Theory of Sensing Response of Nanostructured Tin-Dioxide Thin

Apr 30, 2013 - A theory for sensor response to a reducing hydrogen gas is presented for semiconductor tin dioxide (SnO2) nanoparticle thin films. Pre-...
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Theory of Sensing Response of Nanostructured Tin-Dioxide Thin Films to Reducing Hydrogen Gas Mortko A. Kozhushner,†,‡ Leonid I. Trakhtenberg,†,‡ Aaron C. Landerville,§ and Ivan I. Oleynik*,§ †

Semenov Institute of Chemical Physics RAS, 4 Kosygin Street, Moscow 119991, Russia Moscow Institute of Physics and Technology (State University), Institutskii per. 9, Dolgoprudny, Moscow, Region 141700, Russia § University of South Florida, 4202 East Fowler Avenue, Tampa, Florida 33620-5700, United States ‡

ABSTRACT: A theory for sensor response to a reducing hydrogen gas is presented for semiconductor tin dioxide (SnO2) nanoparticle thin films. Pre-existing oxygen vacancies in SnO2 act as electron donors to the conduction band. Oxygen atoms appearing upon dissociation of atmospheric oxygen at the surface of nanoparticles serve as electron traps, thus, decreasing the concentration of conduction electrons. Sensor response is caused by an increase in the film conductivity upon the addition of the reducing analyte gas H2, which reacts with atomic oxygen at the surface of SnO2 nanoparticles to form water molecules in the gas phase, and is then followed by the transfer of electrons back into the conduction band. The theoretical description of sensor response takes into account the kinetics of surface chemical reactions that both control the concentration of electrons within the conduction band, and the physics of electron transport in nanostructured SnO2. The theory, which couples the electronic response with the microstructure of the film and the chemical environment, predicts sensor sensitivity as a function of temperature, hydrogen pressure, and average size of SnO2 nanoparticles in agreement with experiment.

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the film’s conductivity upon adsorption of reducing gas by invoking the concept of a space charge layer, an electrondepleted region in the subsurface layer of an oxide nanoparticle, created by the surface negative oxygen ions.3,5,11,13 It was thought that the negative surface charge due to both the O− and nonuniform electron charge distributions produce an electric potential barrier of finite thickness, the latter being characterized by the Debye screening radius rD. It is this surface barrier that causes the drop in sensor conductivity. The potential barrier is substantially reduced upon adsorption of the reducing gases at the surface of oxide nanoparticles, which is followed by desorption of oxygen-containing molecules and the release of electrons to the conduction band, resulting in an increase of sensor conductivity. The dominant role claimed for both the space charge layer and the associated surface potential barrier in explaining the sensor effect is not physically justified because such a barrier does not exist in both cases of low and large concentrations of conduction electrons. Consider first the case of low electron concentrations nc in the conduction band, nc ≪ 1016 cm3, corresponding to undoped SnO2 films in ambient atmosphere. For a typical size of SnO2 nanoparticle, d̅ = 100 nm, the conduction band contains one electron. The corresponding Debye screening radius is rD = ((kBTχ)/(4πe2nc))1/2 ≫ d/2 =

INTRODUCTION Nanostructured semiconductor tin dioxide (SnO2) thin films are one of the best sensor materials employed to detect analyte gases.1−5 The sensing films are composed of sintered nanoparticles of an average diameter up to several hundred nanometers, which possess high surface to volume ratio that allows effective surface adsorption of gas phase analyte molecules.2,6−10 The sensor effect is an increase in conductivity upon the addition of reducing gases such as H2 and CO to ambient atmosphere.2,7,11 The excellent stability, high sensitivity, relatively low operating temperatures, and low cost of SnO2 sensors are responsible for their widespread application.1−4,8,12 In spite of extensive efforts in the research and development of tin-oxide sensors, the fundamental mechanisms of coupled chemical and electronic processes are still not fully understood.4,11,13 In particular, the structure−property relationship between the film microstructure and its sensor performance is urgently sought to improve sensor performance via optimization of both the processing conditions at the nanoscale, and the sensor’s operation parameters.5,6,9,10,14 The N-type semiconductor SnO2 material possesses a considerable number of preexisting oxygen vacancies that serve as electron donors to the conduction band.15,16 The sensors operate at temperatures around 300−400 °C, resulting in dissociation of atmospheric O2 molecules on the surface of SnO2 nanoparticles. The oxygen atoms on the SnO2 surface, in turn, capture electrons from the conduction band. Most existing theories of sensor response explain the increase in © 2013 American Chemical Society

