Time-dependent oxygen vacancy distribution and gas sensing characteristics of tin oxide gas sensitive thin films

Sensors and Actuators B 150 (2010) 330–338

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Sensors and Actuators B: Chemical journal homepage: www.elsevier.com/locate/snb

Time-dependent oxygen vacancy distribution and gas sensing characteristics of tin oxide gas sensitive thin films Jianqiao Liu, Shuping Gong ∗, Qiuyun Fu ∗, Yi Wang, Lin Quan, Zhaojie Deng, Binbin Chen, Dongxiang Zhou Department of Electronic Science and Technology, Huazhong University of Science and Technology, Luoyu Road 1037, Hongshan District, Wuhan 430074, Hubei, PR China

a r t i c l e

i n f o

Article history: Received 14 March 2010 Received in revised form 29 June 2010 Accepted 29 June 2010 Available online 7 July 2010 Keywords: Tin oxide thin film Oxygen vacancy Gradient distribution model Non-steady state Time-dependent characteristics

a b s t r a c t The model of gradient distributed oxygen vacancies is extensionally investigated. Based on the dynamics of oxygen vacancies under diffusion and exclusion effects in cooling process, the steady state and nonsteady state solutions of the diffusion equation of the vacancies are solved. The transient distribution of oxygen vacancies is revealed during the idealized cooling process. It is concluded that the exclusion effect dominates the density distribution of oxygen vacancies throughout the crystallite. The time-dependent expressions of potential barrier [qV(x,t)], film resistance (R) and response to reducing gases (S) are formulated on the basis of Schottky model and Poisson’s equation. The simulated gas sensing characteristics are in good correlation with experimental results. The constants and variables are estimated according to the correlation. Presumptions and reservations of the present expressions are also discussed. © 2010 Elsevier B.V. All rights reserved.

1. Introduction As a typical semiconductor gas sensor, tin oxide has been extensively researched since its first production in 1960s [1–3]. Due to the high response and good reliability, they have been widely used in the field of oil exploration [4], environment monitoring [5] and automobile industry [6]. In the later decades, SnO2 gas sensors in form of pellets, thick films, thin films were prepared and investigated [7–12]. Though the production of SnO2 gas sensors is well developed, this type of gas sensor is still of continuous interests and new technologies have been invented. SnO2 nanotubes, nanowires and nanobelts were fabricated [13–15] and sensor arrays were designed based on MEMS technique [16–19]. Unlike the prosperity of the fabrication technique, the gas sensing mechanism of the SnO2 gas sensor has not been completely understood. Generally, the sintered SnO2 material is n-type semiconductive and able to adsorb oxygen ions (typically O− ) on the surface [20]. The adsorbed oxygen seizes electrons, leaving a depletion layer. Thus, Schottky potential barriers for electrons form at grain boundaries [21]. Xu et al. [22] discovered the grain size effects on gas sensitive porous SnO2 elements and proposed a model where the transducer function was operated by a mechanism of grain control, neck control or grain-boundary control, depending on the

crystallite size. In 2001, Sakai et al. [23] established the gas diffusion theory based on the presumption of first order reaction and Knudsen diffusion. The theory attempted at quantitatively explaining the film response to reducing gases. Yamazoe and Shimanoe [24] also quantitatively discussed the power law by combining a depletion theory and gives out the theoretical basis of the power law. On the basis of the power law theory, recently, a modified expression of the gas diffusion theory was proposed and the theory was extended to film response to various reducing and oxidizing gases [25,26]. As summarized by Yamazoe and Shimanoe [27], the gas sensing properties of the SnO2 material were divided into three basic factors as receptor function, transducer function and utility factor. Furthermore, a theory was derived on roles of shape and size of crystals in semiconductor gas sensors [28,29]. The receptor function of small semiconductor crystals with clean and electron-traps dispersed surfaces was continuously discussed [30]. It is noted that the oxygen vacancies in SnO2 material, which form during the sintering stage, play an important role in the device characteristics. The oxygen vacancies act as donors, resulting in the n-type property of the nonstoichiometric system. They form and ionize according to Eqs. (R1) and (R2), respectively. k1

X OO  VOX + k−1

∗ Corresponding authors. Tel.: +86 27 87557447; fax: +86 27 87545167. E-mail addresses: [email protected] (S. Gong), [email protected] (Q. Fu). 0925-4005/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.snb.2010.06.065

k2

1 O2 2

VOX  VO•• + 2e k−2

(R1)

(R2)

J. Liu et al. / Sensors and Actuators B 150 (2010) 330–338

Here k1 , k−1 , k2 and k−2 are rate constants of reactions. VO× represents oxygen vacancies trapping two electrons due to charge neutrality effects. VO•• expresses ionized oxygen vacancies which lose their trapped electrons and show apparent two plus valence. Thus, it is of certainty that the formation and dynamics of these vacancies are correlative with the device performance. Blaustein et al. studied the influence of frozen distributions of oxygen vacancies on the conductance of SnO2 pellets [31]. Also, Shimizu et al. [32] and Zhang et al. [33] investigated the dynamics of point defects in a single crystal during annealing processes. In our previous work [34], the distribution of oxygen vacancies was modulated by sintering with different cooling rates and reannealing technique. Moreover, the oxygen vacancies are found to be responsible for the phenomenon that slow cooling rate in sintering or re-annealing benefits gas sensing characteristics of SnO2 thin films. Successively, a model of gradient distributed oxygen vacancies in SnO2 crystallites was proposed to quantitatively formulate the SnO2 film resistance and response to reducing gases, on the basis of the steady state solution of the diffusion equation of oxygen vacancies. In the present work, the model of gradient distributed oxygen vacancies is extensionally investigated. The time-dependent distribution of oxygen vacancies in SnO2 crystallites before the steady state is discussed via the non-steady state solution of the diffusion equation. The potential barrier, film resistance and response to reducing gases are formulated by employing Schottky model and Poisson’s equation. The simulated gas sensing characteristics are validated by correlating them with experimental results. Limitations and reservations of the present work are also discussed. 2. Model of gradient distributed oxygen vacancies 2.1. Diffusion equation of oxygen vacancies in crystallites As the same as that in the previous work [34], the dynamics of the oxygen vacancy is discussed in an idealized crystallite, which is presumed as a sphere with a radius of RC and has no crystallite boundary in itself. A sphere coordinate is established, whose origin locates at the center of the crystallite, as shown in Fig. 1, where r denotes the distance of a point from the origin. The depletion

