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Conduction models in gas-sensing SnO2 layers: grain-size effects and ambient atmosphere influence
Sensors and Amatom
241
B, 17 (1994) 241-246
Conduction models in gas-sensing SnO, layers: grain-size effects and ambient atmosphere influence N. Bdrsan Instihdeof Physics and Technobgy of Materials, Bucharest-Migwek (RomaniaJ
(Received August 31, 1992, in revised form May 26, 1993; accepted July 1, 1993)
Abstract A model for conduction that fits SnOz polycrystalline layers with a grain size below 50 nm is presented. Quantitative evaluations using bulk parameters, found in the literature, and the assumption that surface oxygen vacancies are sites for atmospheric oxygen chemisorption permit a simple formulation of the problem: conduction by homogeneously distributed electrons along cylindrical filaments. An expression for the mobility is proposed, in which difise surface scattering is taken into account. On its basis, a relationship between the electrical mobility and the concentration of the reducing gas is theoretically derived for the first time. The predictions of the model for the dependence of conductance, concentration of conduction electrons and their mobility on the reducing-gas concentration are compared with experimental data and found to be in good agreement.
1. Introduction Polycrystalline porous SnO, layers are used in the detection of toxic and explosive gases [l-3]. The advantages of gas sensors based on such sensing layers are low cost and high sensitivity. This made them very useful devices and prompted studies devoted to their development. The purpose of these studies is the improvement of sensor performance, especially stability and selectivity. In the range of temperatures in which gas sensors work (400-700 K), the chemical sensing mechanism is the surface reaction between chemisorbed oxygen and reducing gases. The oxygen is chemisorbed as charged species (O,-, O-, 02-) [4-6], inducing the appearance of a surface depletion layer of lower conductivity in comparison with the bulk. The case of O*- is special; it is considered that in this form the oxygen should not be stable at the surface as a chemisorbed species ‘if it does not react immediately or is trapped by an oxygen vacancy’ [6]. One considers that because of the nature of the chemisorption sites, assumed in what follows, one can accept the presence of O*- at the surface as a reactive species. The effect of reducing gases consists in a decrease of the concentration of chemisorbed oxygen, recorded as an increase in conductance. Research in the field of SnO, surface interaction with gases is the subject of both technological and fundamental studies [3, 7, 81. The complex measurements used in these studies impose the elaboration of
0925.4005/94/$07.00 0 1994 Elsevier Sequoia. All rights reserved SSDI 0925-4005(93)00873-W
models that could serve as a guide for the interpretation of the experimental results as a basis for new developments. The aim of this study is the modelling of the electrical condtiction in gas-sensing SnOz layers with average grain sizes below 50 nm. A flat-band hypothesis is justified and assumed. Calculations for electron concentration, mobility and conductance are performed on this basis. The results are compared with experimental data obtained from conductance and Hall-effect measurements [9-111.
2. Conduction model SnO, is a wide band-gap semiconductor, EG = 3.5 eV. Its n-type conductivity is due, when pure, to the presence of bulk oxygen vacancies [12-141.Typical bulk electrical conduction parameters of SnO, single crystals after high-temperature preparation, as used in this paper, are presented in Table 1 [13, 151. TABLE 1. Bulk electrical conduction parameters of Sn02 single crystals ND (me31 N* (m-‘) ED, (ev) ED2(ev) EA (ev)
2.5002x la22 SXlOP 0.04 0.14 0.78
242
As shown elsewhere [M-18], surface oxygen vacancies play the role of chemisorption sites in the presence of atmospheric oxygen. Their concentration, N,, can be evaluated considering that the bulk oxygen vacancies within a sheet of thickness S at the surface are in fact surface vacancies (6 being of the order of magnitude of the SnCl, lattice constant -0.5 nm). For doing this evaluation, one considers that the concentration of oxygen surface vacancies can be obtained as a twodimensional projection of the bulk situation, i.e., one neglects the surface reconstruction. Being interested in the order of magnitude of chemisorption sites, one considers that this hypothesis is an acceptable approximation for the real case. One obtains N,=N,,S(l-S/I&)
(1)
