Hole Trapping at Surfaces of m-ZrO2 and m-HfO2 Nanocrystals - The

Article pubs.acs.org/JPCC

Hole Trapping at Surfaces of m‑ZrO2 and m‑HfO2 Nanocrystals Matthew J. Wolf,† Keith P. McKenna,*,‡ and Alexander L. Shluger*,† †

Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom Department of Physics, University of York, Heslington, York YO10 5DD, United Kingdom

‡

ABSTRACT: We investigate hole trapping at the most prevalent facets of monoclinic zirconia (m-ZrO2) and hafnia (m-HfO2) nanocrystals using first-principles methods. The localization of holes at surface oxygen ions is more favorable than in the bulk crystal by up to ∼1 eV. This is caused mainly by the reduction of the absolute value of the electrostatic potential at the surface ions with respect to the bulk and by the significant surface distortion caused by the hole localization. The mobility of holes at surfaces is much lower than that found in the bulk and is fairly isotropic. Unlike in cubic oxides, such as MgO and CaO, we do not find a significant driving force for preferential trapping of holes at steps on the m-ZrO2 surface. These fundamental results are relevant to mechanisms of water oxidation, photocatalysis, contact charging, and photodesorption.

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INTRODUCTION The trapping and mobility of electrons and holes in metal-oxide materials finds broad interest across the chemistry, physics, and materials science communities, being at the heart of phenomena as diverse as photocatalysis and desorption, the performance of microelectronic devices, magnetism, superconductivity, radiation damage, and tribocharging.1−12 Many of these applications involve polycrystalline samples, powders, or thin films; therefore, the behavior of trapped electrons and holes in oxide nanocrystals is an issue of general importance. For example, the efficiency of oxide-based photocatalysts and solar cells depends crucially on the properties of such trapped charges.13−16 They are also thought to play a key role in photoinduced chemical reactions on interstellar dust grains and in photoinduced desorption of surface atoms.17−20 However, detailed information on the properties and behavior of electrons and holes in oxide nanocrystals is scarce because suitable methods, such as high-resolution femotosecond spectroscopies, are extremely challenging to apply to nanocrystal systems;21−24 hence, to our knowledge, definitive experimental evidence for electron and hole trapping at the surface of an oxide is only available for MgO25−27 and TiO2.28 Unfortunately, there are no general rules which allow one to predict whether electrons or holes can be trapped at the surfaces of other oxides. For these reasons, theoretical calculations have been invaluable. In particular, recently there has been a surge of activity in first-principles-based prediction of the properties of trapped electrons and holes in ideal crystals (self-trapping) or at defects in the bulk.29−32 However, there have been far fewer studies of polaron trapping at oxide surfaces; perhaps the best studied materials in this respect are MgO33−38 and TiO2.39−44 For example, in TiO2 nanocrystals, electrons and holes are predicted to trap both in the bulk and near the surface.42 This propensity for trapping can be explained in part by titania’s very high dielectric constant. In © 2012 American Chemical Society

contrast, electrons and holes do not appear to self-trap in the bulk of MgO, which has a much lower dielectric constant of 9.8, although there is clear evidence that they can be trapped at low coordinated surface sites of MgO nanocrystals, such as corners and kinks.25−27 The preferential trapping at low coordinated sites has been attributed to the reduction of the absolute value of the electrostatic potential with respect to the bulk and increased freedom for short-range ionic distortion. However, this raises some interesting questions for high dielectric constant materials like ZrO2 and HfO2. In particular, given that the volume of material available for polarization is reduced by about half at the surface, will trapped charges prefer to remain in the bulk of the nanocrystal or will they segregate to the surfaces? The current picture is unclear even for the wellstudied surfaces of TiO2, with theoretical calculations suggesting that electrons prefer to localize just below the surface while holes localize at undercoordinated surface oxygen ions.41,43 Another important question is which sites on the surface are the most stable charge traps? These fundamental questions, of relevance to topical problems in clean energy generation and photocatalysis, still remain largely unanswered due to the difficulty of probing the properties and dynamics of electrons and holes at the nanoscale. The aim of this paper is to provide answers to some of these questions by investigating the properties of holes at extended facets and steps of monoclinic (m-)ZrO2 and HfO2 nanocrystals. Although holes have been implicated in the enhanced reactivity of irradiated ZrO2 nanoparticles, e.g., with respect to water oxidation,45 nothing is known about the structure and mobility of holes at surfaces of ZrO2 films and nanoparticles. In this paper, we evaluate the tendency for holes to segregate to Received: September 25, 2012 Revised: November 9, 2012 Published: December 3, 2012 25888

dx.doi.org/10.1021/jp309525g | J. Phys. Chem. C 2012, 116, 25888−25897

The Journal of Physical Chemistry C

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discuss the properties of holes trapped at these facets and their mobility at the surface. Finally, we present two models for step defects to investigate the interaction of trapped holes with topological defects at the surface.

