J. CHEM. SOC. FARADAY TRANS., 1993, 89(3), 559-572 559

Percolation Model of Adsorption-induced Response of the Electrical Characteristics of Polycrystall ine Semiconductor Adsorbents

Valery Ya. Sukharevt L . Ya. Karpov Institute of Physical Chemistry, Obukha St., 10, Moscow 103064, Russia

A percolation model of the adsorption-induced response of the electrical characteristics of semiconductor gas- sensitive adsorbents has been developed. It has been shown that in the case of barrier-disordered poly- crystalline semiconductor adsorbents both the steady-state and transient responses of the electrical conductivity due to adsorption can be explained by alterations in the heights of intergrain barriers as well as by a change in the degree of barrier disorder. On the basis of the approach developed to explain adsorption-related modifi- cation of the height distribution function of the intergrain barriers, we have investigated the adsorption response of oxide composite adsorbents. A qualitative explanation of the anomalously high adsorption sensitivity of s u c h composites characterized by definite composition has been proposed. Our theoretical predictions have been compared with the available experimental data.

Examination of the available experimental data and theoreti- cal results obtained to indicates that the adsorption response of the electrical characteristics of disordered semi- conductor adsorbents is determined by the simultaneous manifestation of two factors : (i) a local adsorption-induced alteration of the electrical characteristics of a single structural element, for instance, an intergrain contact in the case of a polycrystalline semiconductor,8 and (ii) a cooperative adsorption-induced alteration of the structure of a current- conducting network, i.e. an adsorption-related change in the inhomogeneity distribution in the adsorbent. Which of these effects is dominant in the response formation, depends, as will be shown, on the whole set of circumstances. The existence of the above-mentioned factors can be demonstrated by the fol- lowing example of adsorbents based on polycrystalline semi- conductors. Such adsorbents based on various metal oxide semiconductors, widely used as sensing elements for chemical gas sensors, are polycrystalline materials and usually demon- strate many anomalous properties which distinguish them from monocrystalline ones. This is especially true for the sensitivity of their electrical characteristics to various external influences and to the adsorption of various gases in particu- lar. The presence of high energy barriers between the grains which must be surmounted to allow the transport of current carriers, explains the high sensitivity of these characteristics to various external effects, which influence the permeability of these barriers. Similar effects are produced by adsorption which often leads to changes in the height of these intergrain barriers owing to changes in the charge state of the grain surfaces. However, the presence of a wide range of barrier heights in real polycrystalline oxides plays a role as impor- tant as the existence of these barriers. This type of barrier disorder should affect the adsorption processes involving species chemisorbed in a charged form and the influence of these processes on the electrical characteristics of poly- crystalline semiconductor adsorbents. This disorder might be responsible for many features of the detection response of the sensors based on these materials.

In this paper we will attempt to develop a percolation model of the transient and steady-state responses of a semi- conductor adsorbent based on polycrystalline oxides or their composites.

First, the changes in the form of the distribution of the

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intergrain barrier heights taking place during the adsorption of both acceptor and donor particles are described. On the basis of the calculated kinetic behaviour of the barrier heights distribution function, f(5, t) , we have obtained expressions describing adsorption-induced changes in the ohmic conduc- tion and in the slope of the current-voltage (Z-V) character- istics of the polycrystalline semiconductor adsorbents. The changes in the charge state of the grain surface taking place during adsorption have been studied theoretically allowing for a possible adsorption-induced change in the charge trapped in both intrinsic and extrinsic native surface states ( S S ) , which are attributed to the surface lattice atoms and defect^.'.'^

Secondly, the approach developed is extended to cover the adsorption response of adsorbents based on various compos- ites such as mixtures of different oxides. Principles for the selection of the composite compositions with the highest sensitivity are suggested. Theoretical predictions are tested against experimental data on various metal oxide adsorbents.

Model .of Adsorption-related Change in Polycrystalline Semiconductor Conductivity

Barrierdisordered Polycr ystalline Semiconductor

Various suggestions about the necessity of taking into con- sideration the structure or energy disorder of real poly- crystalline semiconductor adsorbents for the correct modelling of their adsorption response have been proposed in many p a p e r l ~ ~ - ~ . ~ However, a comprehensive theory taking into account the wide range of barrier heights existing in real polycrystalline semiconductors' ' has not been devel- oped. This barrier disorder, caused by random crystallo- graphic orientation of the faces at the sites of contact of the grains and by the variations in their sizes and geometry, as well as by the random character of the distribution of surface and bulk impurities and defects, leads to the formation of a percolation structure in the material as it is divided into regions permitted or prohibited to the current carriers.' ' ,12

In ohmic regions, the effective electrical conductivity of a polycrystal is controlled by the height of the critical barriers, t,, determining the percolation level of the random energy relief at the bottom of the conductivity band, see Fig. 1. Current carriers pass only through barriers included in the critical subnet [infinite cluster (IC)], which contains only the barriers with heights < < 5 , . The mean size of this subnet cell

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560 J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89

Ecs

Fig. 1 Energy band diagram with large-scale fluctuations of energy relief. ECB,S and E,,,, are the conduction and valence band edges in the bulk and on a surface, E, is the Fermi-level, 5, percolation level (dotted line), E, interface levels

is determined by the value of the correlation radius, L, which on the other hand determines the mean distance between the critical barriers, which lie in the range from tC to <, + 1,12 Fig. 2. The value of L must obey the requirements L, < 14 L, where 1 is the average grain size and L, is the Debye screening length. One of the most important characteristics of such a material is the height distribution function of the intergrain barriers f(5), which controls both the percolation level 5, , determined by solving the equation

and the correlation radius L x If "(t), and hence, the activa- tion energy and exponential prefactor of the percolation elec- trical conductivity of the medium' 2-14

(1)

Here 0, is a parameter depending on the bulk conductivity of the grain and some characteristics of the contacts between the grains, v is the critical exponent, p c is the percolation threshold, the value depending in our case on the packing density of grains, and f(5,) describes the concentration of critical barriers. Note that throughout this paper 5 is expressed in units of kT, where k is the Boltzmann constant and T is the absolute temperature.

Another important parameter of barrier-disordered poly- crystalline semiconductors determined by the degree of barrier disorder is the slope of the I-V characteristics. Since the change in the electrical conductivity of the barrier- disordered system in strong electric fields is due not only to the lowering of the barriers by the applied field E, but also to the rearrangment of the IC responsible for current trans- port,I2 we can write the following expression for Z-V

0 = 0%. L - exp( - 5,) = 0%. f'(t,)exp( - 5,)

Fig. 2 Graphical illustration of the definition of the infinite cluster as part of random resistor network. The barriers belonging to the IC are represented by the shaded resistors [ R , = R , exp(ti)] in the circu- lar fragment

i.e. the slope of the Z-V curves in ln(Z/Io] - El'('+") coordi- nates is given by the expression

(3)

where C is a constant factor, and q is the electronic charge, 1/(1 + v) = 0.5 in the three-dimensional case, which is valid if the adsorbent thickness, h, obeys the requirement h b L.

Adsorption-induced Changes in a and p: Kinetic Behaviour

It is well known that the formation of various types of surface and bulk states of sorption is possible during interactions between the gas phase and solid surface^.^.^ The presence of various forms of adsorbate-adsorbent interaction, such as physisorption, chemisorption, and formation of surface and bulk defects, shows itself in the character of the adsorption- induced response of the adsorbent electrical properties. l o

Thus, chemisorption of gaseous species may be accompanied by charge transfer in the adsorbate-adsorbent system, condi- tioned by the localization of band carriers at the surface levels during adsorption. This, in turn, is responsible for changes in the electrical characteristics of the subsurface layer. In the case of gas-solid interactions, which lead to surface and bulk defect formation, it is also possible to alter the adsorbent electrical characteristics, by disturbing the equilibrium between the charged and uncharged forms of these defects.

It is assumed that the surface charging contribution is dominant in the response formation in the case of oxide adsorbents characterized by almost stoichiometric composi- tions of their subsurface layers. In contrast, in the case of partially reduced oxides, the main contribution to the adsorption-induced change in their electrical characteristics is conditioned by direct interactions of the gaseous species with surface defects such as oxygen vacancies and interstitial metal atoms, which do not lead to the marked surface band bending effects."