for cylindrical grains and N,=N&(l-6/2r-S2/3rZ)
(2)
for spherical grains, r being the radius in both cases and considering only the lateral surface for the cylindrical grains. The value of N, is about 10x 10” rnP*. The values of the bulk [12, 13, 1.51 and surface parameters of interest in the temperature range 400-700 K are given in Table 2, namely the bulk concentrations of electrons nb, the electrical bulk mobility pb, the effective Debye length AD, the mean free path for electrons A, U, L& AE, (to be defined later in the text) and the radius R of the cylindrical grain for which the condition sN,-VQ,, where s is the lateral surface and v is the volume of the cylindrical filament (see below) is fulfilled. We find for R that 2N,=n,R
(3)
The surface reaction with oxygen and reducing gases may be described by means of a quasi-chemical reaction formalism [6, 19, 201:
(l-~)n,=(cu+pP,“)e
(4)
(l-B)n,2=(a:+PP,)B
(5)
where 0 is the surface coverage of chemisorbed oxygen, n, is the surface concentration of conduction electrons, TABLE 2. Bulk and surface single crystals
T 6) tab
(10’~)
Pb (1O-4 m’lv s) AD (nm) A (nm) R (nm)
AElkr,T AE,kT AEJksT
IYand p are constants and PR is the partial pressure of reducing gases. Equation (4) describes the interaction with O,- chemisorbed species for m = 2 and with Ofor m = 1. Equation (5) describes the interaction with 02- chemisorbed species. In eqns. (4) and (5) 0 and II, are related; there are two problems in order to derive the dependence of the conductance, G, on the concentration of the reducing gas. The first is to find a second relationship between 6 and n, in order to get the dependence of n, on PR and the second is to find the relationship between G and n,, i.e., G and PR. The second relationship between 0 and n, can be obtained from an electrical neutrality equation; its specific form depends on the SnO, layer characteristics. For grains of small average size between 5 and 50 nm, the conduction process seems to take place in long and thin filaments [9, 111. This is the reason why the approximation of cylindrical filaments of length 1 and radius r will be used in the model. One has to evaluate the influence of the surface. Comparing the estimated value of N, and the experimental data for amounts of chemisorbed oxygen [4], we can draw the conclusion that 0, related to N, and not to the total concentration of surface atoms as is usually done [20, 211, is close to unity. As an upper limit for the possible surface influence into the bulk, 8-l is assumed. We are interested in the case in which the energy bands are flat. After ref. 22, this case is obtained if there is enough room on surface levels for all the conduction electrons from the bulk and if r
400 1 178 129 1.96 2.500 0.43 0.23 0.14
parametersof
500 11 a7 43 1.07 230 0.77 0.39 0.26
influence for SnOl
600
58 49 21 0.66 44 1.08 0.56 0.39
700 260 31 11 0.45 2 1.49 0.76 0.51
R=S(l+ zf)( 1+[ 1i$:iJ] The values of interest are presented in Table 2. We have to keep in mind the conditions in which these values are calculated. On one hand the data used in the calculation of R are taken from the literature for a specific single crystal [13,15]; since the concentration of oxygen vacancies in thin or thick porous films can be very different, it is possible that their values do not fit well enough the case of a specific gas sensor. On the other hand, an assumption was made about the value of 0, i.e., 0=1; this means that under the same conditions, namely the same N, and n,,, the value of R is proportional to lower values proportionally to 8. As a conclusion, we can only say that the values of R shown in Table 2 approximate the radius of the cylindrical filament that can be depleted completely of conducting electrons by trapping on surface levels. It is clear that in the range of grain sizes we consider, it is possible to fulfil this condition; it will be assumed
243
in what follows that this is the case. Using this assumption we can write the Poisson equation directly for energy, E, using the Schottky approximation. For our geometry the radial part of the Poisson equation, the interesting part, is
Equations (4), (5) and (13) allow a relationship between n, and PR to be found. Avery simple and easy to use set of equations can be derived if one assumes the fulfillment of eqn. (5) in the whole range of parameters we use:
(7) where q is the electron charge, E is the permittivity and e. is the permittivity of free space. The boundary conditions are
and one obtains from eqns. (4), (5) and (14) n,“+‘=(IY+/3PRm)(nb-n,)
(15)
where E(0) = E,
I
p= 1 for 02- and O-
(?),_,=o r‘ From
eqn.
(7), using
p=2 for 02m=lforO-and@eqn.