331

Fig. 2. Schematic drawing of oxygen vacancy diffusion and exclusion in a SnO2 crystallite.

layer under the surface of the crystallite was marked by a dot line. The symbol w indicates the width of the depletion layer, in which all electrons are considered to be seized by the adsorbed oxygen [20]. The density of the oxygen vacancy (donor) at a point could be expressed as NV (r, , ϕ), which could be simplified as NV (r), considering the symmetry of sphere system. Moreover, the dynamics of the oxygen vacancy is assumed to be independent of its forms (VO× or VO•• ) because the influence of electrons on the thermal immigration of a vacancy would be extremely limited. The dynamics of the vacancy in the presumed crystallite during annealing or cooling process could be accounted for by two factors, namely, the diffusion and exclusion of oxygen vacancies, as shown in Fig. 2. The first one originates from the gradient distribution of oxygen vacancies while the second one is derived from the escaping trend of point defects while thermal vibration in cooling process, which may decrease the energy of whole system. It is easy to formulate the diffusion rate at a point as DV ∂2 NV (r)/∂r2 , where DV is the diffusion coefficient. Similarly, the exclusion rate could be calculated as PNV (r), in which P is the jump frequency of a vacancy moving outwards to a nearby position in unit time. Therefore, the diffusion equation of the oxygen vacancies in SnO2 crystallites is established as Eq. (1). ∂NV (r, t) ∂2 NV (r, t) = DV − PNV (r, t) ∂t ∂r 2

(1)

It is noted that the diffusion coefficient DV is exponential to ED if temperature is fixed, as shown in Eq. (2).

 E  D

DV = D0 exp −

(2)

kTE

Here D0 is the pre-exponential constant and k is Boltzmann constant. ED is the activation energy of diffusion, which represents the required energy that an oxygen vacancy diffuses to a nearby position. TE is the temperature that the diffusion carries on. Similarly, the jump frequency, P, could be expressed as Eq. (3),



P = 0 exp

Fig. 1. Sphere coordinates for a SnO2 crystallite involving a depletion layer with width of w.

−

E − E0 kTE



(3)

where 0 is the thermal vibration frequency of oxygen vacancies. Eϕ and E0 represent the activation energy of oxygen vacancy migration and the unit energy decrease of the system that resulted from the one step exclusion of defects, respectively.

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J. Liu et al. / Sensors and Actuators B 150 (2010) 330–338

The boundary conditions are the same as those of NVst (r) in steady state while the initial condition is set as Eq. (9). NV (r, t) = N0 ,

t=0

(9)

Here N0 represents the uniform density of oxygen vacancies throughout the crystallite at the beginning of cooling process. Thus the transient equation, NVtr (r,t), is established as Eq. (10), whose boundary and initial conditions could be determined by transforming Eqs. (6) and (9) into Eqs. (11) and (12). ∂NVtr (r, t) ∂2 NVtr (r, t) = DV − PNVtr (r, t) ∂t ∂r 2

(10)

 ∂N (r, t) Vtr

= 0, r = 0 ∂r NVtr (r, t) = NVS , r = RC

NVtr (r, t) = N0 − NVst (r),

Fig. 3. Gradient distribution profiles of oxygen vacancies in a 25-nm SnO2 crystallite at steady state for various value of m.

t=0

(12)

To solve the transient equation, the method of separation of variables is used as NVtr (r,t) = f(r)g(t). Here f(r) and g(t) are space and time-dependent functions, respectively. Therefore, Eq. (10) is changed into Eq. (13). dg(t) d2 f (r) = DV −P g(t)dt f (r)dr 2

2.2. The steady state solution At the steady state where the cooling rate is sufficiently small, the steady state solution, NVst (r), is time independent. Thus, the diffusion equation (Eq. (1)) is transformed into Eq. (4). DV

(11)

∂2 NVst (r) = PNVst (r) ∂r 2

(4)

(13)

If both sides of Eq. (13) equal to −, the differential equation of time-dependent function g(t) is formulated as Eq. (14) and its general solution could be easily solved as Eq. (15). It is noted that g(t) is convergent when t reaches infinite only in the case that  is positive. dg(t) + g(t) = 0 dt

(14)

Thus, the general solution of Eq. (4) could be expressed as Eq. (5).

g(t) = exp(−t)

(15)

NVst (r) = C1 exp(mr) + C2 exp(−mr)

For another part of Eq. (13), the space dependent f(r) would be solved by its own differential equation Eq. (16).

(5)

Here, m is defined as m = (P/DV )1/2 while C1 and C2 are integral constants. Based on the conventionally used boundary conditions in Eq. (6), C1 and C2 are determined and then the steady state solution of the diffusion equation is expressed as Eq. (7), where NVS is oxygen vacancy density on crystallite surface.

