +EVBM represent the position of the Fermi level as measured from the VBM where EVBM has been corrected following the procedure outlined in [36]. In the case of oxygen vacancies, the band gap correction has been realized by adding mEg = m (1.184eV ) (where m is the number of electrons in the vacancy induced level having value 2 for VO0 and 0 for VO2+ ) in the calculated formation energy values given by equation (8). As a consequence of adopting band gap correction for oxygen vacancies, the transition levels near the calculated CBM are shifted near the experimental CBM [37]. It is worth noting that due to the anionorbital-like character of the defect states appearing above the VBM in case of the metal atom (Ba and Hf) vacancies, the band gap correction is only applied to the calculated formation energies of oxygen vacancies.
The formation energies ( [X q ]) of neutral and fully charged states of Ba, Hf and O vacancies in BHO have been computed by [28]
3.2.3. Formation energetic.
Xq = Et X q − EtBaHfO3 − µx + q(EF + EVBM )
(8) 6
J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
Figure 5. PDOS plots for pristine and defective BaHfO3 SC showing the orbital contribution of Ba-6s, Hf-5d and O-2p states in the vicinity 0 0 of EF . (a) pristine SC, (b) VBa , (c) VHf and (d) VO0 .
0 formation by hafnium 0 0 energy ( Ba = 2.807 eV) followed ( O = 9.429 eV). ( Hf = 3.963 eV) and oxygen Point B (Ba-poor condition) Ba0 = 0.501 eV, indicates that charge neutral barium vacancy is the most favorable form of vacancy defect in BHO under metal-poor conditions. This also remains true for point X. However, at point C (extreme O-poor condition) the formation energy of neutral oxygen vacancy ( O0 = 3.794 eV) becomes smaller than the 0 0 formation energies of both VBa VHf . Similar behavior0 and 0 is evident at point D where O = 3.874 eV makes VO formation easier compared to neutral Ba and Hf vacancies. A comparison of the formation energies of neutral and fully changed in this work (figure 6), reveals considered vacancies, that Baq and Baq increase, whereas [Oq ] decreases on going from O-rich condition (point-B) to O-poor condition (point-C). Since it is important to verify that the size of SC (2 × 2 × 2) is sufficient for the formation energy calculations presented in this work, we have computed the total energy of oxygen vacancy in a larger (60 atom) SC. Our calculated value for 2.778% VO0 concentration is only 0.066 eV larger than the 0 O value in the 2 × 2 × 2 SC. The Hf-poor O2+ has a lower value condition, 2− . However, even smaller values of compared 0 to Ba Ba indicate the neutral barium vacancy to be the most favorable vacancy defect at point A. On the other hand, both 2− 0 and VBa assume formation energies less than all other VBa
Figure 6. Formation energies (eV/defect) of isolated neutral and fully charged Ba, Hf and O vacancies in barium hafnate at the calculated stability points shown in figure 1.
The formation energies for barium, hafnium and oxygen vacancies in various chemical environments are shown in figure 6. In the case of charge neutral vacancies, it is evident that under O-rich (Hf-poor) conditions, barium has the lowest 7
J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
Figure 7. Variation of [X q ] as a function of Fermi energy (EF ) at stability points A, X, C and D. Charge states of the vacancies are given
as numbers below their respective lines.
C and D VO2+ has the lowest formation energy value near the VBM (EF = 0 eV), whereas at point A (also at point B which is 0 not shown in figure 7) VBa attains the lowest formation energy 4− value for EF = 0 eV. VHf . On the other hand, it has the lowest value of formation energy at the CBM (i.e. EF = 4.8 eV) 2− for points A, X and D, while VBa has the lowest formation energy value for EF = 4.8 eV at point C (Hf-rich condition). The variation of formation energies of fully charged cation and oxygen vacancies with EF , results in negative values inside the band gap which gives us the ‘pinning energy’ Epin [40]. The Epin energy can be defined as a parameter which in the case of oxygen/metal vacancy restricts barium hafnate from assuming a Fermi level which is closer to the VBM/CBM than the value of Epin . It can be seen that for metal-poor conditions (point A and point B) Fermi level pinning does not occur for oxygen vacancies. This means that EF is located inside the valence band which is in accordance with the electronic band structures shown in figures 3(b) and (c). However, at point X, C and D O2+ becomes smaller than the formation energies of fully charged barium and hafnium vacancies for EF values near the VBM. This indicates that oxygen vacancy would compensate anion vacancies from having acceptor-like levels in the band gap [30]. It is worth noting that the vacancy formation in barium hafnate and its corresponding effects reported in this study will be useful in future research work on barium hafnate and might serve as a guide to its device application.
