Ab Initio Studies of Structure, Electronic Properties, and Relative

Article pubs.acs.org/JPCC

Ab Initio Studies of Structure, Electronic Properties, and Relative Stability of SnO2 Nanoparticles as a Function of Stoichiometry, Temperature, and Oxygen Partial Pressure C. A. Ponce,† M. A. Caravaca,‡ and R. A. Casali*,† †

Departamento de Fisica, Facultad de Ciencias Exactas y Nat. y Agr., UNNE, Av. Libertad 5600, C.P. 3400 Corrientes, Argentina Departamento de Fisico-Quimica, Facultad de Ingenieria, UNNE, Av. Las Heras 727, C.P. 3500 Resistencia, Argentina

‡

ABSTRACT: Relative stability of nanocrystals of rutile SnO2 with defined sizes and controlled stoichiometry was studied, allowing the full relaxations of nanoparticle (NP) atoms. Once stabilized in energy, the structure was analyzed by a refinement of the pair distribution function (PDF) of the Sn− Sn and Sn−O pairs. The calculated pair distribution functions Gcalc(r) of our NP’s larger than 3 nm indicate a rutile crystalline core of 2.4 nm size, which are well-compared with recent Gexp(r) determined from XRD experiments. Therefore, distorted surface layers are decreased to 0.25−0.30 nm width, for stoichiometric and oxygen in excess NP’s. This finding is confirmed by the inspection of the core electronic density of states (DOS), determined with the projected density of states (PDOS) of atoms included in radii sizes of 0.5, 0.7, 0.9, 1.1, and 1.5 nm. The deviation of the NP total DOS from the bulk total DOS is analyzed inspecting the PDOS contributions of cation and anion sublattices, for atoms placed in the core, in the distorted layer and in the NP surface. In particular, oxygen atoms at the surface can bond forming dimers and trimers whose electronic states can fix the NP Fermi levels and then act as active centers. The dependence of thermodynamic stability of the NP’s with oxygen pressure and temperature is studied. It was found that at normal O2 pressures and T < 860 K, the more stable case is found to be the NP with O in excess while at low O2 pressures and T > 350 K, the stable NP is the deficient in oxygen one. We also studied the changes of Mulliken populations of Sn and O atoms with respect to the bulk. We found more important variations in those atoms located close or at the surface. These changes strongly depend on NP stoichiometry. This study therefore helps to improve the understanding of the electronic and thermodynamic properties of SnO2 nanocrystals in actual environmental conditions of operation in gas sensor devices.

I. INTRODUCTION Tin dioxide (stannic oxide or SnO2) is a material with important technological applications, such as solar cells, gas sensors, and optoelectronic devices.1−6 It was one of the first oxides considered and presently the most frequently used in high sensibility gas sensors by electrical conductivity variations.3 SnO2 is usually known as a nonstoichiometric oxide, deficient in oxygen. The source of the sensibility to oxygen at the SnO2 surface is attributed to the variable valency of Sn. A partial reduction of Sn4+ to Sn2+ can occur due to a presence of oxygen in the forms of radical ion in different charged states, which are chemisorbed at the surface under variable environmental conditions of temperature and partial pressure of O2.7 Currently, the electronic properties of SnO2 nanodevices with higher sensitivities than their thin-film counterparts8 are heavily influenced by surface processes. Since the chemistry of surface oxygen on SnO2 is modified by the environmental atmosphere, the surface stoichiometric stability in interaction with molecular oxygen attains great practical importance. The dependence on temperature of the structure, composition and electronic properties of three low index surfaces of SnO2 were experimentally investigated with low-energy-ionspectroscopy (LEIS), low-energy-electron diffraction (LEED), © 2015 American Chemical Society

scanning tunneling microscope (STM), and angle-resolved valence band photoemission spectroscopy using synchrotron radiation.9,10 Surface phase transitions from the reduced to stoichiometric form were observed at a 440−520, 610−660, and 560−660 K in (110), (100), and (101) SnO2 surfaces, respectively.11 Theoretical atomistic model has been applied to find the surface relative stability of SnO2 with different Miller indexes and ideal stoichiometric termination for the first time by Mulheran et al.12 Oviedo and Gillan13 have also determined the energetics and structure of a set of surfaces by applying firstprinciples methods and have analyzed the reconstruction of strongly reduced (110) surface, concluding that this (110) surface is the more stable, followed by (100), (101), and (001), respectively. An exhaustive first-principles study of the reduced and stoichiometric SnO2(110) surface with different atomic endings was performed by Manassidis et al.14 Surface electrons that result from the elimination of the neutral oxygen, leaves two electron per oxygen, which distribute in channels passing Received: December 11, 2014 Revised: May 19, 2015 Published: May 27, 2015 15604