 ∂N (r) Vst

= 0, r = 0 ∂r NVst (r) = NVS , r = RC

NVst (r) =

 m=

(6)

NVS cosh(mr), cosh(mRC ) P = DV



0 exp D0



ED − E + E0 2kTE

d2 f (r) P− = f (r) DV dr 2

(16)

It is reminded that the final expression of general solution of Eq. (16) is relative with the value of P − . When P −  > 0 or P = , the solved f(r) is meaningless. Only in case of P −  < 0, the general solution of f(r) could be expressed as



f (r) = A cos

−P r DV





+ B sin



(17)

where A and B are pending constants. According to the boundary conditions f (0) = 0 and f(RC ) = 0, specific solutions of f(r) and  are formulated as Eqs. (18) and (19).

 (7)

f (r) =

∞

An cos

n=0

Thus, a gradient distribution of oxygen vacancies at steady state in the crystallite is acquired in Fig. 3 according to Eq. (7). It is obvious that the distribution of oxygen vacancies is decided by m if NVS is fixed. The physical sense and the value of m have been discussed and estimated previously [34].

n = P +

 2n + 1 2RC

 2n + 1 2 2RC

 r

(18)

2 DV

(19)

Therefore, non-steady state solution of NVtr (r,t) is acquired as Eq. (20). NVtr (r, t) = f (r)g(t) =

2.3. Non-steady state solution

∞

An cos

 2n + 1 2RC

n=0

The whole solution of NV (r,t) involves two parts, NVst (r) and NVtr (r,t), naming steady state solution and non-steady state solution, respectively, as Eq. (8) NV (r, t) = NVst (r) + NVtr (r, t)

−P r DV

(8)

 r exp(−n t)

(20)

According to initial condition of Eq. (12), NVtr (r, 0) =

∞

n=0

An cos

 2n + 1 2RC

 r

= N0 − NVst (r)

(21)

J. Liu et al. / Sensors and Actuators B 150 (2010) 330–338

Fig. 4. Transient of oxygen vacancy distribution profile in a 25-nm SnO2 crystallite from initial time (a) to steady state (e).

Then the pending constant A is determined as



An =

4(2n + 1)NVS 4N0 − (2n + 1) 4m2 RC2 + (2n + 1)2 2



(−1)n

(22)

Hence, the whole solution of the diffusion equation in Eq. (1) is expressed as follows: NV (r, t) = NVst + NVtr (r, t) =

∞

+

An cos

 2n + 1

n=0

 An =

NVS cosh(mr) cosh(mRC )

2RC

4N0 4(2n + 1)NVS − (2n + 1) 4m2 RC2 + (2n + 1)2 2

n = P +

 2n + 1 2 2RC

2 DV

(23)

(−1)n

Fig. 5. Time-dependent density variation of oxygen vacancies at different points of the SnO2 crystallite with diameter of 25 nm.

demonstrated to own a width of 3 nm [22]. The density variation of oxygen vacancies at the verge of depletion layer are shown in Fig. 5(d) (r/RC = 0.88) and it can be speculated that the vacancy density throughout the depletion layer has an incremental trend. Thus in the depletion layer, there is a time increasing gross oxygen vacancy, which is proved to be responsible for the enhancements of film resistance and response to reducing gases in the idealized cooling process. 3. Time-dependent gas sensing characteristics

 r exp(−n t)

333

(24)

(25)

The solution, Eqs. (23)–(25), tells the transient of oxygen vacancy distribution before the steady state in the idealized cooling process, in which the cooling rate is sufficiently small. According to the solution, NV (r,t) is formulated as a function of position (r), time (t), crystallite size (RC ), end temperature of cooling process (m) and so on. Besides, another important factor that determines the value of NV (r,t) is the constant setting of N0 . As illustrated in Fig. 4, the distributions of oxygen vacancies in a typical crystallite with 25-nm diameter at a series of time are simulated by setting the variable and constant as m = 0.03 and N0 /NVS = 0.8. After the idealized cooling process starts, oxygen vacancies are forced from a uniform density (at t = 0) to the gradient distribution (at steady state) under the effects of diffusion and exclusion. For convenience in calculation, the first 20 non-steady state terms of Eqs. (23)–(26) (n = 0–19) are reserved. The approximation leads to the vibration of curve (a) at t = 0 around N0 /NVS . However, the accuracy of later simulation may not be affected since other terms are small enough to be neglected. The time-dependent trends of oxygen vacancy density at different points of the 25-nm crystallite are shown in Fig. 5, with the same variable and constant settings as used in Fig. 4. The vacancy density near the crystallite surface (r/RC = 0.97) reveals a large increase while the one at center (r/RC = 0) descends. It is concluded that the exclusion effect dominates the density distribution of oxygen vacancies throughout the crystallite, especially at the center, as Fig. 5(a) and (b). Depletion layer of a SnO2 crystallite has been

3.1. Potential barrier height and film resistance It is concluded previously that the film resistance and response to reducing gases are proved to have a positive relationship with cooling or annealing time [34]. In the discussion above, Eq. (23) gives out distribution of oxygen vacancy density [NV (r,t)], which is decisive of the film resistance and response. Therefore, the simulations of film resistance and response are necessary. It is possible to make an attempt at quantitatively explaining the time-dependent gas sensing characteristics. As the same as usual, the simulation starts at employing the Schottky model, which is established in one-dimensional coordinates and shown in Fig. 6. In this model, x denotes depth from the surface and the symbol w also indicates the width of the depletion layer. V(x) represents electric potential at given depth. The potential barrier of electrons is expressed as qV(x), where q is electron elemental charge. The qV(x) at the surface (x = 0) is the potential barrier height at crystallite boundaries, named as qVS . The width of depletion layer is denoted by w. It has been proved that almost all the electrons in depletion layer are seized by the adsorbed oxygen (O− ) on the crystallite surface [20,24]. The electric potential, V(x), is commonly considered to correlate with space charge density, (x), according to Poisson’s equation, as shown in Eq. (26). d2 V (x) (x) =− , ε dx2

