Surface and interface electronic properties of tin oxide

Surface and interface electronic properties of tin oxide

5

Celso M. Aldao Institute of Materials Science and Technology (INTEMA), University of Mar del Plata and National Research Council (CONICET), Mar del Plata, Argentina Chapter outline 5.1 Introduction 101 5.2 Schottky barriers  102 5.3 Current transport mechanisms  104 5.4 Tunneling electron transport in SnO2 polycrystalline thick films  107 5.5 Oxygen vacancies density in SnO2 polycrystalline thick films  110 5.6 Barrier height fluctuations  115 5.7 Chemisorption at semiconductor surfaces  120 5.8 Nonparabolic barriers  126 5.9 Conclusions 128 Acknowledgments  129 References  129 Further reading  132

5.1 Introduction Metal oxides constitute an important class of materials, which exhibit diverse properties with a variety of applications [1]. Tin dioxide, in particular, is a wide band-gap semiconductor with high electrical conductivity and optical transparency. It is of interest in optoelectronic applications and also it is used in chemical applications as support of metallic catalysts and for its own catalytic properties. In addition, tin oxide plays a central role in the fabrication of solid-state gas sensors. Many oxides present sensitivity to oxidizing and reducing gases but SnO2 is the most regularly used material in this application [2]. As other oxides, tin oxide has been nanostructured and grown as nanowires presenting high gas sensitivity [3]. In principle, as a catalyst and solid-state gas sensing material, the surface seems to be of relevance while the bulk properties are responsible for making tin oxide a good conductor. However, the presence of interfaces cannot be disregarded in dealing with this transparent conduction oxide. On the other hand, we show that the bulk properties must be considered in determining the materials properties in catalysis and sensing applications. It is important to note that all this involves complex mechanisms that continue being topics of ongoing research and controversy [4, 5]. Tin Oxide Materials­: Synthesis, Properties, and Applications. https://doi.org/10.1016/B978-0-12-815924-8.00005-0 © 2020 Elsevier Inc. All rights reserved.

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Tin oxide combines low resistance with optical transparency in the visible range. These properties make tin oxide and other metal oxides useful in applications related to solar cells and a variety of optoelectronic devices [6, 7]. As many oxides, SnO2 presents a high conductivity due to intrinsic defects, specifically oxygen vacancies. Indeed, in its stoichiometric form tin dioxide is a good insulator but nonstoichiometry, oxygen deficiency, increases its conductivity. Despite spread values, the donor characters of regular native defects have been experimentally confirmed. Kilic and Zunger [8] showed that the formation energy of oxygen vacancies and tin interstitials in SnO2 is very low. Then, these defects form readily and defect levels can be easily ionized. Many oxides are used as a support material for dispersed metal catalysts but tin oxide also exhibits good activity by itself. Transfer of electrons between reacting molecules and solid oxide catalysts can be a complex phenomenon that requires energy levels of the reacting molecule at a region of high density of states of the solid. Molecules can be oxidized by the Mars-van Krevelen mechanism, by consuming lattice oxygen that is reoxidized by the gas-phase oxygen. Thus, the catalyst must be characterized by facile change of the oxidation state of its cations, property of many metal oxides, including tin dioxide [9]. Solid-state gas sensors transduce a chemical quantity, such as a gas concentration, into an electric signal. Metal-oxide semiconductors are the most common inorganic materials used in chemical sensing of gases and vapors. They play a major role in industrial applications such as automotive, aerospace, and food, as well as medical and indoor detection of toxic gases produced by incomplete combustion of heating systems. In particular, polycrystalline SnO2 is an attractive material due to its high sensing response to reducing gases, long-term stability, robustness, and low cost. The exact details of the mechanisms that cause a gas response are still controversial, but it is widely accepted that trapping of electrons at adsorbed molecules and band bending induced by these charged molecules are responsible for a change in conductivity [10–16]. As seen, interfaces play the key role in the most important applications of tin oxide. Thus, the understanding of the surface and interface phenomena becomes crucial and then Schottky-like barriers formed at surfaces and at grain boundaries that determine their properties. We show that the formation and properties of these potential barriers are much more subtle than regularly assumed. In this chapter, we focus on their formation, characteristics, and consequences.

5.2 Schottky barriers A Schottky barrier, named after Walter H. Schottky, is a potential energy barrier for electrons formed at a metal-semiconductor junction [17]. Schottky barriers have rectifying characteristics, and then they are suitable for use as diodes. One of the basic characteristics of a Schottky barrier is its height, regularly denoted by ΦB (see Fig.  5.1). The value of ΦB depends on both, the metal and the semiconductor. The barrier between a metal and a semiconductor has been predicted by the Schottky-Mott

Surface and interface electronic properties of tin oxide103

Fig. 5.1  Schematic representation illustrating of a Schottky barrier at a metal-semiconductor interface. EC, EV, and EF denote the conduction band minimum, valence band maximum, and Fermi level, respectively. Φm is the metal work function, χs the semiconductor electron affinity, ω the thickness of the space-charge layer, eVs the potential barrier, and ΦB the potential barrier height.

rule to be the difference of the metal-vacuum work function and the semiconductor-­ vacuum electron affinity as shown in Fig. 5.1 for an n-type semiconductor [18]

Φ B = Φ m − χs

(5.1)

In practice, however, most metal-semiconductor interfaces do not follow the Schottky-Mott rule. The explanation of the alignment between the energy bands across a metal-semiconductor interface has involved extensive scientific research during decades. Amazingly, a wide range of conflicting opinions and models still coexist in the literature [19]. Despite these inconvenients, the model for a metal-semiconductor interface of Fig. 5.1 is regularly presented. Note that the shape of the potential barrier depends on the charge distribution in the depletion region. Since the electron density reduces exponentially with EC (the conduction band minimum), the shape of the barrier is mostly determined by the spatial distribution of donors. Thus, assuming a constant doping up to the contact with the metal, uncompensated donors determine a uniform space charge in the depletion region. Consequently, the electric field increases linearly from the edge of the depletion region up to the interface and the potential is parabolic. This resulting potential is regularly referred as a Schottky barrier. In the bulk, the position of the Fermi level depends on the bulk doping and, in principle, can lie anywhere in the bandgap for nondegenerate semiconductor. Conversely, the Fermi level at the surface depends on surface states, of intrinsic or extrinsic nature, and their population can be altered by adsorption and illumination. Without making

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a contact, the termination of the translation symmetry, which characterizes a crystal, leads to new states of intrinsic nature localized at the surface with a particular energy distribution and of donor or acceptor nature [20]. On the other hand, surface states of extrinsic nature appear due to adsorption at the surface. Thus, surface states, can result in the trapping of bulk carriers and then in the establishment of a space charge layer similarly to what occurs in a metal-semiconductor contact. The formed barrier has been referred as a Schottky-like barrier or directly a Schottky barrier even though strictly we are not dealing with a metal-semiconductor contact. Following the above discussion, the electric potential (defined for a unit of negative charge) as a function of charge density in the depletion zone can be determined for a one-dimensional model resorting to Poisson equation

ε

d 2V = ρ ( x ) = e  p ( x ) − n ( x ) + N d+ ( x ) − N a− ( x )  , d 2 x2

(5.2)

where e is the electron charge, p and n are the charge carrier densities, and Nd+ and Na− are the densities of singly charged donors and acceptors, respectively, and ε is the material permittivity. Eq. (5.2) can be easily solved by neglecting acceptor and hole concentration (which is acceptable for an n-type semiconductor), assuming a complete ionization of the donors, and considering that n(x) = 0 at the depletion region. Thus, within the depletion region, of thickness ω, the electric field increases linearly and the electric potential quadratically V ( x) =

eN d 2 x , 2ε

(5.3)

where we have considered that the electric potential is zero in the semiconductor bulk, that is, for x ≤ 0. The band bending can be easily calculated as Vs =

eN d 2 ω . 2ε

(5.4)

Due to charge neutrality, the amount of charge at the depletion region must be equal to the surface charge trapped at the interface eNs in coul/m2. This implies that ωNd = Ns and then we can write Vs =

eN s2 . 2ε N d

(5.5)

Note that Eq. (5.4) relates the band bending with the concentration of ionized donors and the depletion width while Eq. (5.5) the Vs is related to the Nd and the surface charge density. The barrier potential energy is eVs = Eb.