The suggested model is related to the case of the metal oxide sintered in oxidizing media and characterized by almost stoichiometric compositions. In this case, the changes in the electrical characteristics of the oxide adsorbents are caused by the fact that the adsorbed species itself acts as a surface donor or acceptor of free electrons. Thus, in the adsorption case for the donor species (R,), taking place on the n-type semiconductor, the adsorbed species injects elec- trons into the conduction band and in this way increases the concentration of free conduction-band electrons

R, + Z( ) % R, % R: + e - (1)

Here, R, and R,f are the neutral and charged forms of chemi- sorbed species, Z( ) is an adsorption site and e is the conduction-band electron. In the case of the acceptor species (A,), the conduction-band electrons are captured by the surface level of adsorption nature, that leads to their concen- tration decrease

A, + Z*( )*A,

A, + e- e A,- (11)

The charge state of the adsorbent surface in both cases is changed, which influences the electrical characteristics of the

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Fig. 3 to, in the case of acceptor-particle adsorption, t , < t , < t ,

Barrier height, 5 , at time t as a function of its initial value,

subsurface layer. The surface band bending value, as well as the width of the charge-space area, is changed. In that case, when the grain contacts are mainly of barrier type, which may be heavily overdepleted necks or places of intergrain joints, the adsorption-induced change in the value of the surface charge causes changes in the heights of the intergrain barriers and therefore, is responsible for the exponential con- tribution to the adsorbent conductance changes.

Note that the suggested model is also applicable to a case where the conductance response caused by the adsorption of reducing gases is conditioned by their reaction with oxygen ionosorbate, 0;-

(IW R, + 0, -, B, + e-

and the product, B,, is a strong enough reducing agent to release the electron to the conduction band.

The change in the state of charge of the surface taking place during adsorption, being caused by the conversion of some of the chemisorbed species into a charged form, pro- duces a change in <. The presence of the correlation between the rate of adsorption change in the barrier height, d</dt, and its initial preadsorption height, to, produces a change in the form off(t). Obviously, the adsorption-induced change in the form and width of the distribution of 5 affects the height and the concentration of the critical barriers, which are observed as changes in both the electrical conductivity and the slope of the I-I/ curves of the adsorbent. Thus, in order to determine the laws of the adsorption-induced changes in a(t) and j ( t ) , we require information on the behaviour off(<, t ) which can be obtained from the dependence of the barrier height < at time t on its initial height to. This may be obtained from the simultaneous solution of (i) Poisson's equation

which relates the value of <(t) with the density of surface charge at time t trapped in both adsorption (n,) and native (n,) surface states (SS) {Q(t) = 4[n,(t) + ns(t)J), and (ii) the equation for the charge-transfer kinetics between the adsorp- tion SS levels and the conduction band, which in the case of acceptor-particle adsorption on the adsorbent of n-type takes the form

and in the donor-particle adsorption case

dn, - = K,:1",(t) - n,lexp( - 0) - n, exP[ - <(t)I) (6) dt

Here N,( t ) is the overall time-dependent concentration of che- misorbed species at time t , n,(t) is the concentration of their charged forms, a and 8 are energy parameters characterizing the depth of the energy levels corresponding to adsorbed species about the Fermi level: a = (Eea - x - 6)/kT and 8 =

(Ei - x - 6)/kT, where E,, is the electron affinity and Ei is the ionisation potential of adsorbed acceptor or donor species, respectively. x is the electron affinity of the crystallite surface, 6 is the bulk doping parameter, K,, is a constant, ND

is the uniform density of shallow donors, ni is the interband equilibrium coeficient, E is the bulk relative permittivity of the adsorbent material and c0 is the permittivity of a vacuum.

In the theoretical model considered the adsorption-induced changes in the semiconductor electrophysical characteristics are determined not only by the changes in a single barrier height but also by the cooperative behaviour of all sets of barriers belonging to the system, or in other words, by adsorption-induced modification of the height distribution function of the barriers. This causes the rearrangement of the IC which is responsible for the current transport. The pro- posed approach explains the power-law relation between the adsorbent conductivity and partial pressure of a surrounding gas and the power-law time dependences of the kinetics of the conductivity and the slope of the I-I/ plot during adsorption. Such behaviour is based on the proportional relationship between the polycrystalline semiconductor conductivity and the part of the barriers included in the IC and participating in the current transport.

As an illustration, let us consider a model of an absorbate- adsorbent system. We consider the model adsorbent as a set of grains characterized by various densities of native SS, DJE), which fill the band gap, E , . Wide-gap n-type impurity semiconductors (SnO, , ZnO, TiO,) are henceforth con- sidered throughout. The main intrinsic donors in such materials may be either oxygen vacancies or interstitial metal ions. In that case, the coefficient K,, from eqn. ( 5 ) and (6) can be found from the diode approximation, which is applicable to the wide-gap oxides with a high shallow donor concentra- tion. So, K,, = oC, n,, where u is the average thermal velocity of the current carriers, C, is the capture cross-section of the carriers on the appropriate surface level, and n, is the bulk concentration of carriers taking part in the conduction current, assumed to be equal to the concentration of shallow donors N , .

Solution of Poisson's equation with the assumption that the change in n,(t) does not cause any considerable changes in the native SS filling [i.e. n,(t) x constant] leads to an expres- sion which relates the barrier height, (, at a moment, t , with the density of the surface charge trapped at the adsorption ss, n,(t)

[ S + exp( - <) - 11 1'2 * = - - " 0 + exp(-<,) - ND LD

(7)

The plus sign denotes the adsorption of donors, the minus sign the adsorption of acceptors. Since > I, and assuming that in the adsorption of donors - Nt > 1, we obtain the following expression for t ( t )

where now the plus sign corresponds to acceptors and the minus to donors. Here and in what follows, the parameters N , and n, are normalized with respect to the product NDLD. Note, as will be shown later, that in the case of a uniform distribution of the native SS energy levels in the energy band gap, the initial barrier height, to, is related to the density of

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562 J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89

surface charge trapped at the native SS by the expression

(9)

where N , = qD,[(E, - 6 ) / 2 ~ ~ ~ ND]1/2. Integration of (5) and (6) after substitution of eqn. (8) gives

a relation between 5, t and to, see Fig. 3 and 4. Thus, for a system far from equilibrium (which is defined either as the complete filling of the SS or as the alignment of the SS energy levels with the Fermi level of the adsorbent), in the case of acceptor adsorption, we find

1 + 2Kn, <A'* exp(- t o ) g,,,) di] (10)

and in the case of donor-particle adsorption

t(t) x - t i ' p , ( z )exp( - y) dr]' (1 1)

where t d = K ; ' exp{ 8). These relations make it possible to find expressions for the

distribution density of barrier heights at time t under adsorp- tion. In the case of adsorption of acceptors,

(12) L(5, t ) =fo(t)C1 - X , P 2 exp(-t5)1-' where X, = 2Kn,N,. This expression is valid for a randomly uniform or fairly smooth initial distribution, fo(to). Note that eqn. (12) has been obtained with the assumption that the transfer of charge to SS formed by adsorption of acceptors is much slower than the adsorption itself, and therefore, the charge transfer is the rate-limiting step in the charging of the surface (see for example, ref. 15), i.e. N,(t) = N , x constant.

From eqn. (12) it follows that at time t the initially lowest barriers have been levelled and begin to increase with equal rates, almost independent of their initial heights, to, i.e. they form a group of levelled barriers, see Fig. 5(a). Thus, eqn. (10) is satisfactorily approximated by two asymptotes

In the adsorption of donors, the distribution density of 5 at time t takes the form

strongly depending on the form of the N,(t) dependence (Fig. 6).