(8) we obtain
for
AE=E(r)-E,,:
(9)
m=2forO,Taking into account the fact that n, +z nb, one obtains for high enough PR values [17]:
n*_pRm/@+l)
or
(16)
2
(10) where kB is the Boltxmarm constant and T is the temperature. AE is the maximum electron energy difference, i.e., that between an electron located on the axis of the cylindrical filament and an electron located on the lateral surface. The flat-band condition, AE/k,T$l, takes the form
The relationship between n, and G via conductivity is a delicate matter. Considering the facts presented above, n, is the concentration of electrons that take part in electrical conduction. These electrons suffer scattering processes in the bulk (along the filament) and at the surface. Using the formula derived by Many etal. [23] for the influence of surface scattering, adapted to our geometry, one gets p=
(11) In Table 2 are presented the values of AE/k,T, which are calculated for grain dimensions of 50 nm and the value of AE,Ik,T and AE,lk,T calculated for grain dimensions of 13 and 6 nm that correspond to the samples for which experimental results are presented in Section 3 [lo, 111. Examining the data presented in Table 2, we see that we can consider the flat-band condition is fulfilled in the range of grain sizes below 50 mu. It is assumed, in what follows, that this is the case. The electrical neutrality condition takes the form BN,s+n,v=n,v
(12)
Equation (12) describes the redistribution of the bulk conduction electrons on surface levels and in the bulk due to the chernisorption of atmospheric oxygen. Then
l+(zvr)
(17)
where Wrepresents the probability of a diffuse scattering process. Comparing h to 2r, one can observe that the influence of the surface cannot be neglected. According to Many et al. [23], W is related to the deviation of the surface from a simple projection of the bulk. For our case this deviation represents the difference between the concentration of scattering centres with which conduction electrons interact when they strike the surface and the concentration of scattering centres with which they interact when they move along the axis of the cylindrical filament. This difference, the extra surface scattering centres, is given by the charged oxygen species chemisorbed on the surface oxygen vacancies. Their concentration being proportional to 6, we propose w-e One obtains for the mobility:
(18)
244
with #=l+
;
and .q=
& b
One can expand eqn. (19) using the fact that qn,lQ 4Cl:
or, with obvious notations: #u=/&(‘yt pPR”I@+l))
(21)
Equation (21) is the first proposed relationship between ,u and PR based on a theoretical model. Thus we have for the conductance (or for the conductivity a) of the cylindrical filament G - bn,(#
+ ~4
(22)
Its dependence on P,,, can take the form G~pRWP+‘)
(23)
3. Discussion There is much information in experimental studies that supports the basic assumptions of the model. First, it is clear from Fig. 1 [9] that there is a critical dimension I
573 K
200
Fig. 1. Correlation between gas sensitivity at 573 K and SnOz crystallite size of pure and doped SnOl gas sensors. The sensitivity is defined as the ratio between the resistance of the sensor in pure air and the resistance. of the sensor in air containing 800 ppm Hz (after [9]).
of the grains in the vicinity of which the dependence of gas-sensor sensitivity on gram dimension changes dramatically for sensors made of pure SnO,. The change corresponds to the extension of surface influence over the whole gram. When only part of the grain is influenced by surface processes, the other part limits the sensitivity. The increase of sensitivity below the critical value can be attributed in part to an increase in mobility, due to the decrease of A/~Yin eqn. (17). The value of the critical dimension (about 9 nm) is lower than the value of about 100 nm (= 2R) derived using the data listed in Table 2, for approximately the same temperature. This is a difference that may be expected, since the values of R presented in Table 2 are calculated using the data listed in Table 1. The data in Table 1 correspond to an acceptor-doped SnO, single crystal, e.g., doped with Al. For a pure material, since the concentration of conduction electrons is higher in comparison with that corresponding to the acceptor-doped material, the value of R is proportionally decreased (see eqn. (5)). Doping with Al (Fig. 1) [9] moves the critical value of the grain dimension above 50 nm (the estimated value for 600 K is 100 nm). Below 50 nm a weak increase of the sensitivity is also observed for the acceptor-doped sample. A possible explanation for this behaviour is the fact that for the relatively high grain sizes for which measurements are available, the influence of surface phenomena on mobility is less important (see eqn. (17)). For donor doping, e.g., with Sb (see Fig. l), the contribution of surface phenomena decreases strongly, and as a consequence, their extension over the whole grain cannot take place, so the sensitivity is not influenced by the decrease in grain size. Figures 2 and 3 present the dependence of the measured electron concentration n, conductivity u and mobility p on reducing-gas concentrations for two different gas sensors [lo, 111. In both cases the grain size is in the range of interest, about 13 nm for the case presented in Fig. 2 and about 6 nm for the case presented in Fig. 3. In both cases the functional dependence is in agreement with that predicted by the model, eqns. (16), (21) and (23). In Fig. 2 in which the data are taken from ref. 10, the change with temperature of the value of the exponent in the power law, eqn. (23), can be explained by the change of the dominating chemisorbed species of oxygen [4-61. The increase in temperature modifies the dominating species from O-, when corresponding to eqn. (23) a linear dependence of u on P, is to be expected, to O*-, when the dependence of o on PR should be sublinear. For the case presented in Fig. 3, which contains data taken from ref. 11, one can say that the interacting oxygen species is O- due to the obtained values of the exponent: l/2 for n and CL,and 1 for u. It is also important to note that the temperatures at which the
523K
2 H2
,
I
4
6
Concentration
1103~
601
I 8
C2H50HConcentrotlon
pm1
I ppm 1
Fig. 3. Conductivity u, carrier concentration n and mobility p of a tin oxide gas sensor, measured at 523 K, as a function of &H50H concentration (after [ll]).