clean nanocrystal surfaces and assess the energetic barriers associated with their diffusion. We note that, for technologically relevant and experimentally addressable nanocrystals, oxide surfaces are likely to be hydroxylated, though previous theoretical investigations of electron trapping at the surface of TiO2 found that the effect of surface hydroxyls on polaron stability is small.41 In any case, the effect of hydroxylation on hole trapping at ZrO2 and HfO2 surfaces will be considered in a separate publication. Previous theoretical modeling31 has shown that holes selftrap on three-coordinated (3C) oxygen ions in the bulk of mZrO2 and m-HfO2 with relatively small trapping energies (∼ 0.1−0.2 eV). Since the 3C oxygen ions are arranged in layers in the monoclinic structure, the hole mobility was found to be high within these layers, which are parallel to the (100) crystallographic plane, but low in the perpendicular direction. Should this anisotropy propagate to the surface, it suggests the interesting possibility that holes would be confined in the onedimensional channels formed by the terminating ions of these planes at the crystal surface. Furthermore, it also suggests that holes created in the bulk of the nanocrystal would rarely segregate to (100) facets as this would require diffusion along the [100] direction. We show in this article that holes self-trap at both 2C and 3C undercoordinated oxygen ions on the three most prevalent surface facets of m-ZrO2 and m-HfO2 nanocrystals. The associated trapping energies are considerably higher than in the bulk, meaning that holes created in the bulk will readily segregate to the surface. The trapping energies show a general dependence upon the coordination, being higher at lower coordinated sites, in line with previous understanding based on the rocksalt-structured oxide MgO. This may be broadly correlated with the lower absolute value of the electrostatic potential at these sites. We also show that, since there is no reduction in the coordination of 2C atoms upon the formation of step edges, the hole trapping energies at and near steps are similar to, and in some cases somewhat lower than, those at terraces. The fact that sites on different terraces exhibit different hole trapping energies indicates that the irradiation of nanoparticles may lead to an inhomogeneous distribution of positive charge. The diffusive hopping of holes between different sites on a given terrace is isotropic and requires overcoming barriers of approximately 0.55 eV, which suggests that holes at surfaces are not very mobile below room temperature. In the remainder of this paper, we focus the presentation and discussion of results on ZrO2 due to its wider range of applications. We find that HfO2 exhibits very similar properties to ZrO2 with repect to hole trapping, and calculated values of some of the key properties are given for comparison. The paper is structured in the following way: we first describe the theoretical methods, including the approach we have employed to ensure accurate prediction of hole localization; this is a challenging problem, as widely employed density functional methods suffer from the self-interaction (SI) error, which tends to delocalize holes.46,47 Here, we employ the recently developed cancellation of nonlinearity (CON) method which ensures the elimination of SI48 and has been employed previously to model hole trapping in the bulk of ZrO2, among other systems.31,48−51 We then begin the presentation of the results with the calculated equilibrium crystal shape of a ZrO2 nanocrystal and the corresponding structural and electronic properties of its most prevalent facets. Next, we

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METHODS Our density functional theory (DFT) calculations are performed using the projector augmented wave (PAW) method, as implemented in the Vienna ab initio simulation package (VASP).52,53 The 4s, 4p, 5s, and 4d (5p, 6s, and 5d) electrons of Zr (Hf) and the 2s and 2p electrons of O are treated as valence electrons and expanded in a plane wave basis with energies up to 300 eV (400 eV for bulk unit cell optimization). For the conventional cells of monoclinic ZrO2 and HfO2, which is the most stable phase in ambient conditions, an 8 × 8 × 8 Monkhorst−Pack k-point grid was used, and structural optimization was performed until forces were less than 0.02 eV/Å . Using the Perdew−Burke− Ernzerhof (PBE) exchange correlation functional,54 we obtained lattice parameters within 1.5% of experiment for ZrO2 (a = 5.218 Å, b = 5.284 Å, c = 5.398 Å, and β = 99.63°) and 0.6% of experiment for HfO2 (a = 5.142 Å, b = 5.192 Å, c = 5.250 Å, and β = 99.65°). These values are also in good agreement with previous theoretical predictions.55−57 For the calculation of surface formation energies we constructed symmetric slabs for all symmetry-inequivalent low-index surface orientations, using the METADISE code,58 and we have used the terminations given in the paper by Christensen and Carter.59 We included a vacuum gap of 10 Å, and the density of k-points within the surface plane was kept the same as for the bulk calculations, with a single k-point perpendicular to it. Surface formation energies, γ, were calculated according to γ=

Eslab − nE bulk 2A

(1)

where Eslab is the total energy of the slab containing n formula units; Ebulk is the bulk cohesive energy; and A is the area of the surface unit cell. The thickness of the slabs was increased until surface energies were converged to within 10 mJ m−2. We also verified that increasing the vacuum gap to 15 Å resulted in negligible change of the surface formation energies (2−3 mJ m−2). To ensure that predictions of hole localization are sufficiently accurate, we employ the recently developed CON method,48 which ensures elimination of SI by applying an occupationdependent potential of the following form Vhs = λhs(1 − nm , σ /nhost)