Received: December 2, 2012 Revised: April 26, 2013 Published: April 30, 2013 11562

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50 nm, where kB is the Boltzmann constant and χ ≈ 13.5 is the dielectric constant. Therefore, at such low electron concentration there are not enough electrons to produce screening, thus, ruling out the existence of an appreciable surface potential barrier. In the opposite case of high electron concentration, nc ≈ 1018−1019 cm−3, which corresponds to another oxide sensor material In2O3, a negatively charged layer might exist. However, according to Gauss’s theorem, the electric field inside the quasispherical nanoparticle produced by the surface O− is zero. Consequently, in both cases of low and high electron concentrations, a surface potential barrier does not exist, hence, its notion can not be invoked to explain the observed sensor effect. Moreover, this is in accord with the standard theory of charge transport in semiconductors, in which the screening of charged impurities is never taken into account.17 In this work we specifically consider the sensor effect produced by SnO2 films, which possess relatively low conduction electron concentration. The conductance of such nanocrystalline oxide films is due to the electron transport through the particles, and is solely determined by the electron concentration in the conduction band nc. The latter depends on the temperature and the concentration of native oxygen donor vacancies, which can accept or donate electrons to the conduction band. The energy difference between the electron level of the oxygen vacancy and the bottom of the conduction band in SnO2 is small enough to produce thermal ionization of the donors at 300 °C, resulting in a considerable concentration of the conduction electrons and appreciable conductivity within the crystalline tin dioxide. In the case of nanocrystalline SnO2 film, an additional factor influences the film’s electrical properties: when exposed to air, a substantial adsorption of atmospheric O2 occurs at the surface of SnO2 nanoparticles. The O2 molecules dissociate at 300−400 °C producing atomic oxygen adsorbates with concentration exceeding that of the oxygen vacancies in the bulk of the oxide. Because the surface oxygen atoms are effective acceptors of electrons, the concentration of the electrons in the conduction band drops considerably, resulting in the reduction of the conductivity within the film. Upon adsorption of the reducing gases (H2 or CO), the oxygen adsorbates react to form H2O or CO2, which then desorb to the gas phase. The removal of surface oxygen acceptors releases the captured electrons back to the conduction band, thus, increasing nc and, consequently, the conductivity within the film. In this paper, a consistent theory of the sensor effect is presented based on a detailed description of the elemental chemical processes taking place at the surface of SnO2 nanoparticles, as well as conductivity mechanisms in the nanostructured films. The quantitative modeling of the sensor effect based on the solution of kinetic equations allows for the prediction of the sensor effect as a function of temperature, pressure of the analyte gas, and the average dimension of the oxide nanoparticles in agreement with experiment.

processing conditions. The contribution of the electrons from oxygen donor vacancies to the conductivity decreases exponentially with the increase in the distance between the donor energy level and the bottom of the conduction band. Therefore, to quantitatively describe the sensor effect, the spectrum of the donor energy levels existing in the real sample is replaced by a single level with average energy −εd (the zero energy is at the bottom of the conduction band). The average energy εd of an oxygen donor vacancy is determined in this work experimentally by measuring the conductivity of the SnO2 film in vacuum as a function of temperature. The film has been annealed in vacuum (10−5 Torr) at 700 K to completely remove adsorbed oxygen from the surface of SnO2 nanoparticles, thus, ensuring that electrons from native oxygen vacancies are the sole source of the film’s conduction. Then, the resistance of the film in vacuum was measured as a function of temperature in the interval 54−395 °C. The results shown in Figure 1 demonstrate that the

Figure 1. Resistance of the SnO2 thin film in vacuum as a function of temperature.

resistance depends exponentially on temperature, R ∼ exp{ER/ kBT}, with activation energy ER = 0.35 eV. In general, the value of ER depends on many factors and can vary from sample to sample.16 For example, in commercial polycrystalline samples of SnO2, ER = 0.15 eV, as determined from the temperature dependence of the resistance measured in the interval 20−500 °C.18 Because the conductivity is proportional to the concentration of the electrons in the conduction band nc, ER is related to the electron energy level εd. As is shown below, ER = εd/2. If the concentration of the oxygen vacancies Nd is known, the dependence of nc on εd is determined using standard statistical mechanics.19 The electronic part of the free energy F is written as F = Fc + Fd