573 K 4.5
sensors presented in Figs. 2 and 3 behave similarly are quite close: 493 K for Fig. 2(a) and 523 K for Fig. 3. Finally, it is interesting that in the case of layers with small grains (typical value 50 nm), the functional dependence of n and G or u on reducing-gas concentration can differ from that obtained for layers with large grain sixes (typical value =l pm) [17]. This interesting feature is a consequence of the conduction mechanism which makes possible a linear relationship between n, and G in the case of layers with large grains. More details will be given in a subsequent paper. H2
(b) 5 ,
I
(4
Cancentrahon I
I
I103
p pm I I
623K
Hq Concentration
I
4. Conclusions
I
(103ppm)
Fig. 2.Normalized conductivities on = u//4, carrier concentrations n.=n/n,,and mobilitiesFL,=d&asafunction of H,concentration. The subscript ‘0’ indicates the absence of reducing gas. The measurements were performed at various temperatures: (a) 493 K, (b) 573 K and (c) 623 K (after [lo]).
The experimental data are in good agreement with the predictions made in the framework of the model presented here. This agreement supports the basic assumption of the model, namely, surface oxygen vacancies are chemisorption sites for atmospheric oxygen. As a consequence, due to the decreasing difference between the number of conduction electrons in the bulk and the number of traps available for them on the surface levels when the dimension of the grain decreases, the energy bands become flat below a critical grain size. In this case the mechanism for electrical conduction is the usual one for semiconductors, a homogeneous concentration of electrons (whose value is directly related to the concentration of the reducing gas) that suffer scattering processes along the axis and at the surface of the cylindrical grain. The fact that the experimentally well-verified eqns. (16), (21) and (23) are derived using the hypothesis 19~1 suggests that it could also be used successfully in the modelling
of other surface processes taking place in SnO, polycrystalline layers. The equations derived for the relationship between n, and 8 and between p and 0 can also be used to analyse the dependence of n, and P on temperature; however, more experimental and theoretical work is needed for this purpose.
References
lo R. Sanjints, V. Demame and F. L&y, Hall effect measure-
14
15 K. Ihokura, Tin dioxide gas sensors for deoxidizing gas, New Mater New Processes Elecfroc!~em. Tech&., 1 (1981) 43-50. J.F. McAleer, P.T. Moseley, J.O.W. Norris, D.E. Williams, P. Taylor and B.C. Tofield, Tin oxide based gas sensors, Mater Chem. Phys., 17 (1987) 571-583. K. Takahata, in T. Seiyama (ed.), Chemical Sensor Technalogy, Vol. 1, Kodansha, Tokyo, 1988, Ch. 3, pp. 39-57. N. Yamaxoc, J. Fuchigami, M. Kishikawa and T. Seiyama, Interaction of tin oxide surface with Ox, Hz0 and Hz, Swface Sci., 86 (1979) 335-344. SC. Chang, Sensing mechanism of thin-film tin oxide, Proc. 1st Meet. Chemical Sensors, Fubroka, Japan, 1983, pp. 78-83. G. Heiland and D. Kohl, in T. Seiyama (ed.), Chemical Sensor Technology, Vol. 1, Kodansha, Tokyo, 1988, Ch. 2, pp. 15-38. K.D. Schierbaum, R Kowalkowski, U. Weimar and W. Opel, Conductance, work function and catalytic activity of SnOxbased gas sensors, Sensors andActuators B, 3 (1991) 205-214. J.N. Zemel, Theoretical description of gas-film interaction on SnO, Thin Solid FiLns, 163 (1988) 189-202. C. Xu, J. Tamaki, N. Miura and N. Ysmaxoe, Grain-size effects on gas sensitivity of porous &Or-based elements, Sensors and Actuators B, 3 (1991) 147-155.