(2)

where nm,σ is the fractional occupancy of sublevel m of spin σ of the oxygen p-orbital. The main advantage of this method is that the SI correction which is applied is determined unambiguously by ensuring that the correct linear behavior of the total energy with respect to fractional occupation number is obtained. This method has been applied previously to model hole polarons in the bulk of ZrO2 and HfO2 where self-consistent values of λhs have been determined (3.4 eV for ZrO2 and 3.8 eV for HfO2),31 and these values were also applied to the surface polaron calculations presented here. 25889



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RESULTS 1. Equilibrium Nanocrystal Shape and Electronic Properties. The relatively complex structure of m-ZrO2 yields nine inequivalent low-index lattice planes, each with at most three possible terminations (though some of these may be symmetrically equivalent and/or lead to polar surfaces). We find that the surface terminations given by Christensen and Carter59 are the most stable. The calculated surface formation energies (eq 1) for all inequivalent low-index surfaces of mZrO2 are summarized in Table 1. For comparison, we also show Table 1. Surface Formation Energies of the Low-Index Surfaces of m-ZrO2a relaxed structure (J m−2) (1̅11) (111) (110) (1̅01) (001) (011) (101) (100) (010)

PBE

PW91

0.94 1.11 1.22 1.23 1.31 1.34 1.42 1.46 1.65

0.87 1.06 1.42 1.23 1.35 1.34 1.52 1.47 1.75

60

LDA

unrelaxed structure (J m−2) 59

1.25 1.54 1.64 1.51 1.80 1.73 1.97 1.83 2.46

PBE

PW9160

LDA59

1.33 1.39 1.77 1.73 1.91 1.93 2.16 1.92 2.52

1.34 1.39 1.89 1.71 1.97 1.89 2.37 1.88 2.53

1.60 1.86 2.25 2.02 2.43 2.31 2.73 2.28 3.50

Figure 1. Equilibrium crystal shape of m-ZrO2 determined using the Wulff construction and the calculated surface formation energies presented in Table 1. The most prevalent crystal facets are those with normals parallel to (1̅11), (111), and (001), which together account for over 95% of the total surface area.

we note that nearest-neighbor bond lengths change by only 0.01 Å. In contrast, ionic displacements are essentially negligible past a depth of approximately 3−4 Å for the (1̅11) and (111) terraces, reflecting the higher stability of these surfaces. We focus further discussion of structure on the surface oxygen ions since these are the sites which may serve as potential hole traps. The (001) terrace has two 2-coordinated (2C) oxygen ions, one of which is derived from the bulk 3C oxygen sublattice and the other from the 4C sublattice; the (1̅11) and (111) terraces have one and two 2C oxygen sites, respectively, all of which are derived from the bulk 3C sublattice, and two 3C surface oxygens each, all of which are derived from the bulk 4C sublattice. Turning to the densities of states for the three terraces (Figure 2d−f), we note that in each case there are states split from the top of the bulk valence band: for the (001) surface, by up to 0.21 eV; 0.31 eV for the (11̅ 1) surface; and 0.10 eV for the (111) surface. A projection of the density of states onto the surface oxygens reveals that these states derive from the 2C surface oxygen ions in each case. This splitting of surface states into the gap is related to the lower absolute value of the electrostatic potential at the surface 2C sites, and both point to the fact that, thermodynamically, holes created in the bulk will tend to segregate to surface oxygen sites. 2. Trapping and Diffusion of Hole Polarons at Surfaces. Earlier we noted that, as hole polarons in the bulk are effectively mobile only within (100) planes, they would very rarely segregate to (100) facets. However, the equilibrium crystal shape shown in Figure 1 has less than 2% of (100) oriented facets by area. Hence, holes in the bulk of the nanocrystal can readily diffuse to reach most of the exposed nanocrystal surface. Therefore, in the following we will investigate the relative stability of holes in the bulk and at the surface and quantify their properties. Structure of Surface Polarons. We have investigated the possibility of hole trapping at the three most prevalent m-ZrO2 terraces. For the (001) surface, we employed a 3 × 3 surface supercell and constructed a slab three layers thick with a 10 Å gap. For the (1̅11) and (111) surfaces, a five-layer slab was used, with a 2 × 2 surface supercell. In all cases the separation

a

The PBE results are from this work, while the PW91 and LDA results are from previous works by other authors.