DOI: 10.1021/jp512343q J. Phys. Chem. C 2015, 119, 15604−15617

Article

The Journal of Physical Chemistry C

using first-principles calculations, a drop in oxygen vacancy formation energy due to the presence of interstitial Sn (VO − Snint), explaining the natural deficiency of O and the nonstoichiometry of bulk SnO2. Regarding structural and electronic properties of small SnO2 clusters, they were first studied by Mazzone using firstprinciples and semiempirical methods.22,23 These works showed that the rutile structure is maintained in the Sn metallic skeleton. The surface reconstruction and the physics of stability resides in the increase of Sn−O and O−O coordinations in the reconstruction of nanoparticles. In their other calculations using DFT it was found that the primary role in cluster stabilization comes from the interaction of the cluster composition and its size.23 The bond energies as a function of particle size, show oscillations that are attributed to different surface atomic endings of the particles, as a result of the trimmings performed during their construction.22 On the other hand, the stoichiometry is a critical parameter, and its variation can cause the bonding energy to increase considerably. It was also found that the energy functional dependence is qualitatively similar but can show noticeable quantitative differences. This shows the central difference between NP and the known cluster properties where the energy is dictated by its size. The aim of the present work is to expand the state of knowledge of the pioneering works made on SnO2 nanograins by Mazzone.22,23 Here, we performed ab initio studies of structure, electronic properties, and relative stability of SnO2 nanoparticles as a function of stoichiometry, temperature, and oxygen partial pressure. The sizes are within the minimum experimental range found in literature. Stability is here determined according to the surface formation energies. Quasi-spherical nanoparticles with ideal stoichiometry, slightly reduced and with excess of oxygen in sizes ranging from 2.6 to 3.23 nm, are studied. Our nanoparticles are isolated and free of stress. Since the standard functionals applied to DFT in this work fail to yield agreement with the correct forbidden energy band gaps,24,25 only the qualitative effects on DOS (changes in bandwidth) states in the forbidden band, oxygen and tin atom contribution according to its depth and oxygen concentrations, is analyzed. In this work, effort has been made to find an adequate Sn pseudopotential, able to deal with the problem of thousands of atoms with a minimum set of atomic bases, which is capable to reproduce efficiently and with lower computational resources, the static and dynamical properties of the monocrystal at almost the same level of accuracy as found in previous calculations on bulk SnO2 under pressures with a more robust basis set.26,27 The work is organized in the following manner: in the next section, the computational methods are described. Then the results of the calculated energies of formations, nanoparticle surface energy, and the effects of temperature and oxygen partial pressure on nanoparticle stability are presented. The reestructurations of oxygen and tin sublattice atoms are analyzed in the whole nanoparticle (core and external layer), upon changes of the stoichiometry. The crystalline core size, its lattice parameter changes, are estimated in the largest NPs. Mulliken populations of atoms in the nanoparticle and partial contributions to the electronic density of states by the atomic species in the core region and the distorted layer are analyzed.

through bridging oxygen sites.14 The associated electronic density can be attributed to the reduction of tin from Sn4+ to Sn2+, only when the charge distribution on Sn2+ is recognized to be highly asymmetric.14 It was also found that the ionic relaxation are moderate for both stoichiometric and reduced surfaces. These results are very similar to those found in rutile TiO2 using pseudopotentials and density functional theory (LDA and GGA).15 Alternatively, Batzill et al.11 have applied DFT to assess the oxidation state and the stability of different surface structures and compositions at various oxygen chemical potentials. In surfaces (110) and (101), theory supports experimental observation that the phase transition is accomplished by removal of a bridging oxygen atom from a stoichiometric SnO2 leaving a SnO surface layer with a 1 × 1 periodicity. The reduction of the SnO2 surface layer is facilitated by the dual valency of Sn, and for all three surfaces, a transition from Sn(IV) to Sn(II) is observed.11 To explain the dependence of low index surface stability to temperature and oxygen partial pressures, studies were carried out using first principle methods. Surface energies for different endings of (110) and (101) surfaces were calculated as a function of temperature and oxygen partial pressures, applying the VASP method (DFT:GGA).16 Bergermayer and Tanaka systematically investigated SnO2 (110) and (101) surfaces with 11 different kinds of terminations for each. They found that the stoichiometric (110) is the most stable surface at high oxygen chemical potential (i.e., low temperatures or at high O2 pressures). At lower chemical potentials, one of the reduced surface with (101) termination is the energetically favorable. The other surface terminations were found to be less stable. It must be noted that in reduced (110) and (101) surfaces, outerlayer relaxations were found very small. Recently, Xu et al.17 have confirmed Bergermayer-Tanaka results and broadens the description in terms of structures and stabilities to other SnO2 surfaces, depending on environmental conditions. It must be noted that this methodology has also been applied to the study of other compound surfaces, as well (i.e., CeO218). On the other hand, crystal defects, valencies of surface atoms, atom/molecules absorption, the formation of oxygen dimers/ trimers, and outer layer structural relaxations are important to characterize since they modify, by the introduction of electronic states, the forbidden energy gaps. The behavior of Sb defect, the Sb-VO and Sn-VO complexes (VO is the oxygen vacancy) in the crystal and on the SnO2 surface have been examined by Slater et al.19 by applying atomistic calculation techniques. They calculated the most favorable complexes and also predicted the surface segregation and defect energies. However, Trani et al.,20 in the context of DFT, have studied in detail surface and subsurface oxygen vacancies of SnO2 (110) and (101). In the mentioned work, it was shown that the oxygen vacancy in the crystal introduces a completely occupied first energy level and localized around 1 eV above the top of the valence level. However, the surface oxygen vacancies are what strongly modifies the density of states with the appearance of intragap states, covering in larger or smaller extent the window of forbidden energy gaps according to the depth of the vacancy with respect to the surface, as found as well by Manassidis et al.14 The oxygen vacancies, according to these authors also account for the electronic affinity variations with respect to stoichiometric surfaces. It must be noted that Kiliç and Zunger21 have found, 15605