50, max 5 0

50, min

Fig. 4 Dependence of the barrier height 5 at time t on its initial value, to, in the donor adsorption case, t , < t , < t , < t,

5

Fig. 5 Shape of the distribution density S,(& t ) at different times during the adsorption of acceptors

This law describes the changes inf(t , t ) up to the time of exhaustion of the empty SS into which electrons can be trapped in the case of acceptor adsorption: t t , = X- '(0 ' I 2

exp[(Jt, + NJ2], or the time of almost complete emptying of donor SS: t x t , , where t , is determined by solving the equation

This depletion regime is possible only for low concentrations of adsorbed species, where alignment of any t with a or 8 is not possible, or in other words, where the energy levels corre- sponding to adsorbed species cannot be aligned with the Fermi level. In the case of adsorption of acceptors, such a condition for small N , is N , < Ja - Jto, min, and in the case of donors it is N , c ,/to, max - JO.

If the opposite inequalities apply the equilibrium state resulting from the complete filling of N , is impossible for any to and the transformation laws, eqn. (12) and eqn. (13), apply up to the times when < begins to be levelled with a in the acceptor case or 8 in the donor one.

Knowing the dependences eqn. (12) and eqn. (13) allows us to determine the kinetics of the changes in t ,(t) andf(t,, t ) resulting from the adsorption of both the acceptor and donor species (in the case of high concentrations) and thus to estab- lish, starting from eqn. (1) and (2), the laws which govern the time dependences of o(t) and &t). For example, in the adsorp- tion of acceptors, in the case N , > Ja - ,/tc0, where tco is the preadsorption height of the critical barriers, we find

- m/( l + v) P(t) = Po( 1 + ;) (16)

Here to, = X, 1tc-t/2 exp(<,,), m x 0.46 for three dimensions ( h > L) and m z 0.15 for two dimensions ( h < L).

h l I

1 \ 1 1 1 Smax(t 1 50, rnax 50, min 8

5 Fig. 6 adsorption of donors in the case of to,

Form of fd(<, t ) at different times ( t 3 > t , > t l ) during the < 0 -= to, max

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For the adsorption of donor species at high concentrations, N , > Jtc0 - JO, the dependences a(t) and P(t) are described by the expressions

c1 - 522n , ( t )1 - vexP~2t~~zn, ( t ) l ; t < to (17) -- 00 - {{[do + nt(t)lz - <co)vexp(<co - 0); t > t o (18)

Here

t o is the time needed for levelling 5, with 8 and is determined by solving the equation J<,, - nt(te) M JO. Possible time dependences of o(t) and P(t) are shown in Fig. 7.

Note that the whole of the above discussion assumes that the initial expressions eqn. (1)-(3) are applied to disordered systems with an exponentially wide spread of local values of the electrical conductivity. In the case considered here, the wide pre-adsorption scatter of ( values is greatly narrowed (see Fig. 5 and 6) by the existing dependences of { ( t ) and to, and at some ratios of the parameters (ao, E,, , Ei , N,) of the adsorbate-adsorbent system this scatter of < can disappear.

A simple analysis for any specific case characterized by given relative values of the parameters of the system yields the conditions and the time ranges for which expressions derived from eqn. (1)-(3) are valid. Thus, in particular, the dependences eqn. (18) and (20) apply as long as the inequality nt(t) < [(tc0 - + 0]1/2 - 01/* holds, which defines the condition for the absence of ICs of levelled barriers. In the time intervals for which the obtained relationships are not applicable the electrical conductivity of the material can be described using the effective medium theory. l 6

At low concentrations of adsorbed species the situation differs markedly from that considered here. Thus, in the acceptor case with Nr < a1/2 - <:!& at long enough times we find a change in the form of the height distribution func- tion f(5, t) [see Fig. 5(b)]. This change in the adsorption kinetics of f(<, t) at times t > tp,min = x-’<;,z?,, exp[(N, + 5:!”,n)2] is due to the almost complete filling of adsorption SS, N , , which originally takes place on the surfaces of the grains with the lowest values of to. So, the rate of growth of the heights of such barriers begins to differ from the rate of height growth of the barriers contained in the group of lev- elled barriers, which is responsible for elimination of such

r- I

1

I t-0.5

0 100 200 300 400 500 600 t f S

Fig. 7 Changes in conductivity (solid lines) and slope of the 1-V curve (dashed line) with time in the donor adsorption case as calcu- lated for Elovich-type adsorption kinetics, N(t ) = N o log[(t + to)/to], and for the following values of parameters: No,”,,!,, = 2, t , = 50 s, t o = 300 S, 0 = 9. (1) rco = 10, (2) tc0 = 16, (3) eco = 25, (4) tc0 = 36

barriers from this ‘group’. Such behaviour off(& t ) can be explained on the basis of a change in the form of the Q t , to) dependence taking place in the depletion regime. In this case, eqn. (5) takes the form

The solution of eqn. (21) is

a t ) = {& + NIC1 - exp(-t/t,)Il2

where t , = 2N,,/(<,)t, and t ,(c0) = x - ~ < ; ’ / ~ exp“

(22)

+ {;/’)’]. At t > t , we have almost complete filling of accep- tor S S in the grain surfaces with original barrier heights less than <# = [ln’/’(K,, t ) - N J 2 , where <# represents < with any index. These barriers do not grow, their heights are equal to < = (J<, + N,)’, and the distribution density of barrier height takes the form

Therefore, in the case of low concentrations of acceptor adparticles ( N , < ,/a - J<,, all barriers can be divided into groups which differ in the form of thef(<, t ) dependence: (a) barriers with + N,)’ < < < ln(K,,t) do not grow and have completely filled acceptor SS; (b) bar- riers wth ln(K,,t) d < < ln[xt,/(ln x t ) ] grow slowly accord- ing to eqn. (22); (c) barriers with ln[xt,/(ln xt)] < < < LY grow according to eqn. (10); (d) barriers with < 2 a are generally unchanged by the adsorption of acceptors with such electron affinities. However, unless adsorption SS, being contained on the grain surfaces with critical barriers, are completely filled [ t < t,, = tp(tc0)], i.e. unless critical barriers are eliminated from a group of levelled barriers, we would obtain a monot- onic decrease in the a(t) and P(t) dependences according to eqn. (15) and (16), respectively. Changes in t in the vicinity of t,, are accompanied by an appreciable additional decrease in the ohmic electrical conductivity and a strong increase in the slope of the I-V curves, see Fig. 8. At times t > t,, the values of a and /? become time independent and may be described by the following expressions

at t > t,,

a z Go( 1 + A ) e x p ( J t c o - ~N,J<,,) (24)

This anomaly in the adsorption kinetics of a(t) and P(t) at t > t,, is due to the effect of the acceptor adsorption on the concentration of the criticaI barriers, f(t,), which determines the value of the conductivity exponential prefactor and the slope of the I-V characteristics. The increase off(<,) resulting from narrowing of the distribution range of < for t < to, and

-1-- ~~

t tpc

Fig. 8 Kinetics of the changes in a(t) [(l) and (3)] and B(t) [(2) and (4)] during the adsorption of acceptors in the case of small [ N , > a’‘’ - (:i2, (1) and (2) and large concentrations of adspecies [ N , > all2

- <:A’ , (3) and (41

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from increasing of the group of levelled barriers for to, < t < t,, is replaced by a fairly large decrease off((,) at t > t,, due to the elimination of (, from this group owing to the almost complete filling of N , [Fig. 5(b)].

In the case of donor adsorption at low concentrations of adspecies ( N , < ,/(O,max - do), the change in the form off((, t ) is due to the existence of a dependence of ( ( t ) on to only. This follows from the solution of Poisson’s equation [eqn. (7) and (S)]. When the system is far from equilibrium, the rate of emptying of donor SS, and so the rate of change of the barrier height, unlike the acceptor adsorption case, is inde- pendent of the original barrier height to [see eqn. (ll)]. In the vicinity of equilibrium, related to complete emptying of donor SS, the solution of eqn. (6) is

where

tdp = l d { l + expro - (J(0 + N t ) 2 1 } - (27)

It follows from eqn. (27) that td, % t, owing to the validity of inequality (,/to + N,)* > 8, and so the dependence of ( ( t ) on to is very weak and has practically no effect on the form of the functionf(5, t ) at long times. In the case where the condi- tion N , < Jt,, - ,/O is valid, the kinetics a(t) and p(t) are described by eqn. (17) and (19) in the whole time range of application of eqn. (1)-(3).