(1)

where Fc is the free energy of the electrons in the conduction band, and Fd is the free energy of the electrons bound to oxygen vacancies. At small electron concentration and relatively large temperatures (equal to or higher than room temperature), the electrons in the conduction band obey Boltzmann statistics;19 therefore,

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EQUILIBRIUM ELECTRONIC PROPERTIES OF NANOSTRUCTURED SNO2 FILM The basic electronic characteristics of SnO2 film are determined by the concentration of the oxygen donor vacancies and corresponding electronic energy levels. Unfortunately, quantitative information on the electronic properties of nanocrystalline SnO2 films is not available. It was shown in ref 16 that both the electronic energy levels and the concentration of oxygen vacancies in crystalline SnO2 are influenced by growth and

Fc = −nckBT[1 + ln(B /nc)]

(2)

where B= 11563

(2m*kBT )3/2 8ℏ3π 3/2

(3)

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and the electron effective mass is m* ≈ 0.33me.18 The free energy due to oxygen vacancies is derived by writing the partition function of the system of (Nd − nc) electrons distributed over the Nd vacancies with energy −εd19 ⎧ ε ⎫ Nd! exp⎨ d ⎬ (Nd − nc) ! nc! ⎩ kBT ⎭

Zd =

section. Therefore, the theoretical conductivity should be 10 times smaller than σth, as specified by expression 7: Σ = 0.1σthA /l

where A is the cross-section of the film between the electrodes and l is the distance between the electrodes. Using expressions 3, 6, 7, and 8, as well as the experimental value for the conductivity of the film at 500 K, Σ = 9.5 × 10−4 Ω−1, we obtain the concentration of oxygen vacancies Nd ≈ 1016 cm−3.

(4)

Using the Stirling formula for large factorials and the definition of the free energy F = −kBT lnZ, the free energy due to oxygen vacancies is written as

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QUANTITATIVE MODEL OF SENSOR EFFECT To provide a quantitative description of the sensor effect, it is necessary to obtain the equilibrium concentration of the conduction electrons nc under a varying concentration of surface oxygen due to chemical reactions with the analyte gas. Because the binding energy of an electron to an oxygen atom depends on the chemical environment of the O atoms adsorbed at the surface, the effective binding energy εO is introduced. To have a substantial sensor effect, atomic O should capture effectively both the conduction electrons and the electrons from oxygen donor vacancies. Evidently, this happens if the electron energy level of O− is lower than that of the oxygen donor vacancy, that is, their binding energies εO > εd (binding energy is always positive). Electron capture by oxygen molecules to form O−2 can be neglected due to their small binding energy: O−2 is thermally ionized with the transfer of an electron back to the conduction band. By introducing the volume concentration of the adsorbed oxygen atoms NO and that of negative atomic ions nO−, the electronic part of the free energy F is written as

⎡ Nd Fd = −(Nd − nc)εd − kBT ⎢Nd ln Nd − nc ⎣ + nc ln

Nd − nc ⎤ ⎥ nc ⎦

(5)

Minimizing the total free energy, eq 1, which includes expressions 2 and 5, over nc under the condition εd ≫ kBT, the equilibrium electron concentration nc is obtained: nc =

⎧ ε ⎫ BNd exp⎨− d ⎬ ⎩ 2kBT ⎭

(6)

n−1 c ,

Because the resistance of the film R ∼ the theoretical (expression 6) and experimental dependence of the resistance on temperature were compared to give εd = 2ER = 0.7 eV. Then, the density of oxygen vacancies Nd is determined by comparing the numerical values of the experimental and theoretical conductivities. To find the latter, we use the fact that the elemental charge transport event is the electron jump between neighboring nanoparticles. Within the Drude model of conductivity, we obtain σth =

1 2 d̅ e nc v ̅ 3 kBT

(8)

F = Fc + Fd + FO−

(9)

where Fc, Fd, and FO are the free energies of the electrons in the conduction band, electrons bound to vacancies, and electrons bound to atomic oxygen, respectively. The expressions 2 and 3 are used for Fc; the free energies Fd and FO− are found in a similar fashion to 4 by calculating the probability of capturing nO− electrons by NO oxygen atoms (one per atom), and the probability of finding Nd − (nc + nO−) electrons on Nd oxygen vacancies (one per O vacancy). Then the corresponding free energies are written as −