vacancy types at the Ba-poor limit (point B). At points X, C and D the smaller values of VO2+ reveal that under ideal growth conditions (i.e. at point X) as well as Ba-rich and Hf-rich conditions, fully charged O vacancy is the most abundant form of vacancy defect in barium hafnate, which is in accordance with experimental observation [17, 38]. The calculated formation energies of fully charged Ba, Hf and O vacancies given in figure 6 can be used for computing Schottky reaction energies which help in understanding the creation of cation and anion vacancies in such a way that stoichiometry of the material remains intact as a whole. The Schottky reaction energies are important in controlling properties of non-stoichiometric perovskite oxides under various growth conditions [39]. We have computed values for the full 2− 4− Schottky reaction VBa + VHf + 3VO2+ and barium/hafnium 2− 4− partial Schottky reaction VBa + VO2+ /VHf + 2VO2+ . The calculated barium/hafnium partial Schottky reaction energies vary between minimum (2.685 eV/4.634 eV) and maximum (3.838 eV/5.402 eV) values such that the average value is 3.262 eV/5.114 eV. Contrary to that, the reaction energy for the full Schottky reaction is 4.316 eV for all stability points. The variation of formation energies of barium, hafnium and oxygen as a function of EF shown in figure 7 clearly suggest that fully charged states of these defects are stable over a wide range of the barium hafnate band gap. At point X, 8
J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
[10] Adler B 2001 J. Am. Ceram. Soc. 84 2117 [11] Woodward D I, Reaney I M, Yang G Y, Dickey E C and Randall C A 2004 Appl. Phys. Lett. 84 4650 [12] Kan D et al 2005 Nat. Mater. 4 816 [13] Ourmazd A and Spence J C H 1987 Nature 329 425 [14] Fleig J 2003 Annu. Rev. Mater. Res. 33 361 [15] Fursenko O, Bauer J, Lupina G, Dudek P, Lukosius M, Wenger C and Zaumseil P 2012 Thin Solid Films 520 4532 [16] Zhao H, Chang A and Wang Y 2009 Physica B 404 2192 [17] Lopez-Garcia A, de la Presa P, Rodriguez A M, Saitovitch H and Silva P R J 1993 Phys. Rev. B 47 84 [18] Dobrowolska A and Zych E 2010 Radiat. Meas. 45 621 [19] Liu Q J, Liu Z T, Feng L P and Tian H 2010 Physica B 405 4032 [20] Yangthaisong A 2013 Phys. Lett. A 377 927 [21] Ni G X and Wang Y S 2009 Chin. Phys. B 18 1194 [22] Evarestov R A 2011 Phys. Rev. B 83 014105 [23] Perdew J P, Burke K and Ernzerhof M 1997 Phys. Rev. Lett. 78 1396 [24] Blaha P, Schwarz K, Madsen G K H, Kvasnicka D and Luitz J 2001 WIEN2K, an Augmented Plane Wave p Local Orbitals Program for Calculating Crystal Properties (Austria: Techn. Universitat Wien) [25] Li L, Kennedy B J, Kubota Y, Kato K and Garrett R F 2004 J. Mater. Chem. 14 263 [26] Zhang S B and Northrup J E 1991 Phys. Rev. Lett. 67 2339 [27] Erhart P and Albe K 2007 J. Appl. Phys. 102 084111 [28] Tanaka T, Matsunaga K, Ikuhara Y and Yamamoto T 2003 Phys. Rev. B 68 205213 [29] Lide D R 2007 (ed) CRC Handbook of Chemistry and Physics (Boca Raton, FL: CRC Press) [30] Liu L G and Bassett W A 1972 J. Geophys. Res. 77 4934 [31] Chase M W Jr 1998 NIST-JANAF Thermochemical Tables Fourth Edition (J. Phys. Chem. Ref. Data, Monograph) (Washington: American Chemical Society and American Physical Society) [32] Hann R E, Suitch P R and Pentecost J L 1985 J. Am. Ceram. Soc. 68 285 [33] Speight J G 2005 Lange’s Handbook of Chemistry 16th edn (New York: McGraw-Hill) [34] Stølen S and Grande T 2004 Chemical Thermodynamics of Materials: Macroscopic andMicroscopic Aspects (Chichester, England: John Wiley) [35] Vali R 2008 Solid State Commun. 147 1 [36] Van de Walle C G and Neugebauer J 2004 J. Appl. Phys. 95 3851 [37] Persson C, Zhao Y-J, Lany S and Zunger A 2005 Phys. Rev. B 72 035211 [38] Lopez-Garcia A, de la Presa P and Rodiguez A M 1991 Phys. Rev. B 44 9708 [39] Lee S, Liu Z-K, Randall and Clive A 2008 Comprehensive linkage of defect equilibrium and phase equilibria through ferroelectric transition behavior in BaTiO3 -based dielectrics ISAF 2008 17th IEEE Int. Symposium on the Applications of Ferroelectrics (Santa Fe, New Mexico, 24–27 February 2008) [40] Zhang S B, Wei S-H and Zunger A 2000 Phys. Rev. Lett. 84 1232
4. Conclusions
The electronic structure and formation energies of isolated vacancies in barium hafnate have been investigated for the first time using first-principles calculations. It is shown that pristine and isolated oxygen vacancy containing barium hafnate are semiconducting, whereas Ba and Hf deficient barium hafnate is found to be hole-doped. The defect level induced by neutral O vacancy appears near the conduction band which is composed of Hf-5d orbitals. On the other hand, Ba and Hf vacancies give rise to unoccupied O-2p states just above the VBM. Neutral Ba vacancy has the lowest formation energy values for metal-poor oxidation conditions, whereas fully charged oxygen vacancies are found to be the most favorable form of vacancy defect in barium hafnate under reduction conditions. The calculated Schottky reaction energies indicate that barium partial Schottky has smaller reaction energy compared to both hafnium partial and full Schottky reactions. A comparison of formation energy values with available experimental and theoretical data of other perovskite oxides confirms the consistency of DFT-GGA results presented in this paper. These findings might help future studies exploring high temperature applications of BHO for electronic and optical devices. Acknowledgments