0
(26)

The space charge density, (x), could be expressed approximately as (x) = q[Nd + (x) − n(x)] in n-type SnO2 material, where Nd + (x) and n(x) are the densities of ionized donors and electrons at given depth, respectively. Since all electrons in depletion layer are seized by adsorbed oxygen, n(x) could be neglected. In another aspect, the role of donors is acted by oxygen vacancies. If all donors are considered to be completely ionized, thus Nd + (x) = NV (x), where NV (x) is the density of oxygen vacancy at given depth. The already

334

J. Liu et al. / Sensors and Actuators B 150 (2010) 330–338

Fig. 6. Schematic drawing of one-dimensional Schottky model and potential energy of electrons.

expressed NV (r,t) in Eq. (23) could be easily transformed into the one-dimensional coordinates by substituting r for x − RC , as Eq. (27). NVS NV (x, t) = cosh[m(x − RC )] cosh(mRC ) ∞

+

An cos

2n + 1 2RC

n=0



(x − RC ) exp(−n t)

(27)

Thus, the Poisson’s equation of Eq. (26) is formulated approximately by using abrupt change model [35], as Eq. (28) qNV (x, t) ∂2 V (x, t) =− , ε ∂x2

0
(28)

Fig. 7. Simulated potential barrier qV(x,t) under the surface of a 25-nm SnO2 crystallite in idealized cooling process by setting the constants and variables as NVS = 5 × 1025 m−3 , w = 3 nm, ε = 10−10 F/m, m = 0.03 nm−1 and N0 /NVS = 0.8.

Here R0 is flat band resistance and T represents temperature. Fig. 7 exhibits the simulated potential barrier qV(x,t) in depletion layer of a 25-nm crystallite from initial time to steady state in the duration of sufficient slow cooling process. Here the constants and variables in Eq. (29) are set as NVS = 5 × 1025 m−3 [23], w = 3 nm [22], ε = 10−10 F/m [36], m = 0.03 nm−1 and N0 /NVS = 0.8. As shown in Fig. 7, qV(x,t) is zero at central bulk because all ionized oxygen vacancies are neutralized by electrons. Since electrons in depletion layer are entirely seized by adsorbed oxygen (O− ), potential barrier starts to establish at the verge of depletion layer (x = w) due to the accumulation of ionized oxygen vacancies. As shown in Fig. 5, the density of oxygen vacancies in depletion layer increases with time. Hence, qV(x,t) is gradually enhanced from initial time until steady state. The potential barrier height (qVS ) and resistance (R/R0 ) at room temperature are also simulated in Fig. 8 by using the same constant and variable settings as above according to Eqs. (30) and

According to the boundary conditions V(w) = 0 and V (w) = 0, the potential barrier qV(x,t) in depletion layer could be solved as Eq. (29). q2 NVS {cosh[m(x − RC )] − m(x − w) sinh[m(w − RC )] − cosh[m(w − RC )]} qV (x, t) = 2 εm cosh(mRC )

⎧

2n + 1 

2n + 1 ⎫ ⎪ ⎪ 2R cos (x − R ) + (2n + 1)(x − w) sin (w − R ) ⎨ ⎬ C C C

2q2 RC 2RC 2RC exp(−n t) A − n

 ⎪ (2n + 1)2 2 ε ⎪ ⎩ −2RC cos 2n + 1 (w − RC ) ⎭ n=0 ∞

(29)

2RC

Successively, the potential barrier height at crystallite surface, qVS (t), is also obtained as Eq. (30). Furthermore, it is commonly considered that the resistance of SnO2 material, R, is exponential to the potential barrier height, as Eq. (31). qVS (t) = qV (0, t) = ∞

+

2q2 RC 2

n=0

(2n + 1) 2 ε

exp(−n t)

εm2

 An

(31). Both of them increase before steady state while R/R0 lifts more significantly because of the amplification by the exponential term.

q2 NVS {cosh(mRC ) − cosh[m(w − RC )] + mw sinh[m(w − RC )]} cosh(mRC )

(2n + 1)w sin

2n + 1 2RC



(w − RC ) + 2RC cos

2n + 1 2RC



(w − RC )

qV (t)  S

= kT 2 q NVS R0 exp{ {cosh(mRC ) − cosh[m(w − RC )] + mw sinh[m(w − RC )]} εm2 kT cosh(mRC )

R = R0 exp

∞

2q2 RC

n=0

(2n + 1)2 2 εkT

+

exp(−n t)}

(30)



An

(2n + 1)w sin

2n + 1 2RC



(w − RC ) + 2RC cos

2n + 1 2RC



(w − RC )

(31)

J. Liu et al. / Sensors and Actuators B 150 (2010) 330–338

Fig. 8. Simulated dynamic potential barrier height (qVS ) and film resistance (R/R0 ) at room temperature for a SnO2 thin film with crystallite size of 25 nm by setting the constants and variables as NVS = 5 × 1025 m−3 , w = 3 nm, ε = 10−10 F/m, m = 0.03 nm−1 and N0 /NVS = 0.8.