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II. METHODS A. Ab Initio Calculations. 1. Consideration about Crystal Calculations. Calculations were performed using SIESTA code that uses local density approximation LDA.28 This method has shown to be successful in predicting ground states, high pressure phases, and elastic properties of group IV oxide crystals HfO2, ZrO2, and TiO2,29−31 and it has recently been used to predict with good agreement changes in elastic properties and the phonons in the SnO2 rutile transition → ClCa2.26 This method consists of calculating self-consistently total energies, Hellman−Feynman forces, and stresses by solving Kohn−Sham equations and the subsequent relaxation of electrons, ions positions, and unit cell parameters. Nonlocal optimized Troullier-Martins pseudopotentials,32 norm-conserving, were expanded in the Kleinman-Bylander form.33 Exchange-correlation potentials were taken in account with the Ceperley-Alder approach as parametrized by Perdew− Zunger. In bulk calculations, the real space cutoff energy Ecutoff corresponding to 175−350 Ry were used to expand electronic wave functions while the reciprocal space includes 567 k-points corresponding to a uniform grid with 20 Å cutoff,28 for the six atoms, rutile unit cell. The total energy and atomic force differences were converged to better than 0.1 meV per formula unit and 0.002 eV/Å per atom, respectively. The basis set is constructed using Sankey-Niklewski type pseudoatomic orbitals (PAO) generalized to include single and double-ζ decay and are used to represent valence wave functions. Pseudopotentials for atoms in the compound and extensive testings to reproduce SnO2 crystal properties were generated and reveal that the inclusion of 4d electrons in the Sn core results in an improvement in equilibrium volume, electronic structures, and phonon properties, compared with when they are included in the valence set.22,23,34,35 The search for the optimized simple basis transferable to nanoparticles should reproduce not only structural properties but also elastic properties and the phonon dispersion curve found in the bulk. The new base results, simple zeta, obtained with a pseudopotential with the same cutoff radius but with a slightly excited configuration for the Sn atom closest to the s1p3 hybrid covalent bond, allow a good convergence of static and dynamic parameters with a lower cutoff energy (175 Ry) and provide similar results (lattice parameters and phonon frequencies) to those obtained with a more complete base as previously described and applied to the bulk.26 Good agreement of our results compared with experiments36,37 provided the necessary confidence to pursue the calculations in nanoparticles. 2. Supercell Construction and Cluster Nanoparticle Tailoring. Atomic Relaxations. First, the 6-atoms rutile cell was optimized with the conjugate gradient technique using the minimum basis set (SZ) and pseudopotential generated with s1p3 configuration for Sn. Then, the relaxed cell parameters and atomic coordinates were used to build a big supercell which contains the nanoparticle. The nanoparticles are then constructed, including the atoms inside a sphere with specified diameter. In the ab initio calculation, to keep the nanoparticles isolated, they were included in a supercell of cubic symmetry with an empty space large enough to avoid any overlap between wave functions of atoms belonging to neighbor cells. It must be noted that the range of interaction defined by any pseudoatom on the surface is established by the cutoff radius rlc of each PAO

orbitals. So the minimum distance between two NP is more than twice the largest cutoff orbital found. In this way, like in molecules or cluster calculations, the electronic structure has to be sampled with one k-point, the k = 0 (Γ point) (no itinerant states, only localized wave functions). This approach was transferred to the study of nanoparticles in the range of thousands of atoms. This minimum base displays a good balance between efficiency and precision as it allows applying quantum mechanics calculations of clusters up to 3.23 nm (1309 atoms) (see Figure 1), close to the experimental scale of 3.5 nm.

Figure 1. Representations of largest nanoparticles here studied, with indication of the number of atoms and stoichiometry. On the upper row, unrelaxed, and in the bottom row, the relaxed NP.

For the relaxation of our nanoparticles, we used molecular dynamic simulations of the annealing type with an initial temperature of 200° K and a target temperature of 0° K. The residual atomic forces were lower than 0.05 eV/Å after the NP relaxations and a typical number of steps between 1000 and 1500 with a time step between them of 1 fs each, where necessary. This approach showed better energy gain by relaxations than the typical conjugate gradient method usually utilized in the search of bulk relaxations. Since residual forces appear in the first steps, the temperature usually increased up to about 800 K, and then as the cluster continued relaxing, forces and total energies monotonically diminished to the target T = 0 K. B. Approach to the Calculation of Nanoparticle Surface Formation Energy. Nanoparticles relative stability was determined with the Gibbs free energy of formation. The energy of formation in general is the result of subtracting the energy obtained from the relaxed structure (nanoparticle total energy) and the chemical potentials of the atomic species, written in an adequate form. In practice, this surface formation energy can be addressed to the energy we need to break the nanoparticle surface atomic bonds. We consider that the NP surface is in contact with an oxygen atmosphere described by an oxygen pressure p and temperature T. This means that the environment acts as a reservoir, because it can give (or take) any amount of oxygen to (or from) the sample without changing the temperature or pressure. The most stable structure is the one that minimizes the surface free energy.38 15606


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Table 1. First-Principles Calculation of the Structural Parameters at p = 0 GPa: Lattice Constants a and c, Oxygen Fractional Coordinate u, Equilibrium Volume V0, Bulk Modulus B0, and Its Pressure Derivative B′a a c u c/a V0 B0 B′

AI-DZ

AI-SZ

FP-LMTOb

calcc

calcd

calce

expf

expg

4.725 3.263 0.308 0.691 72.84 214 4.5

4.720 3.258 0.307 0.690 72.58 211 4.1

4.761 3.184 0.306 0.669 72.17 181 −

4.699 3.165 0.306 0.673 69.88 242.4 4.76

4.715 3.194 0.306 0.677 71.01 221 6

4.58 3.08 0.304 0.672 64.61 212 −

4.737 3.186 0.307 0.673 71.49 205 7.42

4.738 3.186 − 0.673 71.42 212 5.13

a

Experimental data are added for comparison. Lattice constants a and c are in units of Å, V0 is in units of Å3 and bulk modulus B0 in GPa. u is a dimensionless parameter. bRef 43. cRef 44. dRef 45. eRef 35. fRef 46. gRef 47.

γ (T , P ) =

0 potential of oxygen at T = 0 K and p = p0 as 1/2Etotal O2 = μ0(0,p ) . The temperature dependency of oxygen chemical potentials μO(T,P) can be obtained from experimental, thermodynamic tables.39 The oxygen chemical potential dependency on temperature and pressure is written as

1 nano [G (T , P , NSn , NO) − NOμO(T , P) A − NSnμSn (T , P)]

(1)

where A, μO, and μSn are the surface area and chemical potential of O and Sn, respectively. Assuming that the oxidation at the surface takes place with Sn atoms at the surface and O adsorbed atoms, and that the thermodynamic equilibrium (ΔG = 0) is achieved. Then the sum of elemental chemical potentials must be equal to the compound chemical potential SnO2, as was calculated in SnO2.17

μO(T , p) = μOo (T , po ) +

= [μo (0 K) + ΔHO(T ) − T ΔSO(T )] + kT ln(pO /pOo ) 2

2

where ΔHO(T) and TΔSO(T) are enthalpy and entropy corrections to the standard chemical potential μo(0 K) due to pressure and temperature (see the tables of ref 39). In eq 5, the superindex o indicates the standard state for each quantity, pO2 is the partial pressure of oxygen gas, and k is the Boltzmann constant = 8.314 J/(K mol). Following the Bergermayer et al. approximation,16 we consider that pV and TS are similar for different NP surface endings. Thus, we compare only the internal energy contribution to the formation energy of slightly reduced to stoichiometric, when sizes are the same. In principle, one may have to consider the temperature dependence of free energies of solids. However, the major contribution for the temperature dependence of the surface energy, as given by eq 1, comes from the large standard entropy of O2 gas as compared to the solid systems. Moreover, the smaller temperature dependence of chemical potentials of the solid can be canceled in eq 1. Therefore, only the temperature dependence of μO was taken into account in this work.