The authors are grateful to S Nazir for useful discussions. The calculations were performed using computational resources supported by the University of Sargodha. References [1] Lupina G et al 2009 Chem. Vap. Deposition 15 167 [2] Lupina G, Dabrowski J, Dudek P, Kozlowski G, Lukosius M, Wenger C and M¨ussig H-J 2009 Adv. Eng. Mater. 11 259 [3] Wang D, Jiang Z, Yang B, Zhang S, Zhang M, Guo F and Cao W 2014 J. Mater. Sci. 49 62 [4] Maekawa T, Kurosaki K and Yamanaka S 2006 J. Alloys Compounds 407 44 [5] Lupina G, Kozlowski G, Dabrowski J, Wenger C, Dudek P, Zaumseil P, Lippert G, Walczyk C and Mssig H J 2008 Appl. Phys. Lett. 92 062906 [6] Zhang J L and Evetts J E 1994 J. Mater. Sci. 29 778 [7] Van Loef E V, Higgins W M, Glodo J, Brecher C, Lempicki A, Venkataramani V, Moses W W, Derenzo S E and Shah K S 2007 IEEE Trans. Nucl. Sci. 54 741 [8] Tsuruta A, Yoshida Y, Ichino Y, Ichinose A, Matsumoto K and Awaji S 2013 IEEE Trans. Appl. Supercond. 23 8001104 [9] Tilley R J D 2008 Defects in Solids (Hoboken, NJ: John Wiley and Sons)
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First-principles study of thermodynamic stability and the electronic properties of intrinsic vacancy defects in barium hafnate
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 435501 (http://iopscience.iop.org/0953-8984/26/43/435501) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 435501 (9pp)
doi:10.1088/0953-8984/26/43/435501
First-principles study of thermodynamic stability and the electronic properties of intrinsic vacancy defects in barium hafnate S M Alay-e-Abbas1,2 and A Shaukat1 1 2
Department of Physics, University of Sargodha, 40100 Sargodha, Pakistan Department of Physics, GC University Faisalabad, Allama Iqbal Road, 38000 Faisalabad, Pakistan
E-mail: [email protected] Received 13 July 2014, revised 24 August 2014 Accepted for publication 8 September 2014 Published 9 October 2014 Abstract
The formation of intrinsic vacancy defects in barium hafnate, BaHfO3 and their corresponding electronic structures have been investigated using first-principles calculations. The thermodynamics of pristine and vacancy defects containing barium hafnate have been analyzed. Formation energies for neutral and fully charged Ba, Hf and O vacancies have been evaluated for determining their stability with respect to different chemical environments. From the calculated electronic structure and density of states, it is found that cation deficient barium hafnate is hole-doped, while the incorporation of oxygen vacancy retains the insulating nature of this material. The defect reaction energies for partial and full Schottky reactions are also computed, which controls the properties of non-stoichiometric barium hafnate. Keywords: vacancies, theories and models of crystal defects, electronic density of states and band structure (Some figures may appear in colour only in the online journal)
for utilization in positron emission tomography (PET) and computer aided tomography (CAT) scanners [7]. In addition to its applications in electronic and optical devices, BHO has also shown a remarkable increase in the current densities for SmBa2 Cu3 Oy and Gd1 Ba2 Cu3 O7−δ superconductors, where BHO is used as a flux-pinning material, making these compounds suitable for superconducting magnetic energy storage (SMES) and magnetic resonance imaging (MRI) applications [8]. From an application point of view, physical properties of solid state materials crystallizing in perovskite structure can be tuned to desired functionalities by incorporating point defects and impurities. The earliest statistical thermodynamic models of crystal defects considered vacancies in solids to be randomly distributed. However, later progress showed that the lattice energy can be lowered by introducing long range ordering of vacancies [9]. In the case of ABO3 perovskites, the electronic, optical, magnetic and transport properties can be tuned to an extremely large range by introducing dopants and vacancy defects [7, 10]. This can be achieved by
1. Introduction
Barium hafnate, BaHfO3 (BHO), has recently emerged as a potential candidate for utilization in important electronic technologies such as microprocessor and dynamic random access memory (DRAM) [1, 2]. The environmental and health hazards of commercially dominant lead-based PbZr1−x Tix O3 (PZT) have been driving the quest for identifying alternative piezoelectric materials for the electronics industry. In this regard, BHO has found renewed research interest as it has been proposed as a constituent of environmentally friendly piezoelectric ceramics BaTiO3 –CaTiO3 –BaHfO3 [3]. The optically inactive nature, high melting point, large thermal expansion coefficient [4] and sufficiently high dielectric constant [5] make pure BHO an excellent choice for utilization as buffer layers [6]. On the other hand, Ce3+ doping in BHO yields efficient gamma ray scintillation materials with high light output and decay time constants comparable to bismuth germinate [7]. This places BHO among other fast speed transparent scintillation materials that are suitable 0953-8984/14/435501+09$33.00
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J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