Thus, on the basis of the percolation approach developed, we have described the kinetics of the adsorption-induced changes of the electrical conductivity and the slope of the I-V characteristics of polycrystalline semiconductor adsorb- ents. The power-law time dependences of the kinetics of adsorption-induced changes in o(t) and P(t) have been found. A small value of the exponent and correspondingly slow o(t) kinetics in the case of adsorption of acceptor species are caused by an unusual compensation effect consisting of a large increase in the exponential prefactor, when an increase in the activation energy for conductivity takes place, which is produced by a simultaneous increase in the height of the critical barriers and in the density of the current-conducting paths.

The switching effect of the polycrystalline semiconductor Z-V characteristics, predicted by the developed theory, and produced by the adsorption of low concentrations of accep- tor species, which is less the smaller the initial conductivity of the adsorbent, has been supported by e~per iment ,~ see Fig. 9.

We should stress that the model is surprisingly successful in qualitatively explaining the kinetic behaviour of o(t) and p(t) during acceptor species adsorption. It is possible to obtain an attractive fit to the experimental data using a

l- I 1

3i z22.0 L

I !-

I c: 0.4 ? 0 r

0.2 G-

. . . 0 20 60 - a0

t/min Fig. 9 Changes in conductance, G [(l) and (3)], and I-V curve slope [(2) and (4)] with time in an experimental ZnO layer during 0, adsorption. R, = lo8 R, Po* = 5 x Torr [(l) and (2)] and POI = 7 x Torr [(3) and (4)]; 1 Torr = (101 325/760) Pa

100 200 tlmin

Fig. 10 Comparison between the theoretical results for kinetics of the changes in o(t) and P(t) for the case of large concentrations of adspecies with experimental data for 0, adsorption on a ZnO l a ~ e r . ~ ’ ~ ~ R, = 6 x lo9 R, Po2 = 3 x 10’ Torr. The solid lines show the theoretical results obtained from eqn. (15) and (16) with to, = 120 min

correct choice of parameters (see, for example, Fig. 10). Dis- crepancies, which sometimes arise, between our results and experimental data may be connected, for example in the case of thick adsorbents, with diffusion dificulties, which thus break our model requirement for perfect gas permeability of the polycrystalline adsorbents.

In the adsorption of donors, as follows from eqn. (17)-(20), the kinetics of the changes of a(t) and p(t), as mentioned earlier, strongly depend on the form of the chemisorption kinetics, Nt(t) . Comparison between our theoretical results (see Fig. 7) and experimental data, which are shown in Fig. 11, demonstrates that the fit is quite good.

Equilibrium Situation

Concerning the steady-state (post-adsorption) values of oS and p,, we can easily show that the relationships between the original (preadsorption) electrical conductivity of the adsorb- ent, the equilibrium or stationary concentration of chemi- sorbed species, the relative positions of the energy levels of the adsorbate and the adsorbent etc., again play an impor- tant role. In the case of absorption of acceptors, in steady state, the relationship between adsorbent conductivity (0,)

and equilibrium concentration of chemisorbed species ( N J is described by the different expressions, which take forms alter-

2 4 6 8 1 0 t/m i n

Fig. 11 Changes in conductance, G [(l) and (2)], and the slope of I-V curves [(3) and (4)] with time ZnO layers during the adsorption of Zn atoms [(l) and (3)] and H-atoms [(2) and (4)];37 PH2 = 3.5 x lo-* Torr, THsource = 1473 K, TZnsource = 458 K; Go is the initial

conductance

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ing from exponential

a, x Go( 1 - v *)exfi - 2~(5,,)N,,) (28) J r c o

which is valid for low concentrations N , , < J tco , through to the next exponential

- V

ITs = a o ( k ) exP(-N:) (29)

for greater concentrations Jrc0 c N , , < Jct c Jtco, to power law

(301 us = oO(2J(~)Nts)-" exp{tco - a>

which applies as the inequalities ,/a - Jt,, < N , , c Jct

- ,/(a - 5,) are valid. Here 5, is a parameter depending on the form of initial barrier height distribution. In particular, in the case of a randomly uniform initial distribution: 5, = rco - ~ O , m i n . In the case of N,, > Ja - J(a - t,), eqn. (1)-(3)

become confused owing to the presence of an IC of levelled barriers.

It follows from eqn. (28)-(30) that different adsorption iso- therms yield different types of relationship between adsorbent conductivity and external gas pressure, P , which vary from linear to exponential. Thus, when the logarithmic isotherm can be applied ( N , , - In P ) , the power-law dependence a, - P - k may be derived from eqn. (28) and (29). Note that, in the first case, the power-law exponent depends on the degree of barrier disorder. This type of dependence also arises by using eqn. (29) and (30) in the case when the Freundlich isotherm or Henry's law or Rideal-Eley mechanism are valid.

When the inequality N,, < Jtl - Jtco is valid, the I-I/ slope is described by the following expression

N,, v / ( l + v )

P, = Po( 1 + &) (31)

In the case of ,/a - JCc0 < N , , < ,/a - J(ct - t,) the Z-I/ characteristics become

where E , = /3,('fv)[N,,(2,/a - N,,)I2. It follows from eqn. (32) and (33) that for E < E , the I-V slope decreases with increasing N , , , but in this case, the I-I/ curves are linearised in ln(J/J,) - E coordinates. For E > E , the I-V slope regains its preadsorption value, and an increase in N , , leads to a parallel shift of the I-V curves as a whole without affect- ing P. The qualitative picture of os(Nts) and P,(N,,) depen- dences obtained here is illustrated in Fig. 12. For comparison, the experimental data concerning the steady- state values of (T, and & under oxygen adsorption are shown in Fig. 13.

Note that, in some cases, owing to the slow o(t) kinetics observed under acceptor-particle adsorption, the possibility of the experimental recording of the 'quasi-equilibrium ' rela- tion between conductivity and gas particle pressure could not be excluded. In this case, which is most typical for high- Ohmic adsorbents with large to,, proceeding from eqn. (10) and (15), we obtain

t,, = t c o + ln(1 + 1%)

i?,, x ao(l + AtNJ"

(34)

(35)

here A = 2K,, <EL2 exp( - tc0). Thus, the activation energy of such 'quasi-equilibrium' conductivity increases logarithmi- cally with increasing N,, , which corresponds, at an approx- imately linear relationship between chemisorbate coverage and gas partial pressure, to the experimental ~ i tua t ion . '~

0

-8 h

b" 2 v O, -16 -

-24

1 3 5 1 3 5 NtSIN 0 L D

Fig. 12 Dependences of the equilibrium values of (a) os and (b) p , on the concentration of acceptor adspecies, N , , , plotted for several values of tco. (1) tco = 10, (2) tco = 16, (3) tc0 = 25, (4) tco = 30, ( 5 ) tc0 = 35. In all cases: a = 36, v = 0.88

Moreover, at sufficiently high partial pressures of acceptor gas, i.e. at high N,, , eqn. (1 5) at t > to, takes the form

a x ao(t/to,) - " - (tlv,,) - (36)

from which it follows that at longer times, characterized by slow a(t) kinetics, there is a power-law dependence of the quasi-equilibrium conductivity on adspecies concentration and, consequently, on their external pressure in the cases when the Freundlich isotherm or Henry's law are valid. With respect to thick adsorbents, with h > L, we have m z 0.5 and obtain a square-root dependence

In the adsorption of donor species with N,, < Jtc0 - J O we obtain from the time dependences, eqn. (17) and (19), expressions showing further relationships between the adspecies concentration and the steady-state values of (T, and

- P - 1/2.18-20

P,

(37)

For N , , > ,/CEO - JO we find from eqn. (18) and (20)

0, = a o N 2 exP(5,o - 0)

P, = Po(1 + N,$J@) -

(39)

(40) At large enough concentrations of adspecies it follows from eqn. (39) that the steady-state electrical conductivity of

Po -2 -1 0 -2 -1 0

log (PoJTorr)

for a ZnO layer.37 (l), (3): no = 2.25 x 11.8 x 10-9fi-i

Fig. 13 Experimental dependences on 0, pressure of the equi- librium values of the conductance (G) and slope of the I - I / curve

R- ' ; (21, (4): 0 0 =

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barrier-disordered polycrystalline adsorbents increases with N , , by a power law, whereas at low concentrations the increase is exponential, see eqn. (37). In this case, as for the acceptor adsorption, different adsorption isotherms produce different types of dependences of adsorbent conductivity on external gas pressure. Thus, in the case of the applicability of the Freundlich isotherm, which takes place, for instance, in the adsorption of CO on MnO, , 21922 C,H50H on SnO, and Zn0,23 and propane on Sn0,,24 it follows from eqn. (39), that we have a power-law relationship between the steady- state conductivity and reducing gas pressure. Such a relation- ship is also obtained from eqn. (37) in the case of the applicability of the logarithmic isotherm.