2

(2)

where the lower case g denotes a Gibbs free energy per formula unit. Inserting this constraint in eq 1, we obtain the NP energy of formation per unit of surface γ, in thermodynamic equilibrium, γ (T , P ) =

1 nano bulk [G (T , P , NSn , NO) − NSngSnO (T , P ) 2 A + (2NSn − NO)μO(T , P)]

(3)

which now depends only on the oxygen chemical potential. Here Gnano (T,P,NSn,NO) is the Gibbs free energy of the NP surface given by G = U + pV − TS, where U = Enano (T = 0, P = 0, NSn, NO) is the total energy of NP calculated by the DFT bulk method. We also set gbulk SnO2(T,P) = μSnO2(T = 0,P = 0), which is a good approximation in the bulk, assuming the entropy contribution due to temperature is negligible in the range of temperatures studied.38 Given a choice of chemical potentials, we get different responses in the system. Then we must put lower and upper bounds to its values. For instance, if μO becomes too low, all oxygen atoms leave the sample and the Sn atoms crystallize at the surface. This is in connection with the maximum chemical potential of the Sn atom, determined by the condition bulk max[μSn (T , P)] = gSn (T , P )

2

(5)

μSn,surf + μOad = μSn,hcp + 2μ[O] = gSnO ,bulk = μSnO ,bulk 2

1 kT ln(pO /pOo ) 2 2 2

III. RESULTS A. Bulk Results Using SZ and DZ Basis Sets. As found in ref 40, SnO2 structural and elastic properties are wellcharacterized using LDA approximation, as well as lattice dynamical properties under pressures,41 where neither GGA for the central region phonon calculation34 nor the structural band42 give a better description for SnO2. LDA approximation is then chosen since atomic relaxation and reestructurations play an important role in the stabilization of the surface. The results obtained in the study of structural properties, using both SZ and DZ basis set in the SIESTA code, are in very good agreement with experiments, as can be seen in Table 1. We performed further extensive tests of elastic and dynamical properties (i.e., phonons at the center of the Brillouin zone with SZ and found that they are in good agreement with our previous study).26 Therefore, the s1p3 pseudopotential and the SZ basis set are proper to study SnO2 nanoparticles of thousands of atoms.

(4)

which because of eq 2 fixes the lower limit of μO (oxygen poor limit). Given the high temperatures and the variable oxygen pressures in ceramics, it is important to consider including the gaseous phase that interacts with the surface of the particle, from adequate oxygen atom chemical potential.16,17 The gas phase above the SnO2 surface contains O2 molecules. The total energy of the O2 molecule can be related to the chemical 15607


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It can be observed that at temperatures below 290° K and at very low pressures [ln(p/p0) = −80], there is only one stoichiometric state (oxygen in excess). Above this temperature, there are two transformations: one to stoichiometric (290° K < T < 350° K) and the reduced at T > 340° K. The temperatures of these transitions increase with pO2 in a nonlinear form. Finally for pO2 close to 1 atm, the transition temperatures are 850 and 1020° K (577 to 747 °C). If one thinks to use the NPs as constituents of a SnO2 sensor, the suggested temperatures should be close to 577 °C, at high gas pressures, and lower, for lower gas pressures. In practice, the device ranges between 250 to 450 °C, which in this case coincides with the zone of maximum sensibilities for gas pressures of −20 < ln(p/p0) ← 6 (on the order of 0.002 < p/p0 < 250 ppm). We mention a series of experiments (XRD, photoluminiscence spectra, transmittance, and Hall effects) done on polycrystalline thin films of SnO2 deposited on saphire substrate at 450 °C, using laser ablation techique and under O2 pressures of 1, 5, 10, 20 Pa.48 It was observed that the content of SnO (reduced) versus SnO2 was detected at 10 and 20 Pa (XRD). An increase of carrier mobility and a reduction of resistivity of the film from 6 to 0.09 Ω cm, when the oxygen pressure was increased from 1 to 20 GPa was also found. If we trace a line [i.e., of 450 °C (723° K) in our Figure 3], we found transitions at ln(p/p0) = −8, −17, or partial pressures of O2 of 33 and 0.004 Pa, respectively. This range contains the mentioned experimental one of ref 48, so it is an interesting coincidence with our diagram obtained with ab initio calculations. The upper temperature in which the three stoichiometries can be found is 864° K (587 °C degree). At this T, the reduced NP is stable at p < 37 Pa, the stoichiometric one is stable at 37 < p < p0, and at p > p0, the oxygen in excess NP is the more stable form. C. Structure of the NP and Pair Contributions to the Calculated G(r) PDF of the Crystalline Nuclei and the Distorted Layer. The resulting atomic displacements obtained from relaxations and surface reconstructions were analyzed in detail for the largest three particles, according to layers with different radii. The crystalline core size was found by analyzing − the absolute value of relaxation (displacement) Δrj(r) = |rrelax j r0j |, of each atom j with respect to its crystalline, initial position. The coordinates origin is a Sn atom located at the center of the NP. The decompositions of atomic relaxations along the x, y, and z coordinates (directions [100], [010], and [001], respectively) show that in the xy plane, oxygen and Sn displacements are more pronounced than those in the z direction. To recognize the nanoparticle crystalline core size, the atomic displacements were studied as a function of the distance of the nanoparticle center. The region which contains atoms with displacements lower or close to 0.236 Å (Δrj/a0 < 0.05) were considered as a part of the crystalline core.49 This is in correspondence with the Lindemann criterion in which the average displacements 1/2 of atoms due to temperature vibrations that define the order parameter ΔL49 to be between 0.1−0.15, the nearest neighbor distance rNN (in our case, rNN = 2.2 Å, the bond length of Sn−O atoms). In general, it is found that the Sn sublattice acts as a retained rutile skeleton, independent of stoichiometry. The oxygen sublattice is the one that really readjusts to the Sn skeleton, in the core and the nanoparticle surface, as was previously found by Mazzone.22,23