results of previous calculations by other workers using the LDA59 and PW91.60 It is immediately noticeable that the surface formation energies from both sets of GGA calculations are 20−30% smaller than those of LDA, due to the well-known tendency of the LDA to overbind and presumably also to the relatively thin slabs which were used in that study. Also, the ordering of the formation energies is different; in particular, the (1̅01) surface is the fourth lowest in energy in our PBE calculations but the second lowest in LDA.59 There are also differences between our PBE results and the PW91 results of Piskorz et al.60 These are largest for the (110), (101), and (010) surfaces and lead to a difference in ordering of the surface energies; in particular, we predict the (110) surface to be third lowest, while Piskorz et al. place it sixth lowest. We ascribe such differences to these authors holding part of the slab fixed, rather than allowing it to relax fully, as in our calculations; this latter argument is supported by the fact that the unrelaxed energies are in general more similar between the two sets of calculations than the relaxed ones. Also, we used slightly denser k-point meshes in our calculations. In any case, we note that, importantly, both sets of calculations predict that the two lowest-energy surfaces are the (1̅11) and (111). The equilibrium crystal shape, predicted using the calculated surface energies and the Wulff construction, is shown in Figure 1. The most prevalent surface facets are those parrallel to (1̅11), (111), and (001), which together account for over 95% of the total surface area. In the following we describe in detail the atomic and electronic structure of these three surfaces before considering the interaction of holes with them. Figure 2a−c shows the surface structure of the ideal (1̅11), (111), and (001) surfaces. The relaxation of the (001) terrace penetrates rather far into the slab, especially along the 3C sublattice, so that 3C oxygens in the center of the slab, at a depth of 7−8 Å, are displaced by as much as 0.12 Å; however, 25890



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Figure 2. (a−c) Atomic structure of the three most prevalent facets on m-ZrO2 nanocrystals. Zirconium ions, oxygen ions derived from the bulk 4C sublattice, and oxygen ions derived from the bulk 3C sublattice are depicted by gray, red, and green spheres, respectively. All inequivalent undercoordinated surface oxygens are labeled. (d−f) The corresponding density of states highlighting the associated surface states.

between surface oxygen sites and their periodic images was greater than 13 Å. By dilating the nearest-neighbor bond lengths around a particular surface site, removing an electron from the system, and allowing the resultant structure to relax, we found that holes localize at all inequivalent surface oxygen sites, forming polarons. Given the relatively large number of sites considered, we will discuss their properties in general terms, using examples at the (1̅11) terrace to illustrate the trends. Previous studies on holes at surfaces have indicated that oxygen coordination plays a key role, and since we also find this to be the case generally, it is mainly this property that will be used to structure the presentation of the results. Examining the spin densities of the polarons, we found that they may be classified into two types. These two types are exemplified by the polarons at the (1̅11) surface, shown in Figure 3. For the first type (see Figure 3a), the spin density is mainly localized on a p-orbital on the surface oxygen ion, oriented approximately normal to the plane described by the nearest-neighbor bonds, with the remaining spin density distributed fairly evenly among the next-nearest-neighbor oxygen ions. This is similar to the spin density found for bulk polarons, although the onsite value of the spin density (defined as the difference between the spin-up and spin-down electron densities integrated within the radius of the PAW projector) indicates that they are localized more strongly, with values of 0.85 e compared to the 0.77 e found in the bulk. For the second type, the majority of the spin density is again localized on a p-orbital of the surface oxygen, but the remainder is distributed mainly on just one of the next-nearest-neighbor oxygen ions (see Figure 2b,c). In this case, the p-orbital is oriented toward this second oxygen ion, indicating some degree of bonding to have taken place between the two. The onsite value of the spin density for the oxygen ion on which the majority of the spin density is located is dependent upon the specific site, although it is always less than for type 1 polarons, with the values ranging from 0.73 e to 0.82 e.

Figure 3. (a−c) Spin density distributions (left) and associated displacement fields (right) for hole polarons trapped at different oxygen sites on the (11̅ 1) surface. Zirconium ions, oxygen ions derived from the bulk 4C sublattice, and oxygen ions derived from the bulk 3C sublattice are depicted by gray, red, and green spheres, respectively. Spin density isosurfaces (0.14 e A−3) are shown in blue. Arrows depicting the displacements of the ion due to polaron trapping are shown with 5 times their actual magnitude.

The two types of polarons which we have identified are generally correlated with the sublattice from which the oxygen ion derives: type 1 polarons are localized on oxygen ions derived from the 3C sublattice, whereas type 2 polarons are localized on oxygens derived from the 4C sublattice. The exception to this rule is the polaron at the 3C−O3 site on the (111) surface, which is of type 1. The displacement fields associated with the relaxation of the polarons on the (1̅11) are also shown in Figure 3. The major 25891