(7)

where e is the charge of the electron, v ̅ is its average thermal velocity, and d̅ is the average diameter of the nanoparticles. To obtain the total resistance of the sample, it is necessary to take into account the real microstructure of the film. As seen from Figure 2, the average contact area between nanoparticles is approximately ten times smaller than the particles’ cross-

⎡ Nd Fd = −(Nd − nc − nO−)εd − kBT ⎢Nd ln Nd − nc − nO− ⎣ + (nc + nO−)ln

Nd − nc − nO− ⎤ ⎥ nc + nO− ⎦

(10)

⎡ NO FO− = −nO−εO − kBT ⎢NO ln NO − nO− ⎣ + nO ln

NO − nO− ⎤ ⎥ nO− ⎦

(11)

The characteristic time of electronic transitions in the system is several orders of magnitude smaller than the inverse rates of chemical reactions involved. Therefore, the electronic subsystem maintains its state of thermal equilibrium by quickly adjusting to the slow changes in chemical species’ concentrations. Then, the equilibrium electron concentrations nc and nO− are found by minimizing the free energy 9, which includes expressions 1, 10, and 11, over the independent variables nc and nO−:

Figure 2. Microstructure of the SnO2 thin film: transmission electron microscopy photograph. 11564

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⎧ ε ⎫ Nd − nc − nO− = exp⎨ d ⎬ nc(nc + nO−) ⎩ kBT ⎭

⎧ ε − εO ⎫ (Nd − nc − nO)(NO − nO−) ⎬ = exp⎨ d nO−(nc + nO−) ⎩ kBT ⎭

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⎧ εO ⎫ K Orec ≈ a 2νO exp⎨− a ⎬ ⎩ kBT ⎭

(12)

where a is the jump length of an O atom at the surface, a ≈ 3 × 10−8 cm; νO is the vibration frequency of an O atom in its potential well, νO ≈ 1013 s−1; and εOa is the activation energy of the jump. The constant of O2 adsorption is written as

(13)

The density of the ionized vacancies is nc + nO−; therefore, Nd > (nc + nO−) in eqs 12 and 13. In contrast to Nd, which is constant for a given sample, the concentration of the neutral oxygen atoms NO(PH2,T) is a function of both temperature T and the pressure of the analyte gas H2 PH2, (the oxygen pressure PO2 is always constant and is equal to ambient oxygen partial pressure). To find NO(PH2,T), it is necessary to consider the following chemical processes taking place at the SnO2 surface20 des Oad 2 ⇌ O2

(14)

ad Oad 2 → 2O

(15)

Oad + e− ⇌ O(−)ad

(16)

des H ad 2 ⇌ H2

(17)

(−)ad H ad → H 2Odes + e− 2 + O

(18)

K Oad2 =

(19)

(20)

(21)

rec where Kdis O2 and KO are the constants of O2 dissociation and O des recombination at the SnO2 surface, Kad O2 and KO2 are the des constants of adsorption and desorption of O2, Kad H2, and KH2 are the constants of adsorption of H2, KH−O is the reaction constant for the surface reactions with participation of atomic hydrogen, and Nad‑sites and Nad‑sites are the surface concentrations of the O2 O2 H2 and H2 adsorption sites. The O2 dissociation constant Kdis O2 is defined as

⎧ Δ ⎫ ⎬ K Odis2 ≈ νO − O exp⎨− ⎩ kBT ⎭

6kBTmO2

(24)

(25)