5.3 Current transport mechanisms In this section, we discuss the transport mechanisms that determine the conduction properties of a Schottky barrier assuming that the barrier has been established. It has long been recognized that the resistance of polycrystalline semiconductors

Surface and interface electronic properties of tin oxide105

as tin oxide is determined by Schottky barriers at grain surfaces [21]. Indeed, an intergrain can be considered as two Schottky barriers back-to-back. Therefore, the key issues are barrier formation at intergrains and electron transport between grains. Conduction mechanisms have been interpreted in analogy to those in metal-­ semiconductor contact diodes. Accordingly, the electrical properties of polycrystalline semiconductors are usually described with a simple one-dimensional model representing the interface between two grains. As commented, tin oxide behaves naturally as an n-type semiconductor as oxygen vacancies are the dominant defects and they behave as donors. Fig. 5.2 depicts the double Schottky barrier model that is generally accepted for an intergrain. In spite of possible complexities due to the randomness of the structure, usually a so-called bricklayer model with cubic-shaped grains of identical size is assumed [7]. Also, it is regularly considered that a thermionic mechanism is responsible for the sample conductivity. This conduction mechanism is described by Ref. [13] G = G0 exp ( −eVS / kT )

(5.6)

where eVs is the band bending, T the temperature, and k the Boltzmann constant. This equation reflects an activated process due to intergranular barriers. Using Eq. (5.6) (i.e., assuming an Arrhenius relation) the band bending, or the barrier height ΦB, defined as eVS + EC − EF (as shown in Fig. 5.2), is regularly estimated. Indeed, by plotting the conductance (ln G) vs the reverse function of temperature (1/T), an activation energy can be obtained [22]. The transport mechanism given by Eq. (5.6) corresponds to the emission of electrons over the top of the barrier (see Fig. 5.2). This has been modeled adopting the thermionic-emission theory for which the current density from one grain to a contiguous one is given by Ref. [23] J thermoionic = A∗T 2 exp ( −Φ B / kT ) .

(5.7)

Fig. 5.2  Diagram for the intergranular double-Schottky barrier model. The band bending is eVS and ϕΒ the height of the barrier. (A) Indicates emission of electrons from the left grain to the right grain over the top of the barrier, the thermionic contribution to current transport. (B) Indicates quantum mechanical tunneling through the barrier, the thermionic-field emission contribution to electron transport.

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A⁎ is the Richardson constant that it has the value A∗ = 4π m∗ ek 2 / h3 ,

(5.8)

where m* is the effective mass and h the Plank constant. Eqs. (5.6), (5.7) have a direct connection as the difference (EC − EF) affects similarly the conductivity in Eq. (5.6) and the current density in Eq. (5.7). Like in metal-semiconductor junctions, electrons have other ways to be transported in a polycrystal [23]. Indeed, electrons with energies below the top of the barrier can penetrate the barrier and reach the other grain. This is known as quantum-mechanical tunneling, case (B) depicted in Fig. 5.2. For very heavily doped semiconductors at low temperatures, when the semiconductor is degenerated, electrons with energies close to the Fermi level can tunnel the barrier. This mechanism is named as field emission. For not so highly doped semiconductors or by raising the temperature, most electrons tunnel at higher energies. On the one hand, at higher energies the barrier to tunnel becomes lower and thinner and, on the other hand, the number of electrons decreases exponentially with energy. Due to these two effects, most electrons cross the barrier at energies between the bulk conduction band level and the top of the barrier [24]. This mechanism is known as thermionic-field emission. We present the basic steps to calculate the tunneling contribution for a Schottky barrier; the extension for the double barrier of an intergrain is straightforward. Electrons are incident on a parabolic barrier in which the potential is given by 2

x V ( x ) = Vs   , ω 

(5.9)

valid for 0 ≤ x ≤ ω. The transmission probability for the barrier can be determined by means of the Wentzel-Kramers-Brillouin (WKB) approximation [25].  ω  τ = exp  −2 ∫α ( x ) dx  ,  a 

(5.10)

where

α ( x) =

1/ 2 2m eV ( x ) − E   

(5.11)

and 1/ 2

 E  a =   ω.  Eb 

(5.12)

The transmission probability for energies E lower than the barrier potential energy Eb = eVs corresponds to the tunneling contribution to electron transport. For E > Eb the transmission is assumed to be unity and corresponds to the thermionic contribution.

Surface and interface electronic properties of tin oxide107

By performing the integral of Eq. (5.10) for E < Eb, the transmission probability can be determined 1/ 2 1/ 2  2V  ε 1/ 2  m E  E  1 + (1 − E / Eb )  s  τ = exp − ln    1 −  −   N d    Eb  Eb  ( E / Eb )1/ 2   

    .    

(5.13)

Once τ is known, the forward current density, from semiconductor to metal due to tunneling, can be determined A∗T τ ( E ) F ( E ) dE. k ∫0 Eb

Jf =

(5.14)

where F(E) is the Fermi-Dirac distribution. Typical proposed band bendings and dopant concentrations make thermionic-field emission the most relevant conduction mechanism in many cases. However, despite its relevance, most researchers do not include this contribution in their calculations [12]. Next, we present an interesting case that indicates the crucial role of tunneling in electrical conduction.

5.4 Tunneling electron transport in SnO2 polycrystalline thick films We have investigated conduction mechanisms in polycrystalline SnO2 thick films by means of DC electrical resistance during heating-cooling cycles. Samples were maintained at relatively high temperatures in H2 or O2 ambient atmospheres before performing electrical measurements under vacuum or before performing XPS measurements in order to determine band bending. We see that results suggest that Schottky barriers are always present and that tunneling transport through barriers is the dominant mechanism controlling conductivity. Films were prepared as described in detail in Ref. [26]. Commercial high-purity SnO2 (Aldrich, medium particle size 0.4 μm) was ground until a medium particle size of 0.13 μm was obtained. Then, a paste was prepared with an organic binder (glycerol). The solid/organic binder ratio was ½, and no dopants were added. Thick, porous film samples were prepared by painting onto insulating alumina substrates on which Pt electrodes with an interdigitated shape had been deposited by sputtering. Samples were heated up to 380°C using a heating rate of 1°C/min and exposed to dry air at 380°C during 1 h. Some of the samples were kept at 380°C for 4 h in a N2 atmosphere with 5% H2. Samples were labeled SO2 (samples only exposed to dry air) and SH2 (samples exposed to hydrogen). Resistance vs temperature measurements for samples SO2 and SH2 were carried out raising and then decreasing the temperature with different temperature ranges at a rate of ~2°C/min with the samples kept in vacuum (≈10−4 mbar). Later, the resistance of the film labeled SH2 was measured while raising and then decreasing the temperature from room temperature up to 380°C at a rate of ~2°C/min with the sample kept in a dry air atmosphere (atmospheric pressure). Temperature-dependent conductance