As mentioned in many the sublinear response to combustible or reducing gases is the most salient feature of the behaviour of metal oxide semiconductor adsorbents. The power-law response has not been suc- cessfully represented by algebraic mass action functions involving integer or fractional powers of gas concentration or by constant rate parameters of definite order.

From our results we may conclude that calculation of the barrier disorder in a polycrystalline adsorbent offers a con- vincing explanation (in terms of general considerations) of such experimentally observed power-law dependences without the need to postulate interaction of the donor species (reducing gas) with oxygen previously adsorbed in accord- ance with the Freundlich isotherm.25

Effect of Native Surface State Recharging on the Adsorption Response of the Electrical Conductivity of

Polycrystalline Semiconductor Adsorbents The possible recharging of the native surface states induced by adsorption can have a marked effect on both the kinetic and steady-state responses of the electronic properties of semiconductor adsorbents. Indeed, the existence on a real semiconductor surface of a large number of native SS, includ- ing the various structural defects, impurities and previously adsorbed species etc., considerably affects and sometimes defines both the mechanism and kinetics of adsorption pro- cesses. Besides the fact that the above defects can be adsorp- tion centres,’ no less important is the effect of native SS recharging during adsorption, leading to long-time relaxation of some characteristics of the adsorbent which are monitored in adsorption experiments, such as surface conductivity, Hall’s characteristics, the electron work function and others associated with a surface charge, Q, . The semiconductor sensor method, widely used in gas analysis at present, is based on the fact that the nature and magnitude of adsorption-induced changes in the above characteristics of adsorbents make it possible to evaluate the partial concentra- tions of the detected particles in the environment. Changes in Q, due to native SS recharging can superimpose on adsorption-induced changes in Q, and thus mask the true adsorption process. Let us illustrate this phenomenon by an example of the distortion of an adsorption response in the conductivity of a polycrystalline semiconductor due to native SS recharging.

Steady-state Response

The change in the Fermi-level position occurring during adsorption with respect to the energy band edges and local surface levels, due to the conversion of some of the adspecies into the charged form, is responsible for the disturbance in the equilibrium filling of native SS which, in turn, causes an additional change in the value of surface band bending. If on a crystallite surface characterized by the existence of an initial

Ecs

E v s

€CB

EF

EVB

Fig. 14 Schematic diagram of the adsorption-induced change in a band diagram of the adsorbent surface region : acceptor-particle adsorption

barrier (band bending), Fig. 14(a), whose height 5 , is described by eqn. (9), the acceptor-type SS with concentration N,, and electron affinity E,, are formed, the condition for their transfer into the charged form is E,, > x + 6 + V,, (where V,, = <,kT), i.e. as before, charging occurs only at the grains whose surface barrier heights obey the inequality : 5 , < a, Fig. 14(b).

In order to obtain the equilibrium height of the surface energy-barrier, determined by the equilibrium filling of both the native and adsorption SS, we make use of Fermi-Dirac statistics in the absolute-zero approximation. This approx- imation has been shownz7 to hold fairly well for a weakly varying density of SS in the energy band gap.

If we consider the adsorption of acceptor species in the case of a uniform distribution of the native SS density of states, the surface charge is given by the expression

Qs = 4Ds(Eg - 6 - 5 k T ) + qNt, N D L D O(a - 5 ) (41) where D, is the native SS density of states referred to unit surface area and unit energy spacing, O(x) is the unity func- tion. We assume that the surface is neutral before charge trapping by both the native and adsorption SS. Besides that, we assume that the native SS are interface states of acceptor type. The compensating bulk charge, Q,, is determined from the solution of Poisson’s equation [eqn. (4)], from which, neglecting the concentration of the holes (n,lND)’ exp(r), we obtain

Q, = (2~6, ND kT)”2[5 + exp( - 5) - l]’” (42)

Considering, as before, the case 5 > kT, and using the charge neutrality condition Q, = Q, , we obtain an equation for 5

(43) N,(1 - P5) + w, O(a - 5) =

where the following abbreviations have been introduced :

N , = qD,[(Eg - 6)/2&&o P/D] l”

fit = qN,, ND LD/[2&&0 ND(E, - S)]”’ p = k T / ( E , - 6)

N,, as before is normalized with respect to N , LD . It is apparent from eqn. (43) that the height of the equi-

librium surface barrier is defined by the following expressions

5 = P-l({Jcl + 4N,(N, + m1 - 1)/2W2

5 = p-I(J(1 + 4 N 3 - 1)/2NJ2

(44)

which is valid for the barriers with low height 5 < a, and

(45)

for barriers wth 5 > a. Note that eqn. (44) is valid for low barriers for which even a complete filling of adsorption-type SS(N, , ) never leads to E,, and EF levelling. This situation

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takes place when the value of N , fits the inequality

J ( P 4 - f l t N, < N,, = 1 - pa

If the initial barrier height does not satisfy the condition so < a, then the barrier does not change during adsorption and its height is described by eqn. (45). Such a situation occurs when the inequality

J(P4 N, > N,, = - 1 - pci (47)

is valid. The relationship between ( and N , at given N,, is shown in

Fig. 15. Note that in the region N,, < N, < N s 2 , approx- imation of the Fermi function by its T = 0 form does not describe the { ( N , ) dependence. This region with a width of about a few kT, arises owing to the alignment of E,, with the Fermi level, E , .

In the region N, < Nsl, in which we are most interested, where the native SS recharging takes place, we can find from eqn. (44) and (45), a further expression for the relationship between the adsorption-induced barrier height and its initial, pre-adsorption value

(48)

Comparison of this result with the expression obtained in the case of neglecting the native SS recharging [ c % (,/so + N,,)’], makes it possible to conclude that there is a strong

screening effect of native SS on the adsorption-induced charging of the semiconductor grain surface. This effect is shown in Fig. 16(a).

Besides the fact that the native SS recharging impedes the adsorption-induced change in the surface barrier height of each single grain, this phenomenon also affects the adsorption-induced cooperative behaviour of all barrier heights. Indeed, in this case it is easy to find an expression for

tscr % C J t O + Nts(1 - pt0)12

-1 I

Fig. 15 adsorption of acceptors as a function of the native SS density

Height of the equilibrium surface barrier in the case of

50, min a 5 0

0 Ls max 5 0

Fig. 16 Dependence of the barrier height, 5, on its initial value to : (a) adsorption of acceptor particles; (b) donor adsorption: ( 1 ) without and (2) with consideration of the native SS recharging

the post-adsorption distribution density of barrier heights

Comparison of eqn. (49) and (23), obtained with and without consideration of native SS recharging, reveals that the above- mentioned recharging changes the form of fa(<) and impedes the growth of p and the decrease of the exponential pre-factor of 6:

v / ( l + v )

(51) Pscr % PO[ 1 + - Nrs (1 - Pc.~)]

J s c o

Note that these expressions are valid for low equilibrium concentrations of chemisorbed species.