The trend of resistance coincides with the experimental results, which are increasing with cooling or annealing time. 3.2. Response to reducing gases The response of SnO2 material to reducing gases, denoted as S, is commonly defined as the ratio of the resistance in aerial atmosphere (Ra ) to that in a reducing gas (Rg ). When SnO2 crystallites are exposed to reducing gas, part of the electrons seized by adsorbed oxygen is released into the depletion layer. The response of a SnO2 thin film has been formulated to be a function of several parameters, such as film thickness, operating temperature [23], grain size [22] and gas concentration [25]. It could also be influenced by doping, preparation method and procedure. In a microcosmic angle of view, the response of a specific SnO2 thin film at a given temperature is proved to be only decisive by nR , which represents the density of electrons released back to the depletion layer when exposed to reducing gases. Its correlation with oxygen vacancy density has been formulated before as Eq. (32) [34] and the response could be acquired as shown in Eqs. (32) and (33). S=

Ra = exp Rg

nR = 2˛NV =



q2 w2 nR 2εkT

2˛ RC





(32)

335

Fig. 9. Time-dependent response to reducing gases at room temperature of a SnO2 thin film with crystallites of 25 nm under the various reducing atmosphere (˛ = 0.1–0.3), simulated by setting the constants and variables as NVS = 5 × 1025 m−3 , w = 3 nm, ε = 10−10 F/m, m = 0.03 nm−1 and N0 /NVS = 0.8.

ε = 10−10 F/m, m = 0.03 nm−1 and N0 /NVS = 0.8. The idealized cooling process enhances the response and the enhancement is more obvious when it locates in a larger reducing gas concentration, which is represented as a larger value of ˛. 3.3. Film resistance and response under different operating temperature In general, the film resistance and response to reducing gases would be greatly impacted by the operating temperature. As shown in Figs. 10 and 11, the time-variations of film resistance (R/R0 ) and responses (S) under the operating temperature of 25–400 ◦ C are simulated according to Eqs. (31) and (34). Both of the resistances and responses increase with time during cooling process and descend with the operating temperature. It is natural that the film resistance decreases with operating temperature due to its semiconducting properties. However, the film response to reducing gases is commonly found to have an optimum operating temperature and its origin has been formulated by Sakai et al. in the gas diffusion theory [23]. Nevertheless, the present Eq. (34) is derived

RC

NV (x, t)dx 0

(−1)n 4˛ 2˛NVS An exp(−n t) tanh(mRC ) + mRC (2n + 1) ∞

=

(33)

n=0

Here NV is the average density of oxygen vacancy throughout the crystallite. The coefficient ˛ represents how many percents of the seized electrons are released back to depletion layer and locates from 0 (in air) to 1 (in vacuum or other atmosphere without oxygen), being correlative with partial pressure of oxygen and reducing gas. Thus, the final expression of response (S) could be formulated as Eq. (34).



S = exp



q2 w2 εkT

˛NVS tanh(mRC ) + mRC

∞



n

(−1) 2˛ An exp(−n t) (2n + 1)

(34)

n=0

Fig. 9 exhibits the simulated curves of response of a SnO2 thin film with crystallite size of 25 nm at room temperature when exposed to different reducing atmosphere (˛ = 0.1–0.3). The constants and variables are assumed as NVS = 5 × 1025 m−3 , w = 3 nm,

Fig. 10. Time variation of film resistance (R/R0 ) of a SnO2 thin film with crystallites of 25 nm at different operating temperature of 25–400 ◦ C, simulated by setting the constants and variables as NVS = 5 × 1025 m−3 , w = 3 nm, ε = 10−10 F/m, m = 0.03 nm−1 and N0 /NVS = 0.8.

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J. Liu et al. / Sensors and Actuators B 150 (2010) 330–338

Fig. 11. Time variation of film response to reducing gases (S) of a SnO2 thin film with crystallites of 25 nm at different operating temperature of 25–400 ◦ C, simulated by setting the constants and variables as NVS = 5 × 1025 m−3 , w = 3 nm, ε = 10−10 F/m, m = 0.03 nm−1 , ˛ = 0.3, and N0 /NVS = 0.8.

Fig. 12. Correlation between simulated film resistance and actual sample resistance at 100 ◦ C, where the variables and constants are set as NVS = 5 × 1025 m−3 , w = 3 nm, RC = 25 nm, ε = 10−10 F/m, m = 0.01–0.1 nm−1 , N0 /NVS = 0.4–0.6, R0 = 5 × 10−3 .

from a series of identical boundaries interacted by same crystallites, among which the diffusion effects of the reducing gas are ignored. Therefore, the simulation could potentially be improved by combining it with the gas diffusion theory. 4. Validity of expressions As discussed above, the time-dependent film resistance and response to reducing gases are successfully formulated based on the steady and non-steady state solutions of the diffusion equation Eq. (1). Furthermore, the simulated gas sensing characteristics in idealized cooling process are revealed according to Eqs. (30), (31) and (34). It is necessary to check the validity of the expressions by correlating with experimental results. Therefore, a series of un-doped SnO2 thin films were prepared by sol–gel spin-coating technique. The details of the preparation procedure are described elsewhere [34]. The films were sintered at 550 in air and cooled to room temperature. The cooling process is controlled and thus thin films with different cooling rates and time were acquired. The correlations of simulated gas sensing characteristics with experimental data are investigated. The simulated curves and experimental film resistance of thin films with various cooling time at 100 o C are drawn and plotted in Fig. 12, respectively. The variables and constants are set as: NVS = 5 × 1025 m−3 , RC = 25 nm, w = 3 nm, ε = 10−10 F/m, m = 0.01–0.1 nm−1 , N0 /NVS = 0.4–0.6, R0 = 5 × 10−3 . The curves fit the data fairly well and most of the data locate near the curve (a), which represents m = 0.01 nm−1 and N0 /NVS = 0.5. Applying the same values of variables and constants, the simulated curves of film response to reducing gases are correlated with actual response of SnO2 thin films to 13.7 ppm H2 S gas at 100 ◦ C, as shown in Fig. 13. The coefficient ˛ is assumed to be 0.225 for the reducing atmosphere. The correlation seems satisfactory and thus the optimum values of m and N0 /NVS prefer to be 0.01 nm−1 and 0.5, respectively. However, the correlation curves simulated by m = 0.01−1 and N0 /NVS = 0.5 could not cover all of the experimental plots in both Figs. 12 and 13. The deviations may probably be caused by two reasons. For the first, the cooling process in preparation of the actual SnO2 thin films could not satisfy the condition of idealized cooling, which is the basic presumption of the simulated curves. In another aspect, m is assumed to be a constant value in the correlation while actually it is a temperature-dependent variable in the whole pro-