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­ easurements as well as band bending measurements will be used jointly to discrimm inate the dominant electron conduction mechanism. Fig.  5.3 shows the conductance under vacuum of oxygen treated (SO2) and hydrogen treated (SH2) samples vs the inverse of temperature, in cycling experiments. Specifically, conductance was measured by raising (open circles) and then decreasing (filled circles) the temperature under vacuum reaching about 140°C. In this temperature range, the conductivity change for the SH2 sample is quite small (slope of 0.17 eV) and the room temperature value is recovered after the cycle. The SO2 sample shows a conductance change that corresponds to the average activation energy of 0.45 eV. However, note that the room temperature conductance increases about 50% after completing the cycle. Fig. 5.4 also shows a temperature cycle for the SO2 sample under vacuum; in this case, the temperature was raised up to 295°C. As expected, the conductance presents the same behavior as shown in Fig. 5.3 during heating up to 150°C with a 0.45 eV activation energy. Note however that a further increase in temperature results in an activation energy decrease to 0.31 eV. Finally when the cycle is reversed and the temperature is decreased, the conductivity presents higher values, reaching a room temperature value two order of magnitude greater than the initial one. Conversely, a similar cycle for the SH2 (not shown) presents a similar behavior to that of Fig. 5.3, and the room temperature value is recovered at the end of cycle. Results of Fig.  5.3 can be interpreted assuming the thermionic conduction mechanism of Eqs. (5.6), (5.7) and neglecting electron tunneling. The SH2 sample ­conductivity was high due to a low intergrain oxygen amount that is not affected by the

Fig. 5.3  Conductance for the hydrogen (SH2) and oxygen (SO2) treated samples in temperature cycling experiments up to 140°C. The line for the sample SH2 corresponds to the conductivity of a Schottky diode with a barrier height of 0.69 eV and dopant concentrations of 1.8 × 1019 cm−3. The line for the sample SO2 is just a guide to the eye.

Surface and interface electronic properties of tin oxide109

Fig. 5.4  Conductance cycle for the SO2 sample under vacuum up to 295°C. After cooling down, the conductivity adopts a much larger value than at the beginning of the cycle. Lines are a guide to the eye.

heat ­treatment; that is, the oxygen amount at the intergrains remains constant during the temperature cycle. Different conductivities between the SH2 and the SO2 samples would indicate barriers about 0.3 eV lower for the hydrogen treated samples, consistent with the observed slopes. On the contrary, results of Fig. 5.4 cannot be explained by only considering a thermionic mechanism for conduction. With heating, the conductivity increases with a slope corresponding to an activation energy of 0.45 eV. Interestingly, above 150°C, the slope reduces to 0.31 eV indicating that the activation energy, according to Eq. (5.1), must change in the 150–295°C range. First, we can assume that oxygen desorbs as the temperature is increased and then the activation energy would decrease with T. However, with a reducing barrier height, the conductivity should increase faster than at low temperatures, as a low activation energy facilitates conduction. Second, we can assume that the activation energy increases with T. However, after cooling down, the conductivity adopts a much larger value than at the beginning of the cycle, result that is incompatible with a higher barrier. If tunneling contribution to conductivity is relevant, the results of Fig. 5.3 can be interpreted as follows. The SH2 sample is expected to have a high density of oxygen vacancies and then narrow barriers that facilitate tunneling. Thus, the high conductivity and its small temperature dependence would be a direct effect of tunneling through a narrow barrier. The SO2 sample is expected to have a lower density of oxygen vacancies and then wider barriers. Consequently, its conductivity is expected to be much lower than for the SH2 sample and its dependence with temperature higher, as observed. Results of Fig.  5.4 have also a direct interpretation if tunneling is the dominant mechanism in conduction. Indeed, as temperature is raised, oxygen can diffuse out of the SO2 sample increasing the donor density. Thus, tunneling is favored and after

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Fig. 5.5  Sn 3d5/2 core level energy distribution curves for the SH2 and the SO2 samples. Similar binding energy indicates that barrier heights do not differ considerably. Dashed lines indicate a binding energy difference of 0.08 eV.

cooling down the sample presents a much higher conductivity. In a low-doped sample, grains can be completed depleted. As doping increases, grains can have a bulk region in their centers. The slope change in conductivity at about 140°C is an indication that oxygen diffusion led to a doping that narrowed the space charge regions to the point at which grains were not completed depleted [27–29]. By means of XPS core level analysis, band-bending differences can be determined as they are directly related to the XPS peak-binding energy positions. In particular, the Sn3d5/2 peak can be used due to the quality of the signal that can be obtained as presented in Fig. 5.5 [30, 31]. Results obtained for the SH2 and the SO2 samples show that barrier heights do not differ considerably. Interestingly, the barrier for the SH2 sample is slightly higher than that for the SO2 sample. Fittings indicated a difference in binding energy of 0.08 eV, as shown in Fig. 5.5. We determined the binding energy from other core levels of Sn, specifically Sn4d and Sn4s, and also from O1s. Results confirmed a Schottky barrier difference of 0.06 ± 0.04 eV with a slightly higher barrier for the SH2 sample. These results represents strong evidence in favor of a dominant tunneling conduction mechanism as the thermionic contribution only depends on Schottky barrier height and then it could not be responsible for the conductivity experimentally observed.

5.5 Oxygen vacancies density in SnO2 polycrystalline thick films In the discussion of the previous section, changes in the concentration of oxygen vacancies, due to different atmosphere exposures, were proposed as responsible for conductance variations. In this section, we present evidence supporting this interpretation.

Surface and interface electronic properties of tin oxide111

Indeed, from measured capacitance values, variations of the potential barrier widths can be determined. We show that under the presence of an oxygen rich atmosphere, at relatively low temperature, the width of intergranular potential barriers increase to the point that, medium size grains as those studied here become completely depleted of carriers. With subsequent exposure to vacuum, capacitance adopts a higher value, which is indicative for the formation of quasineutral regions at the center of the grains. Oxygen vacancies, being the main dopant in tin oxide and other semiconducting oxides, are also relevant in other related issues. For example, it has been argued that the degradation in metal-oxide varistors is controlled by an electromigration process of defects [32]. Degradation phenomena can appear under continuous, alternating, or pulses of voltage usually combined with thermal treatments [33]. Also, it has been shown that degraded ZnO varistors can recover their properties under an oxygen atmosphere [34]. These type of experiments can be explained as the result of the deformation of Schottky barriers present in the grain boundaries in which oxygen vacancies play a central role [27, 35–38]. As proposed, at temperatures greater than ~200°C changes in conductivity could be attributed to changes in donor concentration implying oxygen diffusion from or into the grains with the consequent creation or annihilation of oxygen vacancies. Consistently, the relevant reactions from the gas phase to the bulk of the grain for oxygen have been proposed to be [2] ′ ″ O2( gas ) ↔ O2′ ( ads ) ↔ Oads ↔ Oads ↔ Oi″ ↔ OO× ,

(5.15)

where O2(gas) refers to oxygen in the gas environment, O'2(ads) to an oxygen molecule adsorbed at the grain surfaces, O'ads and O″ads to singly and doubly ionized monatomic oxygen at the grain surface, O″i to interstitial oxygen, and OO× to oxygen at the tin oxide lattice. The interstitial oxygen O″i migrates from the surface to the bulk annihilating oxygen vacancies, as follow: VO•• + Oi″ ↔ OO× .