where νO2 is the vibration frequency of the adsorbed O2, νO2 ≈ 1013 s−1, and εdes, the binding energy of the molecule to the SnO2 surface. Because of the large binding energy of O atoms to the SnO2 surface, their desorption is not taken into account in eqs 19−21. In addition, the desorption of H2 molecules is also neglected because the hydrogen molecules immediately dissociate upon adsorption, and the binding energy of H atoms to the surface is large. That is why NN2 = NH/2 and = Nad‑sites /2. By the same reason, the fast surface Nad‑sites H2 H reactions of hydrogen with oxygen molecules and atoms involve atomic hydrogen only. Therefore, the net reactions O + 2H → H2O and O2 + 4H → 2H2O are included in the system of kinetic eqs 19−21. For all chemical reactions involving atomic hydrogen, the rate limiting step is the surface diffusion of H atoms; therefore, their rates are the same and equal to the coefficient of H surface diffusion: KH−O ≈ DH. The activation energy εOa of O surface diffusion, as well as those of various reactions in 19−21, are unknown. However, our investigation demonstrates that the theoretical dependence of the SnO2 film sensitivity on temperature, including the temperature Tmax at which the sensor is maximally sensitive, can vary greatly with the choice of the kinetic parameters. Therefore, they are uniquely determined by matching theoretical and experimental dependencies; see below. In eqs 12 and 13 the volume concentration of atomic oxygen NO is used, whereas in 19−21, surface concentrations. A surface concentration NS is converted to volume concentration NV by employing the notion of a specific surface area for the film AV:

K Had2PH2(NHad2‐sites − NH2) − KH − ONONH2 − 2KH − ONO2NH2 =0

αO2Aad

⎧ ε ⎫ K Odes2 ≈ νO2 exp⎨− des ⎬ ⎩ kBT ⎭

K Oad2PO2(NOad2‐sites − NO2) − K Odes2 NO2 − K OrecNO2 − 2KH − ONO2 NH2 = 0

1 Aad αO2uO̅ 2 = 3 kBT

where αO2 is the O2 sticking probability, u̅O2 is the average thermal velocity of O2, Aad is the characteristic adsorption area, and mO2 is the mass of an O2 molecule. The constant of H2 adsorption is written exactly in the same way as in expression 24 with the corresponding quantities αH2, u̅H2, and mO2. The O2 deposition constant is defined as

Because the rates of the chemical reactions are substantially larger than the inverse of the time for sensor response, stationary concentrations are found from the equations of chemical equilibrium for processes 14−18: K Odis2 NO2 − K OrecNO2 − KH − ONONH2 = 0

(23)

NV = N SAV

(22)

(26) 3

where AV is defined as the area per unit volume (cm ). Obviously, AV is a function of the microstructure, and can be determined by using the nanoparticle distribution function over their diameters f(D). Then, assuming the spherical shape of the nanoparticles, their total number N per 1 cm3 is defined as ρf 6/π · N= ρSnO ∫ D3f (D)dD (27) 2

where νO−O is the vibration frequency of an O2 molecule, and Δ is the dissociation energy of O2 adsorbed at the SnO2 surface, Δ being much smaller than the dissociation energy of a gasphase O2 molecule due to the strong binding of oxygen atoms to the surface of the oxide. The recombination reaction of oxygen atoms at the surface is of a diffusive character; therefore, its constant is approximately equal to O diffusion on the SnO2 surface: 11565

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1014 cm−2, Δ = 1.27 eV, εa = 0.15 eV, εd = 0.75 eV, and εO = 0.85 eV are found by optimizing the theoretical Θmax and Tmax to match their experimental values. It is worth noting that these physically reasonable values of the parameters are determined almost uniquely. In fact, the subsequent analysis shows that if a slightly different parameter set is used then the theoretical temperature dependence of sensitivity Θ(T) differs dramatically from experimental Θ(T) obtained at various hydrogen pressures. The final step involves the solution of the full system of eqs 19−21 at varying values of hydrogen analyte pressure to find the surface concentration of O atoms NO(PH2,T). Then, substituting NO(PH2,T)AV as the volume concentration NO in eqs 12 and 13, nc(PH2,T), the concentration of the conduction electrons at a given PH2 is found. Finally, similar to eq 31, the sensor sensitivity is determined

where ρf and ρSnO2 are the mass densities of both film and bulk SnO2. Thus, the specific surface area of the film is AV = N

∫0

∞

πD2f (D)dD ∞

=6

∫0 D2f (D)dD

ρf

· ∞ ρSnO ∫ D3f (D)dD 2 0 ρf 1 =6 · ρSnO deff 2

(28)

where the effective nanoparticle diameter is ∞

deff =

∫0 D3f (D)dD ∞

∫0 D2f (D)dD

(29)

The effective diameter deff is greater or equal to the average nanoparticle diameter d̅ = ∫ 0∞Df(D)dD, deff ≥ d̅, with equality being achieved for nanoparticles of the same diameter. Using the experimental distribution of nanoparticle diameters f(D) for SnO2 films used in ref 20, the calculated specific surface area was found, AV ≈ 1.25 × 105 cm−1.