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The relative stability of the different hole trapping sites is found to be identical with only small differences in the absolute trapping energy. To investigate the role of electrostatics in the hole trapping, we calculated the onsite electrostatic potential for each of the oxygen sites for the ideal surface structure before the hole trapping. We define the onsite potential Φ as the average electrostatic potential integrated within a sphere centered at the ionic core, with the radius being given by the extent of the PAW projector. Figure 4a shows the correlation between the change of the electrostatic potential Φ with respect to that on a bulk 3C oxygen site and the hole trapping energy. The negative trapping energy corresponds to the 4C oxygen site in the bulk of m-ZrO2.31 First, we note a clear correlation between the coordination of a given oxygen ion and the onsite potential. Second, there is also an almost linear relationship between the onsite potential and the trapping energy. There is some spread around the general trend which we attribute to differences in the short-range lattice distortion associated with polaron trapping. This significant short-range distortion plays an important role in hole trapping, but the corresponding trends are more difficult to quantify. Hole polaron formation is associated with the splitting of an unoccupied one-electron state into the band gap, the energetic position of which, relative to the top of the bulk valence band (εhs), gives a first approximation to the optical ionization energy of the hole. Figure 4b summarizes the position of the unoccupied one-electron states for all of the stable polaron traps in the bulk and at each surface. There is a clear correlation between the coordination of the site and the degree to which this level splits from the valence band, which mirrors the trend seen in the trapping energy. Again, very similar properties are found for the m-HfO2 (1̅11) surface (Table 2), in line with expectations on the basis of what is known about the bulk properties of these materials. Hole Polaron Diffusion. Calculations of the trapping energy indicate that holes are most stable at 2C sites on all three surfaces. We now go on to consider the diffusion of polarons between the most stable sites at each surface. We have considered direct hopping both between such sites and via pathways incorporating hops to metastable 2C or 3C sites. We generate configurations along the barriers using a simple linear interpolation scheme

effect is the increase in the nearest-neighbor bonds due to the Coulomb repulsion between the positively charged cations and the hole. There are a few patterns which emerge from a closer analysis: at 2C sites, the bonds generally increase in length by between 0.2 and 0.3 Å; this is larger than the relaxation associated with polaron formation at a bulk 3C site, which is between 0.1 and 0.2 Å. At 3C sites, the relaxation is rather asymmetric, with one bond elongated by approximately 0.1 Å and the other two bonds being elongated by a greater amount, which is as much as 0.52 Å in the case of the 3C−O2 site on the (1̅11) surface, indicative of a broken bond. The elongation of the nearest-neighbor bonds for the other 3C sites is generally more modest, being typically between 0.2 and 0.3 Å. Hole Trapping Energies. We define the hole trapping energy (Et) as the energy gained when a hole, which is delocalized in the bulk crystal, localizes onto a particular oxygen site. In our previous work we showed that the trapping energy for localization on a 3C bulk oxygen site in m-ZrO2 is 0.13 eV, whereas for a 4C bulk oxygen site it is −0.31 eV (i.e., metastable).31 The calculations of these trapping energies incorporated corrections for electrostatic interactions between periodic images.61 However, for surface slabs it is less straightforward to determine such supercell corrections reliably, making calculation of the trapping energy difficult. The approach taken here is to compare the energy of a hole polaron at a 3C oxygen site in the center of the slab with the energy of a polaron at a surface site and assume that the correction to the energy due to electrostatic interactions between periodic images is similar in both cases. We checked that the bulk-like polaron in the slab calculations is very similar to its bulk counterpart, in terms of local ionic relaxation and electronic structure. We also tested the adequacy of the vacuum gap for calculating the trapping energy in this way and found that the difference in total energies increased by only 4−5 meV, when the gap was increased from 10 to 15 Å. Table 2 summarizes the calculated trapping energies for the (11̅ 1) surface. We note that trapping energies are considerably higher at all surface sites than in the bulk. The 2C sites are deeper hole traps than the 3C sites, and this trend is mirrored for the other surfaces considered. For comparison, corresponding properties are also shown for the m-HfO2 (11̅ 1) surface. Table 2. Properties of Hole Polarons at the (1̅11) Surfaces of m-ZrO2 and m-HfO2a Et (eV) bulk 4C bulk 3C 2C−O1 3C−O2 3C−O3

−0.31 0.13 1.18 0.36 0.43

bulk 4C bulk 3C 2C−O1 3C−O2 3C−O3

−0.31 0.18 1.32 0.41 0.45

Φ−Φ3C (V) m-ZrO2 −0.32 0.00 0.87 0.34 0.55 m-HfO2 −0.35 0.00 0.90 0.23 0.38

εhs (eV)

μ (e)

0.75 1.07 2.40 1.61 1.39

0.71 0.77 0.85 0.78 0.81

0.80 1.15 2.20 1.50 1.20

0.71 0.76 0.85 0.79 0.82

R(t ) = (1 − t )R i + t R f

(3)

where Ri and Rf represent the initial and final atomic coordinates and is t is a parameter which varies from 0 to 1. The ions are held fixed when calculating the energies of the intermediate configurations. We find that for the (11̅ 1) and (111) surfaces the lowest barrier is for the direct hop between the most stable site in the surface unit cell and the equivalent site in a neighboring unit cell (see Figure 5). For the (001) surface, hopping is via the metastable site, although the height of the barrier for direct hopping is almost identical. Interestingly, the barriers are similar for each surface, being between 0.55 and 0.6 eV. Furthermore, the barrier is approximately equal for hops in any direction, so that the diffusion of holes at the surface is essentially isotropic, in contrast to hopping in the bulk of the material, which is effectively confined to the 2D planes of 3C oxygen ions. We have also recalculated the lowest-energy barrier obtained via linear interpolation for each surface using the climbing nudged elastic band method,62 using five images, and found that it is reduced by just 0.02−0.05 eV in all cases, which we ascribe to