(5.16)

Therefore, exposing the SnO2 sample to an oxygen-rich atmosphere at high enough T results in a decrease of vacancy concentration. On the other hand, under vacuum the vacancy concentration is expected to increase. The donor density, at which grains become completely depleted, due to their small size, can be relatively high. Assuming a spherical grain, this takes place when. Vs =

eN d R 2 , 6ε 0ε r

(5.17)

where Vs is the band bending, R the grain radius, Nd the dopant concentration, e the electron charge, ε0 the vacuum permittivity, and εr the relative permittivity. In our case, we can estimate that complete depletion occurs for Nd ≤ 7 × 1023 m−3. In order to analyze the film capacitance behavior, it is necessary to consider the shape of the electrodes. According to the geometry of our sensors, the electrical

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c­ apacitance can be related to that of semiinfinite electrodes, coplanar with a gap separating them [39] C0 =

2   2ε 0ε r  w  w l. ln  1 +  +  1 +  − 1  , π a a     

(5.18)

where w is the width of the platinum interdigitated electrodes, 2a is the width of the interelectrode pathway, and l is the length of the electrode. For the dimensions of our set-up w = 0.5 mm, l = 11 cm, a = 10 μm, with Eq. (5.18), we can calculate that C0 is 2.88 pF for εr = 1. Experimentally we found that the capacitance before the incorporation of the sensor film for 1 MHz, C1, is 12 pF. We have to consider that we have two capacitors in parallel, as on one side of the electrodes there is air, with an associated capacitance Cair, and on the other the alumina substrate. Thus, the total capacitance C1 is the sum of both sides’ contributions. Due to the symmetry of the arrange, we can write Cair = C0 / 2.

(5.19)

The contribution to the total capacitance due to the alumina of thick balum, can be expressed, as a good approximation [40]. Calum =

ε 0ε alum l. sinh −1 ( balum / a ) + 0.234  , π

(5.20)

where εalum is the alumina relative permittivity and the measured value of the alumina thickness is balum = 0.41 mm. Using Eq. (5.20) and correcting for the electrode width [40] we could determine that Calum/εalum ≈ 1.41 pF. This value would be 1.44 pF for an infinitely thick alumina substrate implying that the contribution of the air beyond the alumina substrate is Ca = 0.03 pF. Experimentally we found that the capacitance for the sample SO2, C2, was 23.5 pF. The contribution to the total capacitance due to the film of thick bfilm, can be expressed, as a good approximation, as done before for the alumina side contribution, C film =

ε 0ε film π

l. sinh −1 ( b film / a ) + 0.234  ,

(5.21)

where εfilm is the film relative permittivity and the measured value of the film thickness is bfilm = 0.105 mm. With Eq. (5.21) we determined that Cfilm/εfilm = 1.018 pF, the correction for the electrode width is negligible in this case. As above, this value would be 1.44 pF for an infinitely thick film implying that the contribution of the air beyond the film is Cb = 0.42 pF. First, the measured capacitance C1, without the sensor film, can be expressed as C1 = Cair + Calum + Ca .

(5.22)

Surface and interface electronic properties of tin oxide113

Thus, we can determine Calum = 10.53 pF and then the alumina permittivity εalum = 7.5, value consistent with those reported in the literature [25]. Second, the measured capacitance C2, with the sensor film, can be expressed as C2 = Calum + C film + Ca + Cb .

(5.23)

With Eq. (5.23) we determined Cfilm = 12.52 pF and then the film permittivity εfilm = 12.3. This value represents an effective permittivity consistent with the values of 9.6 and 13.5 for the parallel and perpendicular values of the permittivity, ε1 and ε2, respectively, to the regularly named c-axis in rutile [2]. Following Ref. [41], the effective permittivity for the given values of ε1 and ε2 can be calculated and result is 12.4. Note that the found effective dielectric constants validate the performed calculations and the value found for the film permittivity indicates that grains are completely depleted of electrons. We have proposed oxygen diffusion from the bulk of the grains to the intergrains in explaining conductance changes with temperature (Fig. 5.4). This is consistent with a higher dopant concentration due to vacancy formation, which would reflect in a larger capacitance as the grains presents barriers narrow enough to be nonoverlapped, that is, they have quasineutral regions at their centers. After heating, the measured capacitance is not so simple to analyze quantitatively as before, when grains were completely depleted of carriers. Now, electrically, the film consists in a complex network of resistors and capacitors. In addition to the effect of traps, the observed curvature in the capacitance vs frequency dependence at high frequencies has been shown to be the effect of the random character of the network [42]. However, at enough high frequency, one can consider that the impedance of the capacitors is much lower than that of the resistors and then we end up with a random capacitor network. Consequently, capacitance measurements are carried out at about 1 MHz, frequency at which the capacitance is ≈100 pF. Regardless of the exact value, the capacitance increased markedly. For nonoverlapped potential barriers, the capacitance of an intergrain is that of two Schottky barriers in series. In a one-dimensional approximation, the resulting capacitance, C'film, related to that corresponding to a sample with overlapped barriers, Cfilm, can be expressed as C ’film C film

=

d , 2ω

(5.24)

where d is the grain size and ω is the depletion layer width. The ratio between the film capacitances before and after the heating cycle is ≈7.1, using Eq. (5.23). This change can be interpreted with Eq. (5.24): depletion widths become narrower than the grain radius as a consequence of a higher doping. Note that the change in capacitance cannot be due to a barrier lowering, as this would imply a change in conductivity not compatible with the conductivity results of Fig. 5.4. The work function, WF, is the energy needed to extract an electron from a solid, that is, the difference between the vacuum level and the Fermi level, Evac − EF. In a semiconductor, the work function can be modified due to changes in the position of

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the conduction band bottom respect the Fermi level (EC − EF), and/or by modifying surface electric dipoles that control the difference between the vacuum level and the conduction band bottom, that is, the electron affinity χ  = Evac − EC. Thus, the work function can be expressed as the sum of two terms WF = χ + ( EC − EF ) .

(5.25)

For a fixed value of the electron affinity, any shift in the position of the Fermi level with respect to the conduction band will result in an equal and opposite change of the work function. However, the electron affinity can be modified by gas treatments. Indeed, large changes in electron affinity (of about 1 eV) have been attributed to surface termination after oxidizing or reducing annealing [2]. Photoelectron spectroscopy offers a direct manner to determine work functions. Indeed, the secondary electron onset provides a straight measurement of the specimen’s work function. Then, with the help of this technique we can investigate the influence on the work function of the treatments under oxygen and hydrogen. The results of the measurements are shown in Fig.  5.6. The derived work functions are 4.05 and 3.4 eV for SO2 and SH2 samples, respectively. A reduction in the work function, 0.65 eV in the present case, would be expected after the action of a reducing gas. Thus, the observation of a work-function change during an adsorption process does not necessarily correspond to a change in band bending [13]. A variation of the electron affinity can also be involved due to surface atom rearrangement and a dipole layer alteration and this seems to be what occurs in the present case.

Fig. 5.6  Normalized secondary electron cutoff for the SH2 and SO2 samples. As observed, the exposure to hydrogen reduces the work function by 0.65 eV. Core-level energy positions indicate that this is not a consequence of changes in band bending but due to changes in the electron affinity (see Fig. 5.5).