With respect to surface studies, GGA and LDA approximations have been employed.17 It can be noted that the relative surface energies of different indexes stay close in both approximations. In GGA, energies are 20% lower than that obtained by LDA. Since the purpose of this work is to describe the relative stabilities of isolated stress-free nanoparticles according to surface termination, we assume that it is sufficient to apply LDA approximation. B. Nanoparticles Surface Formation Energies. The relative stability of nanoparticles was determined calculating the surface free formation energy as described in the Methods section. In all cases, we considered oxygen partial pressures lower than 1 atm, and the oxygen chemical potentials are the same as those for stoichiometric, reduced, and particles with O in excess. The formation energy γ(T,P) for isolated SnO2 nanoparticles was determined at a given temperature and changing oxygen partial pressures. Therefore, to include the effect of the environment (oxygen partial pressures and temperatures) on reduced particles and with excess of oxygen, the chemical potential was modified, as detailed in eq 5. In Table 2 are shown the calculated formation energies (eq 1) at T = p = 0 for nanoparticles with different stoichiometries and sizes between 2.6 to 3.23 nm. Table 2. Related Parameters in the Study of Quasi-Spherical NPa NP formula

Nattot

γ(T = 0,p = 0)

n = NO/NSn

sizeunrelax

sizerelax

Sn261O516 Sn270O540 Sn321O628 Sn328O656 Sn405O768 Sn413O826 Sn385O924

777 810 949 984 1173 1239 1309

2351.5831 2217.6073 2418.5014 2375.3529 2595.6814 2281.7912 1348.9607

1.98 2.00 1.96 2.00 1.90 2.00 2.40

2.60 2.65 2.77 2.82 2.97 3.00 3.04

2.63 2.70 2.87 2.91 3.02 3.08 3.23

a

Nattot corresponds to the total number of atoms including both species in the nanoparticle, γ(T,P), the formation energy in mJ/m2, n = NO/ NSn, the stoichiometry of the NP, and the unrelaxed and relaxed NP sizes in nanometers.

Figure 2 shows the calculated surface energies for the largest nanoparticles, with 949, 1173, 1239, and 1309 atoms (2.87 to 3.23 nm). The three types of NP can be distinguished by different slopes of the surface energy represented in this case by straight lines as the O chemical potential changes. Stoichiometric particles (n = 2.0) are horizontal since it does not depend on chemical potential (eq 1) and can be distinguished from reduced (n = 1.88) since they have a positive slope and from that with oxygen in excess (n = 2.4) with negative slopes. The slope depends on the temperature. In each graph of Figure 2, panels a, b, c, and d show values of γ(T,p) at temperatures 300, 600, 800, and 1000 K, respectively, where temperature is kept fixed and O2 pressures cover a wide range below 1 atm. Information on surface energy, its temperature, and oxygen partial pressure dependencies can be summarized in the T−P diagram of relative stability Figure 3, where T is made continuous, instead of fixed values, like in Figure 2 (panels a, b, c, and d). Figure 3 represents the relative stability of the NP’s here studied and is built from a projection of a 3D (T,P,GNP) surface, where the GNP is the NP Gibbs free energy are local minima. Then, a different color zone shows stability regions according to the nanoparticle stoichiometry. 15608


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Figure 2. Surface energy as a function of oxygen partial pressure at different temperatures, (a) 300, (b) 600, (c) 800, and (d) 1000 K, and several oxygen concentrations.

1. Relaxations Analysis of 1173 Atoms Reduced Nanoparticle. In Figure 4a are shown the Δrj(r), relative to the lattice constant a0 = 4.73 Å, for the 1173 atoms NP. Note that in the xy plane, oxygen and Sn displacements are greater than those in the z direction (see Figure 4, panels b and c). First, to estimate the relative displacement of oxygens atoms, from inner core of the NP toward the outer shells, the theoretical Sn−O pair distribution functions (see Figure 5) for core sizes of 0.5, 0.7, 0.9, 1.0, 1.1, and 1.5 nm are analyzed. The width of the first three peaks (located at 0.21, 0.36, and 0.39 nm), is correlated with sublattices displacements. We found that atoms, inside of a core size of 1.1 nm, holds a narrow distribution around their initial positions and are considered belonging to the crystalline core. For a core size of 1.5 nm (3.0 diameter), contributions between the narrow peaks become visible, showing an increased disorder in the external layer. In this layer, surface oxygen atoms show larger displacements than the Sn atoms, as found in previous thin film theoretical calculations.14 Theoretical Sn−Sn pair distribution functions (see Figure 6) show less departure than the Sn−O distribution (Figure 5), relative to crystalline peak localization, showing lower distortion for the Sn sublattice. 2. Relaxations Analysis of 1239 Atoms Stoichiometric Nanoparticle. Atomic relaxations of the stoichiometric (Sn413O826) nanoparticle, shown in Figure 7, have a similar behavior to relaxations of the reduced Sn405O768 NP but now with lower distortion at the surface. In the stoichiometric NP (1239at), atomic relaxations are lower in their absolute value (Figure 7a), as well as its Cartesian projections (Figure 7, panels b and c), compared with the reduced NP, shown in Figure 4. Therefore, crystallinity is increased in the stoichiometric case. The radial pair distribution functions (PDF) of Sn−O pairs for radii 0.5, 0.7. 0.9, 1.1, and 1.5 nm (Figure 8) show that, in the stoichiometric nanoparticle, nearest neighbor (NN) atoms and next-nearest-neighbor (NNN) Sn−O atoms are distributed

Figure 3. Temperature and oxygen partial pressure dependence of the SnO2 NP surface energy, in relation with its stoichiometry: with an excess of oxygens at the surface and excess of Sn. In the middle (white), the stoichiometric NP is the more stable.