Et is the trapping energy; Φ−Φ3C is the electrostatic potential on the site in the ideal structure relative to the potential on the bulk 3C site; εhs is the position of the unoccupied hole state above the bulk valence band; and μ is the spin density of the localized hole projected onto the site.

a

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Figure 4. (a) Correlation between oxygen ion coordination, onsite electrostatic potential (Φ), and the polaron trapping energy (Et). Φ3C is the electrostatic potential for the bulk-like 3C oxygen site in the center of the slab. The straight line is shown to guide the eye. (b) Energetic positions of the unoccupied electronic states of stable hole polarons in the bulk and at various sites at the nanocrystal surface.

of steps generally scale with the formation energy of the facet from which the step is formed, we consider a step formed from an intersection between this and the (111) terrace, which has the second lowest formation energy (Figure 6a). However, this does not uniquely define the termination of a monolayer step, so two-step structures have been considered, each formed from one-half of the (111) surface unit cell. Owing to the coordination of the terminating oxygen ions, we will refer to the two steps as 3C- and 2C-terminated (see Figure 6b,c). For the step calculations, we used a supercell corresponding to 2 × 2.5 (1̅11) surface unit cells, having found that including an extra half unit cell is necessary to make the system periodic. The thickness of the slabs was between five and six layers. The lattice parameters were kept fixed in all calculations; however, all ions were allowed to relax, and a 2 × 2 × 1 Monkhorst-Pack k-point mesh was used in all cases. Our results for hole trapping on terraces demonstrate that 2C oxygen sites are the most stable sites for hole polarons, in all cases. Therefore, we focused exclusively on the 2C sites at, or close to, the steps. The supercells used in this study contain four and three inequivalent 2C oxygen sites for the 2C- and 3Cterminated steps, respectively, most of which are derived from the 2C site on the (11̅ 1) surface; we note that, while the 2Cterminated step introduces two new 2C oxygen sites at the step itself, the 3C-terminated step does not. In both cases, the relaxation relative to the bulk-truncated structures is confined to the region immediately surrounding the step and the first layer of the terraces and is essentially negligible in the central region of the slab. The relaxation serves to make the step more rounded, similarly to what has been found previously for MgO and CaO.35 Interestingly, we found that the absolute value of the electrostatic potential for 2C oxygen ions at the step is slightly lower than that for 2C sites on the (1̅11) terrace. This manifests itself in the calculated densities of states as we find there are no new states split from the valence band. On the basis of these observations, we might already suspect holes not to segregate to steps, although as previously noted, more facile relaxation or the opportunity for the hole to form a bond between two oxygen ions may increase the trapping energy. Hole Polarons at and near Steps. As already remarked, the 2C-terminated step supercell contains four inequivalent 2C sites, two of which are created by the formation of the step

Figure 5. (a) Uppermost layer of the (1̅11) surface with an isosurface of the spin density associated with a hole polaron at the 2C−O1 site. Zirconium ions, oxygen ions derived from the bulk 4C sublattice, and oxygen ions derived from the bulk 3C sublattice are depicted by gray, red, and green spheres, respectively. Spin density isosurfaces (0.14 e A−3) are shown in blue. Equivalent sites in the neighboring unit cells are indicated with dashed circles along with the corresponding barrier to diffusion. (b) The same structure viewed from the side.

the low overlap of the displacement fields of the initial and final polaron configurations. The resultant barrier heights of approximately 0.55 eV indicate that hole polarons at surfaces will still be mobile at room temperature, despite their relatively large trapping energy. Therefore, we consider next whether they might be more stable at monolayer steps, as has been shown to be the case for rocksalt-structured MgO. 3. Hole Trapping at Steps. Step Structure. To our knowledge, there have been no studies, either experimental or theoretical, on steps at the surface of m-ZrO2. On the basis of work conducted on other oxides, we constructed two model steps and studied their hole-trapping behavior. The most prevalent surface is the (1̅11), and since the formation energies 25893



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Figure 6. (a) Atomic structure of the (111) surface and (b,c) two proposed monolayer step structures for the (11̅ 1) surface constructed from the intersection of the (111) and (1̅11) surfaces, referred to as 3C and 2C, respectively. Zirconium ions, oxygen ions derived from the 4C sublattice, and oxygen ions derived from the 3C sublattice are depicted by gray, red, and green spheres respectively.

