SnO2: BULK AND SURFACE SIMULATIONS BY AN AB ... - CiteSeerX

SnO2: BULK AND SURFACE SIMULATIONS BY AN AB INITIO NUMERICAL LOCAL ORBITALS METHOD ´ b A.V. POSTNIKOVa,∗, P. ENTELa and PABLO ORDEJON a

Theoretische Tieftemperaturphysik,

Gerhard Mercator Universit¨ at, 47048 Duisburg, Germany b

Institut de Ci`encia de Materials de Barcelona (CSIC),

Campus de la Universidad Autonoma de Barcelona, 08193 Bellaterra, Spain (Received October 30, 2000)

Abstract The results of first-principles crystal structure optimization for the bulk rutile-type tin dioxide and its (110) and (001) surfaces as obtained by applying the siesta code, that incorporates norm-conserving pseudopotentials and strictly localized basis of pseudoatomic orbitals, are summarized. The relaxation near the (110) surface has been studied with the use of small supercells Sn6 O10 and Sn6 O12 , representing reduced and stoichiometric compositions, respectively, and compared with previously known results of other simulations. Further on, the effect of relaxation has been studied for a larger (stoichiometric) supercell Sn10 O20 . While qualitatively the same as obtained for small supercells, the relaxation pattern shows however numerical differences. The relaxation study at the (001) surface shows, as the major effect, the inward relaxation of the upper tin layer and the outward displacement of upper oxygen atoms (by ≈0.3 ˚ A) with respect to it. When going deep into the crystal, the values of Sn–Sn interplane distances and O–Sn interplane displacements fluctuate around the corresponding values in the bulk and decrease. They are not yet stabilized near the 5th tin layer from the surface, implying the need to consider even larger supercells for the accurate numerical estimations of the relaxation at the surface. ∗

Corresponding author. Tel. +49-203-3793323; fax: +49-203-3793665. E-mail: [email protected]

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Keywords:: Surface relaxation, First-principles structure optimization, DensityFunctional Theory

1. INTRODUCTION Tin dioxide is nominally a wide-band insulator which however in normal conditions behaves like an n-type semiconductor, due to the presence of oxygen vacancies. A broad-range change of conductivity depending on ambient conditions allows the use of tin dioxide in gas sensors – see e.g. Watson, Ihokura and Coles (1993). The performance of such sensors is essentially determined by the morphology of, and chemical reactions at, the surface. The properties of SnO2 nanoparticles where the surface effects dominate are therefore of interest for both experimental and theoretical studies. Another field of possible applications of tin dioxide is related to a dependence of its optical band gap on the size of particles. Here again, an insight into electronic structure and its relation to structural properties by means of theory is highly interesting. The electronic structure of ideal bulk SnO2 is well studied by ab initio calculations (Peltzer y Blanc´a, Svane, Christensen, Rodr´ıguez et al., 1993; Goniakowski, Holender, Kantorovich, Gillan et al., 1996), and the phonon dispersion has been recently determined from first principles by Parlinski and Kawazoe (2000). The study of surfaces is quite computationally demanding and advances rather slowly. Rantala, Lantto and Rantala (1994) studied the electronic structure of clean (unrelaxed) surface, also in the presence of adsorbed NO molecules, by a tight-binding cluster method; Manassidis, Goniakowski, Kantorovich and Gillan (1995) calculated surface energy and structure relaxations near the (110) surface by a pseudopotential method. Later on, Rantala, Rantala and Lantto (1999) and Rantala, Rantala and Lantto (2000) studied relaxation of the (110) surface by two different techniques, a semiempirical tight-binding scheme and plane-wave pseudopotential calculations. Recently we became aware that Oviedo and Gillan (2000) performed a more detailed study of several SnO2 surfaces, comparing their energies of formation, and started to look into mechanisms of the formation of oxygen vacancies, making use of the VASP code, that incorporates ultra-soft pseudopotentials and plane waves as basis functions. Our objective is to address the electronic structure, structure relaxation and – in perspective – dynamic properties of tin dioxide on an ab initio basis, making use of a method that would enable to treat bulk systems, surfaces and small clusters on the same footing. For this, the method implemented in the siesta computer code – see Ordej´on, Drabold, Martin and Grumbach (1995), Ordej´on, Artacho and Soler (1996) and S´anchez-Portal, Ordej´on, Artacho and Soler (1997) – seems to be appropriate. The siesta package uses norm-conserving pseudopotentials as a way to cope with deep core states and localized (numerical) basis of pseudoatomic orbitals. The efficiency of basis functions allows to keep their number small but enhance if necessary, 2

without the danger of creating linear dependencies. The semicore states, like e.g. Sn4d, are accounted for by including corresponding pseudoatomic functions in the basis, so that the presence of deep pseudopotentials does not pose a problem. An important feature of the code is that basis functions have strictly limited spatial extension; this makes Hamiltonian and overlap matrices sparse and allows to treat large systems more efficiently than e.g. with a plane-wave pseudopotential method. The siesta code has in the last time been applied to a number of different systems – see Ordej´on (2000) for a review – but not to SnO2 , so preliminary benchmark calculations were necessary. As such, we consider structure optimization of the bulk rutile phase, comparing our results with state-of-art calculations by the full-potential linearized augmented plane wave (FLAPW) method, described in detail by Singh (1994), as implemented in the WIEN97 package by Blaha, Schwarz and Luitz (1997). Later on, we consider the structure relaxation on the (110) and (001) surfaces.