varying the growth conditions which dictate the formation of vacancy defects. For instance, varying the tensile and epitaxial strain during growth can be used for controlling the socalled ‘chemical expansion’ and vacancy types, respectively [10, 11]. Similarly, the thermal radiation treatment of samples is a useful technique for introducing isolated and clustered oxygen vacancies in perovskite oxides. The resulting nonstoichiometric modification of these materials is particularly interesting as, in addition to the possibility of altering their electronic conductivity, high ionic conductivity can be achieved through mobile oxygen vacancies that make defective perovskite oxides functional materials for numerous applications [9]. Depending upon synthesis conditions [10, 11], the type of vacancy and its concentration can lead to charge carriers which are central to achieving optical and electronic properties distinct from those available in the formal stoichiometric material [2, 12, 13]. There is an ever-increasing demand for smarter solid state materials for fulfilling the requirements of sensors and, importantly, different components of solid oxide fuel cells to ensure efficient conversion of chemical energy into electrical energy. This has resulted in an enormous upsurge in research activities focused on defective perovskite oxides [14]. In recent decades, many theoretical and experimental studies exploring the electronic, optical and electrical properties of BHO have appeared in literature [15–18]. Density functional theory (DFT) based first-principles techniques have been effectively utilized by Zhao et al [16], Liu et al [19] and Yangthaisong [20] in an attempt to enrich the information about the electronic, optical, mechanical and bonding properties of bulk barium hafnate. On the other hand, the bare (0 0 1) surfaces of BHO have also been explored in terms of thermodynamic stability and electronic structure [21]. However, intrinsic vacancy defects and the manipulation of electronic structure through vacancy defect incorporation in wide band gap BHO [22] are yet to be explored. In this work, we will perform a series of DFT calculations using the full-potential linear augmented plane wave plus local orbital (FP-LAPW + lo) method for vacancy formation in barium hafnate and investigate its related effects on the electronic structure. The outcome of this study is expected to be important in view of the rapidly increasing use of BHO in electronic, magnetic and optical devices.
18, respectively. The iterative convergence of self-consistent calculations is carried out until the variation in ground state total energy is less than 10−4 Ry. To ensure elimination of residual forces on atoms resulting from vacancy incorporation in BHO, the forces on each atom have been minimized to a value less than 2 mRy/a.u. The cubic perovskite structure of BHO belongs to ¯ crystallographic space group #221 (P m3m) having an experimental lattice parameter of 4.167 Å [25]. In order to compute the electronic structure of pristine and defective BHO, we first performed volume optimization of the perovskite unit cell using PBE GGA and a dense 12 × 12 × 12 k-mesh. Subsequently, a 40 atom 2 × 2 × 2 supercell (SC), having composition Ba8 Hf8 O24 , has been derived from the optimized BHO unit cell for calculations involving pristine as well as defective structures. Neutral and fully charged isolated Ba 2− 4− 0 0 /VBa ), Hf (VHf /VHf ) and O (VO0 /VO2+ ) vacancies in (VBa BHO are realized by removing one Ba, Hf or O atom from 2×2×2 SC. The vacancy concentrations of cation and oxygen vacancy in 2 × 2 × 2 SC correspond to 12.5% and 4.167%, respectively, for which energy and forces minimization are achieved using a 6×6×6 k-mesh. In the current study, we have neglected spin-polarization after performing test calculations for vacancy containing SC, which confirmed that the inclusion of spin-polarization has a negligible effect on the calculated thermodynamic and electronic properties. 3. Results and discussion 3.1. Thermodynamic stability of barium hafnate
In order to obtain the stability diagram shown in figure 1, we first compute chemical potential limits for all atomic species in BHO, which dictate the formation energies of cation and oxygen vacancies [26]. This is because the chemical potential of an isolated atom, µx , is less than the chemical potential of solid/gas that atom in its stable solid or gaseous state (i.e. µx ) [27]: µx = µsolid/gas + µx . x
(1)
In the above equation µx 0 denotes the variation in the chemical potential of species x. For computing valid limits of µx for barium, hafnium and oxygen, the chemical potentials should be varied in such a way that their sum equals the enthalpy of formation of barium hafnate [28]:
2. Method of calculations
The generalized gradient approximation (GGA) parameterization scheme of Perdew, Burke and Ernzerhof (PBE [23]) has been employed for exchange-correlation energy contribution by using the WIEN2k implementation of the FP-LAPW + lo method [24]. A muffin-tin model for wave function, charge density and potential has been assumed within which spherical harmonics and plane wave basis sets are utilized for muffin-tin spheres and interstitial regions, respectively. The muffin-tin radii for Ba, Hf and O atoms are chosen to be 2.5 a.u., 1.98 a.u. and 1.75 a.u. respectively. For all the calculations, plane wave cut off (Ro ×Kmax ) and maximum values of angular momentum (lmax ) and charge density expansion (Gmax ) are set to 7.0, 10 and
HfBaHfO3 = µBa + µHf + 3µo .
(2)
Furthermore, stable production of BHO in equilibrium with binary compounds BaO and HfO2 can be achieved when the (µBa , µHf , µO ) values in equation (2) satisfy: µBa + µO HfBaO
(3)
µHf + 2µO HfHfO2 .
(4)
The conditions in equations (1) through (4) can be met by computing the enthalpies of formations of BaO, HfO2 and 2
J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
Table 1. Comparisons of the calculated lattice parameters (Å) and enthalpy of formations (eV/formula unit) with experimental data.
