Numerical Analysis of Oxygen Adsorption on SnO 2 Surface Using Slab Geometry

Oxidation of thin film SnO 2 layer was simulated. In particular, the evolution of depletion layer was investigated by solving PoissonBoltzmann equation for SnO 2 slab geometry grains. On this basis, the surface energy barrier dependence on layer thickness (30–500 nm) was obtained. The effect of the donor mobility (oxygen vacancies in the bulk) and degree of donor ionization on electric potential inside layer with different thicknesses was discussed. Furthermore, the dependence of per-square conductance on temperature (from 400K to 700K) has been computed. It was assumed that the bulk oxygen vacancies (donors) are singly or doubly ionized and mobile. The temperature variations in the carrier mobility were also taken into account.


Introduction
Tin dioxide (SnO 2 ) is a semiconductor oxide with a wide range of applications.The material is being used as a photocatalyst and a gas sensor and in optical devices [1][2][3].In all these applications, the surface-related phenomena play a major role and become particularly important for nanomaterials, in which surface-to-volume ratio is much higher [4][5][6].Precise numerical model of electrical properties of near-surface layer of SnO 2 grains is fundamental for improving sensitivity and selectivity of SnO 2 based gas sensors [6].Shape and size of SnO 2 nanograins are considered the most important factors for film conductance.It is caused by chemical reactions which take place on the surface of SnO 2 grains.In particular, oxygen adsorption on the surface of SnO 2 (ZnO and TiO 2 ) decreases concentration of electrons in the near-surface layer.Depleted region is formed and conductivity of the layer is decreased.At the same time range of working temperatures of the film is raised from 420 to 720 K. Despite much experimental and theoretical research [7][8][9][10][11][12][13][14][15][16][17][18][19], the role played by adsorbed oxygen in surface and volume phenomena (surface coverage by oxygen ions, band bending, and distribution of mobile donors) occurring in nanolayers of SnO 2 and the influence of these phenomena on conductivity have not yet been explained.Reversible changes in the conductivity of SnO 2 layer have been reported, caused by a change in partial pressure of oxygen for sensor work temperature exceeding 500 K.The observed conductivity changes fulfil the exponential relation  ∼ [ O 2 ]  , where  index assumes values from −0.25 to −0.5 or  = −1/6 for temperatures above 1000 K [12].Band bending caused by oxygen adsorption was reported by Mizsei and Lantto [13] and Semancik and Cox [14].Also, X-ray photoelectron spectroscopy (XPS) was used to demonstrate band bending of 0.2 eV as a result of the interaction between oxygen and nanocrystalline layers of SnO 2 [15].Furthermore, a theoretical analysis of oxygen adsorption on the stoichiometric surface and in the case when surface oxygen vacancies are present was carried out [13,14].Rantala and Lantto [18] calculated the height of the surface energy barrier (below 0.7 eV) for totally depleted grains of various diameters.Three grain geometries were discussed (flat, cylindrical, and spherical).In the calculations, it was assumed that donors are mobile and single or double ionized.As a result of the calculations, it was proved that thin tin dioxide layers placed in synthetic air can be totally depleted in terms of charge carriers.The thickness of these layers cannot exceed 0.25 m and 1 m (at  = 573 K) in the case of, respectively, mobile and single-and double-ionized donors.Thus, modeling sensor structures based on SnO 2 nanolayers requires allowing for the size and shape of grains, which determine, among others, the conductivity of those layers.
In the case of modeling electron properties of thin SnO 2 layers whose thickness is below 500 nm, one should additionally include the phenomenon of moving oxygen vacancies through the volume (the so-called mobile donors), which influence the distribution of potential within the layer.
The purpose of this work was a rigorous theoretical analysis of the influence of surface oxidation on the conductance of SnO 2 nanofilms.Influence on SnO 2 layers electrical properties of the assumption that oxygen vacancies are mobile was studied.From the one-dimensional analytical solution of the Poisson-Boltzmann equation in the case of finite grains with slab geometry, the in-depth profiles of the potential () were obtained for full depleted layers (following [18]).The SnO 2 layer thickness was in the range from 30 nm to 500 nm.Then, the surface energy barrier dependence on temperature (from 400 K to 700 K) was determined.These magnitudes were calculated for different bulk doping, assuming that the bulk oxygen vacancies (donors) are singly or doubly ionized and mobile.These magnitudes were used as input data for computer simulations based on the rate equations proposed by Rantala et al. [4] for the electron transfer between the oxygen surface species and the bulk conduction band to calculate the total coverage by oxygen species and coverages by various oxygen ions.Subsequently, the in-depth profiles of carrier concentrations, distribution of donors, and per-square conductance dependence on temperature were obtained.Furthermore, the influence of partial oxygen pressure on the SnO 2 nanofilm conductance was studied.