16
17
23
ments in SnO, film sensors exposed to reducing and oxidizing gases, Thin Solid Fibns, 1931194 (1990) 935-942. H. Ogawa, M. Nishikawa and A. Abe, Hall measurements studies and an electrical conduction model of tin oxide ultrafine particle films, J. Appl. Phys., 53 (1982) 4448-4454. S. Samson and C.G. Fonstad, Defect structure and electronic donor levels in stannic oxide crystals, 1. Appl. Phys., 44 (1973) 461u621. Z.M. Jarzebski and J.P. Marton, Physical properties of SnO, materials - Electrical properties, L Elecfrochem. Sot., 123 (9) (1976) 299c-31oc. J. Maier and W. Gopel, Investigations of the bulk defect chemistry of polyctystahine tin(IV) oxide, J. SolidStateChem., 7.2 (1988) 293-302. KD. Schierbaum, Elektrische und spektroskopische Untersuchungen an Diinnschicht-SnO,-Gassensoren, Ph.D. Thesis, Eberhard Karls University, Ttibingen (1987). D.F. Cox, T.B. Fryberger and S. Semancik, Oxygen vacancies and defect electronic states on the SnOr(ll0) 1 x 1 surface, Phys. Rev. B, 38 (3) (1988) 207&2083. N. B&man and R. Ionescu, The mechanism of the interaction between CO and the SnOa surface - the role of water vapour, Sensors and Actuators B, 12 (1993) 71-75. A. Vancu, R. Ionescu and N. Barsan, in P. Ciureanu and S. Middelhoek (eds.), Thin-FilmR&tive Sensors,IOP, Bristol, 1992, Ch. 6. S. StriisIer and A. Reis, Simple models for n-type metal oxide gas sensors, Sensors and Actuators, 4 (1983) 465-472. D.E. Williams, in P.T. Moseley and B.C. Tofield (eds.), Solid State Gas Sensors, Adam Hilger, Bristol, 1987, Ch. 5, pp. 71-123. S.R. Morrison. The Chemical Phvsics of Surfaces. Plenum. New York, 1977, p. 29. ’ ’ ’ ’ J.W. Orton and M.J. Powell, The Hall effect in wlvcrvstalline and powdered semiconductors, Pep. bag. Phys.,bJili) (1980) 1263-1305. A. Many, Y. Goldstein and N.B. Grover, Semiconductor &faces, Interscience, New York, 1965, p. 308.
241
B, 17 (1994) 241-246
Conduction models in gas-sensing SnO, layers: grain-size effects and ambient atmosphere influence N. Bdrsan Instihdeof Physics and Technobgy of Materials, Bucharest-Migwek (RomaniaJ
(Received August 31, 1992, in revised form May 26, 1993; accepted July 1, 1993)
Abstract A model for conduction that fits SnOz polycrystalline layers with a grain size below 50 nm is presented. Quantitative evaluations using bulk parameters, found in the literature, and the assumption that surface oxygen vacancies are sites for atmospheric oxygen chemisorption permit a simple formulation of the problem: conduction by homogeneously distributed electrons along cylindrical filaments. An expression for the mobility is proposed, in which difise surface scattering is taken into account. On its basis, a relationship between the electrical mobility and the concentration of the reducing gas is theoretically derived for the first time. The predictions of the model for the dependence of conductance, concentration of conduction electrons and their mobility on the reducing-gas concentration are compared with experimental data and found to be in good agreement.
1. Introduction Polycrystalline porous SnO, layers are used in the detection of toxic and explosive gases [l-3]. The advantages of gas sensors based on such sensing layers are low cost and high sensitivity. This made them very useful devices and prompted studies devoted to their development. The purpose of these studies is the improvement of sensor performance, especially stability and selectivity. In the range of temperatures in which gas sensors work (400-700 K), the chemical sensing mechanism is the surface reaction between chemisorbed oxygen and reducing gases. The oxygen is chemisorbed as charged species (O,-, O-, 02-) [4-6], inducing the appearance of a surface depletion layer of lower conductivity in comparison with the bulk. The case of O*- is special; it is considered that in this form the oxygen should not be stable at the surface as a chemisorbed species ‘if it does not react immediately or is trapped by an oxygen vacancy’ [6]. One considers that because of the nature of the chemisorption sites, assumed in what follows, one can accept the presence of O*- at the surface as a reactive species. The effect of reducing gases consists in a decrease of the concentration of chemisorbed oxygen, recorded as an increase in conductance. Research in the field of SnO, surface interaction with gases is the subject of both technological and fundamental studies [3, 7, 81. The complex measurements used in these studies impose the elaboration of
0925.4005/94/$07.00 0 1994 Elsevier Sequoia. All rights reserved SSDI 0925-4005(93)00873-W
models that could serve as a guide for the interpretation of the experimental results as a basis for new developments. The aim of this study is the modelling of the electrical condtiction in gas-sensing SnOz layers with average grain sizes below 50 nm. A flat-band hypothesis is justified and assumed. Calculations for electron concentration, mobility and conductance are performed on this basis. The results are compared with experimental data obtained from conductance and Hall-effect measurements [9-111.