Experiment ¯ BaO (space group # 225, F m3m) ao 5.539a Hf −5.680b HfO2 (space group # 14, P 21 c) ao , bo /ao , co /ao 5.117, 1.011, 1.034c Hf −11.864d ¯ BaHfO3 (space group # 221, P m3m) ao 4.167e Hf −18.933f a
BaHfO3 from DFT total energy calculations by using the following equations:
Hf hcp
HfHfO2 = EtHfO2 − Et
1 − 2 EtO2 2
5.577 −5.554 5.137, 1.014, 1.039 −11.268 4.208 −19.129
[30], b [31], c [32], d [33], e [25], f [34].
agreement with the stability region that can be obtained from the experimental Hf . The chemical potential values at point A and point B represent the extreme O-rich (oxidation) condition. Point A also represents the extreme Hf-poor condition, whereas the extreme Ba-poor condition is represented by point B. The O-poor/Hf-rich and Ba-rich (reduction) conditions are represented by point C and point D, respectively. The comparison of the oxygen chemical potentials of all the points shown in figure 1 depicts a large variation (−5.635 µO 0). Moreover, a series of lines parallel to A–B in the direction of the arrow pointing to the O-rich condition (µO = 0) also indicates the possibility of achieving a large variation in the chemical potentials of barium and hafnium. Therefore, the chemical potentials of Ba and Hf can also be tuned over a large range. In figure 1 the (µBa , µHf , µO ) coordinates of point X are also provided which is located in the center of the stability region (in between O-rich and O-poor conditions). The stability points shown in figure 1 provide us with the range of chemical potentials which dictate the formation of vacancy defects in barium hafnate.
Figure 1. Thermodynamic stability diagram of BaHfO3 showing chemical potential values at points A, B, C, D and X as an inset. The stability of barium hafnate is confined to the gray region enclosed in quadrangle ABCD. The O-rich condition is satisfied at both points A and B and for all points on lines drawn parallel to A–B away from the origin, while point C represents the extreme O-poor condition where the oxygen chemical potential is −5.635 eV.
1 HfBaO = EtBaO − EtBa fcc − EtO2 2
Calculated
(5) (6)
1 − 3 EtO2 (7) 2 where the minimum total energies of face centered cubic (fcc) Ba, hexagonal close packed (hcp) Hf, BaO, HfO2 and BHO for fcc hcp optimized unit cell volumes are represented by EtBa , EtHf , BaHfO HfO 3 EtBaO , Et 2 , and Et , respectively. On the other hand, O2 Et , is the minimum total energy of a relaxed O2 dimer. The calculated lattice parameters and enthalpies of formation for barium oxide, hafnium oxide and barium hafnate are listed in table 1. One can see that these values are in good agreement with experimental data and confirm the reliability of the formation energetics of vacancy defects in barium hafnate, as will be discussed later. A comparison of calculated and experimental enthalpy of formation values clearly indicates that HfBaHfO3 < HfBaO + HfHfO2 , is in accordance with the requirement for stable production of BHO from its binary constituents BaO and HfO2 [29]. Using the enthalpy of formation values presented in table 1, we have drawn the thermodynamic stability diagram shown in figure 1 where the equilibrium of BHO is maintained without any secondary phases for the (µBa , µHf , µO ) coordinates inside the quadrangle ABCD. From figure 1, it is evident that above the lines AD and BC, the chemical potential region for stable production of BHO is limited by BaO and HfO2 , respectively. The stability region of BHO is thus confined to the gray area which is in good Hf hcp
HfBaHfO3 = EtBaHfO3 − EtBa fcc − Et
3.2. Isolated neutral vacancies in BaHfO3
The structurally optimized pristine and defective SCs shown in figure 2 provide a visual description of the relaxation of atomic positions caused by the incorporation of anion and cation vacancies. After the minimization of Hellman–Feynman forces on the atoms surrounding vacancy sites, both bond lengths and bond angles of nearest neighbor (NN) and next-nearest neighbor (NNN) atoms deviate from their ideal values. The calculated NN and NNN distances in pristine as well as neutral cation and oxygen vacancies in a SC of BHO are compared in table 2. 0 0 In the case of VBa /VHf , the first NN O atoms move away from the vacancy site, whereas the second NN Hf/Ba atoms move inward. The outward relaxation of the NN O atom in the 0 0 case of VBa is smaller (2.967%) compared to VHf (4.363%). 0 0 In the case of NNN Hf/Ba atoms around VBa /VHf , vacancy, the inward relaxation towards the vacancy site amounts to 1.250%/5.135%. These values are in accordance with the stronger covalent bonding between linearly coordinated O and Hf atoms. In addition to the above effects, the Ba–O and 0 Hf–O bond lengths near the VBa site increase to 2.977 Å and 2.131 Å, whereas these bond lengths decrease to 2.837 Å and
3.2.1. Structural and bonding properties.
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J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
Figure 2. (a) Pristine and defective 2 × 2 × 2 SC of BaHfO3 . The relaxation of atomic positions resulting from incorporation of charge neutral Ba, Hf and O vacancies are shown in (b), (c) and (d), respectively. Ba, Hf and O atoms are represented by black, gray and red spheres, respectively.