To recognize the tetragonal phase in the relaxed nanoparticle crystalline core, a refinement of the atomic radial pair distribution function PDFij(r), between different species i, j needs to be applied. These functions are related to the G(r) PDF calculated from X-ray and electron diffraction experimental data, currently used to find the nanoparticle structure.50−52 Its knowledge is very important to determine the nanostructure of matter.53 Thus our PDF can not only be compared with available experimental data of G(r)50,51 but also be used to obtain, from first-principles, the effective structural parameters of the NP: oxygen fractional coordinate u(O), lattice constants a, c, and c/a relation. The last three parameters are estimated from the second and third Sn−Sn sublattice neighbors of our theoretical PDF(Sn−Sn), which, as it will be seen below, were found to be in agreement with recent experimental assessments.51,52 15609


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Figure 6. Pair distribution function of Sn−Sn as a function of cut radii in the stoichiometric 3.02 nm (Sn405O768) NP.

Figure 4. Atomic displacements of the reduced (Sn405O768) nanoparticle, (a) absolute relaxation, (b) relaxations along the y axis, and (c) relaxations along the z axis.

Figure 5. Pair distribution function of Sn−O for different cut radii of the stoichiometric 3.02 nm (Sn405O768) nanoparticle.

Figure 7. Atomic displacements of the stoichiometric (Sn413O826) nanoparticle. (a) Absolute relaxation, (b) relaxations along the y axis and (c) relaxations along the z axis.

in a narrow position around the unrelaxed one, except when one considers the region between 1.3 < r < 1.5 nm, which contributes to widen the distribution. Also there appear some contributions between the first peak located at 0.22 nm and the second at 0.36 nm. Similar behavior is found in the Sn−Sn PDF

shown in Figure 9. The comparison with the PDF of the reduced nanoparticle shows again more crystallinity. 3. Relaxations Analysis of 1309 Atoms with Oxygen in Excess Nanoparticle. In the case of 1309 atoms NP, both Sn and O sublattices held low relaxations for r < 1.25 nm but they increase noticeably, in the O sublattice, for 1.25 < r < 1.6 nm 15610


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Figure 8. Pair distribution function of Sn−O for different cut radii of the stoichiometric 3.08 nm (Sn413O826) nanoparticle.

Figure 10. Atomic displacements in the 3.23 nm nanoparticle Sn385O924. (a) Absolute relaxation, (b) relaxation along the y axis and (c) relaxation along the z axis.

Figure 9. Pair distribution function of Sn−Sn as a function of cut radii in the stoichiometric 3.08 nm (Sn413O826) NP.

(see Figure 10a). The excess of oxygen in the layer close to the surface induce somewhat large displacements in these O atoms. However, the Sn sublattice is much less relaxed. Again here, in Figure 10 (panels b and c), atomic relaxations along the (010) direction are slightly greater than in the (001) one. The pair distribution functions in this oxygen-in-excess NP reveal crystallinity inside a core of 1.1 nm, as seen both in Sn−O and Sn−Sn distribution functions (Figures 11 and 12). In Figure 13, we represent the theoretically calculated Gcalc(r), using the program ISAACS53 and our relaxed atomic coordinates for the three largest NPs, compared with available experimental G(r) [Gexp(r)], estimated from XRD experiments50,52 in nanometric SnO2 powder. Gcalc(r) was obtained without any smoothing. There is an overall agreement until r = 1.5 nm, for the positions of the main and secondary G(r) peaks of our Gcalc and Gexp, although little better agreement between the case of the nanoparticle with oxygen in excess (1309 atoms) with the resulting Gexp of ref 50 using a Qmax = 28 Å−1. Note that Gexp of ref 52 was determined with a lower resolution, Qmax = 20 Å−1. From the G(r) of Figure 13, we determine a lattice constant c to be 3.71, 3.70, and 3.68 Å for Gcalc[1309at], refs 52 and 50, respectively. The lattice parameter a estimated from Figure 13 are a = 4.73, 4.76, and 4.74 Å for Gcalc[1309at], refs

Figure 11. Pair distribution function of Sn−O at different cut radii in the oxygen in excess of 3.23 nm (Sn385O924) NP.

52 and 50, respectively. Finally, from the Sn−O distances dSn−O = 2.07, 2.07, and 2.032 Å, we get u(O) = 0.309, 0.3075, and 0.303 for Gcalc[1309at], refs 52 and 50. D. Electronic Properties and Mulliken Charge Population in the Crystalline Core and the Distorted Layer of the SnO2 Nanoparticle. Dependence with Its Stoichiometry. To ensure that the averaged atomic displace15611


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the contribution of PDOS of all atoms included in the investigated region. These DOS inside the NP were compared to the bulk density of states. Therefore, concentric layers of accumulated density of electronic states were analyzed for the quasi-spherical form, for stoichiometric and those with excess oxygen cases (Figure 14, panels a and b). Atom contributions are distinguished according to their concentric location and for some specific radii belonging to crystalline core, the procedure similar to the one done by Williamson et al.,54 in order to distinguish the states introduced by the NP external atoms. It is found that the NP electronic density of states are similar to bulk density of states, for cores sizes which agree with structural criteria described previously. Studies of charge populations were performed on relaxed NPs, with different atomic species surface concentrations. In general, the region referred to as crystalline core presents a constant charge population, which does not depend on which species predominate in the surface. It is in the distorted and surface layers, where the electronic charge population on the atoms changes noticeably. In Figure 15 (panels a−c), particle charge populations are shown for (a) slightly reduced, (b) stoichiometric, and (c) with oxygen in excess NPs. The difference between the Mulliken population of each atom in nanoparticle, and its value in a pure SnO2 crystal, was obtained. These differences are shown in Figure 15 (panels a−c), as a function of the atom distance to the nanoparticle center. Charges are in units of e− and the distance to center r in nanometers, respectively. 1. Particle with Sn Atoms in Excess (n = 1.89): 3.02 nm or 1173 Atoms. As in all other nanoparticles, in the crystalline core region the charge differences between the atom in the NP and in the bulk, Q−Qbulk, is almost constant and below 0.1e− for O and 0.2e− for Sn. When we approach the layer close to the surface region and inside of it, a gain of electronic charge in tin atoms is observed, which increases as the surface boundary is approached. The amount of charge loss in oxygen atoms is much smaller than that gained by tin, in this case. A marked charge accumulation is established on the surface boundary with an approximate width of 0.15 nm (see Figure 15a). This creates a double electric layer (of positive ions on the exterior and negative ions on the interior) that generates a dipolar moment aiming outward of the NP, causing a possible downward shift of the Sn electronic states that contribute to the valence band.