Surface and interface electronic properties of tin oxide115

5.6 Barrier height fluctuations Most researchers assume that electrical conduction is dominated by thermionic emission and then the conductance is generally described by Eq. (5.6). With Eq. (5.6) (i.e., assuming an Arrhenius relation) the barrier height ΦB, defined as eVS + EC − EF, can be estimated. Thus, the conductance (ln G) vs the reverse function of temperature (1/T) should show a straight line [22]. Interestingly, experimental results on polycrystalline semiconductors do not regularly show such behavior as Arrhenius plots deviate from straight lines, especially at low temperatures [24]. To explain the observed temperature dependence of the electrical conductivity, a variety of reasons have been proposed. Among them, it has been claimed that the Arrhenius plots are curved due to fluctuations in barrier heights [43, 44]. As described, it is customary to model a Schottky barrier with a one-dimensional quadratic potential, as done in Section 5.2, obtained from a jellium of charge in the depletion region. However, the punctual character and random nature of the impurity positions lead to inhomogeneities that can significantly affect the conductivity. In Fig. 5.2 we depict the double Schottky barrier model that is generally accepted. Interestingly, the underlying assumption is that the potential barrier has the same height and shape along the whole interface. Strictly speaking, this is an incorrect picture because the potential barrier arises from the Coulomb potential due to all present charges and then fluctuations must be present. The relevance of these fluctuations will be analyzed after presenting some experimental results. Samples were prepared as described in Section  5.4. Then, samples were placed in the measuring cell, and inside an isolating recipient with liquid nitrogen for cooling purposes. Gaseous nitrogen was flown the entire time to ensure no contact with ambient oxygen. The lowest temperature reached was −123°C and then increased separating the sample from the liquid nitrogen. External heaters were used for higher temperatures, without removing the sample. Measurements were made after reaching steady state at different temperatures. Fig. 5.7 shows the conductance measured in nitrogen of the oxygen treated (SO2) and hydrogen treated (SH2) samples vs the inverse of temperature, from 150 up to 330 K. In this temperature range, the conductivity change for the SH2 sample as a function of temperature is relatively small; the activation energy at the highest temperatures can be estimated 0.19 ± 0.01 eV. The SO2 sample shows a much lower conductance and the activation energy at the highest temperatures corresponds to an activation energy of 0.41 ± 0.02 eV. Higher temperatures were not explored to avoid gas adsorption/desorption that can take place during the heating and cooling processes. Both Arrhenius plots show a curvature at low temperatures. Similar results have been obtained by many authors and for a variety of polycrystalline semiconductors [45–51]. Many researchers assume that the fluctuations of barrier heights present a Gaussian distribution of the form [45, 52–57] P (φ ) =

1 u 2π

exp

− (φB − φ0 ) 2u 2

2

,

(5.26)

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Fig. 5.7  Conductance for the hydrogen (SH2) and oxygen (SO2) samples as a function of temperature from 150 up to 330 K. Lines are fittings to the experimental results assuming thermionic emission conduction.

where ϕ0 is the mean value and u the standard deviation of the barrier height. Thus, the electrical conductance, assuming a thermionic emission mechanism, can be calculated with ∞

G = G0 ∫ P (φB ) F (φB ) dφ ,

(5.27)

Φm

where F(ϕΒ) is the Fermi-Dirac function evaluated at the energy corresponding to the barrier height ϕΒ. The lower limit of integration ϕm is the minimum barrier height, for the flat band condition, that is, VS = 0. Fig. 5.7 shows the fitting of the experimental results assuming thermionic emission conduction by means of Eqs. (26), (27) and taken into account the geometry of our films (see Ref. 4 for details). The mean value of the barrier height ϕ0 and its standard deviation u are adjustable parameters. Reported fittings correspond to ϕ0 = 0.95 eV and u = 0.143 eV for the sample SH2, and ϕ0 = 0.95 eV and u = 0.126 eV for the sample SO2. Results are very sensitive to the fitting parameters to the point that uncertainties in their determination are below 1%. Interestingly, experiments are reproduced by changing its standard deviation and using the same mean value of the barrier height [33]. The model indicates that larger deviations in the barrier height directly affect the conductance, especially at low temperatures. The reason is that, as the barrier height deviation increases, the transfer of electrons across the interface predominantly occurs at regions with lower barriers. This can be clearly detected by plotting the integrand in Eq. (5.27). In Fig. 5.8, curves (A) correspond to T = 333 K and curves (B) to T = 150 K. It is clearly seen that electrons mostly overcome the barrier at places where the barrier

Surface and interface electronic properties of tin oxide117

Fig. 5.8  Integrand P(ϕΒ)F(ϕΒ) of the Eq. (5.27) as a function of barrier height for T = 333 K (A), and T = 150 K (B). Sample conductivity is proportional to the integral of P(ϕΒ)F(ϕΒ). Note that electrons mostly overcome the barrier at places of the interface with barrier heights much lower than its mean value. At low temperatures most electrons cross the interface at places were there in no barrier.

is remarkably smaller than its mean value. The product P(ϕΒ)F(ϕΒ) presents a maximum at ϕmax that can be analytically determined

φmax = φ0 −

u2 . kT

(5.28)

For large values of the dispersion u and/or low temperatures, Eq. (5.28) predicts a negative ϕmax, which is mathematically correct but it has no physical sense. This result indicates that most electrons overcome the barrier at its lower possible values. In fitting experimental data, as we did in Fig. 5.7, two main assumptions are taken for granted: (A) the thermionic emission is the dominating electrical conduction mechanism, and (B) fluctuations of barrier heights present a Gaussian distribution. In principle, a Gaussian distribution seems a sound choice. However, this distribution is only valid for independent events; this is not the present case. We were then motivated to check the validity of this assumption as described in what follows. We estimated the inhomogeneities of the potential intergranular barriers due to the punctual character and random nature of the impurity positions in depletion regions. This can significantly affect the potential at polycrystalline semiconductors and, in general, of potential barriers at any semiconductor interface. Following Mahan [58], we built a parallelepiped of size x × y × z; the width of the parallelepiped z is 2ω, were ω is the depletion region. We randomly distributed a finite number of donors inside the parallelepiped in −ω ≤ z ≤ ω except in the plane z = 0 that corresponds to the grain boundary. Then, we associate to each charge an equal and opposite charge in the interface, at the grain boundary; with this arrangement there is charge neutrality. For our simulations we use a mesh of 200 × 200 × 100 and distribute approximately 1150 charges that corresponds to a band bending of 0.8 V and a doping concentration Nd = 1024 m−3 which are typical values in tin oxide.

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The band-bending height for a given position in the plane of the barrier is defined as the maximum electrostatic potential in the direction normal to this plane (z direction) for every point in the parallelepiped. An approximate potential can be generated by cutting off the 1/r potential at a distance Rs, which is close to the Bohr radius [59, 60]. In our simulations, we replace the potential with a constant inside this radius. The choice of Rs was made by using the next argument. If we consider the charge density as constant within Rs, the potential will behave inversely with the distance outside Rs, and quadratically inside Rs. Therefore, we choose Rs such that the constant potential value inside is approximately the potential reached if the charge were distributed uniformly in Rs. We found that Rs must be approximately 14 Å for tin oxide. Taking into account this cut off, we compute the electrostatic potential in every point of the system as the sum of the Coulomb potentials generated by all charges. It is expected that this approximation lead to an error by excess, enhancing fluctuations. A sample subsection of the resulting band bending as a function of position at the interface is shown in Fig. 5.9. Other doping concentrations, doubly charged dopants, other sizes of the parallelepiped and barrier heights were tested to check the finite size effects and the mesh sensitivity of the results; we found that fluctuations present always a similar pattern. The band-bending height frequency or normalized probability density, for 20 observations, is shown as empty circles in Fig. 5.10. To do this, data were binned in intervals of 1 meV. The probability density has a maximum close to 0.8 eV, the expected value for a jellium of charge, and a standard deviation of ≈0.1 eV due to fluctuations. At first sight, it seems that the probability density can be fitted with a Gaussian distribution that only fails for small and large values of the barrier heights but reproduces quite well the central part of the distribution. However, Fig. 5.10B shows the probability density for the band-bending height in a log-linear scale, which makes more evident the difference between numerical simulation results and the Gaussian approximation. In particular, the simulation corresponds approximately to an interface equivalent to a square with a side of 0.6 μm, which is a much larger surface than the typical intergrains of our films. Interestingly, the band-bending height is never below 0.53 eV or above 1.31 eV.