2. CALCULATIONAL SETUP AND REULTS FOR THE BULK RUTILE PHASE Norm-conserving pseudopotentials for valence configurations Sn4d10 5s2 5p2 and O2s2 2p6 have been prepared along the scheme by Troullier and Martins (1991), using the following cutoff radii (for l=0 to 3): 2.50, 3.20, 1.70, 2.20 Bohr for Sn, and 1.15 Bohr (for all l channels) for O. A pseudocore correction has been included in the Sn pseudopotential according to the prescription of Louie, Froyen and Cohen (1982) with the core correction radius of 0.85 Bohr. The basis set included “double-zeta” functions with angular moments up to l=2 (Sn) and l=1 (O) and a d shell to polarize the p channel for both atoms. A finite spatial extent of basis functions is achieved by calculating them as “pseudoatomic wavefunctions” of a compressed atom, i.e., solutions corresponding to a pseudopotential in question in an additional infinitely deep potential well of a certain radius. The requirement that all eigenvalues get the same energy shift by imposing such boundary conditions leads to different basis functions having different spatial extent, as described by Artacho, S´anchez-Portal, Ordej´on, Garc´ıa et al. (1999); the effect of the latter on the calculation results has been analyzed (see below). In a subsequent slab calculation for non-polar surfaces, the separation through the vacuum gap between the most protruded atoms at both sides of the repeated slab must be large enough to prevent direct overlap of basis functions. The calculations have been done in the local density approximation (LDA), parameterized according to Perdew and Wang (1992). Total densities of states in SnO2 as calculated by siesta and WIEN97 are shown in Fig. 1. Small differences are due to the fact that the tetrahedron integration was used in WIEN97 but the k-points sampling in siesta; the agreement in energy positions and relative intensities of different DOS fea3

tures is quite good, including the structure of the semicore Sn4d band. The latter is typically attributed to deep core in pseudopotential calculations, but one can see that the energy dispersion within this band and its hybridization with O2s states is not negligible. The results of structure optimization with siesta are shown in Table 1 in comparison with earlier pseudopotential results. One can see that whereas the internal coordinate u was perfectly reproduced in all calculations, the agreement with the experimental data in absolute values of lattice parameters, and particularly of the c/a ratio, is better in our present calculation. It could be in part due to more accurate treatment of semicore Sn4d states which were explicitly attributed to the valence band. Since the finite extension radius of basis functions remains a distinctive feature of the siesta formalism and has to be externally set, we checked how strongly the structural information depends on its value. We list in table 2 the (relative) values of equilibrium volume, calculated with fixed c/a, along with the corresponding values of the bulk modulus. The value calculated by the FLAPW method (using the LDA, the same as in the siesta calculation) are also shown. One can see that equilibrium volume remains largely unaffected , and the accuracy in reproducing the bulk modulus is quite satisfactory. This justifies our subsequent application of the siesta code with the calculation setup as described above for the study of structure relaxation at the surfaces.

3. RELAXATION ON THE (110) AND (001) SURFACES The search for the equilibrium structure at the surface has been done in the slab geometry, with the smallest unit cell possible in the surface plane and the necessary number of atomic levels in the normal direction to sum up to the Sn10 O20 composition in a supercell. We considered (110) and (001) surfaces, which in a sequence of recent plane-wave pseudopotential calculations by Oviedo and Gillan (2000) were shown to be the most favorable and the next favorable surfaces, according to their formation energies. The (110) surface seems to be the most extensively studied one in different simulations to date. Fig. 2 depicts two largest supercells used in our simulation for both surfaces in question. The crystal structure parameters have been taken as optimized for the bulk material (see previous section), the length of the supercells was fixed at 27.8 ˚ A for the (110) and at 23.1 ˚ A for the (001) surfaces, just in order to avoid direct overlap of the compact basis functions across the vacuum with those in a translated supercell. The coordinates of all atoms were free to relax. The (001) supercell shown in Fig. 2 represents the final convergent structure. For the sake of more detailed discussion, we show in Fig. 3 the smallest supercell we considered for the simulation of the (110) surface, the stoichiometric Sn6 O12 . The removal of the topmost (bridge) oxygen atom on each side leaves the reduced Sn6 O10 supercell. The choice of these 4