itself, the other two being on the (1̅11) terraces. The cell for the 3C-terminated step contains three 2C sites, all of which are on the (1̅11) terrace regions. We found that the polarons on the (1̅11) terrace regions are qualitatively similar to those described in the previous section and so do not require further description. Therefore, we will describe in detail only the polarons at the ″new″ sites, i.e., those associated with the formation of the 2C step, which are labeled in Figure 6c. The spin density and ionic relaxation associated with holes trapped at the 2C-terminated step are shown in Figure 7. Though the 2C−O1 site effectively derives from the 2C−O1 site on the (111) terrace, the polaron trapped there is quite different. The relaxation associated with hole trapping at site 2C−O1 involves the hole-bearing oxygen moving parallel to the (111) face of the step. This is in contrast to the 2C−O1 site on the (111) terrace, which displaces essentially perpendicular to the (111) plane. The onsite spin density at the 2C−O1 site is 0.82 e, with most of the remainder, 0.15 e, on a threecoordinated oxygen ion directly behind it on the top of the step. Thus, it is a ″type 2″ polaron. The spin density for the 2C−O2, shown in Figure 7c, is more similar to that of its 2C−O2 counterpart on the (111) surface, being oriented almost perpendicular to the nearest-neighbor bonds; the onsite value of the spin density is 0.82 e, with the remainder being distributed fairly evenly over the nearestneighbor oxygen ions. Despite the fact that the spin density distribution is fairly similar, the relaxation due to the polaron at the 2C−O2 site on the step, which is shown in Figure 7b, is rather different from that of the 2C−O2 ion on the (111) terrace, whereas in the latter case, the nearest-neighbor zirconium ions undergo the major displacement, with the oxygen remaining almost in its initial position, at the step the oxygen moves furthest, in the plane of the (111) step face. We find that the trapping energies at all but two sites are similar, being between 1 and 1.1 eV. The two exceptions are the site on the face of the 2C step, at which the trapping energy is

Figure 7. (a,b) Spin density distributions (left) and associated displacement fields (right) for the two stable hole polarons trapped at the 2C step. Zirconium ions, oxygen ions derived from the 4C sublattice, and oxygen ions derived from the 3C sublattice are depicted by gray, red, and green spheres, respectively. Spin density isosurfaces (0.14 e A−3) are shown in blue. Arrows depicting the displacements of the ion due to polaron trapping are shown with 5 times their actual magnitude.

0.46 eV, and the site on the terrace immediately below the 3C step, at which the trapping energy is 1.19 eV. We note that this is almost identical to the trapping energy at the 2C site on the (1̅11) terrace calculated in the previous section. This indicates that there is no significant driving force for preferential trapping of hole polarons at the step and is consistent with the fact that the absolute value of the electrostatic potential is not decreased at the step. 25894



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potential (i.e., λhs) also eliminates self-interaction exactly for holes at the surface. This issue is not limited to the CON approach but is present for other popular methods which attempt to remove the self-interaction error in DFT; for example, both the DFT+U method and hybrid functionals, which include nonlocal exchange, involve a parameter (or parameters) which must be defined. Whether parameters appropriate for the bulk are transferable to the surface is uncertain and in general very difficult to prove. Another challenging issue for the prediction of hole trapping at surfaces stems from the use of periodic boundary conditions. Most commonly, DFT calculations are performed using threedimensional periodic boundary conditions, as we have done in this case. Therefore, spurious electrostatic interactions between periodic images affect the calculated total energies. In the bulk, there are well-defined approaches for correcting for these interactions, which depend on the dielectric properties of the material. At or close to surfaces, such corrections become very difficult to apply since the dielectric response is highly inhomogeneous, and there is also the possibility of dipole formation. In this paper, we defined the energies of holes localized at surfaces with respect to holes localized in the bulklike region of the slab, partly sidestepping this problem; essentially, we assume that the errors are of a similar order of magnitude regardless of the location of the localized hole. This approximation may introduce some small errors in the absolute trapping energy but should be quite accurate for predicting the relative trapping energies of surface sites. We believe that these issues do not affect our qualitative predictions that holes prefer to localize at surfaces of ZrO2 and HfO2 nanoparticles. Direct experimental observation of the behavior of electrons and holes in oxide nanoparticles is no less challenging than theoretical simulation. Siedl et al.63 have carried out EPR experiments on photoexcited powders of very small (average size 7 nm) m-ZrO2 nanocrystals and detected high concentrations of trapped electrons at the surface. However, the corresponding holes were not detected. Formation of O− species at surfaces of photoexcited ZrO2 nanoparticles, which would correspond to trapped holes, has been observed in ref 45. More detailed studies are hampered by the recombination of holes with electrons and by the presence at surfaces of water and other species. In particular, our recent calculations64 demonstrate that the hole recombination with electrons in the bulk of HfO2 may lead to formation of self-trapped excitons, which in turn can decay nonradiatively or radiatively, producing characteristic luminescence.65 Importantly, our results predict that the trapping energy at the most stable site on a given surface is rather strongly surface dependent, meaning that inhomogeneous charging of nanoparticles is expected to occur. This may lead to an enhanced reactivity of a given surface for example, as the density of O− species on that surface will be higher. Owing to the anisotropic diffusion of holes in bulk m-ZrO2, we predict that the (100) surface should exhibit a far weaker tendency to accumulate holes, for example, under irradiation, than other low index surfaces. Preparation of films or nanocrystals which expose the different surface orientations preferentially would permit a quantitative investigation of this effect. We also predict barriers for hole diffusion at surfaces, which could be probed by temperature-dependent femtosecond time-resolved spectroscopic studies. In summary, we have studied hole localization at surfaces of monoclinic zirconia and hafnia crystals. Since there is little