Figure 12. Pair distribution function between Sn−Sn atoms at different cut radii in the oxygen in excess of 3.23 nm (Sn385O924) NP.

Figure 13. Pair distribution function Gcalc(r) for the largest nanoparticles with 1173, 1239, and 1309 atoms compared with experimental G(r) determined from X-ray experiments.50,52

ment criteria of 0.05a (0.236 Å), that is used to define the crystal core size to be correct, the density of electronic states (DOS) from atoms inside the NP core was also analyzed in each nanoparticle. In this case, the calculated DOS comes from

Figure 14. Density of states at different core sizes, for nanoparticles, (a) stoichiometric and (b) with oxygen excess. In each graph, total DOS of bulk SnO2 is added for comparison. 15612


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Figure 15. Variations of the atomic Mulliken electronic populations of the atoms in nanoparticles (a) reduced, (b) stoichiometric, and (c) with oxygen excess, with respect to the SnO2 bulk. Red and black dots corresponds to O and Sn atoms, respectively. Charges are in units of e−.

Figure 16. Projected density of states for atoms located close to the axis which pass through the center of the NP, along the z direction. (a) PDOS of tin atoms and (b) PDOS of oxygen atoms. The upper PDOS of (b) corresponds to the atom O18 (dimer) and lower PDOS to the O1169 (middle atom in the trimer O−O−O). The total atom counting of the NP is 1173.

If the Sn atoms partial density of states localized in the crystalline core region is analyzed, similarities can be observed, especially in the deep region, from −21 to −17 eV, as in the upper region, from −10 to −2.5 eV, or at the upper valence band (Fermi level is at 0 eV). Analyzing contributions of Sn atoms located at the subsurface layer (second layer in Figure 16a), similar characteristics to inner atoms can be observed. In case of atoms in the upper limit (Sn8 and Sn1173 of Figure 16a), specific contributions to PDOS in the range of −15 to −22 eV can be observed and add laterally to the lower valence band (−17 to −21 eV). These peaks come from the contribution to the Sn site mainly from oxygen 2s electronic states whose coordination number is smaller than that of the crystal. In the PDOS of the oxygen atoms (Figure 16b), the O external atoms (labeled as

O1169 and O18) add a couple of peaks near the Fermi level (one of them of O18, coincides with Ef) and other localized peaks at −15 and −21 eV (O18) and −15 and −22 eV (O1169). Note that the Mulliken populations of O1169 and O18 are 6.56 and 6.24 electrons, respectively. The interatomic distances for the dimer O18−O9 is 1.54 Å and for the trimer O1171-O1169-O1164 1.5 and 1.74 Å, which are higher than the interatomic distance of 1.20 Å of the O2 molecule. In Figure 17 can be seen detailed the positions of surface atoms O18 and O9, forming a dimer and in Figure 18 details of the positions of surface atoms O1171, O1169, and O1164, forming a trimer bounded to Sn surface atoms. These contributions are not present in subsurface oxygens (Oss), nor in internal oxygen’s (Oint), but it should be noted that in Oss, a couple of small localized contributions at −1 eV 15613


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from more outer oxygen atoms is observed. Mulliken’s population analysis of both atomic species shows that, within a sublayer with thickness of the order of 0.25 nm, atomic Sn charges undergo an increase of fractions of e−, while oxygen atoms lose electrons, in less degree. This effect can be verified in the PDOS of the oxygen atoms. These PDOSoxygen (see Figure 19b) does not undergo major changes through the surface layer toward nanoparticle inner atoms. However, in the PDOSSn can be observed that outer Sn atoms (Sn20 and Sn1196) are distinguished from the inner Sn atoms through the appearance of localized electronic states peaks, in the valence band, at −1 eV from Ef (Figure 19a). This band is slightly wider and with a center of gravity slightly displaced toward the upper levels. The integration of PDOSSn up to the Fermi level Ef yields the atoms total charge, which in this case is greater than the bulk, thus there will be a charge increase at Sn in the surface. External layer Sn atoms contribute to the unoccupied electronic state part of DOS, with a pair of peaks located at +3.5 and +4 eV. As expected, internal Sn atoms barely contribute to DOS at Ef and the forbidden energy band (Figure 19a). 3. Nanoparticle with an Excess of Oxygen (n = 2.4): 3.23 nm or 1309 Atoms. Because of the importance of O atoms in the surface, on the electronic DOS of NP, we have investigated the possible formation of oxygen complexes in the largest particles and divide them in groups according to their coordination number. In the first group, enter the monatomic oxygen bounded with one, two, and three Sn atoms. The second group is composed by two O atoms, and they are coordinated with Sn atoms in four different ways. In the third group, with 3 oxygen atoms, we found two forms. In the fourth group, complexes of four O, they are connected in two ways. In the largest group, with five O atoms, we found two complexes. Here, the crystalline core region shows a Mulliken charge population similar to the previously discussed particles (Figure 15c). However, in the distorted layer is where a different behavior is found. By analyzing the charge difference of the atoms located inside the layer, visible changes in atomic charges of oxygen can be seen. Inside the 0.4 nm subsurface layer and outside of the NP, oxygens lose part of their charge (δQ up to −0.7e−) in order to become more neutral (partial charge compensation) for a good number of them (more in the outer ones). This suggests that oxygens on the surface change their charge state from O2− to states between O−1.5 and O−1.3.

Figure 17. Details of the positions of surface atoms O18 and O9 forming a dimer and bounded together to Sn8. Arrow points to O18. Total atom counting of the NP is 1173.