2. Conduction model SnO, is a wide band-gap semiconductor, EG = 3.5 eV. Its n-type conductivity is due, when pure, to the presence of bulk oxygen vacancies [12-141.Typical bulk electrical conduction parameters of SnO, single crystals after high-temperature preparation, as used in this paper, are presented in Table 1 [13, 151. TABLE 1. Bulk electrical conduction parameters of Sn02 single crystals ND (me31 N* (m-‘) ED, (ev) ED2(ev) EA (ev)
2.5002x la22 SXlOP 0.04 0.14 0.78
242
As shown elsewhere [M-18], surface oxygen vacancies play the role of chemisorption sites in the presence of atmospheric oxygen. Their concentration, N,, can be evaluated considering that the bulk oxygen vacancies within a sheet of thickness S at the surface are in fact surface vacancies (6 being of the order of magnitude of the SnCl, lattice constant -0.5 nm). For doing this evaluation, one considers that the concentration of oxygen surface vacancies can be obtained as a twodimensional projection of the bulk situation, i.e., one neglects the surface reconstruction. Being interested in the order of magnitude of chemisorption sites, one considers that this hypothesis is an acceptable approximation for the real case. One obtains N,=N,,S(l-S/I&)
(1)
for cylindrical grains and N,=N&(l-6/2r-S2/3rZ)
(2)
for spherical grains, r being the radius in both cases and considering only the lateral surface for the cylindrical grains. The value of N, is about 10x 10” rnP*. The values of the bulk [12, 13, 1.51 and surface parameters of interest in the temperature range 400-700 K are given in Table 2, namely the bulk concentrations of electrons nb, the electrical bulk mobility pb, the effective Debye length AD, the mean free path for electrons A, U, L& AE, (to be defined later in the text) and the radius R of the cylindrical grain for which the condition sN,-VQ,, where s is the lateral surface and v is the volume of the cylindrical filament (see below) is fulfilled. We find for R that 2N,=n,R
(3)
The surface reaction with oxygen and reducing gases may be described by means of a quasi-chemical reaction formalism [6, 19, 201:
(l-~)n,=(cu+pP,“)e
(4)
(l-B)n,2=(a:+PP,)B
(5)
where 0 is the surface coverage of chemisorbed oxygen, n, is the surface concentration of conduction electrons, TABLE 2. Bulk and surface single crystals
T 6) tab
(10’~)
Pb (1O-4 m’lv s) AD (nm) A (nm) R (nm)
AElkr,T AE,kT AEJksT
IYand p are constants and PR is the partial pressure of reducing gases. Equation (4) describes the interaction with O,- chemisorbed species for m = 2 and with Ofor m = 1. Equation (5) describes the interaction with 02- chemisorbed species. In eqns. (4) and (5) 0 and II, are related; there are two problems in order to derive the dependence of the conductance, G, on the concentration of the reducing gas. The first is to find a second relationship between 6 and n, in order to get the dependence of n, on PR and the second is to find the relationship between G and n,, i.e., G and PR. The second relationship between 0 and n, can be obtained from an electrical neutrality equation; its specific form depends on the SnO, layer characteristics. For grains of small average size between 5 and 50 nm, the conduction process seems to take place in long and thin filaments [9, 111. This is the reason why the approximation of cylindrical filaments of length 1 and radius r will be used in the model. One has to evaluate the influence of the surface. Comparing the estimated value of N, and the experimental data for amounts of chemisorbed oxygen [4], we can draw the conclusion that 0, related to N, and not to the total concentration of surface atoms as is usually done [20, 211, is close to unity. As an upper limit for the possible surface influence into the bulk, 8-l is assumed. We are interested in the case in which the energy bands are flat. After ref. 22, this case is obtained if there is enough room on surface levels for all the conduction electrons from the bulk and if r
400 1 178 129 1.96 2.500 0.43 0.23 0.14
parametersof
500 11 a7 43 1.07 230 0.77 0.39 0.26
influence for SnOl
600
58 49 21 0.66 44 1.08 0.56 0.39
700 260 31 11 0.45 2 1.49 0.76 0.51
R=S(l+ zf)( 1+[ 1i$:iJ] The values of interest are presented in Table 2. We have to keep in mind the conditions in which these values are calculated. On one hand the data used in the calculation of R are taken from the literature for a specific single crystal [13,15]; since the concentration of oxygen vacancies in thin or thick porous films can be very different, it is possible that their values do not fit well enough the case of a specific gas sensor. On the other hand, an assumption was made about the value of 0, i.e., 0=1; this means that under the same conditions, namely the same N, and n,,, the value of R is proportional to lower values proportionally to 8. As a conclusion, we can only say that the values of R shown in Table 2 approximate the radius of the cylindrical filament that can be depleted completely of conducting electrons by trapping on surface levels. It is clear that in the range of grain sizes we consider, it is possible to fulfil this condition; it will be assumed
243
in what follows that this is the case. Using this assumption we can write the Poisson equation directly for energy, E, using the Schottky approximation. For our geometry the radial part of the Poisson equation, the interesting part, is
Equations (4), (5) and (13) allow a relationship between n, and PR to be found. Avery simple and easy to use set of equations can be derived if one assumes the fulfillment of eqn. (5) in the whole range of parameters we use:
(7) where q is the electron charge, E is the permittivity and e. is the permittivity of free space. The boundary conditions are
and one obtains from eqns. (4), (5) and (14) n,“+‘=(IY+/3PRm)(nb-n,)
(15)
where E(0) = E,
I
p= 1 for 02- and O-
(?),_,=o r‘ From
eqn.