Fig. 5.9  A sample portion of the band bending as a function of the position for a back-to-back Schottky barriers. The doping concentration is Nd = 1024 m−3 and the average band bending is 0.8 eV.

Surface and interface electronic properties of tin oxide119

Fig. 5.10  The normalized barrier height density distribution in eV−1, as a function of barrier height in a linear-linear scale (A) and in log-linear scale (B). The solid line is the fitting with a Gaussian function.

In nature, many outcomes that depend on the sum of independent events approximate the Gaussian distribution. This is valid if the sample is obtained containing a large number of observations, each observation being randomly generated in a way that does not depend on the values of the other observations and then they are uncorrelated. This is not valid in our case since the barrier potential arises from summing Coulomb potentials. Our results show that the probability density for the barrier heights differs significantly from a Gaussian as |ϕ − ϕ0| increases and very low barriers are not present; see Fig. 5.10B. As a consequence, the inclusion of low barrier values as dictated by the Gaussian distribution, even though with low probability, can notably affect the conductance leading to huge errors. The introduced simple model captures the essence of the problem showing that fluctuations of barrier heights due to the punctual character and random nature of the impurity positions at depletion regions do not present a Gaussian distribution. With the found type of fluctuations, it is not possible to fit experimental results assuming thermionic emission as the dominating electrical conduction mechanism. As mentioned, a double Schottky barrier model is widely accepted to describe polycrystalline semiconductor intergrains. However, many researchers consider grain boundaries of essentially zero width, while others take into account a nonnegligible disordered layer at the grain boundaries, such that the electron transport occurs in two steps. Since the main conclusions will not differ, for the sake of simplicity we will adopt here the second assumption. Then, we calculated the expected conductivity corresponding to a parabolic barrier without fluctuations but including the tunneling contribution. We adopted typical values of the relevant parameters for tin oxide: m = 0.27 m0 for the electron effective mass, ε = 12.3ε0 for the electric permittivity Fig. 5.11 shows the expected conductivity for several dopings. In all cases shown the tunneling contribution is the dominant conduction mechanism in the studied temperature range. Indeed, at T = 333 K the thermionic contribution is about 200 times smaller than that due to tunneling for the lowest studied doping. In Fig. 5.11 we also include our experimental data of Fig. 5.7.

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Fig. 5.11  Arrhenius plots of electrical conductivity in which thermionic and thermionic-field contributions are included. We considered parabolic barriers without fluctuations with height equal to 0.95 eV. Experimental results of Fig. 5.7 are also shown with fittings corresponding to dopings equal to 1.1 × 1025 for SH2 and 7 × 1024 m−3 for SO2.

In calculating the electrical conductance assuming thermionic emission, see Eq. (5.27), the Fermi-Dirac function is multiplied by P(ϕΒ), which is the distribution of barrier heights due to fluctuations. Similarly, in Eq. (5.14), the Fermi-Dirac function is multiplied by τ(E), which is the transmission probability, that is, the probability of tunneling through the barrier. Interestingly, τ(E) presents a dependence with E very similar to a Gaussian. In particular, at low temperatures most electrons overcome the barrier at energies below the barrier top. Thus, by using a Gaussian as a weighting function for the barrier heights assuming thermionic emission, the observed temperature dependence of Arrhenius plots can be fitted. However, the tunneling contribution is the dominant conduction mechanism for large doped samples, especially at low temperatures, and also reproduces the observed Arrhenius plots.

5.7 Chemisorption at semiconductor surfaces The interaction of molecules with solid surfaces has been of central interest as adsorption plays a key role in many fields related to surface science. Some of the relevant species at the surface come from the gaseous phase. In addition, other species come from the bulk, such as electrons and holes, and, as we already have seen, can also be atoms or ions. Thus, we have to keep in mind that the phenomena that take place at a surface can have a three-dimensional character.

Surface and interface electronic properties of tin oxide121

In metal-oxide film sensors, it is usually considered that the active sensing element has a receptor and a transducer functions. The former is related to the interaction of the surface with the analyte, whereas the latter refers to the transduction of this interaction into a macroscopic signal, the change of the electrical resistance. Thus, the sensor film transforms chemical information, originated in the interaction of the gas molecules to be detected with the gas-sensitive material, into a convenient electrical signal [15, 16]. Interaction of gases with semiconductor surfaces is a wide field. We will restrict here to the adsorption of simple gases on an ideal semiconductor surface. The aim of this section is to understand and describe the basic adsorption-desorption mechanisms at semiconductor surfaces. A commonly used approach to model chemisorption on semiconductors is based on Wolkenstein theory [61]. In particular, we reexamine the expression of the adsorption isotherm for dissociative chemisorption. From our derivation, we show that the expression most commonly applied in the literature is incorrect. In particular, we investigate the effects of oxygen chemisorption on n-type SnO2. On clean surfaces, the termination of the periodic potential, dangling bonds and/or lattice defects originate intrinsic surface states. On real surfaces, extrinsic states, due to adsorption of foreign species are regularly dominant. This is the case of oxygen that can ionosorb at the oxide surface. Ionosorbed oxygen acts as an electron acceptor due to the relative energy position of its LUMO (lowest unoccupied molecular orbital) with respect to the Fermi level (Fig. 5.12). Electrons that ionize oxygen come from the semiconductor bulk, contributing to the formation of the space-charge layer. As already discussed, the positively charged depleted region and the negatively charged surface are responsible for the band bending depicted in Fig. 5.12 and then a potential barrier of height eVs that electrons will have to overcome to go from the bulk to the surface. The equilibrium of a semiconductor in contact with a gaseous medium is established when the number of molecules passing from the gaseous phase to the surface per unit time equals the number of molecules leaving the surface by desorption [2]. We assume independent adsorption centers with the same adsorption characteristics

Fig. 5.12  Band bending at an n-type semiconductor surface due to surface sites. EC, EV, and EF denote the energy of the conduction band minimum, the valence band maximum, and the Fermi level, respectively. EA is the surface site energy and eVs the potential barrier or band bending. The superscripts “b” and “s” denote bulk and surface, respectively.