Fig. 13. Correlation between simulated film response and actual sample response to 13.7 ppm H2 S gas at 100 ◦ C, where the variables and constants are set as NVS = 5 × 1025 m−3 , w = 3 nm, RC = 25 nm, ε = 10−10 F/m, m = 0.01–0.03 nm−1 , N0 /NVS = 0.4–0.6, ˛ = 0.225.

cess. Besides the two main reasons, the setting of constants other than m and N0 /NVS may also affect the accuracy of the correlation. 5. Discussion In the discussion above, the transient of density distribution of oxygen vacancy has been revealed based on the model of gradient distributed oxygen vacancy for crystallites of SnO2 thin films. The time-dependent gas sensing characteristics to reducing atmosphere are formulated by employing one-dimensional Schottky model and Poisson’s equation. The simulated film resistance and response correlate with the experimental results fairly well. In the discussion, however, there are several presumptions, which are necessarily indicated and reserved for successive investigations. Firstly, all of the oxygen vacancies are treated as completely ionized in solving Poisson’s equation. However, the complete ionization would never be achieved. The ionization of the oxygen vacancies obeys the Boltzmann distribution and is influenced by the energy level of oxygen vacancies and Fermi energy. The energy level of the oxygen vacancy could be relative with its position in crystallites and would descend when it is near the surface. Amendments on the Poisson’s equation, Eq. (28), may improve the accuracy of

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expression, provided that there is a sufficient support regarding the quantitative understanding on the energy level of the oxygen vacancy in SnO2 crystallites. It is noted that two kinds of oxygen vacancies (VO× and VO•• ) could change their forms according to Eq. (R2). However, for convenience in the dynamics discussion, the forms of oxygen vacancies are not taken into consideration because the diffusion and exclusion effects are mainly dominated by thermodynamic factors. Furthermore, the simulation of potential barrier, qV(x,t), around the verge of depletion layer (x = 3 nm) is not of complete precise according to Eq. (29). The ionized oxygen vacancy is gradiented distributed and could not be wholly neutralized by the electrons, which are free to form a uniform density throughout the crystallite. Thus, a contribution of ionized oxygen vacancies is additionally accumulated on qV(x,t). On the contrary, the contribution may be eliminated by the electric field produced by ionized oxygen vacancies in gradient distribution, which could force the electrons to reduce the additional qV(x,t). Moreover, the abrupt model is employed to estimate the space charge density around the verge of depletion layer. The approximation may directly influence the simulation of qV(x,t) though its effects on other gas sensing characteristics are limited. In addition, both the steady state and non-steady state solutions of the diffusion equation Eqs. (1)–(3) are solved on the presumption of idealized annealing process, in which the cooling rate is sufficiently small. An improved diffusion equation that is applicable for any cooling rate may be formulated by substitute ˇt for TE in Eqs. (1)–(3), where ˇ and t are the cooling rate and the time from the cooling starts, respectively. Several attempts have been made on the solution of the new diffusion equation. It seems that analytic solution is impossible to be solved. Nevertheless, numerical analysis would also be a feasible method for further researches. Moreover, the temperature-dependent variable m is treated as a constant when correlating simulated film resistance and response to reducing gases with experimental results. As discussed in the previous work [34], m changes little with temperature in the concerned range of 25–550 ◦ C if it locates near 0.01. Thus, it is a tolerable approximation of constant m in correlation, though the treatment deviates slightly from precise. The density of oxygen vacancy on surface (NVS ) is another variable that is treated as constant in simulation and correlation. Actually, NVS is a time-dependent variable because it could be lifted by the exclusion of vacancies in cooling process. It is also temperature dependent since the rate constant k1 and k−1 are relative with reacting temperature. In another angle of view, the variation of NVS could be eliminated by the reversible reaction. Further investigations on oxygen vacancies at the crystallite surface in SnO2 thin films are still needed for a better explanation of the gas sensing mechanism of this series of gas sensors. Finally, the simulation of the film response to reducing gases could be hardly suitable for various operating temperatures because the gas diffusion effect is not taken into consideration. The present work focuses on the dynamics of oxygen vacancies in a crystallite during the cooling process. The time-dependent potential barrier height, film resistance and response to reducing gases are formulated based on the presumption of identical idealized crystallites and boundaries. However, for a film consisting of a large quantity of crystallites, the reducing gas is proved to form a gradient profile of concentration in the porous film [23]. Thus, the simulation results of the film response to reducing gases may alter, especially the response at various operating temperatures. The gas diffusion theory provides ways to formulate the response as a function of operating temperature [23]. It is expected to combine the present expressions with the gas diffusion theory in further studies. Moreover, the expression of the film response to reducing gases is only validated by the experimental response to the H2 S gas. Its validity