small supercells was motivated by the fact that such geometry was earlier addressed by Rantala, Rantala and Lantto (1999) in a first-principles planewave calculation, so that one could compare the results obtained by different methods. The values of structural parameters as introduced in Fig. 3 are listed in Table 3 according to our present calculation and two earlier ones. In all cases, our calculations essentially confirm the plane wave pseudopotential results by Rantala, Rantala and Lantto (1999), even when they differ from tight-binding calculation results by Godin and LaFemina (1993). In particular, the tin atom between two bridging oxygen atoms (in the middle of the unit cell’s upper face) relaxes slightly outwards, and the outward relaxation of the upper in-plane oxygen is larger in the relaxed supercell than in the stoichiometric one. However, the difference in the magnitude of the corresponding parameter ∆1⊥2 between relaxed and stoichiometric cases is not so large in our calculation as in that by Rantala et al. Whereas qualitatively the relaxation pattern remains the same in a larger Sn10 O20 supercell, actual numbers of relaxation are somehow affected by the supercell size. For instance, both ∆1⊥1 and ∆1⊥2 parameters become nearly 0.28 ˚ A. The relaxation goes deep inside the crystal, so that the supercells we consider were actually not yet large enough to achieve a convergency towards bulk crystal structure values in the middle of the slab. The same applies to the calculation representing the (001) surface. In this case (see Fig. 2, right panel), the distances between adjacent Sn planes on going inwards change as 1.27. 1.85; 1.48; 1.72 and 1.53 ˚ A, fluctuating around, but only slowly converging to, the bulk value of c/2=1.62 ˚ A. With respect to every corresponding Sn plane, the oxygen atoms displace outwards (+) or inwards (−) as +0.33; −0.24; +0.14; −0.10; +0.08 and −0.07 ˚ A (that must become zero in the bulk). Therefore, the treatment of larger supercells, feasible with the siesta code, is necessary for the reliable evaluation of surface relaxation.

4. SUMMARY We studied the electronic structure and dynamic relaxation of tin dioxide in its bulk (rutile) phase and for two different types of surface-representing slab geometries with the aim to test the accuracy of the tight-binding ab initio siesta code and to simulate from first principles the structure relaxation near the surfaces. Based on benchmark calculations for the bulk material, the accuracy of the method was found sufficiently high for a subsequent use in structure relaxations for large supercells. For the smallest supercell representing the (110) surface, our results agree reasonably well with those previously obtained by a plane-waves calculation. For the largest supercells we considered for both (110) and (001) surfaces, Sn10 O20 , the relaxation pattern does not yet sufficiently converge towards the middle of the slab, indicating the need to consider somehow larger supercells for reliable quantitative estimation of the relaxation at the surface. 5

Acknowledgements One of the authors (V. A. P.) would like to acknowlege financial support by the DFG-sonderforschungsbereich “Nanopartikel aus der Gasphase: Entstehung, Struktur, Eigenschaften”. References Artacho, E., D. S´anchez-Portal, P. Ordej´on, A. Garc´ıa et al. (1999). Linear-scaling ab-initio calculations for large and complex systems. Physica Status Solidi (b), 215, 809. Blaha, P., K. Schwarz and J. Luitz (1997). WIEN97, Vienna University of Technology. Improved and updated Unix version of the original copyrighted WIEN-code, which was published by P. Blaha, K. Schwarz, P. Sorantin, and S. B. Trickey, in Comput. Phys. Commun., 59, 339 (1990). Godin, T. J. and J. P. LaFemina (1993). Surface atomic and electronic structure of cassiterite SnO2 (110). Phys. Rev. B, 47, 6518. Goniakowski, J., J. M. Holender, L. N. Kantorovich, M. J. Gillan et al. (1996). Influence of gradient corrections on the bulk and surface properties of TiO2 and SnO2 . Phys. Rev. B, 53, 957. Hazen, R. M. and L. W. Finger (1981). Bulk moduli and high-pressure crystal structures of rutile-type compounds. J. Phys. Chem. Solids, 42, 143. Louie, S. G., S. Froyen and M. L. Cohen (1982). Nonlinear ionic pseudopotentials in spin-density-functional calculations. Phys. Rev. B, 26, 1738. Manassidis, I., J. Goniakowski, L. N. Kantorovich and M. J. Gillan (1995). The structure of the stoichiometric and reduced SnO 2 (110) surface. Surf. Sci., 339, 258. Ordej´ on, P. (2000). Linear scaling ab initio calculations in nanoscale materials with siesta. Physica Status Solidi (b), 217, 335. Ordej´ on, P., E. Artacho and J. M. Soler (1996). Self-consistent order-N densityfunctional calculations for very large systems. Phys. Rev. B, 53, R10441. Ordej´ on, P., D. A. Drabold, R. M. Martin and M. P. Grumbach (1995). Linear system-size scaling methods for electronic-structure calculations. Phys. Rev. B, 51, 1456. Oviedo, J. and M. J. Gillan (2000). Reduction and oxidation processes at the SnO2 (110) surface. Ψk -2000 conference “Ab initio (from electronic structure) calculation of complex processes in materials”, Schw¨ abisch Gm¨ und, Germany, August 22–26, 2000, see : Programs and Abstracts, p. 185.