In the adsorption of donor species characterized by the ionisation potential, Ei , it is also easy to obtain an expression describing the post-adsorption interface barrier height in the case of allowing for the recharging of the native SS

N , < N,, =- Jpe (52) 1 - p e

1 sscr

(54)

These expressions testify that significant screening of the adsorption-induced charging of grain surfaces by the native SS recharging also takes place in the case of adsorption of donor species [Fig. 16(b)]. The expression for the post- adsorption barrier height distribution takes the form

and equilibrium values of B and /3 determined by donor adsorption in the case of possible native SS recharging are described by the expressions

o,, ,cr = 6 0 ( 1 + - $:o - 61rN,sJsco)Y

p s , scr % PO( 1 + Jr,, Nts - 6rNtsJtc0)

x exPC2NrsJtcJ.l - ~ i c o ) / ( l + ~ t c 0 ) I (56)

(57) - v / ( l + v)

Comparison of the expressions obtained, eqn. (50), (51) and (56), (57), with the corresponding expressions, eqn. (28), (31) and (37), (38), obtained without consideration of native SS recharging, indicates that such recharging plays an important role in the adsorption-induced charging of grain surfaces.

Thus, the native SS recharging in some cases can be responsible for the low sensitivity of the semiconductor elec- trical properties to adsorption. Note that the recharging may be one of the reasons for the significant impact of the methods of adsorbent production and surface treatment on the magnitude and kinetics of the adsorption response.

However, the direct measurement of equilibrium post- adsorption electrical characteristics of the adsorbent, fol- lowed by simple comparison with their preadsorption values,

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does not allow evaluation of the native SS recharging contri- bution to the adsorption response. Such evaluation is pos- sible if the investigations are carried out on monocrystal adsorbents with well known surface conditions” or during investigation of the charging kinetics of semiconductor sur- faces during adsorption. In the latter case, we are sometimes able to separate the contributions of the adsorption-induced charging and the native SS recharging to the electrical properties, owing to the different rates of these p roces~es .~ .~*

Transient Response

Let us consider the case, when adsorption of gaseous species takes place on a wide-gap n-type impurity semiconductor with a depleted surface region due to presence of mono- energetic acceptor-like native SS. We shall assume that before adsorption, the native SS are in a state of charge equilibrium with the bulk. This equilibrium state is characterized by a charge on the native SS, n,,, and a corresponding surface barrier, to, resulting from the surface band bending, q,, which can be determined by solving Poisson’s equation for the near-surface region with corresponding boundary condi- tions. Chemisorption in the charged form affects the surface band bending and hence the position of the surface local energy levels relative to the Fermi level of the adsorbent, i.e. it disrupts the equilibrium in the filling of the native SS.

Thus in general, the determination of the charging kinetics of the surface involves a self-consistent calculation of the kinetics of the filling of surface states of adsorption type and of recharging (either filling or emptying) of native SS. The linking equation in this calculation is the Poisson equation which relates the barrier height at time t to the concentration of charged SS of both types.

In the adsorption of acceptor species, a system of equations describing surface charging can be written

dnt - = K,,((N, - n,)ex~[: - &t)I - nr ~ X P ( - &>> (58) dt

[t + exp( - 5) - 1) li2 = n,(t) + n,(t) (60)

where all the notations have the same meanings as earlier. As for eqn. (59), here N , is the total concentration of native SS, n, is the concentration of its charged form, y = ( E n - x - 6)/kT, where En is the local energy level of the native SS,

and K,, = vn,C, ; here C, is the capture cross-section for electrons of the native SS.

Assuming, as before, that the rate-limiting step in the surface charging is the formation of the charged form of the chemisorbed species, we can suggest that the dependence of the density of adsorption SS on time can be expressed as

0; t < O N , = const.; t 2 0 N,( t ) =

By solving eqn. (58)-(60) with the initial conditions, n,(O) = 0, n,(O) = n,,, where n,, is given by solving the equation ( N , - n,,)exp( -TI,’,) = n,, exp( - y), it is easy to show that if the condition p = x exp(5,) > 1 applies, where x = x,/(x, + xt) and x,, is given by x, = 2vn,C.,(N, - n,,) and x,, as

before, x, = 2un, C, N , , the dependence of the barrier height on time takes the form

t(t) x 5 , + 1nCl - (1 - ~)exp(-t/tJl (62)

where t, = [x, <A’2 exp( - to)] - ’. The condition /? = x exp(5,) > 1 imposes an upper limit on the rate of charging of

the native SS, and also on the maximum achievable change in the barrier height: 5 , < 25,. In other words, in this case, the ‘fast’ subsystem consists of native SS and the adsorption SS form a ‘slow’ subsystem. When /? + 1, i.e. when x, > x,, there is practically no change in the charge state of the surface during adsorption, because of the very fast relaxation of the charge in the native SS. For the opposite condition, p = x exp(5,) < 1, the parameter 5 , can differ markedly from to, and the following expressions for ( ( t ) apply

t o + lnC1 + 2(1 - x)x,, 5;” exp( - to)] ; t < t o (63)

ln{2( 1 - x)x,, 1nli2[2( 1 - x)x,, t]} ; to < t < t , (64)

t > t , (65)

where, as previously, to = x,- ‘56 1/2 exp(5,) and x,, = x, + x,. It is clear from these expressions that for t < t , the

surface charge kinetics are determined mainly by the adsorp- tion SS parameters and in the limit of x, + 0 it transforms to the kinetics described by eqn. (lo), which we had obtained by stipulating a non-relaxing native SS charge state. The <(t) kinetics for these two cases are shown in Fig. 17 (curves 1 and 2). To test these results, the system in its general form, eqn. (58)-(60), was solved numerically for the case x, + x,. The solution (Fig. 18) shows that for t d t , the kinetics of Q,(t) z n,(t) + n,(t) are determined by the rate of charging of the

Fig. 17 Kinetic curves for the charging of a semiconductor surface during adsorption. Acceptors: ( 1 ) -fl Donors: (3) N , < N , ; (4) N , > N ,

1 ; (2) f l < 1 ; (2’) x, = 0.

1 4k-

I I

tn 5 10 15 r/104 s

Fig. 18 Numerical solution of the system eqn. (58)-(60) for the fol- lowing parameters: x, = 2 x s - ’ ; N , = 1 0 ’ ~ cm-2; a = 44; x, = 1.4 x lo6 s- ’ ; N , = l O I 3 cm-2; y = 24. (1) Q,(t), (2) n,(t),(3) n,(t)

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adsorption S S , and the contribution from the relaxation of the charge state of the native S S is negligible.

In the adsorption of donor species with an ionisation potential small compared with the electron work function of the adsorbent the system of equations describing the surface charging kinetics takes the form

[{ + exp(-t) - I l l i2 = nn(t) - n,(t) (68)

The solution of eqn. (66)-(68), when the assumption N: c to is applied, i.e. at concentrations of adsorbed species much lower than the concentrations of the charged native S S , leads to the conclusion that surface charging kinetics are character- ized by non-monotonic behaviour (Fig. 17, curve 3)

[ + ln[exp(2Nt - exp(-t/tn)]; t > t , (70)

where, as previously, t d = Kn, exp(8), j i = (1 + tn/t,) and t , = t,,/2N,,/tO. Such behaviour derives from the fact that the presence of the surface barrier in this case has practically no effect on the charging kinetics of the adsorption S S , but suffi- ciently affects the recharging kinetics of the native SS. This causes rapid emptying of the adsorption S S followed by slow rearrangement of the charge in the native states. We may conclude from eqn. (69) and (70) that, at times t < t d , the kinetics of the change in the barrier height are controlled by the charging kinetics of the adsorption S S in the adsorption of donor species. By using typical values of the parameters for the present adsorbate-adsorbent systems we conclude that the emptying of the donor levels takes place in times of the order of seconds, whereas the relaxation of the charge in the native S S , characterized by the parameter tn , is completed after several hours.

In the opposite case of large adspecies concentrations, when the barrier is completely removed, the extremal behav- iour of the t ( t ) kinetics is replaced by a monotonic decrease in 5 (curve 4 of Fig. 17), accompanied by a decrease in the rate of charging of the adsorption S S and an increase in the rate of recharging of the native S S .