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on other reducing gases such as CO and H2 is still needed for further checking. 6. Conclusion In the present work, the model of gradient distributed oxygen vacancies is extensionally investigated. On the basis of dynamics of oxygen vacancies in the model, the steady state and non-steady state solutions of the diffusion equation are solved employing the method of separation of variables. The time-dependent distribution of oxygen vacancies is revealed in the idealized cooling process. The transient of vacancies shows they are forced from a uniform density to the gradient distribution under the effects of diffusion and exclusion after the idealized cooling process starts. The vacancy density near the crystallite surface reveals a large increase while the one at center descends. The conclusion is drawn that the exclusion effect dominates the vacancy density distribution throughout the crystallite, especially at the center. The dynamic expressions of potential barrier [qV(x,t)], film resistance (R) and response to reducing gases (S) are formulated based on one-dimensional Schottky model and Poisson’s equation. The simulated curves exhibit an increasing trend of gas sensing characteristics in the sufficiently slow cooling process. The enhancement could be ascribed to the lift of oxygen vacancy density in the depletion layer. Furthermore, the validity of obtained expressions is checked by the correlation of the simulated film resistance and response with experimental results. The simulated curves fit the actual data fairly well and the values of m and N0 /NVS prefer to be 0.01 nm−1 and 0.5. Finally, several presumptions and limitations of the present expressions and correlations are discussed and reserved. Numerical analysis is potentially feasible for the improved diffusion equation, which is applicable for any cooling rate in practical operation. References [1] T. Seiyama, A. Kato, K. Fujiishi, M. Nagatani, A new detector for gaseous components using semiconductive thin films, Anal. Chem. 34 (1962) 1502–1503. [2] T. Seiyama, S. Kagawa, Study on a detector for gaseous components using semiconductive thin films, Anal. Chem. 38 (1966) 1069–1073. [3] N. Taguchi, Published patent application in Japan, S37-47677, October 1962. [4] S. Wang, Y. Zhao, J. Huang, Y. Wang, H. Ren, S. Wu, S. Zhang, W. Huang, Lowtemperature CO gas sensors based on Au–SnO2 thick film, Appl. Surface Sci. 253 (2007) 3057–3061. [5] R. Rella, P. Siciliano, S. Capone, M. Epifani, L. Vasanelli, A. Licciulli, Air quality monitoring by means of sol–gel integrated tin oxide thin film, Sens. Actuators B: Chem. 58 (1999) 283–288. [6] E. Comini, G. Faglia, G. Sberveglieri, Z. Pan, Z.L. Wang, Stable and highly sensitive gas sensors based on semiconducting oxide nanobelts, Appl. Phys. Lett. 81 (2002) 1869–1871. [7] I. Kocemba, S. Szafran, J.M. Rynkowski, T. Paryjczak, Sensors based on SnO2 as a detector for CO oxidation in air, React. Kinet. Catal. Lett. 72 (2001) 107–114. [8] D. Koziej, K. Thomas, N. Barsan, F. Thibault-Starzyk, U. Weimar, Influence of annealing temperature on the CO sensing mechanism for tin dioxide based sensors—Operando studies, Catal. Today 126 (2007) 211–218. [9] L.A. Patil, D.R. Patil, Heterocontact type CuO-modified SnO2 sensor for the detection of a ppm level H2 S gas at room temperature, Sens. Actuators B: Chem. 120 (2006) 316–323. [10] G. Sakai, N.S. Baik, N. Miura, N. Yamazoe, Gas sensing properties of tin oxide thin films fabricated from hydrothermally treated nanoparticles dependence of CO and H2 response on film thickness, Sens. Actuators B: Chem. 77 (2001) 116–121. [11] G. Korotcenkov, V. Brinzari, Y. Boris, M. Ivanov, J. Schwank, J. Morante, Influence of surface Pd doping on gas sensing characteristics of SnO2 thin films deposited by spray pyrolysis, Thin Solid Films 436 (2003) 119–126. [12] Y. Wu, M. Tong, X. He, Y. Zhang, G. Dai, Thin film sensors of SnO2 -CuO-SnO2 sandwich structure to H2 S, Sens. Actuators B: Chem. 79 (2001) 187–191. [13] G.X. Wang, J.S. Park, M.S. Park, X.L. Gou, Synthesis and high gas sensitivity of tin oxide nanotubes, Sens. Actuators B: Chem. 131 (2008) 313–317. [14] J.D. Prades, R. Jimenez-Diaz, F. Hernandez-Ramirez, S. Barth, A. Cirera, A. Romano-Rodriguez, S. Mathur, J.R. Morante, Equivalence between thermal and room temperature UV light-modulated responses of gas sensors based on individual SnO2 nanowires, Sens. Actuators B: Chem. 140 (2009) 337–341. [15] S. Lettieri, A. Bismuto, P. Maddalena, C. Baratto, E. Comini, G. Faglia, G. Sberveglieri, L. Zanotti, Gas sensitive light emission properties of tin oxide and zinc oxide nanobelts, J. Non-Cryst. Solids 352 (2006) 1457–1460.