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Parlinski, K. and Y. Kawazoe (2000). Ab initio study of phonons in the rutile structure of SnO2 under pressure. Eur. Phys. J. B, 13, 679. Peltzer y Blanc´a, E. L., A. Svane, N. E. Christensen, C. O. Rodr´ıguez et al. (1993). Calculated static and dynamic properties of β-Sn and Sn-O compounds. Phys. Rev. B, 48, 15712. Perdew, J. P. and Y. Wang (1992). Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B, 45, 13244. Rantala, T. S., V. Lantto and T. T. Rantala (1994). A cluster approach for modelling of surface characteristics of stannic oxide. Physica Scripta, T54, 252. Rantala, T. T., T. S. Rantala and V. Lantto (1999). Surface relaxation of the (110) face of rutile SnO2 . Surf. Sci., 420, 103. Rantala, T. T., T. S. Rantala and V. Lantto (2000). Electronic structure of SnO 2 (110) surface. Materials Science in Semiconductor Processing, 3, 103. S´anchez-Portal, D., P. Ordej´on, E. Artacho and J. M. Soler (1997). Densityfunctional method for very large systems with LCAO basis sets. Int. J. Quant. Chem., 65, 453. Singh, D. J. (1994). Planewaves, pseudopotentials and the LAPW method. Kluwer, Boston. Troullier, N. and J. L. Martins (1991). Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B, 43, 1993. Watson, J., K. Ihokura and G. S. V. Coles (1993). The tin dioxide gas sensor. Measurement Science & Technology, 4, 711.

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Table 1: Crystal structure parameters of SnO2 in the rutile structure optimized with siesta and in plane-wave pseudopotential calculation by Goniakowski, Holender, Kantorovich, Gillan et al. (1996), where Sn4d states were attributed to the deep core. Deviations from experimental values are given in brackets.

siesta (present calculation) PW-pseudopotential Expt.

˚) a (A 4.714 (−0.5%) 4.645 (−1.9%) 4.737

˚) c (A 3.241 (+1.7%) 3.060 (−4.0%) 3.186

u 0.307 0.307 0.307

Table 2: Equilibrium volume and bulk modulus as calculated with siesta for different values of energy shift that determines the spatial extension of basis functions. The values calculated with the WIEN97 package (FLAPW method) are shown for comparison. siesta Energy shift (Ry) 0.005 0.010 0.015 0.020

Basis extension (Bohr) Sn O 6.95 4.69 6.44 4.35 6.13 4.04 5.98 3.93

FLAPW Expt.a a

(V −Vexp. )/Vexp. 1.040 1.028 1.015 1.007 0.996

Hazen and Finger (1981).

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Bulk modulus (MBar) 2.000 2.119 2.275 2.434 2.33 2.18

Table 3: Relaxation parameters ∆ specified in Fig. 3 (in ˚ A) for the (110) surface as calculated by siesta, in comparison with earlier calculations by ab initio plane-wave method (PW) and by a tight-binding method (TB).

∆1⊥1 ∆1⊥2 ∆1⊥3 ∆1⊥4 ∆1y a b

Sn6 O10 (reduced) siesta TBa PWb +0.140 −0.05 +0.11 +0.292 +0.28 +0.40 1.207 2.430

1.29 2.58

1.33 2.45

Sn6 O12 siesta 0.172 0.248 1.327 1.239 2.453

(stoichiometric) TBa PWb 0.10 0.21 0.29 0.22 1.37 1.41 1.18 1.22 2.53 2.50

Godin and LaFemina (1993) Rantala, Rantala and Lantto (1999)

20

10

FLAPW

SIESTA 10

0

5

−20

−10 Energy (Ry)

0

10

0

−20

−10 Energy (Ry)

0

10

Figure 1: Total densities of states as calculated by WIEN97 (left panel, states per Ry and unit cell) and siesta (right panel, states per Ry and per formula unit). Occupied states are shown as shaded areas.

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(110)

(001)

Figure 2: Stoichiometric Sn10 O20 supercells used in the slab calculations for the (110) and (001) surfaces.

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∆1y

-

∆1⊥3 ∆1⊥2

?

?

6

6

?

?

6

6

∆1⊥1

∆1⊥4

Figure 3: Side view of the Sn6 O12 supercell representing slab geometry for the (110) surface, with characteristic distances indicated.

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