Thus our theoretical analysis suggests that the charging of the adsorbent surface resulting from adsorption leads to the relaxation of the charge state of the pre-existing native SS and can influence the kinetics of the conversion of adspecies into the charged form. However, in the chemisorption of active species having short charging time constants (as com- pared with the charge rearrangement in the native S S ) , the initial rate and the magnitude of the change in the surface charge and therefore also in the electrical conductivity, work function etc. are initially determined by the charging kinetics of the adsorbate. The conditions governing this effect are as follows :

(Nn - fi0)Cn ex~(tc0) < Nt '2

t < t n c = Xn t c o exp(tc0)

(71)

(72) - 1 112

in acceptor adsorption, and

in the adsorption of donor species. t < t d = K , exp(8) (73)

Large effects associated with the recharging of the native S S are observed only at long times, often not experimentally accessible, and they usually produce stationary values of surface charge lower than those found on an a priori clean surface. Thus, the data on the initial rates of change of the above mentioned electrophysical parameters of the adsorbent can be used to determine the concentrations of the gaseous species on the adsorbent surface and in the gas phase.

Adsorption-related Response of Adsorbents based on Polycrystalline Metal Oxide Composites

The approach developed to model the adsorption response of the electrical characteristics of polycrystalline adsorbents looks promising for understanding the sensing properties of mixed oxides, which are widely used for chemical sensors.29 It has been shown that, in some cases, it is the mixed oxides that possess the maximum sensitivity to various gases3' Fur- thermore, the adsorption-related behaviour of the electrical parameters of such composites are characterized by some a n ~ m a l i e s , ~ ~ ' ~ ~ which are typical for only definite composi- tions of these adsorbents. The existence of such anomalies looks promising for the design of sensors with the ability to detect only strong concentrations of given g a ~ e s . ~ , . ~ ~ Many composite semiconducting systems, which are mixtures of various polycrystalline semiconductors, reveal complex and sometimes anomalous dependences of (preadsorption) con- ductivity on composition, which are characterized by the presence of limiting points, regions of rapid and almost step- wise alteration, as well as slow changes in (properties of such systems are described in detail in ref. 34). Note that quite complex electroconductivity-omposition dependences are also seen in mixtures whose components interact poorly, if-at all, i.e. those which dilute each other mechanically, for instance Si0,-TiO, .35

Now we will attempt to show that both of these types of anomalous dependences of electroconductivity and its adsorption response on the composite composition can be related to the barrier disorder existing in the considered poly- crystals. Mixed polycrystalline semiconducting composites can be treated as mixtures of two or more barrier-disordered components differing in the dispersion amplitudes of the intergrain barrier heights. Thus, in the case of binary systems, addition of one such barrier-disordered component to another causes a considerable change in the function of the intercrystallite barrier height distribution. Naturally, this gives rise to changes in the percolation level, t c , and corre- lation radius, L, which are responsible for the alteration of the effective conductivity of polycrystal. For a mixture of two components in which the volume fraction of the poorly con- ducting component (1) is (1 - y ) and that of the highly con- ducting one (2) is y , then of the total number of bonds presented in the system, the 1-1 bonds, i.e. the bonds between the grains of the poorly conducting component, form a part, P , , = (l-y),. Part P , , = y 2 is the fraction of 2-2 bonds, i.e. the bonds between grains of the highly conducting com- ponent, and part P , , = 2y(l-y) refers to the mixed bonds of 1-2 type.

Let us introduce laws and ranges of barrier height distribu- tion, F,{t) and [O; ti], ti % 1, where j = 1, 2 or 3 (index 3 corresponds to 1-2 bonds) and assume that there is no corre- lation between F 3 ( t ) on the one hand and F , ( < ) and F,(<) on the other. The effective conductivity of the single j component is given as previously [see eqn. (l)] by the expression

(74)

The function of the barrier height distribution in the y com- position is F ( t , y ) and, therefore, the expression for the frac-

oj = o,L,r1 exp( --tCj)

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tion of bonds, x(5, y), containing barriers lower than 5, depends on the correlation between the C j and rcj parameters. Thus, in the case of t2 < t3 < c1 the distribution density F(5, y) is described by the expressions

Y2F2(5) + 2Y(l - Y)F3(5) + (1 - Y)2F1(5);

0 < 5 < 5 2

2Y(l - Y)F3(5) + (1 - Y)2Fl(5); 5 2 d 5 < 5 1

(1 - Y)2Fl(t); 5 b 5 1 1 (75)

F(5, Y) =

This results in qualitatively different variations of the <,( y) and L(y) dependences at the various correlations between the parameters mentioned. Indeed, from the expressions for the magnitude of the percolation level

5 J Y )

P c = F(5, Y) d< (76)

and for the correlation radius

5 d Y ) + 1

L(Y) = l l P - p c r v !z l l Jc (y ) F(5, Y) d i l - ' (77)

and taking into account the normalization conditions

it can easily be shown that in the case of r2 < t3 < t1 growth of y results in a rather monotonic decline of <,(y), while in the case of t2 < < t3 , a maximum is seen on the tc(y) depen- dence that is due to the initially higher growth of the fraction of high barriers. Note that for simplicity we assume that the mean sizes and forms of the grains of the different com- ponents are more or less alike, i.e. the mixture percolation threshold does not depend on the composition. The behav- iour of L depending on the mixture composition is, in all cases, of a quite non-trivial character and depends greatly on the form of F(5, y). Regions of rapid changes in the depen- dence of F(<, y) on t, related to the existence of limiting heights of intergrain barriers, can give rise to abruptly non- monotonic behaviour of L(y). Thus, for instance, when the inequality 1 4 c2 < t3 < tCl < t1 is satisfied, the dependence of L(y) at the (y*, y;) region [where y* and y; are found from the equations tC(y,) = t3 and c,(y',) = t3 - 13 is

We can see that if F3(5) drops abruptly in the vicinity of r 3 , changes in the effective conductivity of the mixture, a(y) - L- '( y)exp[ - <,( y)], in the (y, , y;) region can be determined by the behaviour of the exponential prefactor, i.e. L( y). In this case, o(y) also behaves similarly in the (y,,, y;,) interval. The boundaries of this region are found from the equations

To illustrate this, Fig. 19 and 20 show t,(y), L(y) and a(y) curves with randomly uniform barrier height distributions, F,( t ) = <,:'. Fig. 19 pertains to the case when r 2 < r3 < tcl < t1 and Fig. 20, when t2 < tCl < t1 < t3 .

Abrupt alterations of a in the vicinity of certain concentra- tions of y are due to abrupt alterations of the exponential prefactor in the expression for a(y) which is conditional on

5c(Y**) = r 2 and <C(Yi*) = 5 2 - 1.

sc ( L I Ino

sc1

5 ,

5c2

Ino2

b S -

Ino,

0 y. y.* 1 Y

Fig. 19 (1) tC(y) , (2) uy) and (3) o ( y ) dependences in the case of 5 2 < 5 3 < tCl < 5 1

the rapid alteration of F(5, y) in the vicinity of given y. In the case of the uniform distribution of FA{), it is caused by stepped alterations in F(5, y). In real systems, a stepped behaviour of FA<) is most unlikely. However, thanks to the available data on the existence of limit barriers (in particular, the so-called Weisz barriers,36 maximum barriers resulting from surface charging due to filling of the surface states), we can expect quite an abrupt decrease in F A [ ) at 5 exceeding some which is maximum for this component. There- fore, from the above it is easy to conclude that variations in the barrier height distribution function caused by changing the composite composition are responsible for some features of the composite electroconductivity behaviour.

Similar effects can also be responsible for abrupt alter- ations in the conductivity of composites under various exter- nal influences which cause the form of the distribution function F(5 , y) to change. Indeed, it is the form of F(5, y) that determines the values of the 'critical' fractions of the highly conducting component y, or y,, . Any external influ- ence that alters F(5, y) also alters the values of y, or y,, . At some intensity, this causes y, (or y,.,.) to attain a y-value characteristic of the mixture composition selected. As has been shown earlier, this may result in an abrupt alteration in 0. Such behaviour of the composite conductivity is possible when gas particles are chemisorbed due to the adsorption obeying the form of F(5, y).