Θ(PH2 , T ) =

nc(PH2 , T ) nc(0, T )

(32)

The calculated sensitivity Θ(PH2,T) as a function of temperature for a sensor with d̅ = 100 nm at several values of hydrogen pressure PH2 are shown in Figure 3. As seen in the figure, the

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MODELING OF THE SENSOR EFFECT: COMPARISON OF THEORY WITH EXPERIMENT To model the sensor effect using the kinetic scheme outlined above, the following procedure is implemented. First, in the absence of the analyte gas, such that PH2 = 0 and NH2 = 0, the system of eqs 19−21 is solved to obtain the concentration of atomic oxygen NO(0, T ) =

K Odis2

NOad2‐sites

· K Orec 1 +

K Odes2 K Oad2PO2

(30) Figure 3. Calculated dependence of the SnO2 sensor sensitivity on temperature at various pressures of H2 analyte. The effective diameter of nanoparticles is deff = 100 nm.

where the rate constants in eq 30 are specified in eqs 22−25. Then, the volume concentration of O atoms, NO(0,T)AV is substituted into eqs 12 and 13, and the system is solved to obtain the concentration of the electrons in the conduction band nc(0,T). Second, the concentration of the conduction electrons is determined for a very large concentration of H2 analyte. Under such conditions, the SnO2 surface is free of adsorbed oxygen, because it reacts with the hydrogen, and the H2O products are released into the gas phase. Therefore, expression 6 specifies the value of nc(PH2 = ∞,T). The maximum sensitivity Θmax = Θ(∞,T) is determined as the ratio of the film’s conductivities at very large and zero H2 pressures, or according to eq 7, as the ratio of the corresponding concentrations of the electrons in the conduction band Θmax = Θ(∞ , T ) =

nc(∞ , T ) nc(0, T )

maximum sensitivity is reached at PH2 = 10−2 atm. Upon further increase of PH2, the sensitivity saturates: the curves Θ(T) at higher pressures practically coincide with the curve for PH2 = 10−2 atm. Although a quantitative agreement between experiment and theory is difficult to achieve due to uncertainties related to the specific preparation conditions of the samples, our calculations display a qualitative correspondence to experimental observations in spite of the fact that only the maximum sensitivity at one specific pressure PH2 = 10−2 atm is used to fix the kinetic parameters of the model. Specifically, both experiment21 and theory exhibit asymmetric bell-shaped dependence for Θ(PH2,T) on temperature at various hydrogen pressures. Although the theoretical widths at the half-maximum for the Θ(PH2,T) curves are greater than that of ref 21 they match the widths of Θ(PH2,T) obtained in recent experiments,22 for which the average size of SnO2 nanoparticles is close to that used in our simulations. Last, but not the least, both theory and experiment exhibit the same trend of increasing temperature

(31)

Third, the parameters of rate constants eqs 22−25 are obtained by using a very limited amount of experimental information: the value of maximum sensitivity Θmax ≈ 15 and the temperature Tmax ≈ 700 K at which this maximum is observed in the case of a film with an average diameter of nanoparticles d̅ = 250 nm.21 Then, the following values αO2 = 3 = 3 × 1013 cm−2, Nad‑sites =2× × 10−3, αH2 = 5 × 10−2, Nad‑sites O2 H2 11566

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Tmax of the maximum of Θ(PH2,T), with increasing hydrogen pressure PH2.