Figure 18. Details of the positions of surface atoms O1171, O1169, and O1164 forming a trimer, bounded to Sn1113 and Sn1119. Arrow points to O1169. Total atom counting of the NP is 1173.

from the Fermi level appear. These contributions disappear for Oint. In conclusion, the studied reduced particle, Ef, is dominated by states that come from Osup atoms. Note that the PDOS described is performed on atoms chosen from a line parallel to the Z axis and covers the totality of the particle. 2. Stoichiometric Nanoparticle (n = 2): 3.08 nm or 1239 atoms. The Mulliken charge population of this crystalline core is similar to the reduced particle. However, in the distorted layer, with an extension of 0.25 nm, a larger charge transfer

Figure 19. Projected density of states for atoms located close to the axis which pass through the center of the NP, along the z direction. (a) PDOS of tin atoms and (b) PDOS of oxygen atoms. NP with 1239 atoms. 15614


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Figure 20. Projected density of states for atoms located close to the axis which pass through the NP along the z direction. (a) PDOS of tin atoms and (b) PDOS of oxygen atoms. NP with 1309 atoms.

Sn atoms show smooth variations of their charge states with respect to the values in the crystal, even when they are located in the distorted layer region (although there are a couple of atoms with δQ = 0.3e− and δQ = 0.5e−, the majority does not take more than δQ = 0.1e−). It can also be noted in some of the Sn, for r between 1.0 nm and 1.4 nm, a minor charge loss that does not exceed δQ = 0.05e−. In this particle with an excess of oxygen, PDOS for Sn atoms from the center of the particle are identical among themselves and do not exhibit, in practice, any contribution to states at the Fermi level. What can be found is a broadening of the valence band of PDOS of Sn5 and Sn1048 (external atoms in Figure 20a), coming from electrons lost by their neighboring oxygens. The appearance of a pair of additional peaks located in the −15 eV zone and others at +1.5 eV, already in the conduction band, must be noted in these atoms (Sn5 and Sn1048). The PDOS analysis of oxygen atoms show that Osup (O1100 and O1168) contribute with localized states at −22, −15 eV (O electrons 2s), in the Fermi level (O1100) as well as in the vicinity of Ef (O1168). Morover, Oss (O68 and O1076) shows states located slightly below Ef and lateral peaks in the deep region of valence band, located between −20 and −17 eV. In this nanoparticle, Ef is fixed by the oxygen atoms located in the surface region and shows a shift in the valence band toward higher energies (see PDOS of O1100 Figure 20b).

is 50% greater than the distorted layer volume in nanoparticles of 3.08, 3.23 nm, explaining why the nanoparticles of these size ranges are distinguishable with the XRD technique. The effects of temperature and oxygen gas partial pressures were included in the NP Gibbs free energy. With G(p,T), we determined the conditions of stability for size and stoichimetry as a function of T, p(O2). For instance, at low T and high O2 partial pressures, the more stable form corresponds to NP with oxygen in excess at the surface. The studied nanoparticles transit from stoichiometric to the reduced surface, at conditions of oxygen pressures and temperature similar to that used in gas sensor device operations. Theoretically, the maximum temperature at which the three states can be found is about 864° K (587 °C). This is above the usual temperature (450 °C) of gas sensors made of SnO2. At this last temperature, our study predicts oxygen partial pressures in the range of 0.004 to 33 Pa at which the transition occurs, in agreement with the device working conditions. Thus, our work allows one to build a diagram of “states” which indicates how the concentration n = NO/NSn changes in a wide range of temperatures and oxygen partial pressures. These results show the reliability of the method in predicting the operating regime of these sensors. The analysis of different contributions to the electronic density of states (DOS) reveals, in the relaxed NP, the formation of surface oxygens forming dimers and trimers. These types of defects at the surface, are the main contribution to peaks of PDOS close to the NP Fermi level, therefore they are candidate to surface active sites. Another analysis of particle DOS shows a clear difference in the total DOS between core radii of 1.1 and 1.5 nm. While the lower radii show similar DOS to the bulk states, the r = 1.5 nm fills the band gap with states supplied by Sn and O surface atoms. Analyzing electronic Mulliken population changes at the nanoparticle, the charge populations for atoms in the core are similar to atoms of the bulk. However, close to the distorted shell and at the surface, there are charge redistributions which strongly depend on the surface termination, that is, the stoichiometry. In the case of the reduced particle, the Sn atoms at the surface, gain electronic charge, suggesting an electronic transition from Sn4+O2 to Sn2+O2, which somehow reproduces experimental interpretations. However, in the case of the particle with oxygens in excess, or experimentally the case of a rich O2 atmosphere, a loss of

IV. CONCLUSIONS Relative stability of rutile-SnO2 nanocrystals, with defined sizes and controlled stoichiometry was ab initio studied, allowing the nanoparticle full relaxation. Once minimized their total electronic energies, the structure was analyzed by inspection of atomic displacements of Sn and O sublattices and a refinement of the Sn−Sn and Sn−O pair distribution functions (PDF). Atomic displacement analysis show that reduced and stoichiometric NPs (3.02 and 3.08 nm) have a distorted surface layers width of 0.25 nm, while the oxygen in excess NP (3.23 nm) has a distorted layer width of 0.30 nm. The PDF functions show that a rutile crystalline core of significant size remain in NPs of about a 3 nm diameter. In other words, the critical size of the crystalline core turned out to be about 2.5−2.6 nm and is consistent with the values determined from the inspection of the electronic density of states of concentric layers inside the NP. These results shows that the volume of the crystalline core, 15615


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negative charge is verified on the surface, which is lost mostly by surface oxygen atoms (transition O−2 to O−). The stoichiometric nanoparticle charge is an intermediate case between the two previous cases.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We like to thank support from the ANPCYT, Argentina, for the grants received under Project PICT-CNPq Nr. 066/2008. Also support from the Projects supported by SECYT-UNNE. C.A.P. would like to thank the CONICET for the fellowships received during the development of this work. ANPCYT, SECYTUNNE, and CONICET are Academic-Scientific Organizations belonging to the Argentine Republic.

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REFERENCES

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