Thus, by choosing various compositions which differ slightly from 'critical' ones, we may obtain gas-sensing adsorbents which will be characterized by pronounced changes in electroconductivity under the adsorption of given concentrations of a certain gas. Furthermore, we can comment on the interconnection between the two above- mentioned types of influence on the composite conductivity, i.e. the adsorption-induced influence and that related to the

5c1

5 C

sc2

O Y- 1 Y

Fig. 20 (1) <,(y), (2) y y ) and (3) o ( y ) dependences in the case of 5 2 < 5 c l < < 53

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changes in composite compo~ition.~ ' The conductivity of such composite adsorbents in equilibrium with the gas-phase is described by the expression

(80) 0 = 0"FVC5(Y, P) , Y, PIexpC-t,(y, PI1 where F ( { , , y, P ) is the density of barrier height distributions and P is the partial pressure of the gas being detected. If the adsorbent composition, y , belongs to the composition region where changes in the effective conductivity of the composite are determined by the behaviour of the exponential prefactor, then, as has been shown in ref. 37, the expressions for the changes in conductivity take the following forms

in the case when the conductivity alteration is caused by the composition changing, dy, and

under adsorption due to the partial pressure alternation, 6P. In both cases, the magnitude of A ~ J varies directly as the derivative of the barrier height distribution density with respect to the barrier height and may be large in the vicinity of the 'critical points' (y, or y**). This correlation suggests that the maximum adsorption sensitivity would indicate the composites with compositions corresponding to regions in the electronconductivity-composition plot with the largest daldy. The experimental data shown in Fig. 21 provide support for this ~uggestion.~' Fig. 21 contains curves showing the behaviour of the initial (pre-adsorption) electro- conductivity of the ZnO-SnO, composite, oo, the sensitivity of the electroconductivity of such composites to adsorption of CO, s = [(do/dt)a- l],.+O, and the catalytic activity for CO oxidation on such composite adsorbents (a = [CO,]/[CO]) as functions of the composite composition. Analysis of these data enables us to conclude that there is a correlation between the rate of conductivity alteration caused by the changes in composite composition (do,/dy) and the magni- tude of the conductivity response caused by adsorption (s).

h

0.0 0.2 0.4 0.6 0.8 1.0 ZnO (znO),-, (Sn02)x Sn02

Fig. 21 Initial conductivity (oo), sensitivity to CO adsorption (o-' da/dt It,o) and the catalytic activity for CO oxidation to CO, of the composite adsorbent (ZnO), -,(SnO,), as functions of its composi- tion ( x ) ~

Note that the small variations in the catalytic activity of such adsorbents taking place when adsorbent composition is changing do not correlate with the changes in its gas sensi- tivity. Furthermore, X-ray analysis of such composites has shown that we dealt with simple mechanical mixtures of the oxides chosen, and (in the range of experimental accuracy) there is no new phase which may be responsible for the men- tioned conductivity alterations related to carbon monoxide adsorption. Therefore, the theoretical analysis performed has given conclusive evidence of the validity of our suggestion ; that is, for composite adsorbents which are the mixtures of barrier-disordered polycrystal components, the adsorption response is determined by the cooperative behaviour of the whole system of inter-grain barriers, i.e. by adsorption-related transformation of the current-conducting network.

In many cases the consistent inclusion of this cooperative factor leads to unexpected conclusions which make possible a new view of previous results. Thus, for instance, in ref. 38 it has been shown that the adsorption-reduced response of the electrical conductivity of sintered polycrystal adsorbents characterized by the existence of necks between the grains, is described by a logarithmic dependence of the partial pressure of the gas being detected, P

(83) AtJ - z A + B In P 6 0

which is valid in the case of a log-normal distribution of the neck thicknesses. Note that eqn. (83) is derived in the case where the Freundlich or Henry adsorption isotherms are applicable. In the same paper,38 it has been shown that addi- tion of a non-conducting component to the gas-sensitive oxide with finite conductivity causes an increase of the gas sensitivity, which results from a decrease in the amount of conducting necks. In the limiting case, when the composite composition falls to a scaling region ( l y - y , I 4 y,), its gas sensitivity increases sharply. This phenomenon is related to the fact that in the vicinity of the percolation threshold, y,, our system takes the features of a one-dimensional system and thus the removal of only a few bonds (necks) leads to an interruption of the current-conducting path. Note that such a sharp increase in the sensitivity to methane adsorption of layers of semiconducting oxide Sr - ,Ca,Fe03 - when a mineral insulating binder with concentrations between 30 and 40 vol.% was added, has been found in ref. 39.

Conclusion A model based on a percolation approach to the adsorption- related response of the electrical characteristics of poly- crystalline oxide-semiconductor adsorbents accounts for the essential features of the experimental results. However, in some cases, this model must be complicated. It is related pri- marily, to the adsorbents based on ceramic or thick oxide films. In this case, diffusion of the gas being detected into the adsorbent bulk through the empty spaces between the grains plays an important role in the adsorption-related response.4o This may cause a disturbance resulting from the assumption of the model of free permeability of the adsorbent layers to the gases being detected, which is connected with the assump- tion regarding neglection of the kinetics of adsorption site formation. In later work, we will consider this question.

Apart from the diffusion contribution to the kinetics of the response formation, some complicated factors must be taken into account in the case of composite adsorbents. Among them are the possible variations in sizes and geometry of grains belonging to the different components of the con- sidered composite ;41 the possibility of interrelations between the distributions of the heights of intergrain barriers of pure

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572 J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89

components, on the one hand, and barriers formed when grains belonging to the different components come into contact, on the other.42

Besides the wide scatter of the heights of the intergrain bar- riers that has been taken into consideration, these types of barrier may also differ qualitatively, since the intergrain bar- riers, in some cases, constitute heterojunctions, formed when grains already possessing surface barriers come into ont tact.^^,^^ By virtue of the random nature of the grains in contact, such as the electron work function, the surface barrier height, the electron affinity, the characteristics of the heterojunctions produced are also random quantities fluctu- ating from contact to contact. This also may complicate our percolation model because ‘directed percolation ’ must be applied in this case.44

Note that only the direct process of the formation of the electrical characteristics of the barrier-disordered semicon- ductors caused by the adsorption of donor and acceptor gas species has been investigated in the suggested model. The kinetic equations that have been solved describe the time dependence of the concentration of the charged form of surface states formed under adsorption. The same equations, but with other initial conditions which correspond to the desorption area of a phase trajectory (dnJdt < 0), also describe the reverse process, the transformation of the adsorbed species from the charged form to the neutral one. However, the solution of these equations in the mentioned area will describe the desorption kinetics of the conductance change only in the case where the transformation ‘charged form-neutral form’ is the limiting stage. It is possible for valence-saturated species. In contrast, the dissociation of surface adsorption complexes resulting from such surface processes as desorption, surface recombination, and chemical reactions with molecules and radicals from the gas phase, may be the limiting stage in the case of adsorption- desorption cycles, where atomic or radical species take part.45 A comprehensive description of the desorption- induced behaviour of the conductance of barrier-disordered polycrystalline semiconductor adsorbents will be published later.

The effects predicted by the developed theory are in fair agreement with the experimental results. From our point of view, this is conditioned by the fact that in the suggested model we have taken into account not only the effect of adsorbent disorder on the adsorption-induced phenomena, but also the reverse effect of adsorption on the disorder of the adsorbents. Only such a self-coordinated consideration enables one to develop a comprehensive theory of gas- sensitive semiconductor layers.

4 5

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12, p. 144.

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38 39 The author thanks his colleagues, in particular, Prof. I. A.

Myasnikov and Dr. B. Sh. Galyamov for helpful discussions, and Prof. G. Heiland, Prof. S. R. Morrison and Drs. J. Gardner, J. Mizcei and P. T. Moseley for interest in the results of this work. Most of the experimental data are based on measurements by Mr. V. V. Chistyakov.

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42 43 44 45

References 1

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F. F. Volkenstein, Electron Proceses on Semiconductor Surfaces during Chemisorption, Nauka, Moscow, 1987. V. Ya. Sukharev, Russ. J . Phys. Chem., 1989,63,366. V. Ya. Sukharev and 1. A. Myasnikov, Phys. Stat. Sol. A, 1987, 100,277. Paper 2/01976K; Received 15th April, 1992

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