338

J. Liu et al. / Sensors and Actuators B 150 (2010) 330–338

[16] D.C. Meier, J.K. Evju, Z. Boger, B. Raman, K.D. Benkstein, C.J. Martinez, C.B. Montgomery, S. Semancik, The potential for and challenges of detecting chemical hazards with temperature-programmed microsensors, Sens. Actuators B: Chem. 121 (2007) 282–294. [17] V.V. Sysoev, J. Goschnick, T. Schneider, E. Strelcov, A. Kolmakov, A gradient microarray electronic nose based on percolating SnO2 nanowire sensing elements, Nano Lett. 7 (2007) 3182–3188. [18] K.E. Kramer, S.L. Rose-Pehrsson, M.H. Hammonda, D. Tillett, H.H. Streckert, Detection and classification of gaseous sulfur compounds by solid electrolyte cyclic voltammetry of cermet sensor array, Anal. Chim. Acta 584 (2007) 78–88. [19] B. Guo, A. Bermak, P.C.H. Chan, G.-Z. Yan, A monolithic integrated 4 × 4 tin oxide gas sensor array with on-chip multiplexing and differential readout circuits, Solid State Electron. 51 (2007) 69–76. [20] N. Yamazoe, J. Fuchigami, M. Kishikawa, T. Seiyama, Interactions of tin oxide surface with O2 , H2 O and H2 , Surface Sci. 86 (1979) 335–344. [21] S.R. Morrison, Selectivity in semiconductor gas sensors, Sens. Actuators 12 (1987) 425–440. [22] C. Xu, J. Tamaki, N. Miura, N. Yamazoe, Grain size effects on gas sensitivity of porous SnO2 -based elements, Sens. Actuators B: Chem. 3 (1991) 147–155. [23] G. Sakai, N. Matsunaga, K. Shimanoe, N. Yamazoe, Theory of gas-diffusion controlled sensitivity for thin film semiconductor gas sensor, Sens. Actuators B: Chem. 80 (2001) 125–131. [24] N. Yamazoe, K. Shimanoe, Theory of power laws for semiconductor gas sensors, Sens. Actuators B: Chem. 128 (2008) 566–573. [25] J. Liu, S. Gong, J. Xia, L. Quan, H. Liu, D. Zhou, The sensor response of tin oxide thin films to different gas concentration and the modification of the gas diffusion theory, Sens. Actuators B: Chem. 138 (2009) 289–295. [26] S. Gong, J. Liu, J. Xia, L. Quan, H. Liu, D. Zhou, Gas sensing characteristics of SnO2 thin films and analyses of sensor response by the gas diffusion theory, Mater. Sci. Eng. B 164 (2009) 85–90. [27] N. Yamazoe, K. Shimanoe, New perspectives of gas sensor technology, Sens. Actuators B: Chem. 138 (2009) 100–107. [28] N. Yamazoe, K. Shimanoe, Roles of shape and size of component crystals in semiconductor gas sensors I. Response to oxygen, J. Electrochem. Soc. 155 (2008) 85–92. [29] N. Yamazoe, K. Shimanoe, Roles of shape and size of component crystals in semiconductor gas sensors II. Response to NO2 and H2 , J. Electrochem. Soc. 155 (2008) 93–98. [30] N. Yamazoe, K. Shimanoe, Receptor function of small semiconductor crystals with clean and electron-traps dispersed surfaces, Thin Solid Films 517 (2009) 6148–6155. [31] G. Blaustein, M.S. Castro, C.M. Aldao, Influence of frozen distributions of oxygen vacancies on tin oxide conductance, Sens. Actuators B: Chem. 55 (1999) 33–37. [32] Y. Shimizu, N. Kobayashi, A. Uedono, Y. Okada, Improvement of crystal quality of GaInNAs films grown by atomic hydrogen-assisted RF-MBE, J. Cryst.Growth 278 (2005) 553–557. [33] M. Zhang, C. Lin, H. Weng, R. Scholz, U. Gösele, Defect distribution and evolution in He+ implanted Si studied by variable-energy positron beam, Thin Solid Films 333 (1998) 245–250. [34] J. Liu, S. Gong, L. Quan, Z. Deng, H. Liu, D. Zhou, Influences of cooling rate on gas sensitive tin oxide thin films and a model of gradient distributed

oxygen vacancies in SnO2 crystallites, Sens. Actuators B: Chem. 145 (2010) 657–666. [35] S.M. Sze, Semiconductor Devices—Physics and Technology, John Wiley & Sons, Inc., New York, 1985. [36] C. Malagù, V. Guidi, M. Stefancich, M.C. Carotta, G. Martinelli, Model for Schottky barrier and surface states in nanostructured n-type semiconductors, J. Appl. Phys. 91 (2002) 808–814.

Biographies Jianqiao Liu is pursuing his Ph.D. degree in Huazhong University of Science and Technology (HUST) after acquiring his bachelor’s degree in 2007. His research interests include preparation of nanocrystalline SnO2 thin film sensors and investigation of sensing mechanism. Prof. Shuping Gong received her bachelor’s degree in electronic materials and components from Huazhong Institute of Technology in 1970. After graduation, she became a teacher in the Department of Solid-State Electronics. For many years, her research programs have concentrated on functional ceramics, sensor technology and its applications. At present, Prof. Gong assumes the chief engineer in Engineering and Researching Center of Functional Ceramics of MOE. Qiuyun Fu received the Ph.D. degree in electronic engineering at Dresden University of Technology, Germany, in 2005. She is currently working at Huazhong University of Science and Technology as associate professor. Her current research interests include semiconductor sensors, wireless RF SAW devices and RFID system. Yi Wang acquired his B.E. degree in 2007 and now is engaging for his M.E. degree in microelectronics and solid-electronics. Lin Quan acquired her bachelor’s degree in HUST in 2006 and is pursuing her Ph.D. degree. Her research interests concentrate on the preparation of semiconducting sensing materials. Zhaojie Deng acquired her B.E. degree in 2008 and now is engaging for her M.E. degree in microelectronics and solid-electronics. Binbin Chen acquired his B.E. degree in 2009 and now is engaging for his M.E. degree in microelectronics and solid-electronics. Prof. Zhou Dongxiang is a seasoned expert in the studies of solid-state electronics and micro-electronics. He acquired his master degree in Huazhong Institute of Technology and used to engage in researching information functional ceramic in Shonan Institute of Technology in Japan. He has continuously conducted systematic research into theories and application technologies of semiconductor ceramics physics, functional materials design and virtual instruments, etc. His publications include “PTC Materials & Applications”, “Semiconductor Ceramics and Applications”, “Electronic Materials and Components Testing Technology”.