should increase the characteristic response time of the sensor due to the resulting increase in the diffusion pathways along the surface of the nanoparticles. It should be noted that only experimental values of Θmax and Tmax at PH2 = 10−2 atm are used to fit the kinetic parameters of the model. The temperature dependence of such parameters as sticking probability and characteristic adsorption area are not considered in the present work. In addition, the parameters of the model may also depend on conditions of sensor film preparation. Therefore, only a qualitative prediction of the sensor effect, including its dependence on gas pressure, temperature, and average nanoparticles size, is expected from the theory. Only stationary concentrations of adsorbates are considered in this work because the characteristic times of concentration change, which are inversely proportional to the reaction rates in the kinetic system eqs 22−25, are small τr ≤ 10−10 s. At the same time, the minimum sensor response time currently achieved in experiment is much larger, τresp ≈ 1 s. Such a time scale for sensor response is explained by the fact that τresp is mostly determined by the diffusion time of the hydrogen analyte through the interior of the film to the surface of the nanoparticles. The recovery to the operational state of the sensor after removal of H2 analyte from the environment is an even slower process that lasts several minutes. This recovery time scale is determined by the diffusion of the oxygen to the surface of the nanoparticles. Due to the substantially larger diameter of O2, its diffusion coefficient is much smaller than that for H2. In summary, a theory of sensor response to a reducing hydrogen analyte has been developed for nanostructured tin dioxide thin film. It is shown that the sensor response constitutes the increase of electrons in the conduction zone upon the addition of H2 analyte, followed by the reaction with and removal of adsorbed O atoms at the surface of the nanoparticles. The predicted dependence of sensor sensitivity on temperature, H2 pressure, and the average diameter of the nanoparticles is found in agreement with experiment.

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DISCUSSION AND CONCLUSIONS In this section we provide a physical interpretation of the theoretical results. The aforementioned saturation of sensitivity at H2 pressure exceeding 10−2 atm can be explained by the fact that all adsorbed oxygen reacts with hydrogen releasing products to the ambient atmosphere. In this case, the film conductivity is the same as that in vacuum. Further increase of H2 pressure has no effect on the sensitivity. The temperature dependence of Θ at large PH2 displays a sharp maximum, which is due to the exponential temperature dependence of all the processes responsible for changes to the electron concentration in the conduction band, and the rates of almost all surface chemical reactions including adsorption and desorption. The theoretical and experimental positions of the maximum coincide because this requirement is used to determine the unknown parameters of the model; see previous section. In contrast to the high pressure case, two maxima, lowtemperature and high-temperature, appear at pressures below 10−2 atm; see Figure 3. Although the second high-temperature maximum was not observed in the experiment of ref 21 due to the range of temperatures being limited to 750 K, an indication of the existence of both maxima appeared in another experiment; see ref 23. The nonmonotonic temperature dependence of Θ exhibiting two maxima is explained by a complicated dependence of NO(PH2,T) on temperature obtained from the solution of the kinetic system eqs 22−25. As the pressure PH2 is decreased, the low-temperature maximum of Θ(PH2,T) is shifted to the left in agreement with experiment.21 The microstructure of the film, as exemplified by the average diameter of the nanoparticles d̅, influences substantially the sensor sensitivity. As d̅ becomes small, the specific area of the film AV, and, according to eq 26, the volume concentration of O, which effectively captures the electrons, both become large. Therefore, the conductivity of the film in air becomes smaller, and the maximum sensitivity Θmax, in accordance with eq 31, becomes larger. Figure 4 displays the calculated maximum sensitivity Θmax 31 as a function of the effective diameter of the nanoparticles deff, eq 29. As seen in the Figure 4, Θmax increases with the decrease of deff, which is in agreement with experiment.6,8 In addition, the decrease of nanoparticle size

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation (Grant No. CMMI-1030715) and the Russian Foundation for Basic Research (Grant Nos. 13-03-00447 and 13-07-00141).

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REFERENCES

(1) Gopel, W.; Schierbaum, K. Sens. Actuators, B 1995, 26, 1−12. (2) Shimizu, Y.; Egashira, M. MRS Bull. 1999, 24, 18−24. (3) Yamazoe, N. Sens. Actuators, B 2005, 108, 2−14. (4) Korotcenkov, G. Mater. Sci. Eng., B 2007, 139, 1−23. (5) Yamazoe, N.; Shimanoe, K. Sens. Actuators, B 2009, 138, 100− 107. (6) Xu, C.; Tamaki, J.; Miura, N.; Yamazoe, N. Sens. Actuators, B 1991, 3, 147−155. (7) Williams, D. E. Sens. Actuators, B 1999, 57, 1−16. (8) Lu, F.; Liu, Y.; Dong, M.; Wang, X. P. Sens. Actuators, B 2000, 66, 225−227. (9) Rothschild, A.; Komem, Y. J. Appl. Phys. 2004, 95, 6374−6380.

Figure 4. Calculated maximum sensitivity Θmax as a function of the effective diameter of nanoparticles deff. 11567

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The Journal of Physical Chemistry C

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dx.doi.org/10.1021/jp311847j | J. Phys. Chem. C 2013, 117, 11562−11568