The variations in the bond lengths around the vacancy sites are also accompanied by changes in the cation–O–cation bond angles (Ba–O–Ba = 90◦ and Hf–O–Hf = 180◦ in bulk 0 BHO) which can also be seen in figure 2. For VBa both Ba–O–Ba and Hf–O–Hf bond angles around the vacancy site change to 91.761◦ and 172.932◦ . On the other hand, only the Ba–O–Ba bond angles decrease to 89.718◦ while Hf–O–Hf 0 . These changes in remains unchanged in the case of VHf the bond angles are in accordance with the relaxations and coordination environments of atoms surrounding the vacancy sites. The changes in the bond angles around the vacancy site, brought about by the incorporation of VO0 , are very small (Ba–O–Ba = 89.898◦ and Hf–O–Hf = 179.677◦ ). This is due to simple 2-fold linear coordination and 4-fold square planar coordination of O atoms with Hf and Ba atoms, respectively. In figure 3 the charge density contour plots are displayed, which reveal the nature of bonding between Ba/Hf and O atoms and the changes in bonding nature and charge distribution resulting from vacancy defects. It is evident that Hf–O bond is more covalent compared to the Ba–O bond. This can be attributed to the relatively larger electronegativity value of 0 Hf on the Pauling scale. The charge density plots for VBa 0 and VHf (figures 3(b) and (e)) compared to pristine BHO, demonstrate that the O-2p orbitals surrounding the vacancy sites become localized, thus giving rise to acceptor-like levels near the valence band maximum (VBM). A comparison of the charge density plots around the cation/anion sites in the
Table 2. Nearest neighbor distances (Å) for pristine (atom–atom) and defective (vacancy site–atom) BaHfO3 SC. In the case of defective 2 × 2 × 2 SC, distances larger/smaller than their corresponding pristine values depict outward/inward relaxation of the nearest neighbor atoms relative to the vacancy site. The neighboring atom, along with its coordination number, is shown in parenthesis.
Ba Pristine BaHfO3 0 VBa Hf Pristine BaHfO3 0 VHf O Pristine BaHfO3 VO0
NN
NNN
(O × 12) 2.976 3.067 (O × 6) 2.104 2.200 (Hf × 2) 2.104 2.108
(Hf × 8) 3.644 3.599 (Ba × 8) 3.644 3.466 (Ba × 4), (O × 8) 2.976, 2.976 2.980, 2.966
0 2.008 Å in the case of VHf . In accordance with the electrostatic interactions among cation and anion, the incorporation of VO0 results in an outward relaxation of NN and NNN cations, while the NNN O atoms undergo an inward relaxation towards the oxygen vacancy site. Consequently, the Ba–O/Hf– O bond length near the oxygen vacancy site decreases to 2.974 Å/2.010 Å. This may be further understood from the analysis, which is given below, of the bonding nature in Ba–O and Hf–O bonds.
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J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
Figure 3. The charge density contour plots for pristine and barium, hafnium and oxygen vacancy containing BaHfO3 . The top and bottom panels display the charge densities in BaO and HfO planes, respectively. Ba, Hf and O atoms are represented by black, gray and red spheres, respectively.
BaO and HfO planes for pristine and cation/anion deficient cases shows maximum charge localization/delocalization. For instance, it is clear that more charge localization occurs for hafnium vacancy compared to barium vacancy. This means that the presence of electrons in the unoccupied states above 4− would Fermi level (EF ) for a charged hafnium vacancy VHf 2− . result in higher positioning of these states compared to VBa A comparison of charge densities for pristine (figures 3(a) and (d)) and VO0 containing BHO (figures 3(c) and (f )) depicts the elimination of the covalent bonding between the metal atom and O vacancy site. In the case of oxygen vacancy, the charge density shows more delocalization in the HfO plane, which can be attributed to dangling Hf-5d orbitals on the creation of VO0 . Consequently, this yields an occupied defect level near the conduction band minimum (CBM) in the band structure of oxygen deficient BHO (see next section).
are located 1.160 eV above the occupied defect level at the symmetry point. The shifting of the EF near the CBM occurs to conserve the semiconductor nature of barium hafnate, whereas bulk-like states of valence band appearing around −3 eV in figure 4(d) are separated from this defect level by an energy difference of 2.478 eV. In order to elucidate the contribution of cation and anion orbitals for pristine and defective BHO in the vicinity of EF , figure 5 shows the calculated partial density of states (PDOS) corresponding to the band structure diagrams (figure 4). It can be seen that the valence band states near EF are predominantly made up of the O-2p orbitals with insignificant contributions from Ba and Hf, while the CBM is dominated by the Hf-5d orbitals. Consequently,the direct (–) band gap appears between the occupied O-2pand unoccupied Hf−5d orbitals. In the case of cation vacancies, the defect levels appearing just above the VBM at point assume an anion orbital-like character, resulting in the creation of triply degenerate O-2p hole states. These hole states appear 0.117 eV and 0.186 eV 0 0 above the EF for VBa and VHf , respectively. The higher position 0 of the VHf unoccupied hole-doped states is in accordance with its higher charge state and lower oxygen coordination. Until the synthesis of polycrystalline barium hafnate by Maekawa et al in 2006 [4], only a few studies were available in literature addressing its physical properties. For this reason, theoretical investigations exploring the electronic properties of BHO like the ones by Vali [35] and Yangthaison [20] assumed that the experimental band gap of BHO would be around 6.0 eV. However, in recent spectroscopy ellipsometry and refractometry studies of BaHfO3 thin films grown on TiN and Si revealed a fundamental band gap of 4.8 eV [15]. As
The band structures of pristine and defective 2×2×2 barium hafnate SCs calculated using the PBE GGA are displayed in figure 4. The VBM and CBM of pristine BHO are both located at the symmetry point and have a band gap (Eg ) amounting to 3.616 eV which is comparable with previous pseudopotential calculations (3.783 eV [35] 0 0 and VHf in and 3.610 eV [20]). The incorporation of VBa BHO gives rise to unoccupied energy levels which are located slightly above the EF as evident from figures 4(b) and (c). These unoccupied energy levels lead the cation deficient BHO into a hole-doped state. On the other hand, neutral oxygen vacancy yields a defect level, occupied by 2 electrons, which appears below the CBM. Consequently, EF is shifted near the CBM such that the unoccupied conduction band states 3.2.2. Electronic properties.