(7), using
p=2 for 02m=lforO-and@eqn.
(8) we obtain
for
AE=E(r)-E,,:
(9)
m=2forO,Taking into account the fact that n, +z nb, one obtains for high enough PR values [17]:
n*_pRm/@+l)
or
(16)
2
(10) where kB is the Boltxmarm constant and T is the temperature. AE is the maximum electron energy difference, i.e., that between an electron located on the axis of the cylindrical filament and an electron located on the lateral surface. The flat-band condition, AE/k,T$l, takes the form
The relationship between n, and G via conductivity is a delicate matter. Considering the facts presented above, n, is the concentration of electrons that take part in electrical conduction. These electrons suffer scattering processes in the bulk (along the filament) and at the surface. Using the formula derived by Many etal. [23] for the influence of surface scattering, adapted to our geometry, one gets p=
(11) In Table 2 are presented the values of AE/k,T, which are calculated for grain dimensions of 50 nm and the value of AE,Ik,T and AE,lk,T calculated for grain dimensions of 13 and 6 nm that correspond to the samples for which experimental results are presented in Section 3 [lo, 111. Examining the data presented in Table 2, we see that we can consider the flat-band condition is fulfilled in the range of grain sizes below 50 mu. It is assumed, in what follows, that this is the case. The electrical neutrality condition takes the form BN,s+n,v=n,v
(12)
Equation (12) describes the redistribution of the bulk conduction electrons on surface levels and in the bulk due to the chernisorption of atmospheric oxygen. Then
l+(zvr)
(17)
where Wrepresents the probability of a diffuse scattering process. Comparing h to 2r, one can observe that the influence of the surface cannot be neglected. According to Many et al. [23], W is related to the deviation of the surface from a simple projection of the bulk. For our case this deviation represents the difference between the concentration of scattering centres with which conduction electrons interact when they strike the surface and the concentration of scattering centres with which they interact when they move along the axis of the cylindrical filament. This difference, the extra surface scattering centres, is given by the charged oxygen species chemisorbed on the surface oxygen vacancies. Their concentration being proportional to 6, we propose w-e One obtains for the mobility:
(18)
244
with #=l+
;
and .q=
& b
One can expand eqn. (19) using the fact that qn,lQ 4Cl:
or, with obvious notations: #u=/&(‘yt pPR”I@+l))
(21)
Equation (21) is the first proposed relationship between ,u and PR based on a theoretical model. Thus we have for the conductance (or for the conductivity a) of the cylindrical filament G - bn,(#
+ ~4
(22)
Its dependence on P,,, can take the form G~pRWP+‘)
(23)
3. Discussion There is much information in experimental studies that supports the basic assumptions of the model. First, it is clear from Fig. 1 [9] that there is a critical dimension I
573 K
200
Fig. 1. Correlation between gas sensitivity at 573 K and SnOz crystallite size of pure and doped SnOl gas sensors. The sensitivity is defined as the ratio between the resistance of the sensor in pure air and the resistance. of the sensor in air containing 800 ppm Hz (after [9]).