Analytical Procedure
The in-depth potential profiles in the depletion region induced by gas adsorption and surface energy barrier heights were computed for one-dimensional SnO 2 grains with slab geometry (i.e., layer thickness 2 ×  is much lower than other dimensions; see Figure 1(a)) and n-type doping (donor concentration   ) [20].The electric potential () was obtained by solving the Poisson-Boltzmann equation [18]: where  is the distance from the slab center,  is the elementary charge,  = /  ,  = 1 for single donors and 2 for double donors,  0 is the semiconductor permittivity, and  0 and  0 are the donor and electron concentrations in the center ( = 0) of the slab, determined by the following formulas: where   and   are the donor and electron concentrations at the surface and   is the surface potential.
The first and second boundary conditions are ( = 0) = 0 and / = 0 in the slab center  = 0 [18].The first term in (1) represents the contribution of the mobile donors at the temperature , whereas the second term, including the charge distribution of electrons, was neglected because of the total depletion assumption [18].The analytical solution of (1) gives the following expression for the electric potential: and for the distribution of bulk donor concentration it gives where Assuming that donor concentration   is uniform in the bulk in no oxygen-containing atmosphere (before surface reaction with oxygen), we can write It should be noted that in our approach, the surface donor concentration   was not a parameter, contrary to the assumption in [18].
The surface potential   for different temperatures, layer thicknesses, and donor concentration   was obtained by solving the system of ( 2) and ( 4)- (7).The   values were then applied as input data in the calculations of the surface total coverage with oxygen ions, coverage with various oxygen ions, and thus carrier concentration in-depth profile in the depletion layer [20].These simulations were based on the solving of rate equations following [4].In our approach based on the adsorption-desorption model proposed by Rantala et al. [4], all possible surface reactions and phenomena, that is, multistep oxygen adsorption, dissociation, recombination, and desorption, were taken into account.As a result of adsorption, the negative charge trapped in oxygen species causes an upward band bending and thus a reduced conductivity, when compared with that in the flat band condition.The data concerning the surface coverage by oxygen ions O 2 − and O − needed for simulations were taken from [4,21].
The sample conductance  ◻ was determined from the classical formula [9,22] by using the obtained carrier concentration in-depth profiles: where () and () are the electron and hole concentrations.
Because of the high doping of n-type SnO 2 , the hole concentration in (8) was neglected.Furthermore, the gas pressure sensitivity  was calculated from the standard formulas.The calculations were carried out for n-SnO 2 layers with different thicknesses (2 × ), from 30 nm to 500 nm, and doping level (single or double mobile bulk oxygen vacancies [23]) from   = 1 × 10 24 m −3 to   = 2.5 × 10 25 m −3 and within the temperature range from 300 K to 700 K.In the calculations, the carrier parameters, electron mobility   = 150 cm 2 /(Vs) at temperature 300 K and effective mass of electron equal to 0.3  , were assumed.The temperature variations in the carrier mobility were taken from [24,25].