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J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
0 0 Figure 4. Calculated electronic band structures of (a) pristine BaHfO3 SC and defective SCs containing (b) VBa , (c) VHf and (d) VO0
vacancies.
expected from GGA band structure calculations, our calculated band gap is underestimated compared to the experimental data. Calculations of formation energy for anion defects in perovskite require band gap correction for predicting correct trends of vacancy formation [28, 36]. Consequently, a rigid shift of the GGA CBM (∼1.184 eV) is required as band gap correction places the defect level for VO0 containing BHO at 3.662 eV above the bulk-like valence band states shown in figure 4(d). This demonstrates that BHO is inactive in the visible region in accordance with experimental observations [15] supporting the band gap correction for formation energy calculations of oxygen vacancies in the next section.
where EtBaHfO3 and Et [X q ] are the minimum total energies of pristine and defective SCs. The vacancy type and its charged state are represented by X and q, respectively. The quantity µx is the chemical potential of species X which can be varied according to the stability points shown in figure 1. EF +EVBM represent the position of the Fermi level as measured from the VBM where EVBM has been corrected following the procedure outlined in [36]. In the case of oxygen vacancies, the band gap correction has been realized by adding mEg = m (1.184eV ) (where m is the number of electrons in the vacancy induced level having value 2 for VO0 and 0 for VO2+ ) in the calculated formation energy values given by equation (8). As a consequence of adopting band gap correction for oxygen vacancies, the transition levels near the calculated CBM are shifted near the experimental CBM [37]. It is worth noting that due to the anionorbital-like character of the defect states appearing above the VBM in case of the metal atom (Ba and Hf) vacancies, the band gap correction is only applied to the calculated formation energies of oxygen vacancies.
The formation energies ( [X q ]) of neutral and fully charged states of Ba, Hf and O vacancies in BHO have been computed by [28]
3.2.3. Formation energetic.
Xq = Et X q − EtBaHfO3 − µx + q(EF + EVBM )
(8) 6
J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
Figure 5. PDOS plots for pristine and defective BaHfO3 SC showing the orbital contribution of Ba-6s, Hf-5d and O-2p states in the vicinity 0 0 of EF . (a) pristine SC, (b) VBa , (c) VHf and (d) VO0 .
0 formation by hafnium 0 0 energy ( Ba = 2.807 eV) followed ( O = 9.429 eV). ( Hf = 3.963 eV) and oxygen Point B (Ba-poor condition) Ba0 = 0.501 eV, indicates that charge neutral barium vacancy is the most favorable form of vacancy defect in BHO under metal-poor conditions. This also remains true for point X. However, at point C (extreme O-poor condition) the formation energy of neutral oxygen vacancy ( O0 = 3.794 eV) becomes smaller than the 0 0 formation energies of both VBa VHf . Similar behavior0 and 0 is evident at point D where O = 3.874 eV makes VO formation easier compared to neutral Ba and Hf vacancies. A comparison of the formation energies of neutral and fully changed in this work (figure 6), reveals considered vacancies, that Baq and Baq increase, whereas [Oq ] decreases on going from O-rich condition (point-B) to O-poor condition (point-C). Since it is important to verify that the size of SC (2 × 2 × 2) is sufficient for the formation energy calculations presented in this work, we have computed the total energy of oxygen vacancy in a larger (60 atom) SC. Our calculated value for 2.778% VO0 concentration is only 0.066 eV larger than the 0 O value in the 2 × 2 × 2 SC. The Hf-poor O2+ has a lower value condition, 2− . However, even smaller values of compared 0 to Ba Ba indicate the neutral barium vacancy to be the most favorable vacancy defect at point A. On the other hand, both 2− 0 and VBa assume formation energies less than all other VBa
Figure 6. Formation energies (eV/defect) of isolated neutral and fully charged Ba, Hf and O vacancies in barium hafnate at the calculated stability points shown in figure 1.
The formation energies for barium, hafnium and oxygen vacancies in various chemical environments are shown in figure 6. In the case of charge neutral vacancies, it is evident that under O-rich (Hf-poor) conditions, barium has the lowest 7
J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
Figure 7. Variation of [X q ] as a function of Fermi energy (EF ) at stability points A, X, C and D. Charge states of the vacancies are given
as numbers below their respective lines.