of the grains in the vicinity of which the dependence of gas-sensor sensitivity on gram dimension changes dramatically for sensors made of pure SnO,. The change corresponds to the extension of surface influence over the whole gram. When only part of the grain is influenced by surface processes, the other part limits the sensitivity. The increase of sensitivity below the critical value can be attributed in part to an increase in mobility, due to the decrease of A/~Yin eqn. (17). The value of the critical dimension (about 9 nm) is lower than the value of about 100 nm (= 2R) derived using the data listed in Table 2, for approximately the same temperature. This is a difference that may be expected, since the values of R presented in Table 2 are calculated using the data listed in Table 1. The data in Table 1 correspond to an acceptor-doped SnO, single crystal, e.g., doped with Al. For a pure material, since the concentration of conduction electrons is higher in comparison with that corresponding to the acceptor-doped material, the value of R is proportionally decreased (see eqn. (5)). Doping with Al (Fig. 1) [9] moves the critical value of the grain dimension above 50 nm (the estimated value for 600 K is 100 nm). Below 50 nm a weak increase of the sensitivity is also observed for the acceptor-doped sample. A possible explanation for this behaviour is the fact that for the relatively high grain sizes for which measurements are available, the influence of surface phenomena on mobility is less important (see eqn. (17)). For donor doping, e.g., with Sb (see Fig. l), the contribution of surface phenomena decreases strongly, and as a consequence, their extension over the whole grain cannot take place, so the sensitivity is not influenced by the decrease in grain size. Figures 2 and 3 present the dependence of the measured electron concentration n, conductivity u and mobility p on reducing-gas concentrations for two different gas sensors [lo, 111. In both cases the grain size is in the range of interest, about 13 nm for the case presented in Fig. 2 and about 6 nm for the case presented in Fig. 3. In both cases the functional dependence is in agreement with that predicted by the model, eqns. (16), (21) and (23). In Fig. 2 in which the data are taken from ref. 10, the change with temperature of the value of the exponent in the power law, eqn. (23), can be explained by the change of the dominating chemisorbed species of oxygen [4-61. The increase in temperature modifies the dominating species from O-, when corresponding to eqn. (23) a linear dependence of u on P, is to be expected, to O*-, when the dependence of o on PR should be sublinear. For the case presented in Fig. 3, which contains data taken from ref. 11, one can say that the interacting oxygen species is O- due to the obtained values of the exponent: l/2 for n and CL,and 1 for u. It is also important to note that the temperatures at which the
523K
2 H2
,
I
4
6
Concentration
1103~
601
I 8
C2H50HConcentrotlon
pm1
I ppm 1
Fig. 3. Conductivity u, carrier concentration n and mobility p of a tin oxide gas sensor, measured at 523 K, as a function of &H50H concentration (after [ll]).
573 K 4.5
sensors presented in Figs. 2 and 3 behave similarly are quite close: 493 K for Fig. 2(a) and 523 K for Fig. 3. Finally, it is interesting that in the case of layers with small grains (typical value 50 nm), the functional dependence of n and G or u on reducing-gas concentration can differ from that obtained for layers with large grain sixes (typical value =l pm) [17]. This interesting feature is a consequence of the conduction mechanism which makes possible a linear relationship between n, and G in the case of layers with large grains. More details will be given in a subsequent paper. H2
(b) 5 ,
I
(4
Cancentrahon I
I
I103
p pm I I
623K
Hq Concentration
I
4. Conclusions
I
(103ppm)
Fig. 2.Normalized conductivities on = u//4, carrier concentrations n.=n/n,,and mobilitiesFL,=d&asafunction of H,concentration. The subscript ‘0’ indicates the absence of reducing gas. The measurements were performed at various temperatures: (a) 493 K, (b) 573 K and (c) 623 K (after [lo]).
The experimental data are in good agreement with the predictions made in the framework of the model presented here. This agreement supports the basic assumption of the model, namely, surface oxygen vacancies are chemisorption sites for atmospheric oxygen. As a consequence, due to the decreasing difference between the number of conduction electrons in the bulk and the number of traps available for them on the surface levels when the dimension of the grain decreases, the energy bands become flat below a critical grain size. In this case the mechanism for electrical conduction is the usual one for semiconductors, a homogeneous concentration of electrons (whose value is directly related to the concentration of the reducing gas) that suffer scattering processes along the axis and at the surface of the cylindrical grain. The fact that the experimentally well-verified eqns. (16), (21) and (23) are derived using the hypothesis 19~1 suggests that it could also be used successfully in the modelling
of other surface processes taking place in SnO, polycrystalline layers. The equations derived for the relationship between n, and 8 and between p and 0 can also be used to analyse the dependence of n, and P on temperature; however, more experimental and theoretical work is needed for this purpose.
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14
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