Results and Discussion
An influence of temperature (from 400 K to 700 K) and doping on the surface energy barrier are shown in Figure 1 for a relatively high donor concentration (2.5 × 10 24 -2.5 × 10 25 m −3 ), if the thickness of the SnO 2 plate is 2 ×  = 30 nm.On the other hand, in the presence of oxygen, in temperatures ranging from 400 K to 700 K, the band bending increases from 0.24 eV to 0.35 eV (for single-ionized oxygen vacancies:  = 1) and from 0.17 eV to 0.26 eV (for double-ionized oxygen vacancies:  = 2), if donor concentration is   = 1 × 10 25 m −3 and 2 ×  = 30 nm (Figure 1(b)).The surface barrier grows along with the rise in donor concentration   throughout the investigated temperature range for various thicknesses of the layer.
The calculated energy barrier values are in line with the results obtained by Rantala and Lantto [18].The height of the barrier is influenced both by the thickness and the degree of donor ionization (Figure 2).For example, at 700 K, a change in the thickness of the layer from 15 nm to 150 nm is accompanied by an increase in eVs from 0.20 eV to approximately 0.74 eV, when  = 1 (Figure 2(a)), or an increase from 0.18 eV to 0.45 eV, when  = 2 (Figure 2(b)).The energy barrier for double-ionized donors is lower than that in the case of single-ionized donors (at the same temperature, for equal   concentrations, and the same layer thickness).The results of the calculations were compared with the ones obtained by Rantala and Lantto [18], who assumed a constant value of donor concentration near the surface   = 10 26 m −3 .Differences between the respective curves (Figure 2,  = 600 K) result from different beginning conditions in calculations, namely, each continuous curve point in Figure 2 corresponds to the same value of donor concentration   (uniform in the bulk before surface reaction with oxygen); however, every point of the intermittent curve corresponds to different donor concentration values   in a sample of a given thickness.The calculated total coverage by oxygen species and the coverages of the oxygen ions O − and O 2 − versus temperature in dry synthetic air, for mobile single donors (  = 2.5 × 10 24 m −3 ,  = 250 nm) and for mobile double donors (  = 2.5 × 10 25 m −3 ,  = 15 nm), are shown in Figure 3.The calculations suggest that the coverages of the atomic and molecular ionic species are equal at a temperature of about 445 K.The calculated temperature corresponds to the transition temperature of 423 K, found by Chang [8,26], as a result of measurements of ∼100 nm thick SnO 2 nanolayers, deposed on Al 2 O 3 using a method of reactive deposition from the target containing powdered tin dioxide.Compatible experimental results were reported by Rembeza et al. [27] while investigating nanocrystalline layers with grain diameter < 20 nm (layer thickness > 1 m) formed by means of magnetron cathode deposition on glass substrates.The dynamics of the changes of total coverage by oxygen ions depends on grain thickness and the degree to which the donors are ionized and, in the case of very thin layers, is negligible (see Figure 3(b)).Total coverage by oxygen ions is strongly dependent on temperature and donor concentration (Figure 3(a)) if the thickness of a totally depleted layer is relatively high ( > 150 nm). Figure 4 shows the charge distribution in the case of mobile single and double donors at temperature  = 600 K if the content of oxygen in nitrogen is  = 4%, and half of the layer thickness  = 100 nm.From the calculations, it is observed that SnO 2 thin films may be depleted from electrons in the atmospheric air in the case of mobile donors.
The surface space charge region, induced by adsorbed ions, contributes significantly to the conductance of the semiconducting SnO 2 layers.The sample per-square conductance was determined from the classical formula [9,22], using the obtained in-depth carrier profiles (Figure 4).The dependence of the per-square conductance on temperature for different partial pressures of oxygen and slab thicknesses is shown in Figure 5.The calculations show that the maximum persquare conductance shifts in the direction of lower temperatures if the partial pressure of oxygen decreases or when   increases (Figure 5(a)).Experimentally, the maximum conductance was observed at 450 K for nanocrystalline layers of SnO 2 with grain diameter < 20 nm and between 550 K and 600 K for polycrystalline, thin layers less than 500 nm and average grain size 15 nm [28].The results of conductivity measurements of SnO 2 layers may vary considerably; they depend on the surface structure (its stoichiometry and defectiveness), the layer smoothness, the type of the substrate, and the different methods of layer forming.For the slabs with dimensions smaller than the space charge region and relatively low oxygen concentration, a strong change in the conductance is observed (Figure 5(b)).If the thickness of SnO 2 sample decreases, the ratio of the thickness of depletion layer and thickness of the sample increases; the sample of SnO 2 thin layer can be depleted as a whole and the conductance of the sample is small [22,29].
From Figure 6(a), it is evident that the per-square conductance depends on the layer thickness.The calculations show that the maximum per-square conductance shifts in the direction of lower temperatures if the thickness of the slab increases.Experimentally, within the temperature range between about 420 K and 723 K, the maximum conductance was observed at temperature above 600 K in the case of thin films [28,30,31] and at lower temperatures (555 K) in the case of thick films [30][31][32].The sensitivity with respect to the partial pressure of oxygen (power coefficient ) versus temperature is independent of the degree of donor ionization for the slab thickness (2 × ) lower than 150 nm, as it is observed in Figure 6(b).The value of the coefficient  changes in the range from −0.55 ( < 70 nm) to −0.34 ( = 200 nm) for the sensor working temperature between 500 K and 700 K.This result is consistent with [33], where the calculations performed on the basis of acting mass low give  = −0.5 for the constant concentration of oxygen lattice defects (oxygen vacancies).Experimentally, the values of  were between −0.25 and −0.5 [11].For thick, porous layers of SnO 2 ,  = −0.38 was obtained [11]; in the case of thin, polycrystalline layers, for which the changed grain size was ranging from 100 nm to 500 nm,  coefficient value was −0.5 [10].

Summary and Conclusions
A numerical analysis of oxygen adsorption on the surface of flat-geometry grains was carried out.It was proved that the depth profiles of potential and the value of surface energy barrier are strongly influenced by the temperature, the layer thickness, and the degree of donor ionization.With the use of computer simulations, it was reported that the coverage by ionized oxygen species (O 2 − and O − ) and layer conductivity strongly depend on its thickness (from 30 nm to 500 nm).
It was demonstrated that the presence of oxygen in ambient air leads to a decrease in the conductivity of the nanolayer as the partial pressure of oxygen increases.In a range of temperatures, from 500 K to 700 K, the found  index values (sensitivity in relation to oxygen partial pressure) are in conformity with the theoretical and experimental data obtained as a result of the interaction of the surface with oxygen.Theoretical analysis of oxygen adsorption on the surface of SnO 2 , in a single-dimension case for flatgeometry grains (the so-called slab model), is an accurate approximation of electron properties of surface and nearsurface region of thin epitaxial layers and small grains (SnO 2 layer based sensors), for which the thickness of the depleted layer is comparable with the grain radius.

Figure 6 :
Figure 6: (a) Per-square conductance versus temperature for different half of the layer thickness.(b) Sensitivity with respect to partial pressure of oxygen in ambient atmosphere (power coefficient ) versus temperature,   = 2.5 × 10 24 m −3 , for different half of the layer thickness.