C and D VO2+ has the lowest formation energy value near the VBM (EF = 0 eV), whereas at point A (also at point B which is 0 not shown in figure 7) VBa attains the lowest formation energy 4− value for EF = 0 eV. VHf . On the other hand, it has the lowest value of formation energy at the CBM (i.e. EF = 4.8 eV) 2− for points A, X and D, while VBa has the lowest formation energy value for EF = 4.8 eV at point C (Hf-rich condition). The variation of formation energies of fully charged cation and oxygen vacancies with EF , results in negative values inside the band gap which gives us the ‘pinning energy’ Epin [40]. The Epin energy can be defined as a parameter which in the case of oxygen/metal vacancy restricts barium hafnate from assuming a Fermi level which is closer to the VBM/CBM than the value of Epin . It can be seen that for metal-poor conditions (point A and point B) Fermi level pinning does not occur for oxygen vacancies. This means that EF is located inside the valence band which is in accordance with the electronic band structures shown in figures 3(b) and (c). However, at point X, C and D O2+ becomes smaller than the formation energies of fully charged barium and hafnium vacancies for EF values near the VBM. This indicates that oxygen vacancy would compensate anion vacancies from having acceptor-like levels in the band gap [30]. It is worth noting that the vacancy formation in barium hafnate and its corresponding effects reported in this study will be useful in future research work on barium hafnate and might serve as a guide to its device application.
vacancy types at the Ba-poor limit (point B). At points X, C and D the smaller values of VO2+ reveal that under ideal growth conditions (i.e. at point X) as well as Ba-rich and Hf-rich conditions, fully charged O vacancy is the most abundant form of vacancy defect in barium hafnate, which is in accordance with experimental observation [17, 38]. The calculated formation energies of fully charged Ba, Hf and O vacancies given in figure 6 can be used for computing Schottky reaction energies which help in understanding the creation of cation and anion vacancies in such a way that stoichiometry of the material remains intact as a whole. The Schottky reaction energies are important in controlling properties of non-stoichiometric perovskite oxides under various growth conditions [39]. We have computed values for the full 2− 4− Schottky reaction VBa + VHf + 3VO2+ and barium/hafnium 2− 4− partial Schottky reaction VBa + VO2+ /VHf + 2VO2+ . The calculated barium/hafnium partial Schottky reaction energies vary between minimum (2.685 eV/4.634 eV) and maximum (3.838 eV/5.402 eV) values such that the average value is 3.262 eV/5.114 eV. Contrary to that, the reaction energy for the full Schottky reaction is 4.316 eV for all stability points. The variation of formation energies of barium, hafnium and oxygen as a function of EF shown in figure 7 clearly suggest that fully charged states of these defects are stable over a wide range of the barium hafnate band gap. At point X, 8
J. Phys.: Condens. Matter 26 (2014) 435501
S M Alay-e-Abbas and A Shaukat
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4. Conclusions
The electronic structure and formation energies of isolated vacancies in barium hafnate have been investigated for the first time using first-principles calculations. It is shown that pristine and isolated oxygen vacancy containing barium hafnate are semiconducting, whereas Ba and Hf deficient barium hafnate is found to be hole-doped. The defect level induced by neutral O vacancy appears near the conduction band which is composed of Hf-5d orbitals. On the other hand, Ba and Hf vacancies give rise to unoccupied O-2p states just above the VBM. Neutral Ba vacancy has the lowest formation energy values for metal-poor oxidation conditions, whereas fully charged oxygen vacancies are found to be the most favorable form of vacancy defect in barium hafnate under reduction conditions. The calculated Schottky reaction energies indicate that barium partial Schottky has smaller reaction energy compared to both hafnium partial and full Schottky reactions. A comparison of formation energy values with available experimental and theoretical data of other perovskite oxides confirms the consistency of DFT-GGA results presented in this paper. These findings might help future studies exploring high temperature applications of BHO for electronic and optical devices. Acknowledgments
The authors are grateful to S Nazir for useful discussions. The calculations were performed using computational resources supported by the University of Sargodha. References [1] Lupina G et al 2009 Chem. Vap. Deposition 15 167 [2] Lupina G, Dabrowski J, Dudek P, Kozlowski G, Lukosius M, Wenger C and M¨ussig H-J 2009 Adv. Eng. Mater. 11 259 [3] Wang D, Jiang Z, Yang B, Zhang S, Zhang M, Guo F and Cao W 2014 J. Mater. Sci. 49 62 [4] Maekawa T, Kurosaki K and Yamanaka S 2006 J. Alloys Compounds 407 44 [5] Lupina G, Kozlowski G, Dabrowski J, Wenger C, Dudek P, Zaumseil P, Lippert G, Walczyk C and Mssig H J 2008 Appl. Phys. Lett. 92 062906 [6] Zhang J L and Evetts J E 1994 J. Mater. Sci. 29 778 [7] Van Loef E V, Higgins W M, Glodo J, Brecher C, Lempicki A, Venkataramani V, Moses W W, Derenzo S E and Shah K S 2007 IEEE Trans. Nucl. Sci. 54 741 [8] Tsuruta A, Yoshida Y, Ichino Y, Ichinose A, Matsumoto K and Awaji S 2013 IEEE Trans. Appl. Supercond. 23 8001104 [9] Tilley R J D 2008 Defects in Solids (Hoboken, NJ: John Wiley and Sons)
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