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Tin Oxide Materials: Synthesis, Properties, and Applications

and each center can hold one gas atom or molecule and adsorbed particles do not interact among themselves, as proposed by Langmuir adsorption theory [62]. Also, the density of adsorption sites is independent of their occupancy and it does not vary with temperature, and no activation energy for adsorption was considered. We first analyze oxygen nondissociative adsorption. The rate of adsorption is given by Ref. [63] Ra = s (1 − Θ )

p 2π mkT

,

(5.29)

where Θ is the surface coverage, p is the gas-phase pressure, m is the mass of the gas molecule, k is the Boltzmann constant, T is the temperature, and s is the sticking coefficient. In nonactivated adsorption, the sticking coefficient is temperature independent and it is usually equal to unity. Depending on the Fermi-level position respect to energy of a surface state, the adsorbed particle can be neutral, negative, or positive. In the case of oxygen, it can be considered that it behaves only as an acceptor as the donor level is regularly far below the Fermi level. For the sake of simplicity, we will not include in our analysis intrinsic surface states. Fig. 5.12 shows the energy band diagram for the case of chemisorption of an acceptor-like particle on an n-type semiconductor. We focus on this case, considering only one type of adsorbate and assuming that chemisorption is the only source of surface charging. Following Wolkenstein [61], a particle from the gas phase may become a neutral adsorbate, process in which the system gains q0, that is, the binding energy. If the adsorbed particle presents an acceptor level, it can be occupied by an electron from the conduction band gaining an extra energy ECb − EA and, consequently, the total adsorption energy is q0 + (ECb − EA). The above result can also be derived in the following manner. When an electron is transferred to an acceptor level, it departs from its reservoir, which is the Fermi level. Consequently, there is an energy gain of (EF − EA). In the case of semiconductors, the charging of surface levels is accompanied by the broadening of the depletion region, and consequently there will be a lower number of occupied levels at the conduction band in the bulk. One less electron at the conduction band implies an energy gain of (ECb − EF) and, therefore, the total energy gain after filling an acceptor level at the surface is (ECb − EA). Wolkenstein proposed that both adsorbed type of particles, neutral and charged particles, can desorb. However, it is important to note that, in both cases, the desorption product is a neutral particle. If it is assumed that desorption of a charged particle occurs in a single elementary step, in which the electron returns to the conduction band while the adsorbate desorbs to the gas phase, the rate of desorption can be written as  q0   q0 + ∆E  ∗ − − Rd = N ∗Θ 0ν 0 exp  − + Θ ν exp N (5.30)  − , kT   kT   where N⁎ is the number of adsorption sites per unit surface area, with N0 and N− the number of neutral and charged adsorbates per unit surface area, respectively, Θ0 = N0/N⁎ is the partial coverage of neutral adsorbates and Θ− = N−/N⁎ is the partial coverage of charged adsorbates, k is the Boltzmann constant, T is the temperature, and

Surface and interface electronic properties of tin oxide123

ν0 and ν− are the oscillation frequencies of the neutral and charged adsorbates. The equilibrium coverage of chemisorbed particles as a function of the partial pressure of the adsorbate, that is an adsorption isotherm, can be determined equating the rates of adsorption and desorption, Eqs. (5.29), (5.30). Desorption is an activated process in which an electron, in principle, must overcome the energy barrier ΔE = (ECs − EA) to return to the bulk. The occupation probability of surface states induced by chemisorption is given by the Fermi-Dirac distribution function f f =

N− = N + N− 0

1 .  E − EF  1 + exp  A   kT 

(5.31)

With Eqs. (5.29)–(5.31), the adsorption isotherm can be derived for nondissociative chemisorption

Θ=

βp 1+ β p

  EF − E A   1 + exp  kT     β = β0   ECs − EF  1 + exp  −  kT  

(5.32)

where, following Refs. [62, 63], we used ν0 = ν and

β0 =

s N ∗ν 0 ( 2π mkT )

1/ 2

 q0  exp  .  kT 

(5.33)

It is worth mentioning that authors in the literature adopt different values for ΔE, specifically ΔE = (ECs  − EA) or ΔE = (ECb−EA), indicating that the electron is considered to come from the bottom of the conduction band at the surface or at the bulk [64–70]. At odds with Wolkenstein, it can be considered that the only relevant reaction is that of the gas with the neutral adsorbed species, with the subsequent ionization of the adsorbed particles as dictated by the Fermi statistics [71, 72]. This implies that adions do not desorb directly in one step, but they first must become neutral by losing their charges. Consequently, by eliminating the second term in Eq. (5.30), β adopts the following form:   E − EA  β = β 0 1 + exp  F (5.34)  ,  kT    which differs from Eq. (5.32) only by the term 1 + exp.(ECs − EF)/kT in the denominator. Interestingly, in practice, Eqs. (5.32), (5.34) are not substantially different because the second term in the denominator of Eq. (5.32) is regularly negligible, except for very high dopings.

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At temperatures above 150°C, O2 dissociates, leading to two atomic adsorbates [9]. Therefore, we must take into account that the gas-phase particles are dioxygen molecules, while the surface adsorbates consist of single oxygen atoms, which can be neutral or charged. The rate of adsorption takes now the following form: Ra = s (1 − Θ )

2

p 2π mkT

.

(5.35)

The square in the term (1 − Θ)2 takes into account that an impinging molecule needs two adjacent unoccupied sites to adsorb atomically. If all possible pairs of adsorbed particles can desorb, the rate of desorption should be 2 2  q0   q 0 + 2∆E  ∗ − − Rd = N ∗ Θ 0 ν 0 exp  −  + N Θ ν exp  −  kT  kT    0   + ∆ q E + 2 N ∗Θ 0Θ −ν exp  − , kT  

( )

( )

(5.36)

where, as before, the extra energy ΔE is the energy needed to return an electron to the bulk. Note that when one electron is involved an energy ΔE is required, while when two electrons return to the bulk the energy needed is 2ΔE. In this case q0 is the adsorption energy corresponding to an oxygen molecule and ν0, ν−, and ν are the oscillation frequencies of the corresponding adsorbates. By equating Eqs. (5.35), (5.36), the adsorption isotherm for dissociative chemisorption can be derived. With ΔE = (ECs − EA) and assuming that all oscillation frequencies of neutral and charged adsorbates are the same,

Θ=

βp 1+ β p 2

  EF − E A   1 + exp  kT     β = β0  . 2 s   EC − EF   1 + exp  −  kT    

(5.37)

If only neutral species desorbs, by eliminating the second and third terms of the right-hand side in Eq. (5.36), β takes the form 2

  E − EA  (5.38) β = β 0 1 + exp  F  ,  kT    a result with similar energy dependence to that for the nondissociative case, Eq. (5.34). As for the nondissociative case, the exponential in the denominator of Eq. (5.37) is regularly negligible and then Eqs. (5.37), (5.38) lead to similar results. Isotherms, Θ = Θ(p,T), can be determined with the electroneutrality condition: the positive charge at the space charge must be the same than the negative charge at the

Surface and interface electronic properties of tin oxide125

surface. For a homogeneous semiconductor, with a uniform doping concentration, the charge due to the uncompensated doubly charged donors is given by Ref. [73]

σ b = ( 2eε N d )

1/ 2

kT   2Vs − e (1 − exp(−eVs / kT )   

1/ 2

,

(5.39)

where ε is the semiconductor permittivity and Nd the dopant concentration. On the other hand, the negative charge at the surface is given by

σ s = eN ∗Θ − .

(5.40)

The position of the Fermi level at the surface depends on the extent to which the surface is covered by charged chemisorbed particles. Simultaneously, the surface coverage of neutral and charged particles and σb depend on the band bending. Thus, this has to be solved self-consistently. We adopted typical values of the relevant parameters for tin oxide: N* = 1019 m−2 for the adsorption sites per unit surface, and ν 0 = ν− = 1013 s−1 for the oscillation frequencies of the adsorbates. Following Ref. [64], we fixed the energy level of chemisorption-induced states at ECs − EA = 1 eV, and the adsorption energy for a neutral particle as q0 = 0.1 eV [74, 75]. In Fig. 5.13 we present the resulting band bending for nondissociative and dissociative chemisorption, which only includes desorption of neutral particles. Considering desorption of charge particles as proposed by Wolkenstein leads to similar results. Note that dissociation leads to noticeable band bendings from lower pressures as desorption requires two contiguous adsorbates to occur, while in nondissociative chemisorption desorption in proportional to coverage.