4. DISCUSSION AND CONCLUSIONS Much of what is currently understood about the interaction of electrons and holes with oxide surfaces is based upon just a few systems which have been studied either experimentally or theoretically. However, there is a widespread perception that charge should preferentially segregate to topological features on the surface such as steps and kinks, due to the presence of undercoordinated ions. Our calculations partly support this idea, insomuch as there is a clear correlation between the coordination of an oxygen site and its propensity to trap holes in m-ZrO2 (m-HfO2). However, our results also highlight an important factor which is often not considered. In bulk m-ZrO2 there are both 3C and 4C oxygen sites, and it is the 3C site which traps holes most strongly. At low index surfaces of mZrO2, one finds both 2C and 3C oxygen ions which can trap holes more strongly than bulk oxygen sites. However, the steps at the m-ZrO2 surface which we have studied introduce no lower coordinated sites than at the terraces, and therefore there is no strong tendency for them to trap holes preferentially. Furthermore, if steps or kinks did expose lower coordination ions (i.e., 1C), they would be rather unstable, either to reactions with surface adsorbates or to desorption. It is interesting to consider if any general trends emerge from these results when taken together with previous calculations on other oxide materials. In this respect, it appears that oxygen ion coordination (and the associated variation of onsite electrostatic potential) is perhaps a more important factor for determining where holes trap in oxide nanocrystals than the long-range polarization of the dielectric by the trapped hole. Therefore, one can make useful predictions concerning where holes will trap in any oxide nanocrystal simply by considering the coordination of ions at its prevalent surface features, including terraces, steps, and kinks. As exemplified by MgO, materials which have highly coordinated ions in the bulk crystal are more likely to exhibit preferential trapping at steps, corners, and kinks at the surface since the electrostatic potential varies strongly with coordination. On the other hand, for materials with lower ionic coordination in the bulk, such as m-ZrO2, trapping is more likely to take place on extended terraces. Furthermore, given the strong relationship between ionicity and bulk coordination number, these observations can also be directly translated into ones relating the location of trapped charges and the ionicity of the material. Regarding polaron diffusion, in the bulk we found that hopping is essentially confined to two-dimensional planes of 3C oxygen ions, despite the three-dimensional nature of the bulk crystal. The extension of this behavior to the surface would be the confinement of polarons to the 1D terminating rows of such planes. However, due to the lower symmetry of the surface, there is typically only one inequivalent oxygen ion per surface unit cell at which the hole is most favorably trapped, as opposed to four in the bulk unit cell. This results in diffusion being essentially isotropic as the hops are over longer distances. Accurate theoretical prediction of hole trapping at oxide surfaces remains an extremely challenging problem. In particular, there are two issues that deserve further discussion. The first concerns the correction of self-interaction at surfaces. In this paper, we have applied the CON method with a λhs value which guarantees the exact elimination of self-interaction for hole polarons in the bulk. This was necessary to be able to compare bulk and surface polarons consistently. However, it is unclear whether the same parametrization for the hole state 25895



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experimental data on the surface structure of these crystals, we first calculated formation energies of all low-index surfaces and found that the most prevalent surfaces were those parallel to (1̅11), (111), and (001), in that order, the first two of which agree with previous calculations. Our electronic structure calculations demonstrated that in all three cases 2C surface oxygen ions formed a narrow band of occupied states split into the bulk band gap, indicating that holes will segregate to the surface. Modeling of hole polarons located at all undercoordinated surface oxygen sites shows that holes do indeed prefer to trap at surfaces, with an energy gain with respect to the bulk of up to 1.05 eV. In all cases they are strongly localized at 2C surface sites. Our results show how the interplay between the electrostatic potential and local relaxation leads to variations in hole trapping energies at different surface sites. The low coordination of surface sites leads to splitting of the surface states into the band gap, reducing the kinetic energies of surface holes, whereas the large distances between equivalent surface sites lead to small hopping integrals and large adiabatic barriers for hole diffusion, also facilitating hole localization.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS MJW was partially supported by Pacific Northwest National Laboratory with funds provided by the Department of Energy, Divisions of Chemical Sciences of the Office of Basic Energy Sciences. Pacific Northwest National Laboratory is operated for the U.S. Department of Energy by Battelle. We are grateful to S. Lany and A. Zunger for discussions and help in using the CON method. We are also grateful to O. Diwald for numerous stimulating discussions of properties of ZrO2 nanopowders and to O. Diwald, K. Hermansson, N. G. Petrik, and P. V. Sushko for helpful comments on the manuscript. This work made use of the facilities of HECToR, the UK’s national highperformance computing service, via our membership of the UK’s HPC Materials Chemistry Consortium, which is funded by EPSRC (EP/F067496). A portion of the research was performed using EMSL, a national scientific user facility sponsored by the Department of Energy’s Office of Biological and Environmental Research and located at Pacific Northwest National Laboratory.

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Hole Trapping at Surfaces of m-ZrO2 and m-HfO2 Nanocrystals - The

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