Fig. 5.13  (A) Nondissociative chemisorption-induced band bending as a function of semiconductor doping Nd and oxygen pressure for T = 600 K. (B) Dissociative chemisorptioninduced band bending as a function of Nd and oxygen pressure assuming that only neutral adsorbates can desorb for T = 600 K. (Similar results are obtained if desorption of charged species are incorporated.)

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5.8 Nonparabolic barriers As discussed in the previous sections, it is generally accepted that oxygen adsorption at the grains surfaces directly reflects on band bending by assuming parabolic barriers due to a constant density of donors along the depletion region. However, the equilibrium oxygen vacancies density can be far from constant and then surface barriers are not parabolic. This can be demonstrated with more than an argument, as shown below. The oxygen exchange equilibrium between SnO2 with the gas phase is regularly written as [76] 1 O0 ⇔ VO·· + 2e − + O2 , 2

(5.41)

where O0 is neutral oxygen at the crystal, VO·· is a doubly ionized oxygen vacancy, e− is an electron, and O2 an oxygen molecule at the gas phase. Eq. (5.41) implies several mechanisms, as the density of vacancies must be able to evolve to reach equilibrium (e.g., after a change of the oxygen pressure or temperature). These mechanisms can be the following: neutral oxygen in the crystal leaves its position to create a doubly ionized vacancy and a doubly ionized interstitial, oxygen interstitials migrate to the surface, ionized oxygen at the surface becomes neutral, and finally adsorbed oxygen desorbs to the gas phase. Regardless of the specific mechanisms involved, under thermodynamic equilibrium, the corresponding mass action law for Eq. (5.41) is K = VO··  n 2 p ( O2 ) . 1/ 2

(5.42)

Square brackets denote concentration, n is the electron density, and p(O2) is the oxygen partial pressure. Eq. (5.42) implies that the product [VO··]n2 remains constant for a given value of oxygen pressure. This relation has tremendous implications because the vacancy concentration near the surface can be very different from that at the bulk due to band bending. The constant K does not depend on the relative position of the Fermi level respect to the bands, but [VO··] does because of n that is proportional to exp[−(Ec − EF)/kT]. Thus, Eq. (5.42) clearly implies that, at a fixed O2 partial pressure and in equilibrium, the oxygen vacancies density increases exponentially as EF goes down in bandgap, which results in the impossibility to have, in equilibrium, a constant doping level along the whole space charge region. As an alternative approach, we can consider that the density of lattice vacancies is directly related to its formation energy, Efor, as [77]

[Vo ] ∝ exp ( − E for / kT ) .

(5.43)

Efor is the energy needed to take an atom from a lattice site inside the crystal to the vapor phase. However, the resulting vacancy in the crystal behaves as a donor that ionizes giving two electrons away. Since electrons relaxed to the Fermi level, the formation energy of an ionized donor decreases as EF moves away from the conduction band.

Surface and interface electronic properties of tin oxide127

Based on the above discussions, for a given gas-phase pressure, [VO··] can be expressed as a function of band bending as VO··  = N d exp 2 ( Ec ( x ) − Ecb ) / kT  ,

(5.44)

where Nd denotes the vacancy concentration in the bulk, Ec(x) is the bottom of the conduction band at position x and Ecb the bottom of the conduction band at the bulk. Interestingly, Lantto and coworkers raised the issue long time ago [78]. He originally considered a temperature high enough allowing sufficiently mobility to ionized oxygen vacancies. If so, they will tend to move toward the surface due to the present electric field, which rearrange the vacancies. Eventually, there will be a concentration gradient responsible for a diffusion of vacancies in the opposite direction than that originated by the electric field. Then, assuming that oxygen vacancies are the only relevant charged particles (strictly, this is only valid for fully depleted slabs [79]) by resorting to the Poisson’s equation the potential V(x) can be deduced to be V ( x) =

 x kT  ln 1 + tan 2   2e   2 LD

   ,  

(5.45)

where LD = (εkT/4e2Nd)1/2 is the Debye length. In Fig.  5.14 we plot V(x) for tin oxide for an oxygen concentration at the bulk Nd = 1023 m−3 that are doubly charged. This figure shows a potential corresponding to a constant density of oxygen vacancies leading to a parabolic barrier, and a potential

Fig. 5.14  Electric potential for a bulk oxygen vacancies of Nd = 1023 m−3. In the case of constant doping, V(x) shows a parabolic shape. For doubly mobile donors, V(x) presents the Gouy-Chapman potential shape.

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Tin Oxide Materials: Synthesis, Properties, and Applications

given by Eq. (5.45), denote as Gouy-Chapman potential that radically differs from the parabolic-shaped barrier. The point is that the shape of the intergranular barriers has a strong effect on the conduction properties and then on the sensitivity of sensors based on this type of materials [80]. In particular, Gouy-Chapman potentials are much sharper than parabolic potentials and then the contribution of tunneling through the barrier has an even larger contribution.

5.9 Conclusions In this chapter we have analyzed Schottky-like barriers formed at surfaces and at grain boundaries of tin oxide. We have seen that these potential barriers can have a much more intricate behavior than regularly though, as many nonsimple mechanism are involved. We showed that polycrystalline tin dioxide films present intergranular barriers that can be affected by thermal treatments at moderate temperatures. Interestingly, XPS results show that samples with different conductivities can present similar barrier heights. Thus, a thermionic mechanism cannot be responsible for the observed differences. Experiments indicate that oxygen diffusion into and out of the grains can change the doping to the point that they can present fully developed Schottky barriers, with quasineutral regions at their centers, or be completely depleted of carriers. We specially focused on capacitance values, which were analyzed in detail considering the geometry involved to quantify doping changes due to in- and out-diffusion that lead to barriers overlapping. Also, we presented UPS results that show that exposing the SnO2 samples to oxidizing or reducing environments modifies the work function without modifying the band bending. The experimental conductivity for polycrystalline tin oxide can be fitted assuming thermionic emission conduction at grain boundaries with a Gaussian distribution of barrier height fluctuations. However, resorting to a computational numerical model, we found that spatial fluctuations in barrier heights due to the discreteness of the donors and their statistical distribution at the depletion region differ from a Gaussian distribution. The type of obtained fluctuations, considering thermionic emission conduction, cannot explain the Arrhenius plots for the electrical conductivity found experimentally, especially at low temperature. Conversely, the tunneling contribution to conduction, without resorting to fluctuations, presents the observed trends. We have derived a corrected form of adsorption isotherms for dissociative chemisorption of gases on a semiconductor surface using the Wolkenstein theory. In particular, it was shown that a dissociative chemisorption corresponds to a second-­ order reaction involving two electrons and, since desorption is an activated process, electrons must overcome a surface barrier to return to the bulk. Then, results were used to determine band bendings and adsorbate coverages as a function of pressure and doping. It is important to emphasize that these calculations were carried out by assuming a constant doping along the depletion region in the semiconductor [29]. In fact, the equilibrium oxygen vacancies density can be far from constant and then

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surface barriers are not parabolic. Similar calculations using the equilibrium doping distribution, which leads to the so-called Gouy-Chapman potential, can lead to very different outcomes opening a wide range of theoretical and experimental work.

Acknowledgments This work was partially supported by the National Council for Scientific and Technical Research (CONICET) of Argentina and the National University of Mar del Plata (Argentina).

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